16915892 theory of electromagnetic wave propagation

THEORY OFELECTROMAGNETICWAVE PROPAGATION

CHARLES HERACH PAPASPROFESSOR OF ELECTRICAL ENGINEERINGCALIFORNIA INSTITUTE OF TECHNOLOGY

DOVER PUBLICATIONS, INC., NEW YORK

Copyright @ 1965,1988 by Charles Herach Papas.All rights reserved under Pan American and International

Copyright Conventions.Published in Canada by General Publishing Company, Ltd., 30

Lesmill Road, Don Mills, Toronto, Ontario.Published in the United Kingdom by Constable and Company,

Ltd., 10 Orange Street, London WC2H 7EG.This Dover edition, first published in 1988, is an unabridged and

corrected republication of the work first published by the McGraw-Hill Book Company, New York, 1965, in its Physical and QuantumElectronics Series. For this Dover edition, the author has written anew preface.

Manufactured in the United States of AmericaDover Publications, Inc., 31 East2nd Street, Mineola, N.Y. 11501

Library of Congress Cataloging-in-Publication Data

Papas, Charles Herach.Theory of electromagnetic wave propagation / Charles Herach

Papas.p. em.

Reprint. Originally published: New York : McGraw-Hill,cl965. (McGraw-Hill physical and quantum electronics series)With new pref.Includes index.ISBN 0-486-65678-0 (pbk.)1. Electromagnetic waves.~. Title.

QC661.P29 1988 "i'

530.1'41-dcI9 88-12291CIP

To RONOLDWYETH PERCIVALKING

Gordon McKayProfessor of Appl1e~Physics,

Harvard University

Outstanding Scientist, Inspiring Teacher,

and Dear Friend

"""',.~,"

PrefaceThis book represents the substance of a course of lectures Igave during the winter of 1964 at the California Institute ofTechnology. In these lectures I expounded a number ofnewly important topics in the theory of electromagnetic wavepropagation and antennas, with the purpose of presenting acoherent account of the subject in a way that would revealthe inherent simplicity of the basic ideas and would place inevidence their logical development from the Maxwell fieldequations. So enthusiastically were the lectures receivedthat I was encouraged to put them into book form and thusmake them available to a wider audience.The scope of the book is as follows: Chapter 1 provides the

reader with a brief introduction to Maxwell's field equationsand those parts of electromagnetic field theory which he willneed to understand the rest of the book. Chapter 2 presentsthe dyadic Green's function and shows how it can be used tocompute the radiation from monochromatic sources. InChapter 3 the problem of radiation emitted by wire antennasand by antenna arrays is treated from the viewpoint of anal-ysis and synthesis. In Chapter 4 two methods of expandinga radiation field in multipoles are given, one based on theTaylor expansion of the Helmholtz integrals and the otheron an expansion in spherical waves. Chapter 5 deals withthe wave aspects of radio-astronomical antenna theory andexplains the Poincare sphere, the Stokes parameters, coher-ency matrices, the reception of partially polarized radiation,the two-element radio interferometer, and the correlationcoefficients in interferometry. Chapter 6 gives the theoryof electromagnetic wave propagation in a plasma mediumand describes, with the aid of the dyadic Green's function,the behavior of an antenna immersed in such a medium.Chapter 7 is concerned with the covariance of Maxwell's

vii

Preface

equations in material media and its application to phenomena such asthe Doppler effect and aberration in dispersive media.The approach of the book is theoretical in the sense that the subject

matter is developed step by step from the Maxwell field equations.The advantage of such an approach is that it tends to unify the varioustopics under the single mantle of electromagnetic theory and servesthe didactic purpose of making the contents of the book easy to learnand convenient to teach. The text contains many results that canbe found only in the research literature of the Caltech AntennaLaboratory and similar laboratories in the U.S.A., the U.S.S.R., andEurope. Accordingly, the book can be used as a graduate-level text-book or a manual of self-instruction for researchers.My grateful thanks are due to Professor W. R. Smythe of the Cali-

fornia Institute of Technology, Professor Z. A. Kaprielian of theUniversity of Southern California, and Dr. K. S. H. Lee of the Cali-fornia Institute of Technology for their advice, encouragement, andgenerous help. I also wish to thank Mrs. Ruth Stratton for herunstinting aid in the preparation of the entire typescript.

Charles Herach Papas

Preface to the Dover EditionExcept for the correction ofminor errors and misprints, this edition ofthe book is an unchanged reproduction of the original.My thanks are due to my graduate students, past and present, for the

vigilance they exercised in the compilation of the list of corrections, andto Dover Publications for making the book readily available once again.

Charles Herach Papas

viii

Contents

Preface vii

Preface to the Dover Edition viii

1 The electromagnetic field 1

1.1 Maxwell's Equations in Simple Media 11.2 Duality 61.3 Boundary Conditions 81.4 The Field Potentials and Antipotentials 91.5 Energy Relations 14

2 Radiation from monochromaticsources in unbounded regions 19

2.1 The Helmholtz Integrals 192.2 Free-space Dyadic Green's Function 262.3 Radiated Power 29

3 Radiation from wire antennas 37

3.1 Simple Waves of Current 373.2 Radiation from Center-driven Antennas 423.3 Radiation Due to Traveling Waves of Current,

Cerenkov Radiation 453.4 Integral Relations between Antenna Current

and Radiation Pattern 483.5 Pattern Synthesis by Hermite Polynomials 503.6 General Remarks on Linear Arrays 563.7 Directivity Gain 73

4 Multipole expansionof the radiated field 81

4.1 Dipole and Quadrupole Moments 814.2 Taylor Expansion of Potentials 864.3 Dipole and Quadrupole Radiation 894.4 Expansion of Radiation Field in Spherical Waves 97

5 Radio-astronomical antennas 109

5.1 Spectral Flux Density 1115.2 Spectral Intensity, Brightness, Brightness Temperature,

Apparent Disk Temperature 1155.3 Poincare Sphere, Stokes Parameters 1185.4 Coherency Matrices 1345.5 Reception of Partially Polarized Waves 1405.6 Antenna Temperature and Integral Equation

for Brightness Temperature 1485.7 Elementary Theory of the Two-element

Radio Interferometer 1515.8 Correlation Interferometer 159

6 Electromagnetic wavesin a plasma 169

6.1 Alternative Descriptions of Continuous Media 1706.2 Constitutive Parameters of a Plasma 1756.3 Energy Density in Dispersive Media 1786.4 Propagation of Transverse Waves in Homogeneous

Isotropic Plasma 1836.5 Dielectric Tensor of Magnetically Biased Plasma 1876.6 Plane Wave in Magnetically Biased Plasma 1956.7 Antenna Radiation in Isotropic Plasma 2056.8 Dipole Radiation in Anisotropic Plasma 2096.9 Reciprocity 212

7 The Doppler effect 217

7.17.27.37.47.57.6

Covariance of Maxwell's Equations 218Phase Invariance and Wave 4-vector 223Doppler Effect and Aberration 225Doppler Effect in Homogeneous Dispersive Media 227Index of Refraetion of a Moving Homogeneous MediumWave Equation for Moving Homogeneous Isotropie Media

230233

Index 24i

Theelectromagnetic 1

field

In this introductory chapter some basic relations and con-cepts of the classic electromagnetic field are briefly reviewedfor the sake of easy reference and to make clear the signifi-cance of the symbols.

1.1 Maxwell's Equations inSimple Media

In the mks, or Giorgi, system of units, which we shall usethroughout this book, Maxwell's field equationsl are

av x E(r,t) = - iii B(r,t)

av x R(r,t) = J(r,t) + iii D(r,t)

V. B(r,t) = 0

v .D(r,t) = p(r,t)

(1)

(2)

(3)

(4)

where E(r,t) = electric field intensity vector, volts per meterR(r,t) = magnetic field intensity vector, ainperes per

meter

1 See, for example, J. A. Stratton, "Electromagnetic Theory,"chap. 1, McGraw-Hill Book Company, New York, 1941.

1

Theory of electromagnetic wave propagation

D(r,t) = electric displacement vector, coulombs per meter2B(r,t) = magnetic induction vector, webers per meter2J(r,t) = current-density vector, amperes per meter2p(r,t) = volume density of charge, coulombs per meter3r = position vector, meterst = time, seconds

The equation of continuity

aV' • J(r,t) = - at p(r,t) (5)

which expresses the conservation of charge is a corollary of Eq. (4) andthe divergence of Eq. (2).The quantities E(r,t) and B(r,t) are defined in a given frame of

reference by the density of force f(r,t) in newtons per meter3 acting onthe charge and current density in accord with the Lorentz forceequation

f(r,t) = p(r,t)E(r,t) + J(r,t) X B(r,t) (6)

In turn D(r,t) and H(r,t) are related respectively to E(r,t) and B(r,t) byconstitutive parameters which characterize the electromagnetic natureof the material medium involved. For a homogeneous isotropiclinear medium, viz., a "simple" medium, the constitutive relations are

D(r,t) = EE(r,t)

1H(r,t) = - B(r,t)IJ.

(7)

(8)

where the constitutive parameters E in farads per meter and IJ. in henrysper meter are respectively the dielectric constant and the permeabilityof the medium.In simple media, Maxwell's equations reduce to

aV' X E(r,t) = - J.L at H(r,t)

aV' X H(r,t) = J(r,t) + E at E(r,t)

2

(9)

(10)

V. H(r,t) = 0

V • E(r,t) = ~ p(r,t)E

The electromagnetic field

(11)

(12)

The curl of Eq. (9) taken simultaneously with Eq. (10) leads to

02 0V X V X E(r,t) + ILE ot2E(r,t) = -IL at J(r,t)Alternatively, the curl of Eq. (10) with the aid of Eq. (9) yields

02V X V X H(r,t) + ILE i)t2H(r,t) = V X J(r,t)

(13)

(14)

The vector wave equations (13) and (14) serve to determine E(r,t) andH(r,t) respectively when the source quantity J(r,t) is specified andwhen the field quantities are required to satisfy certain prescribedboundary and radiation conditions. Thus it is seen that in the case ofsimple media, Maxwell's equations determine the electromagneticfield when the current density J(r,t) is a given quantity. Moreover,this is true for any linear medium, i.e., any medium for which therelations connecting B(r,t) to H(r,t) and D(r,t) to E(r,t) are linear, beit anisotropic, inhomogeneous, or both.To form a complete field theory an additional relation connecting

J(r,t) to the field quantities is necessary. If J(r,t) is purely an ohmicconduction current in a medium of conductivity u in mhos per meter,then Ohm's law

J(r,t) = uE(r,t) (Hi)

applies and provides the necessary relation. On the other hand, ifJ(r,t) is purely a convection current density, given by

J(r,t) = p(r,t)v(r,t) (16)

where v(r,t) is the velocity of the charge density in meters per second,the necessary relation is one that connects the velocity with the field.To find such a connection in the case where the convection current ismade up of charge carriers in motion (discrete case), we must calculate

3


the total force F(r,t) acting on a charge carrier by first integrating theforce density f(r,t) throughout the volume occupied by the carrier, i.e.,

F(r,t) = ff(r + r',t)dV' = q[E(r,t) + v(r,t) X B(r,t)J (17)

where q is the total charge, and then equating this force to the force ofinertia in accord with Newton's law of motion

dF(r,t) = dt [mv(r,t)] (18)

where m is the mass of the charge carrier in kilograms. In the casewhere the convection current is a charged fluid in motion (continuouscase), the force density f(r,t) is entered directly into the equation ofmotion of the fluid.Because Maxwell's equations in simple media form a linear system,

no generality is lost by considering the "monochromatic" or "steady"state, in which all quantities are simply periodic in time. Indeed, byFourier's theorem, any linear field of arbitrary time dependence can besynthesized from a knowledge of the monochromatic field. To reducethe system to the monochromatic state we choose exp (- iwt) for thetime dependence and adopt the convention

G(r,t) = Re {G",(r)e-i",t} (19)

where G(r,t) is any real function of space and time, G",(r) is the con-comitant complex function of position {sometimes called a "phasor"),which depends parametrically on the frequency f( = w/27r) in cyclesper second, and Re is shorthand for "real part of." Application of thisconvention to the quantities entering the field equations (1) through(4) yields the monochromatic form of Maxwell's equations:

v X E",(r) = iwB",(r)

V X H.,(r) = J",(r) - i~D",(r)

V • B",(r) = 0

V. D",(r) = p",(r)

4

(20)

(21)

(22)

(23)


In a similar manner the monochromatic form of the equation ofcontinuity

V' • J",(r) = iwp",(r) (24)

is derived from Eq. (5).The divergence of Eq. (20) yields Eq. (22), and the divergence of

Eq. (21) in conjunction with Eq. (24) leads to Eq. (23). We inferfrom this that of the four monochromatic ~Iaxwell equations only thetwo curl relations are independent. Since there are only two inde-pendent vectorial equations, viz., Eqs. (20) and (21), for the deter-mination of the five vectorial quantities E",(r), H",(r), D",(r), B",(r) ,and J",(r) , the monochromatic Maxwell equations form an under-determined system of first-order differential equations. If the system isto be made determinate, linear constitutive relations involving the con-stitutive parameters must be invoked. Oneway of doing this is first toassume that in a given medium the linear relations B",(r) = aH",(r),D",(r) = iSE",(r), and J",(r) = 'YE",(r) are valid, then to note that withthis assumption the system is determinate and possesses solutionsinvolving the unknown constants a, is, and 'Y, and finally to choose thevalues of these constants so that the mathematical solutions agree withthe observations of experiment. These appropriately chosen valuesare said to be the monochromatic permeability /-l"" dielectric constantE"" and conductivity u'" of the medium. Another way of defining theconstitutive parameters is to resort to the microscopic point of view,according to which the entire system consists of free and bound chargesinteracting with the two vector fields E",(r) and B",(r) only. Forsimple media the constitutive relations are

B",(r) = /-l",H",(r)

D",(r) = E",E",(r)

J",(r) = u",E.,(r)

(25)

(26)

(27)

In media showingmicroscopicinertial or relaxation effects,one or moreof these parameters may be complex frequency-dependent quantities.For the sake of notational simplicity, in most of what follows we

shall drop the subscriptw and omit the argument r in the mono-

5


chromatic case, and we shall suppress the argument r in the time-dependent case. For example, E(t) will mean E(r,t) and E will meanEw(r). Accordingly, the monochromatic form of Maxwell's equationsin simple media is

v X E = iW,llH

V X H = J - iweE

V.H=O

1V.E=-pe

1.2 Duality

(28)

(29)

(30)

(31)

In a region free of current (J = 0), Maxwell's equations possess acertain duality in E and H. By this we mean that if two new vectorsE' and H' are defined by

and H' = + iE (32)

then as a consequence of Maxwell's equations (source-free)

V.E=O

V X H = -iweE

V.H=O

V X E = iW,llH(33)

it follows that E' and H' likewise satisfy Maxwell's equations (source-free)

V X H' = -iweE'

V. E' = 0 V. H' = 0(34)

and thereby constitute an electromagnetic field E', H' which is the"dual" of the original field.This duality can be extended to regions containing current by

employing the mathematical artifice of magnetic charge and magnetic

6


current.l In such regions Maxwell's equations are

VXH=J-iwEE v X E = iWIlH

V.H=O1V.E=-pE

(35)

and under the transformation (32) they become

(36)V X H' = -iWEE'v X E' = :t ~ J + iWIlH'

V • H' = :t! ~pIl '\j;V. E' = 0

Formally these relations are Maxwell's equations for an electro-magnetic field E', H' produced by the "magnetic current" +: viIIIE Jand the "magnetic charge" :t vip.1 E p. These considerations suggestthat complete duality is achieved by generalizing Maxwell's equationsas follows:

V X H = J - iWEE. V X E = -Jm + iwp.H

1V.E=-pE

(37)

where Jm and Pm are the magnetic current and charge densities.Indeed, under the duality transformation

E' = :t~H H' = +: ~~E J' = :t ~~ Jm

J;" = +: ~~J p' = :t ~~ Pm p;" = -~+ ;P

V X E' = -J;" + iWIlH' V X H' = J' - iWEE'

V. E' = !p'E

H' 1,V. =-pp. m

(38)

(39)

1 See, for example, S. A. Schelkunoff, "Electromagnetic Waves," chap. 4,D. Van Nostrand Company, Inc., Princeton, N.J., 1943.

7


Thus to every electromagnetic field E, H produced by electric current Jthere is a dual field H', E' produced by a fictive magnetic current J~.

1.3 Boundary ConditionsThe electromagnetic field at a point on one side of a smooth interfacebetween two simple media, 1 and 2, is related to the field at the neigh-boring point on the opposite side of the interface by boundary condi-tions which are direct consequences of Maxwell's equations.We denote by n a unit vector which is normal to the interface and

directed from medium 1 into medium 2, and we distinguish quantitiesin medium 1 from those in medium 2 by labeling them with the sub-scripts 1 and 2 respectively. From an application of Gauss' divergencetheorem to Maxwell's divergence equations, V'. B = Pm and V'. D = P,it follows that the normal components of Band D are respectively dis-continuous by an amount equal to the magnetic surface-charge density7/m and the electric surface-charge density 7/ in coulombs per meter2:

(40)

From an application of Stokes' theorem to Maxwell's curl equations,V'X E = -Jm + iWJ.lH and V'X H = J - iweE, it follows that thetangential components of E and H are respectively discontinuous byan amount equal to the magnetic surface-current density Km and theelectric surface-current density K in amperes per meter:

(41)

In these relations Km and K are magnetic and electric "current sheets"carrying charge densities 7/m and 7/ respectively. Such current sheetsare mathematical abstractions which can be simulated by limitingforms of electromagnetic objects. For example, if medium 1 is aperfect conductor and medium 2 a perfect dielectric, Le., if 0"1 = 00

and 0"2 = 0, then all the field vectors in medium I as well as 7/m and Km

vanish identically and the boundary conditions reduce to

(42)

s


A surface having these boundary conditions is said to be an "electricwall." By duality a surface displaying the boundary conditions

n' Bz = 7]m n X H2 = 0 (43)

is said to be a "magnetic wall."At sharp edges the field vectors may become infinite. However, the

order of this singularity is restricted by the Bouwkamp-Meixner1 edgecondition. According to this condition, the energy density must beintegrable over any finite domain even if this domain happens toinclude field singularities, i.e., the energy in any finite region of spacemust be finite. For example, when applied to a perfectly conductingsharp edge, this condition states that the singular components of theelectric and magnetic vectors are of the order o-~,where /) is the dis-tance from the edge, whereas the parallel components are alwaysfinite.

1.4 The Field Potentials andAntipotentials

According to Helmholtz's partition theorem2 any well-behaved vectorfield can be split into an irrotational part and a solenoidal part, or,equivalently, a vector field is determined by a knowledge of its curl anddivergence. To partition an electromagnetic field generated by a cur-rent J and a charge p, we recall Maxwell's equations

V' X H = J - iwD

V' X E = iwB

(44)

(45)

1 C. Bouwkamp, Physica, 12: 467 (1946); J. Meixner, Ann. Phys., (6) 6: 1(1949).

2 H. von Helmholtz, Uber Integrale der hydrodynamischen Gleichungen,welche den Wirbelbewegungen entsprechen, Crelles J., 55: 25 (1858). Thistheorem was proved earlier in less complete form by G. B. Stokes in his paperOn the Dynamical Theory of Diffraction, Trans. Cambridge Phil. Soc., 9: 1(1849). For a mathematically rigorous proof, see O. Blumenthal, Uber dieZerlegung unendlicher Vektorfelder, Math. Ann., 61: 235 (1905).

9


V.D=p

V.B=O

and the constitutive relations for a simple medium

D = eE

B = ,uH

(46)

(47)

(48)

(49)

From the solenoidal nature of B, which is displayed by Eq. (47), itfollowsthat B is derivable from a magnetic vector potential A:

B=VxA (50)

This relation involves only the curl of A and leaves free the divergenceof A. That is, V . A is not restricted and may be chosen arbitrarily tosuit the needs of calculation. Inserting Eq. (50) into Eq. (45) we seethat E - iwA is irrotational and hence derivable from a scalar electricpotential et>:

E = -Vet> + iwA (51)

This expressiondoes not necessarily constitute a complete partition ofthe electric field because A itself may possess both irrotational andsolenoidal parts. Only when A is purely solenoidal is the electricfield completely partitioned into an irrotational part Vet> and a sole-noidal part A. The magnetic field need not be partitioned inten-tionally because it is always purely solenoidal.By virtue of their form, expressions (50) and (51) satisfy the two

Maxwell equations (45) and (47). But in addition they must alsosatisfy the other two Maxwell equations, which, with the aid of theconstitutive relations (48) and (49), become

!V X B = J - iweE,u

and V.E = pie (52)

When relations (50) and (51) are substituted into these equations, the

10


following simultaneous differential equations are obtained,l relating cPand A to the source quantities J and p:

V2cP - iwV. A = -p/E

V2A+ k2A = -jLJ + V(V. A - iWEjLcP)

(53)

(54)

where k2 = W2jLE. Here V . A is not yet specified and may be chosen tosuit our convenience. Clearly a prudent choice is one that uncouplesthe equations, Le., reduces the system to an equation involving cPalone and an equation involving A alone. Accordingly, we chooseV • A = iWEjLcP or V • A = O.

If we choose the Lorentz gauge

V • A = iWEjLcP

then Eqs. (53) and (54) reduce to the Helmholtz equations

V2cP + k2cP = - p/E

V2A + k2A = -jLJ

(55)

(56)

(57)

The Lorentz gauge is the conventional one, but in this gauge the elec-tric field is not completely partitioned. If complete partition isdesired, we must choose the Coulomb gauge2

V.A = 0 (58)

1Also the vector identity V X V X A = V(V. A) - V2A is used. Thequantity'V2A is defined by the identity itself or by the formal operationV2A= L V2(eiAi),where the Ai are the components of A and the ei are the

iunit base vectors of the coordinate system. The Laplacian V2operates onnot only the Ai but also the ei' In the special case of cartesian coordinates,the base vectors are constant; hence the Laplacian operates on only the Ai,that is, V2A= L eiV2Ai. See, for example, P. M. Morse and H. Feshbach,

i"Methods of Theoretical Physics," part I, pp. 51-52, McGraw-Hill BookCompany, New York, 1953.

2See, for example, W. R. Smythe, "Static and Dynamic Electricity,"2d ed., p. 469, McGraw-Hill Book Company, New York, 1950.

11


which reduces Eqs. (53) and (.54)to

1V2q, = - - P

E

(59)

(60)

We note that Eq. (59) is Poisson's equation and can be reduced nofurther. However, Eq. (60) may be simplified by partitioning J intoan irrotational part Ji and a solenoidal part J., and by noting thatthe irrotational part just cancels the term involving the gradient. Toshow this, J is split up as follows: J = Ji + J., where by definitionV X Ji = 0 and V. J. = O. Since Ji is irrotational, it is derivablefrom a scalar function !J;, viz., Ji = V!J;. The divergence of this rela-tion, V. Ji = V2!J;, when combined with the continuity equationV • J = V. (Ji + J.) =V. Ji = iwp, leads to V2!J;= iwp. A com-parison of this result with Eq. (59) shows that !J; = -iWEq, and henceJi = V!J; = -iWEVq,. From this expression it therefore follows that- IJoJi - iWEIJoVq, vanishes and consequently Eq. (60) reduces to

(61)

Thus we see that in this gauge, A is determined by the solenoidal partJ. of the current distribution and q, by its irrotational part Ji. Sinceq, satisfies Poisson's equation, its spatial distribution resembles that ofan electrostatic potential and therefore contributes predominantly tothe near-zone electric field. It is like an electrostatic field only in itsspace dependence; its time dependence is harmonic.In regions free of current (J = 0) and charge (p = 0) we may

supplement the gauge V . A = 0 by taking q, == O. Then Eq. (53) istrivially satisfied and Eq. (54) reduces to the homogeneous Helmholtzequation

(62)

In this case the electromagnetic field is derived from the vector poten-tial A alone.Let us now partition the electromagnetic field generated by a mag-

netic current Jm and a magnetic charge Pm. We recall that Maxwell's

12


equations for such a field are

V' X H = -iwD

V' xE = -Jm +iwB

V'.D=O

V'.B=Pm

(63)

(64)

(65)

(66)

and, as before, the constitutive relations (48) and (49) are valid.From Eq. (65) it follows that D is solenoidal and hence derivable froman electric vector potential A.:

D = -V' X A. (67)

In turn it follows from Eq. (63) that H - iwA. is irrotational and henceequal to - V'cPm, where cPm is a magnetic scalar potential:

H = -V'cPm + iwA. (68)

Substituting expressions (67) and (68) into Eqs. (64) and (66), we get,with the aid of the constitutive relations, the following differentialequations for A. and cPm:

V'2cPm- iwV' • A. = - ~-Pmp.

V'2A. + k2A. = -elm + V'(V' • A. - iwp.EcPm)

If we choose the conventional gauge

V' • A. = iwp.EcPm

then cPm and A. satisfy

(69)

(70)

1V'2cPm + k2cPm = - - Pm (71)

p.

V'2A. + 1c2A. = -EJm (72)

In this gauge cPm and A. are called "antipotentials." Clearly we may

13


also choose the gauge V'. Ae = 0 which leads to

1V'2cf>m = - - Pm

/L (73)

where JmB is the solenoidal part of the magnetic current; this gaugeleads also to cf>m = 0 and

(74)

for regions where Jm = 0 and Pm = O.If the electromagnetic field is due to magnetic as well as electric cur-

rents and charges, then the field for the conventional gauge is given interms of the potentials A, cf> and the antipotentials Ae, cf>m by

E = - V'cf>+ iwA - !V' X Aet

B = V' X A - /LV'cf>m + iW/LAe

1.5 Energy Relations

(75~

(76)

The instantaneous electric and magnetic energy densities for a losslesemedium are defined respectively by

We = J E(t) . ft D(t)dt and Wm = J H(t) . ft B(t)dt (77)

where E(t) stands for E(r,t), D(t) for D(r,t), etc. In the presentinstance these expressions reduce to

We = ~tE(t) . E(t) and Wm = ~ILH(t) • H(t) (78)

Both We and Wm are measured in joules per meter3• To transformthese quadratic quantities into the monochromatic domain we recall

14


that

E(t) = Re {Ee-u.t} and H(t) = Re {He-i.,t} (79)

where E is shorthand for E.,(r) and H for H.,(r). Since E can alwaysbe written as E = E1 + 'iE2, where E1 andE2 are respectively the realand imaginary parts of E, the first of Eqs. (79) is equivalent to

E(t) = E1 cos wt + E2 sin wt

Inserting this representation into the first of Eqs. (78) we obtain

(80)

(81)

which, when averaged over a period, yields the time-average electricenergy density

where the bar denotes the time average. Since

where E* is the conjugate complex of E, we can express 'II'.in the equiv-alent form

We = %eE. E* (83)

By a similar procedure it follows from the second of Eqs. (78) and thesecond of Eqs. (79) that the time-average magnetic energy is given by

Win = %J.tH.H*

The instantaneous Poynting vect~r S(t) is defined by

(84)

S(t) = E(t) X H(t) (85)

where S(t) stands for S(r,t) and is measured in watts per meter2• With

15


the aid of expressions (79), the time average of Eq. (85) leads to thefollowing expression for the complex Poynting vector:

S = ~E X H* (86)

If from the scalar product of H* and V' X E = iWJLH the scalar prod-uct of E and V' X H* = J* + iweE*(e is assumed to be real) is sub-tracted, and if use is made of the vector identity

V' . (E X H*) = H* .V' X E - E . V' X H*

the following equation is obtained:

V' . (E X H*) = - J* . E + iW(JLH • H* - eE . E*) (87)

which, with the aid of definitions (83), (84), and (86), yields the mono-chromatic form of Poynting's vector theorem!

V'. S = -~J*. E + 2iw(wm - 1V.)

The real part of this relation, i.e.,

V' • (Re S) = Re ( - ~J* . E)

(88)

(89)

expresses the conservation of time-average power, the term on the rightrepresenting a source (when positive) or a sink (when negative) and cor-respondingly the one on the left an outflow (when positive) or an inflow(when negative).In Poynting's vector theorem (88) a term involving the difference

wm - W. appears. To obtain an energy relation (for the monochro-matic state) which contains the sum Wm + lb. instead of the differencewm - W. we proceed as follows. From vector analysis we recall that.the quantity

V' . (oE X H* + E* X oH)ow ow

1 F. Emde, Elektrotech. M aschinenbau, 27: 112 (1909).

16

(90)


is identically equal to

H* • V' X oE - oE . V' X H* +~!:!.V' X E* - E* . V' X oH (91)ow ow ow ow

From Maxwell's equations V' X E = iWJLH and V' X H = J - iwEE itfollows that

V' X oH = ~ (V' X H) = ~. (J - iWEE) = oj _ iEE _ iWEoE _ iwE OEow ow ow ow ow ow

and V' X H* = J* + iWEE*

Substituting these relations into expression (91) we obtain the desiredenergy relation

V' . (oE X H* + E* X ~!!)= i [o(WJL) H • H* + O(WE)E . E*JoW ow ow ow

_ oE. 1* _ E* . oj (92)ow ow

which we call the "energy theorem." Here we interpret as the time-average electric and magnetic energy densities the quantities

(93)

which reduce respectively to expressions (83) and (84) when the mediumis nondispersive, i.e., when OE/OW = 0 and OJL/ow = o.

17

Radiation frommonochromatic

sources inunbounded regions 2

The problem of determining the electromagnetic field radi-ated by a given monochromatic source in a simple,unbounded medium is usually handled by first finding thepotentials 'of the source and then calculating the field froma knowledge of these potentials. However, this is not theonly method of determining the field. There is an alter-native method, that of the dyadic Green's function, whichyields the field directly in terms of the source current. Inthis chapter these two methods are discussed.

2.1 The Helmholtz Integrals

We wish to find the vector potential A and the scalarpotential cP of a monochromatic current J, which is confinedto a region of finite spatial extent and completely surroundedby a simple, lossless, unbounded medium. For this pur-pose it is convenient to choose the Lorentz gauge

V' • A = iWEfJ.cP (1)

In this gauge, cP and A must satisfy the Helmholtz equations(see Sec. 1.4)

1V'2et>(r) + k2cP(r) = - - per)

E

V'2A(r) + k2A(r) = -fJ.J(r)

(2)

(3)

19


Since the medium is unbounded, q, and A must also satisfy the radiationcondition. In physical terms this means that q, and A in the far zonemust have the form of outwardly traveling spherical (but not neces-sarily isotropic) waves, the sphericity of the waves being a consequenceof the confinement of the sources p and J to a finite part of space.Let us first consider the problem of finding q,. We recall from the

theory of the scalar Helmholtz equation that q, is uniquely determinedby Eq. (2) and by the radiation condition 1

lim l' (uq, - ikq,) = 0r....•'" ur

(4)

where r = (yr' r) is the radial coordinate of a spherical coordinatesystem r, e, 1/;. To deduce from this radiation condition the explicitbehavior of q, on the sphere at infinity, we note that the scalar Helm-holtz equation is separable in spherical coordinates and then write q,in the separated form q,(r) = f(e,1/;) u(1'), where f is a function of theangular coordinates and u is a function of l' only. Clearly the radiationcondition (4) is satisfied by u(1') = (1/1') exp (ik1') and accordingly atgreat distances from the source the behavior of q, must be in accordwith

eikrlim q,(r) = f(e,1/;) -T-+OO r

(.5)

That is, the solution of Eq. (2) that we are seeking is the one that hasthe far-zone behavior (5).Since the scalar Helmholtz equation (2) is linear, we may write q, in

the form2

q,(r) = ~ f p(r')G(r,r')dV' (6)

1This is Sommerfeld's "Ausstrahlungbedingung"; see A. Sommerfeld, DieGreensche Funktion del' Schwingungsgleichung, JahTesbericht d. D. Math.VeT., 21: 309 (1912).

2 From the point of view of the theory of differential equations, the solutionof Eq. (2) consists of not only the particular integral (6) but also a comple-mentary solution. In the present instance, however, the radiation conditionrequires that the complementary solution vanish identically.

20

Monochromatic sources in unbounded regions

where G(r,r') is a function of the coordinates of the observation pointr and of the source point r', and where the integration with respect tothe primed coordinates extends throughout the volume V occupied byp. The unknown function G is determined by making expression (6)satisfy Eq. (2) and condition (5). Substituting expression (6) into Eq.(2) we get

f p(r')(V2 + lc2)G(r,r')dV' = - p(r) (7)

where the Laplacian operator operates with respect to the unprimedcoordinates only. Then with the aid of the Dirac 0 functionl whichpermits p to be represented as the volume integral

p(r) = fp(r')o(r - r')dV' (r in V) (8)

we see that Eq. (7) can be written as

f p(r')[(V2 + k2)G(r,r') + o(r - r')]dV' = 0 (9)"

From this it follows that G must satisfy the scalar Helmholtz equation

V2G(r,r') + k2G(r,r') = - o(r - r') (10)

Since G satisfies Eq. (2) with its source term replaced by a 0 function,G is said to be a Green's function2 of Eq. (2).The appropriate solution of Eq. (10) for r ~ r' is

eiklr-r'lG(r,r') = a I 'I (11)r-r

1The 0 function has the following definitive properties: oCr - r') = 0 for

r ~ r' and = 00 for r = r'; /.f(r)o(r - r')dV = fer') forr' in V and = 0for r' outside of V wherefis any well-behaved function. See P. A. M. Dirac,"The Principles of Quantum Mechanics," pp. 58-61, Oxford UniversityPress, London, 1947. See also L. Schwartz, TMorie des distributions,Actualites scientijiques et industrielles, 1091 and 1122, Hermann et Cie, Paris,1950-51.

2 See, for example, R. Courant and D. Hilbert, "Methods of MathematicalPhysics," vol. 1, pp. 351-388, Interscience Publishers, Inc., New York, 1953.

21


where a is a constant. It becomes clear that this solution is compatiblewith the requirement that the form (6) satisfy condition (5) when werecall the geometric relation

Ir - r'l = vr2 + r'2 - 2r' r' = r VI + (r' /r)2 - 2r' . r/r2 (12)

where r2 = r . rand r'2 = r' . r/, and from this relation find the limitingform

eiklr-r'l eikrlim G(r,r') = lim a I /1 ~ a - exp (-ikr' . r/r) (13),-....+00 r~c:o r - r r

To determine the constant a, expression (11) is substituted into Eq. (10)and the resulting equation is integrated throughout a small sphericalvolume centered on the point r = r". It turns out that a must beequal to 7i7l", and hence the Green's function is

eiklr-r'lG(r,r') = 4 I '17I"r - r

(14)

Therefore, since the form (6) satisfies Eq. (2) and condition (5) whenG is given by expression (14), the desired solution of Eq. (2) can bewritten as the Helmholtz integral

1J eiklr-r'lq,(r) = - p(r') 4 I '1dV'

E 7I"r - r(15)

Now the related problem of finding A can be easily handled. Clearly,the appropriate solution of Eq. (3) must be the Helmholtz integral

J eiklr-r'lA(r) = p. J(r') 471"Ir_ r/I dV' (16)

because it has the proper behavior on the sphere at infinity and it sat-isfies Eq. (3). To show that it satisfies Eq. (3), one only has to operateon Eq. (16) with the operator (V' + k2) and note that J(r') depends onthe primed coordinates alone and that the Green's function (14) obeysEq. (10).When in addition to the electric current J there is a monochromatic

magnetic current distribution Jm of nnite spatial extent, the antipoten-tials q,m and Ae should be iJ?-voked. The magnetic scalar potential cPmand the electric vector potential Ae satisfy the Helmholtz equations

22

Monochromatic sources in unhounded regions

(see Sec. 1.4)

(17)

(18)

where V' • A. = iWP.E!Pm and V'. Jm = iwPm. A procedure similar to theone we used in obtaining the Helmholtz integrals for !Pand A leads tothe following Helmholtz integrals for !Pmand A.:

1 f eiklr-r'l!PmCr) = - Pm(r') 4 I 'IdV'p. 11'1'-1'

f eikjr-r'l

A.(r) = E Jm(r') 411'11' _ 1"1dV'

(19)

(20)

From a knowledge of !P,A, !Pm, A. the radiated electric and magneticfields can be derived by use of the relations (see Sec. 1.4)

E = - V'!p+ iwA - !V' X A.E

H = !V' X A - V'!pm+ iwA.p.

(21)

(22)

It is sometimes desirable to eliminate !Pand !Pmfrom these relations andthereby express E and H in terms of A and A. only. This can be donewith the aid of

t-V'!p = -V'(V'.A)WEP.

and i-V'!pm = - V'(V. A.)WEP.

(23)

which follow from the gradients of the Lorentz conditions V'. A = iWEP.!Pand V' • A. = iwp.!Pm' Thus relations (21) and (22) may be written asfollows:

E = iw [A + b V'(V' • A) ] ....• ~ V' X A.

H = ; V' X A + iw [ A. + b V'(V' • A.) ]

(24)

(25)

23


To enable us to cast A + ~ V(V. A) and A. + k V(V. A.) into the

form of an operator operating on A and A., we introduce the unit dyadicu and the double-gradient dyadic VV which in a cartesian system ofcoordinates are expressed by

m,=3n=3

U = I I emenOmnm,=ln=l

m=3n=3\' \' a avv = L. L. emen ax:' aXnm,=ln=l

(26)

(27)

where Xi (i = 1, 2, 3) are the'cartesian coordinates, ei (i = 1, 2, 3) arethe unit base vectors, and the symbol omn is the Kronecker delta, whichis 1 for m = nand 0 for m ;;e n. The properties of u and \7V that wewill need are u. C = C and (VV) . C = V(V. C), where C is any vectorfunction. These properties can be demonstrated by writing C in com-ponent form and then carrying out the calculation. Thus

= L L L eme" . epCpOmn = L L L emOnpCpOmn = L epCp = C (28)mn.p mll.p P

where e" . ep = Onp, and

With the aid of these results, relations (24) and (25) become

E = iw (u + ~ VV) . A - 1 V' X A.k- €

24

(30)


(31)

Using the Helmholtz integrals (16) and (20) and taking the curl oper-

ator and the operator u + iz V'V'under the integral sign, we get

J ( 1 ) [ eiklr-r'I]E = iwp, u + k2 V'V' • J(r') 41l"lr _ r'l dV'

J [ eiklr-r'l ]- V' X Jm(r') 41l"jr _ r'l dV'

J [ eiklr-r'I]H = V' X J(r') 41l"lr _ r'l dV'

J ( 1 ) [ eiklr-r'l ]+ iWE U + k2 V'V' • Jm(r') 41l"tr _ r'l dV'

(32)

To reduce these expressions we invoke the following considerations.If a is a vector function of the primed coordinates only and w is a scalarfunction of the primed and unprimed coordinates, then

= (, )' e e ~-~"!!-).a = (V'V'w) . a (34)'-' L. m -n OXmOXnm n

and

V' X (aw) = (I emo~JX aw = (I em :~) X a = V'w X a t35)m m .

In view of identities (34) and (35), expressions (32) and (33) reduce tothe following:

J [( 1 ) eiklr-r'l ]E = iwp, U + k2 V'V' 41l"lr _ r'l • J(r')dV'

J ( eiklr-r'l)~ V' 41l"lr _ r'l X Jm(r')dV' (36)

25


J ( eiklr-r'l)H = V 41J"lr_ r'l X J(r')dV'

J [( 1 ) eiklr-r'l ]+ iWE U + k2 VV 41J"lr_ r'l • Jm(r')d.V'

Since the quantity

eiklr-r'lG(r,r') == 4 I 'I1J"r - r

(37)

(38)

(39)

is known as the free-space scalar Green's function, it is appropriate torefer to the quantity

r(r,r') == ( U + b VV) 4:1:lr=r'~'1 == ( U +b VV) G(r,r')

as the free-space dyadic Green's function. Using (38) and (39) we canwrite expressions (36) and (37) as follows:

E(r) = iwp.fr(r,r') • J(r')dV' - fVG(r,r') X Jm(r')dV'

H(r) = fVG(r,r') X J(r')dV' + iWEfr(r,r') . Jm(r')dV'

(40)

(41)

These relations formally express the radiated fields E, H in terms of thesource currents J and Jm.1

2.2 Free-space Dyadic Green'sFunction

In the previous section we derived the free-space dyadic Green's func-tion using the potentials and antipotentials as an intermediary. Inthis section we shall derive it directly from Maxwell's equations.We denote the fields of the electric current by E', H' and those of the

magnetic current by E", H". The resultant fields E, H are obtained

1 If the point of observation r lies outside the region occupied by the source(which is the case of interest here), then Ir - r'l ~ 0 everywhere and theintegrals are proper. On the other hand, if r lies within the region of thesource, then Ir - r'l = 0 at one point in the region and there the integralsdiverge. This improper behavior arises from interchanging the order ofintegration and differentiation. See, for example, J. Van Bladel, SomeRemarks on Green's Dyadic for Infinite Space, IRE Trans. Antennas Propa-gation, AP-9 (6): 563-566 (1961).

26


by superposition, i.e., E = E' + E" and H = H' + H". Let us con-sider first the fields E', H' which satisfy Maxwell's equations

V' X H' = J - iWEE' and V' X E' = iWJLH' (42)

From these equations it follows that E' satisfies the vector Helmholtzequation with J as its source term:

(43)

In this equation, E' is linearly related to J; on the strength of this lin-earity we may write

E'(r) = iWJLfr(r,r') . J(r')dV' (44)

where r is an unknown dyadic function of l' and 1". To deduce thedifferential equation that r must satisfy we substitute this expressioninto the vector Helmholtz equation. Thus we obtain

V'X V'X fr(r,r') . J(r')dV' - k2 fr(r,r'). J(r')dV'

= fu.J(r')Il(r - r')dV' (45)

Noting that the double curl operator may be taken under the integralsign and observing that V' X V' X (r. J) = (V' X V' X r) .J, we getthe following equation:

f[V' X V'X r(r,l") - k2r(r,r') - uo(r - 1")]. J(r')dV' = 0 (46)

Since this equation holds for any current distribution J(r'), it followsthat r(r,r/) must satisfy

(curl curl - k2)r(r,r') = ull(r - 1") (47)

Now we construct a dyadic function r such that Eq. (47) will besatisfied and expression (44) will have the proper behavior on thesphere at infinity. One way of doing this is to use the identity curlcurl = grad div - V'2 and write Eq. (47) in the form

(V'2+ k2)r(r,r') = - ull(r - 1") + V'V'• r(r,r') (48)

27

Theory of electromagn~tic wave propagation

!<'romEq. (47) it follows that \7 . r(r,r') = - ~2 \7o(r - r'). With the

aid of this relation, Eq. (48) becomes

(\72+ k2) r(r,r') = - ( u + :2 \7\7) o(r - r')

Clearly this equation is satisfied by

r(r,r') = ( u + ;2 \7\7) G(r,r')

where G(r,r') in turn satisfies

(\72 + k2)G(r,r') = - o(r - r')

(49)

(50)

(51)

To meet the radiation condition, the solution of Eq. (51) must be

iklr-r'lG(r,r') = 4 el 'I.1rr-r

Thus the desired dyadic Green's function is

(1 ) eiklr-r'l

r(r,r') = u + k2 \7\7 41r/r _ r'l

The fields E", H" satisfy Maxwell's equations

(52)

(53)

\7 X H" = - iwtE" and \7 X E" = -Jm + iw~H" (54)

from which it follows that H" satisfies the vector Helmholtz equationwith Jm as its source term:

\7 X \7 X H" - k2H" = iwd",

As before, if we write

II" = iwtfr(r,r') . Jm(r')dV'

28

(55)

(56)


then Eq. (55) and the radiation condition will be satisfied when r isgiven by expression (53). That is, the dyadic functions in the inte-grands of Eqs. (44) and (56) are identical.

Since H' = ~ V' X E' and E" = - J:- V' X H" it follows from2W~ 2WE'

expressions (44) and (.'56) that

H' = V' X fr(r,r') . J(r')dV' = f(V' X r(r,r')] . J(r')dV' (57)

E" = - V' X fr(r,r') . Jm(r')dV' = - f[V' X r(r,r')] . Jm(r')dV' (58)

But r(r,r') = (u + ~ V'V') G(r,r') and consequently

(V' X r(r,r')]. J(r') = V'G(r,r') X J(r')

In view of this, Eqs. (57) and (58) become

H' = fV'G(r,r') X J(r')dV'

E" = - fV'G(r,r') X Jm(r')dV'

(59)

(60)

Combining expressions (44) and (56) with (60) and (59) respectively,we get

E = E' + E" = iw~fr(r,r') . J(r')dV' - fV'G(r,r') X Jm(r')dV' (61)

H = H' + H" = iWEfr(r,r') . Jm(r')dV' + fV'G(r,r') X J(r')dV' (62)

These expressions are identical to expressions (40) and (41).

2.3 Radiated Power

For the computation of the power radiated by a monochromatic elec-tric current, the complex Poynting vector theorem (see Sec. 1.5)

V' • S = - HJ* . E + 2iw(wm - We) (63)

29


can be used as a point of departure. The real part of this equationwhen integrated throughout a volume V bounded by a closed surfaceA, which completely encloses the volume Vo occupied by the current J,yields

Re Iv V'. S dV = -72 Re Ivo J*. E dV (64)

Converting the left side by Gauss' theorem to a surface integral overthe closed surface A with unit outward normal n, we get

Re f n' S dA = -72 Re f j*. E dVA Vo

(65)

The right side gives the net time-average power available for radiationand the left side the time-average radiated power crossing A in an out-ward direction. In agreement with the conservation of power this rela-tion is valid regardless of the size and shape of the closed surface A aslong as it completely encloses Vo. Thus we see that the time-averageradiated power can be computed by integrating - (72) Re (j* . E)throughout Vo or, alternatively, by integrating Re (n' S) over anyclosed surface A eIl:closing Vo• In one extreme case, A coincides withthe boundary Ao of Vo; in the other, A coincides with the sphere atinfinity, Aoo'

The imaginary part of Eq. (63) when integrated throughout V yields

1m f n' S dA = -72 1m f J*. E dV + 2", f (wm - w.)dV (66)A Vo v

As before, A is an arbitrary surface completely enclosing Vo• WhenA coincides with Ao this equation becomes

1m lAO n . S dA = -72 1m IVo j* •E dV + 2",(Wmint - W.int) (67)

where

W int = I w dVm Vo m W.int = f w. dV

Vo(68)

denote the (internal) time-average magnetic and electric energies

30


stored inside Vo. When A coincides with Aoo it becomes

1m J n' S dA = - ~ 1m J J*. E dVA. Vo

where

W ex = J 11) dVm V-Vo mWex = J 11) dV

• V - Vo •(70)

denote the (external) time-average magnetic and electric energiesstored outside Vo. In the far zone, S is purely real and consequentlyEq. (69) reduces to

From this relation it is seen that the volume integral of - (~) 1m (J*. E)throughout Vo gives 2w times the difference between the time-averageelectric and magnetic energies stored in all space, i.e., inside Vo and out-side Vo• A relation involving only the external energies is obtained bysubtracting Eq. (71) from Eq. (67), viz.,

(72)

Now, in accord with the left side of relation (65), we shall find thetime-average radiated power by integrating Re (n' S) over the sphereat infinity. As was shown in Sees. 2.1 and 2.2, the electric field Eproduced by a monochromatic current J is given by

E(r) = iwj.£ J r(r,r'). J(r')dV'Vo

where

(1 ) eiklr-r'l

r(r,r') = u + k2 VV 41rJr_ r'l

(73)

(74)

31


Since

eiklr-r'l eiklr-r'lV . - V' ~._--411"lr- r'l - - 411"lr- r'l

we may write expression (74) in the form

( 1) eiklr-r'lr(r,r') = u + k2 V'V' 411"lr- r'r

(75)

(76)

with the double gradient operating with respect to the primed coordi-nates only. In the far zone, which is defined by

1'» 1" and kr» 1 (77)

where l' = yr-=r and r'= yr'~, the following approximation isvalid:

(78)

where ere= rlr) is the unit vector in the direction of r. In this approxi-mation we may replace exp (iklr - r'/) by exp [ik(r - er • r')] andl/lr - r'l by 1/1'. Accordingly Eq. (76) reduces to

(1 ) ikrr(r r') = u + - v'v' ~ e-ike,'r', k2 411"1'

(79)

in the far zone. The double gradient V'V' operates on e-ike,.r' only,and since

(80)

we have

(81)

With the aid of this relation, expression (79) for the far zone r becomes

eikrr(r,r') = (u - erer) 4- e-ike,'r'11"1'

32

(82)


Substituting this expression into Eq. (73) and using the vector identity(u - ercr)' J = J - cr(er• J) = -Cr X (cr X J) we obtain the follow-ing representation for the far-zone electric field:

eikr [J ]E(r) = -iwJ.l.

41Trer X Cr X voe-;ke,'r'J(r')dV' (83)

The far-zone magnetic field is found by taking the curl of Eq. (83) in

accord with the Maxwell equation H = J:- yo X E. Thus1.wJ.l.

eikr [J ]-H(r) = ik - er X e-ike,.r'J(r')dV'

41Tr Vo(84)

Comparing expressions (83) and (84) we see that the far-zone E and Hare perpendicular to each other and to Cr, in agreement with the factthat any far-zone electromagnetic field is purely transverse to thedirection of propagation, viz., in the far zone

or H = r~(cr X E)'1M (85)

is always valid. Expressions (83) and (84) yield the following expres-sion for the far-zone Poynting vector:

(86)

The notation ICl2 where C is any vector means C. C*. From expres-sion (86) we see that S is purely real and purely radial, i.e., directedparallel to Cr' The element of area of the sphere over which S is to beintegrated is r2 dn, where dn is an element of solid angle. Hence, thetime-average radiated power P is given by

P = JA~ n . S dA = J Cr' Sr2 dn

= I~-,~ f dn I c, X f e-ike,.r'J(r')dV'12

(87)'1f 321T2 Vo

33


This way of calculating the radiated power is called the "Poyntingvector method."A formally different way of calculating the radiated power consists in

integrating throughout Vo the quantity - (72) Re (J* . E) in which Eis taken to be the radiative electric field as given by Eq. (73). Thisalternative procedure, which was proposed by Brillouin,l yields

p = -72 Re J J*. E dVvo

= WIJ. J J J*(r). (u + .!- vv) sin (klr ~ r'D . J(r')dV' dV (88)81T vo VO k2 Ir - r I

and is called the "emf method" since it makes use of the inducedelectromotive force (emf) of the radiative electric field.Although representations (87) and (88) of these two methods are

apparently different, they nevertheless yield the same result for P andin this sense are consistent. To exemplify this we now apply these twomethods to the relatively simple case of a thin straight-wire antenna.The antenna has a length 2l and lies along the z axis of a cartesiancoordinate system with origin at the center of the wire. Since the wireis thin, the antenna current is closely approximated by the filamentarycurrent

J = e.Ioo(x)o(y)f(z) (89)

where lois the reference current, e. is the unit vector in the z direction,and fez) is generally a complex function of the real variable z. By useof this current we get

Jvo e-ike,'r' J (r')dV' = e.lo J~l e-ik., coe ef(z')dz' (90)

where (J is the colatitude in the spherical coordinate system (r,(J,It»defined by x = r sin (Jcos cP, Y = r sin (Jsin cP, and z = r cos (J. Denot-ing the unit vectors in the r, (J, and cP directions respectively by er, ee,and eq,and noting that er X e. = -eq, sin (J, we find from Eq. (90) and

1L. Brillouin, Origin of Radiation Resistance, Radioelectricite, April, 1922.

34


expression (87) that the Poynting vector method yields

p = ~ ~ 101* I I I I f(z')f*(z)dz dz' r'" eik(z-z') C08 8 sin3 0 dO'\j"i 1611" 0 -I -I Jo (91)

Moreover, by substituting the current as given in Eq. (89) into expres-sion (88), we see that the emf method yields

p = WJJ. 101* II II f(z')f*(z) (1 + ~~) sin (klz - z'l) dzdz' (92)811" 0 -I -I k2 az2 Iz - z'l

To show that expressions (91) and (92) are equivalent, we invoke thefollowing elementary results:

r,.. eik(z-z') COB 8 sin3 0 dO = i. (~~~:U- cos u)Jo ~ u

(1 + !~)sin (klz - z'\) _ 2k (sin 'u _ )

k2 a 2 I 'I - 2 cos UZ z-z U U

where u = k(z - z'). With the aid of these results and the introduc-tion of the new variables ~ = kz and 1] = kz' expressions (91) and (92)pass into the 'common form

p = ~ ~ 10Iri I kl I kl d1]d~ f(1])f*W [sin (~ - 1]) - cos (~- )J

411"'\j"i -kl -kl (~ - 1])2 ~ - 1] 1]

(93)

Thus we see that the Poynting vector method and the emf methodultimately lead to the same formula (93) for the time-average radiatedpower P and hence are consistent with each other.From a practical viewpoint, formula (93) as it stands is too clumsy

to use, owing to the presence of the double integral. However,Bouwkamp by successive transformations succeeded in reducing thedouble integral to a repeated integral and then finally to an elegantform involving only single integrals. To demonstrate the capabilitiesof this form he applied it to several "classical" cases which had been

35


handled previously by the Poynting vector method. For details werefer the reader to his original paper.!The present discussion may be extended to the case of magnetic cur-

rents by using the duality transformations of Sec. 1.2. For example,if we replace J, E, H respectively by - vi EI IL Jm, - vi ILl E H, vi EI IL Ein the far-zone field formulas (83) and (84), we obtain the correspondingformulas for the far-zone field of a monochromatic magnetic currentdensity:

eikr (f )H = -iWE 41rr er X er X vo e-ike,'r'Jm(r')dV'

e~( f )E = -ik - er X e-ike,.r'Jm(r')dV'41rr Vo

The Poynting vector of this far-zone electromagnetic field is

(94)

(95)

S = ;YzE X H* = er I"! ~ I e, X f e-ike,.r'Jm(r')dV'12

(96)"J IL 321r2r2 Vo

and consequeritly the time-average radiated power is

P = f. n' S dA = f e. Sr2 dnA.. A•• r

= I~~ f dn Ier X f e-ike,.r'Jm(r')dV'12

(97)"J IL 32712 Vo

! C. J. Bouwkamp, Philips Res. Rept., 1: 65 (1946).

36

Radiationfrom wire 3antennas

As a practical source of monochromatic radiation the wireantenna plays an important role. The field radiated bysuch an antenna can be obtained from a knowledge of itscurrent distribution by using the formulas derived in theprevious chapter. Although the determination of theantenna current is a boundary-value problemof considerablecomplexity, a sufficiently accurate estimate of the currentdistribution can be obtained in the case of thin wires byassuming that the antenna current is a solution of the one-dimensional Helmholtz equation and hence consists of anappropriate superposition of simple waves of current. Thissimplifying approximation yields satisfactory results for thefar-zone field and for those quantities that depend on thefar-zone field, e.g., radiation resistance and gain, becausethe far-zone field in almost all directions is insensitive tosmall deviations of the current from the exact current. Theradiation properties of thin-wire antennas and their arraysare discussed in this chapter.

3.1 Simple Waves of CurrentWe consider a straight-wire antenna lying along the z axisof a cartesian coordinate system with one end at z = -l andthe other at z = l, as shown in Fig. 3.1. Since the wire is

37


z

x

y

Fig. 3.1 Coordinate sys-temfora straight-wire antenna ex-tending from z =-Z to z = z. Qis observationpoint. Q' is pro-jection of Q inx-y plane.

thin and since we wish to calculate only the far-zone field it is apermissible mathematical idealization to assume that the antennacurrent density is the filamentary distribution

J = ezo(x)o(y)j(z) (1)

The total current is the integral of this distribution over the crosssection of the wire:

I(z) = ezI(z) = IJ dx dy = e.j(z)Jo(x)o(y)dx dy = ezj(z) (2)

(3)k = wlc = 2'Tr/'A

It is supposed that the wire is cut at some cross section z = 1'/ and amonochromatic emf is applied across the gap. The current is neces-sarily a continuous function of z, but the z derivative of the currentmay be discontinuous at the gap. The antenna is said to be "center-fed" when 1'/ = 0 and "asymmetrically fed" when 1'/ ;;c O.For a center-fed antenna, j(z) is a symmetrical function of z and

satisfies the one-dimensional Helmholtz equation 1

crJ + k2j = 0dz2

as well as the end conditions

feZ) = f( -l) = 0 (4)

1 It appears that Pocklington was the first to show that the currents alongstraight or curved thin wires in a first approximation satisfy the Helmholtzequation. See H. C. Pocklington, Proc. Cambridge PhiZ. Soc., 9: 324 (1897).

38

Radiation from wire antennas

The two independent solutions of Eq. (3) are the simple waves eikz ande-ikz• Accordingly a general form for the current is

I(z) = Aeikz + Be-ikz (5)

where A and B are constants. Writing this form for the two segmentsof the antennas, we have

I1(z) = Aleikz + Ble-ikz

I2(z) = A2eikz + B2e-ikzfor 0 ::::;z ::::;l

for -l ::::;z ::::;0(6)

When applied to these expressions,the end conditions (4) yield

(7)

from which it follows that BIIAI = _e2ikl, BdA2 = _e-2ikl• Withthese results, Eqs. (6) become

I1(z) = -2iA1eikl sin k(l- z) for 0::::;z ::::;l

I2(z) = 2iA2e-ikl sin k(l + z) for -l ::::;z ::::;0(8)

The continuity condition 11(0) = 12(0) requires that A I and A2 berelated by

(9)

In view of this connection between Al and A2 it followsfrom Eqs. (8)that the current distribution, apart from an arbitrary multiplicativeconstant 10, is given by the standing wavel

I(z) = 10 sin k(l - jz/) (10)

which, for several typical cases, is displayed in Fig. 3.2. This "sinus-oidal approximation" is adequate for the purpose of computing thefar-zone radiation pattern of a center-fed straight-wire antenna,provided the antenna is neither "too thick" nor "too long." A closerapproximation to the true current may be obtained heuristically byadding to the sinusoidal current a quadrature current, which takes into1 J. Labus, Z. Hochjrequenztechnik und Elektrokustik, 41: 17 (1933).

39

(a)

(b)

kl=1r

(c)

Fig.3.2 Radiation patterns of a center-driven thin-wire antenna ofcurrent distribution shown by dotted lines.

40

kl •• 7-rrj6

(d)

various lengths shown by solid lines. Assumed sinusoidal

41


account the reaction of the radiation and the ohmic losses on thecurrent. 1 Since the antenna is center-fed, the principal alteration thatsuch a quadrature current can make on the far-zone radiation pattern isthe presently negligible one of relaxing the intermediate nulls of thepattern. 2

However, for an asymmetrically fed antenna (1] ;t. 0) the radiationpattern calculated solely on the basis of a simple standing wave ofcurrent can be in serious error due to the presence of traveling waves ofcurrent. The standing wave, which for arbitrary values of 1] has theform3

1(z) = 10 sin k(l + 1]) sin k(l - z)

1(z) = 10 sin k(l- 1]) sin k(l + z)

for 1] ::; z ::; l

for -l ::;z ::; 1]

(11)

always gives rise to a radiation pattern that is symmetrical about theplane (J = 7r/2. On the other hand, a traveling wave produces anasymmetrical radiation pattern, viz., a pattern tilted toward the direc-tion of the traveling wave. Accordingly the traveling waves tend totilt the lobes of the pattern and to change their size. Thus if thetraveling waves are appreciable, marked changes in the shape of theradiation pattern can occur. Generally the problem of finding theradiation pattern of an asymmetrically driven antenna cannot behandled adequately within the framework of the simple wave theory,except in those cases where either the standing wave or the travelingwaves dominate the pat.tern.

3.2 Radiation from Center-drivenAntennas

As indicated by Eqs. (83), (84), and (86) of Chap. 2, the calculation ofthe far-zone radiation emitted by a distribution of monochromatic

1Ronold King and C. W. Harrison, Jr., Proc. IRE, 31: 548 (1943).2 C. W. Harrison, Jr., and Ronold King, Proc. IRE, 31: 693 (1943).3 See, for example, S. A. Schelkunoff and H. T. Friis, "Antennas: Theory

and Practice," chap. 8, John Wiley and Sons, Inc., New York, 1952.

42


current centers on the evaluation of the so-called radiation vector Ndefined by the integrall

N = J e-ike,'r'J(r')dV'Vo

(12)

where er is the unit vector pointing from the origin to the point ofobservation and r' is the position vector extending from the origin tothe volume element dV'. The required information on J can beobtained either by solving the boundary-value problem which theanalytical determination of J poses or by choosing the current onempirical grounds. In the present case of a thin-wire antenna, thelatter alternative is adopted, according to which it is alleged that asufficiently accurate representation of the antenna current can bebuilt from simple waves to agree with the results of measurement.Accordingly, let us consider the case of a center-driven thin-wire

antenna lying along the z axis with one end at z = -l and the other atz = l. It is known a posteriori that the current distribution alongsuch an antenna may be approximated, insofar as the far-zone radiationis concerned, by the sinusoidal filamentary current

J(r) = e.Ioo(x)o(y) sin k(l - Izl) (13)

Substituting this assumed current into definition (12) and performingthe integrations with respect to x' and y', we get the one-dimensionalintegral

N = ezlo J~l e-ikz' cos 9 sin k(l - Iz'l)dz'

which by use of the integration formula

J ~EeaE sin (b~ + c)d~ = a2 + b2 [a sin (b~ + c) - b cos (b~ + c)]

yields

N = ez2Io cos (kl cos.O) - cos kl

k sm2 0

(14)

(15)

(16)

1S. A. Schelkunoff, A General Radiation Formula, Proc. IRE, 27: 660-666(1939).

43


With the aid of this result and the vector relations er X e. = -eq, sin (),er X (er X e.) = eB sin (), it follows from Eqs. (83), (84), and (86) ofChap. 2 that the far-zone electric and magnetic fields are

E _ i I~eikr 10cos (kl co~ () - cos kl

'.IB = ~ E 27l"r ----s~m-()---

and

. eikr I cos (kl cos ()) - cos klHq, = -2 211'1' 0 sin ()

and that the radial component of the Poynting vector is

8r= I~-.!.L [cos (kl co~ ()) - cos kl]2

~f 811'2r2 sm ()

In these expressions, the common factor

F«() = cos (kl co~ () - cos klsm()

(17)

(18)

(19)

(20)

is the radiation pattern of the antenna. Since the radiation pattern isindependent of q, it is said to be "omnidirectional." When the antennais short compared to the wavelength (kl« 1) the radiation patternreduces to!

F«() = }-2(kl)2 sin () (21)

From this we see that the radiation pattern of a short wire antenna con-sists of a single lobe that straddles the equatorial plane () = 11'/2 andexhibits nulls at the poles () = 0 and ()= 11'. As kl increases up tokl = 11' the lobe becomes narrower and more directive. As kl exceedskl = 11' and approaches kl = 311'/2, two side lobes appear, graduallygrowing in size and ultimately becoming larger than the central lobeitself. (See Fig. 3.2.)Since F«() is an even function of () - 11'/2 that vanishes at () = 0 and

1 In the case of a Hertzian dipole F«() = kl sin (). To show this, we recallthat the current density of a Hertzian dipole of length 2l, located at the originof coordinates and directed parallel to the z axis, is defined as J = e,loo(x)o(y),then note that for this current N = e,21I 0 and hence Sr = VPJE (I 02 j811'2r2)(kl sin ()2.

44


8 = 1r, it may be expanded I in a Fourier series of the form

..F(8) = L b2n+l sin (2n + 1)8

n=O

where

b2n+l = ~ (" F(8) sin (2n + 1)8 d81r 10

(22)

(23)

For small kl, all the higher-order coefficients are, to a good approxi-mation, negligible compared to the first coefficient

2/"bl = - [cos (kl cos 8) - cos kl]d8 = 2Jo(kl) - 2 cos kl1r 0 (24)

Although this simple approximation deteriorates as the length of theantenna increases, for a half-wave dipole (kl = 1r/2) it is stilI satis-factory and yields

(1r )cos 2 cos 8

F(8) = . 8 ~ 0.94.5 sin 8sm (25)

From a practical viewpoint this approximate representation of the dis-tant field of a half-wave dipole provides a useful simplification. Forexample, it enables one to obtain a convenient expression for the radia-tion resistance of certain linear arrays of half-wave dipoles. 2

3.3 Radiation Due to TravelingWaves of Current, Cerenkov

Radiation

In the previous section we noted that the far-zone radiation field of acenter-fed thin-wire antenna is determined with sufficient accuracy by

1 R. King, The Approximate Representation of the Distant Field of LinearRadiators, Proc. IRE, 29: 458-463 (1941); C. J. Bouwkamp, On the EffectiveLength of a Linear Transmitting Antenna, Philips Res. Rept., 4: 179-188(1949).

2 C. H. Papas and Ronold King, The Radiation Resist,ance of End-fire andCollinear Arrays, Proc. IRE, 36: 736-741 (1948).

45


using the standing-wave part of the antenna current and ignoring thetraveling-wave part. In the present section we shall discuss the con-verse state of the antenna, wherein the traveling-wave part of the cur-rent is dominant and the standing-wave part is quite negligible. Sucha state can be achieved by the proper excitation and termination of theantenna.lAccordingly we assume that the current distribution along a thin-

wire antenna is the traveling wave

J(r) = ezIoo(x)o(y)eipkz (-l~z~l) (26)

Here the index p is the ratio of the velocity of light to the velocity ofthe current wave along the antenna. This index, which is equal to orgreater than unity, depends on the degree to which the antenna isloaded. If the antenna is unloaded, i.e., if the antenna wire is bare, pis approximately equal to unity. Then as the loading is increased2

there is a corresponding increase in p.Substituting expression (26) into definition (12) we get

11. . • sin [kl(p - cos 0)]N = ezIo e-,kZCOB 6e,pkz dz = ez2Io --------I k(p - cos 0)

(27)

This expression for the radiation vector, when introduced into Eqs.(83), (84), and (86) of Chap. 2, yields the following nonvanishing com-ponents of the far-zone fields and the Poynting vector:

E _ ~H _ .~~ eikrI . o sin [kl(p - cos 0)]6 - - <I> - -2 -- osm ---=----~ ~ 27rr p - cos 0

S - ~~ 1 I 2 • 20 sin2 [kl(p - cos 0)]r - - -- 0 sm -------'~-------'~ 87r2r2 (p - cos 0) 2

(28)

(29)

From these expressions it follows that the radiation pattern of the

1A practical example of such an antenna is the "wave antenna" or "Bever-age antenna." See H. H. Beverage, C. W. Rice, and E. W. Kellogg, TheWave Antenna, a New Type of Highly Directive Antenna, Trans. AlEE, 42:215 (1923).

2 The loading may take the form of a dielectric coating or a corrugation ofthe surface.

46

(30)

~ Radiation from wire antennas

---Direction of travelingcurrent wave

Fig.3.3 Typical radiation pattern for traveling wave of current.

traveling wave of current (26) is

F(e) = sin e sin [kl(p - cos 0)]p-cose

When the antenna is short (kl« 1), the radiation pattern (30)reduces to

F(e) = kl sin e (31)

Comparing radiation patterns (31) and (21) we see that for shortantennas (kl « 1) the radiation pattern (21) of the standing wave ofcurrent (13) has the same form (sin e) as the radiation pattern (31) ofthe traveling wave of current (26).However, for longer antennas the patterns (30) and (20) differ mark-

edly, the essence of the difference being that the pattern (20) of thestanding wave is symmetrical with respect to the equatorial planee = 17"/2 whereas the pattern (30) of the traveling wave is asymmetrical.The maxium radiation of the traveling wave appears as a cone in theforward direction, i.e., in the direction of travel of the current wave;the half-angle of the cone decreases as p increases or as kl increases (seeFig. 3.3). This type of conical beam radiation resembles the Cerenkovradiation 1 from fast electrons.

1 P. A. Cerenkov, Phys. Rev., 52: 378 (1937). 1. Frank and 1. Tamm,Comptes rendus de ['Acad. Sci. U.R.S.S., 14: 109 (1937). See also, J. V.Jelley, "Cerenkov Radiation and Its Applications," Pergamon Press, NewYork, 1958.

47


3.4 Integral Relations betweenAntenna Current and RadiationPattern

Again we study the thin-wire antenna, but in this instance we do notspecify the current distribution. That is, we restrict the current dis-tribution only to the extent of postulating a monochromatic current ofthe form

J(r) = e.6(x)6(y)f(z) (Izj ~ l) (32)

where the functionf(z) may be complex. From Eq. (83) of Chap. 2, itdirectly follows that the far-zone electric field of this current distribu-tion is

eikr JIE B = - iwp. - sin 0 e-ik• COB Bf(z )dz

411"T -I(33)

Since the O-dependent factor of this expression is, by definition, theradiation pattern, we have

F(O) = sin 0 JI e-ikzcOBBf(z)dz-I

(34)

This integral relation shows that when f(z) is given in the interval(izi ~ l), the radiation pattern F(O) is uniquely determined for all realangles in the interval (0 ~ 0 ~ 11").To proceed toward a relation that would yield f(z) from a knowledge

of F(O), we cast Eq. (34) into the form of a Fourier integral and thenfind its mate. Accordingly, the finite limits on the integral in Eq. (34)are replaced with infinite ones by assuming thatf(z) vanishes identicallyoutside the interval (jzl ~ l), i.e.,

j(z) = 0 for (35)

With f(z) so continued, Eq. (34) can be written as

F. (0) = J '" e-ikzc08 Bj(z )dzsmO -'"

48

(36)


or, in terms of the new variable 7]( = k cos 8), as

(-k::; 7]::; k) (37)

Now the range of validity of Eq. (37) is extended from (-l~ ::; 1]::; k)to (- 00 ::; 1]::; (0) by letting 8 trace the contour C in the complex 8plane (Fig. 3.4). Such an extension of Eq. (37) leads to the Fourierintegral

F(cos-I1]/k) f'" ..----.-~ = e-'~'f(z)dzVI - 1]2/k2 -'"(38)

By the Fourier integral theorem, the mate of Eq. (38) is

(- 00 ::; z::; (0) (39)

Transforming to the complex 8 plane (by use of 1]= k cos 8) and explic-itly taking into account the requirement (35) that f(z) vanish for

Fig. 3.4 Trace of con-tour C in thecomplex 8plane.

IIIIIIIIII

o

49


Izi ~ l, we obtain the desired relation

fez) = ~ ~ F(B)eikzcoge dB (izi ~ l) (40)211" c

and the side condition

o = ~ ~ F(B)eikzco9 e dB (izi ~ l) (41)211" c

From Eq. (40) it is clear that F(B) must be known along the entire con-tour C before fez) can be evaluated from it. Moreover, since F(B)must satisfy the side condition (41), it cannot be chosen arbitrarily.Nevertheless, it seems possible] to find an F(B) which satisfies Eq. (41)and closely approximates. a prescribed radiation pattern in the range ofreal values (0 ~ B ~ 11").

3.5 Pattern Synthesis by HermitePolynomialsIn connection with the antenna of the previous section we now brieflysketch the approximation method of Bouwkamp and De Bruijn, ~whichenables one to calculate a current distribution that will produce a pre-scribed radiation pattern, or, in other words, enables one to synthesizea given radiation pattern.The point of departure is the integral relation (34) connecting the

radiation pattern F(B) to the current distribution fez). For conven-ience, however, we express this relation in terms of the dimensionlessvariables t = cos B, ~ = kz and the dimensionless constant a = kl.Thus, Eq. (34), apart from an ignorable constant, is written first as

(-l~t~l) (42)

1 For a heuristic discussion of such a possibility, see P. M. Woodward andJ. D. Lawson, The Theoretical Precision with which an Arbitrary RadiationPattern may be obtained from a Source of Finite Size, J. Inst. Elec. Eng., 95(part III); 363-370 (1948).

2 C. J. Bouwkamp and N. G. de Bruijn, The Problem of Optimum AntennaCurrent Distribution, Philips Res. Rept., 1: 135-138 (1946).

50


and then, by use of the shorthand

G(t) = ~VI - t2

as

(-1 S t S 1)

(43)

(44)

Referring to this integral equation, we see that the synthesis problemconsists in finding fW when G(t) is given in the interval -1 S t S 1.By virtue of a theorem due to Weierstrass, I we may approximate the

given function G(t) by a polynomial pet) of sufficiently high degree N:

G(t) = pet) == 'Yo + 'Ylt + ... + 'YNtN

Moreover, we may invoke unknown functions fn(~) such that

(45)

(46)

Substituting expressions (45) and (46) into the integral equation (44),we see that functions fn(~) for n = 0, 1, ... , N must satisfy

(-1 S t S 1) (47)

To find the functionsfn(O, we introduce the Hermite polynomials Hn(u)defined by2

(n = 0, 1,2, ... ) (48)

From this formula, the following result can be verified by repeated

I See, for example, R. Courant and D. Hilbert, "Methods of MathematicalPhysics," vol. 1, p. 65, Interscience Publishers, Inc., New York, 1953.

2 This definition agrees with that of E. T. Whittaker and G. N. Watson,"A Course of Modern Analysis," p. 350, Cambridge University Press, London,1940. Hn(u) = e"'/4Dn(u), where Dn(u) is that given by Whittaker andWatson.

51


partial integrations:

(-I~t~l)

(49)

When the arbitrary positive constant A is large, the factor exp(-A2~2/2)Hn(A~) decreases rapidly to zero as I~I~ 00. Hence, thecontribution of the integration beyond a certain range (say, I~I> a) isnegligible. Also, when A is large, the factor exp ( - t2/2A 2) approachesunity. Accordingly, if we choose A sufficiently large, then Eq. (49)closely approximates

(-I~t~l) (50)

Comparing Eqs. (50) and (47), we see that the functionsfnW are givenby

(51)

Substituting this result into Eq. (46) We get the formal solution ofintegral equation (44):

(A large) (52)

As an application of the above method we now synthesize the radia-tion pattern

F(O) = sin2NH 0

Since t = cos 0, then F(t) = VI - t2 (1 - t2)N and hence

G(t) = (1 - t2)N

52

(53)

(54)


By the binomial theorem, we have

(55)

where the binomial coefficients are given by

(N) N!l = (N - l)!l!

Comparing expansions (55) and (45) we see that

-Y21 = (~) (_i)21

and substituting these values into Eq. (52) we get

The arbitrary positive constant A is chosen such that

{1A=-a

(56)

(57)

(58)

where (3 is greater than the largest root of H 21(U) = o. With the use ofexpression (58), the current distribution (57) is transformed to

(59)

Thus, corresponding to N = 0, N = 2, N = 4, we have the radiationpatterns

F=sine F = sin5 e F = sin9 e (60)

53


a

-1.00

Fig. 3.5 Current distributions along antenna for N = 0 and N = 2.Length of antenna is approximately quarter wave.

and the respective current distributions that produce them:

(61)

(62)

(63)

54


According to the calculations of Bouwkamp and De Bruijn, these dis-tributions. are explicitly given by

fo(0.8,~,4) = 2.394rI2.oEI

f2(%A,6) = 5.175 X 104e-aW(1 - 128.663p + 1379.59~4)

f4(7r/4,~,9) = 1.4208 X lQlle-6oP(1 - 526P + 34,559~4

- 605,706~6 + 2,843,678~8)

In Figs. 3.5 and 3.6, curves of foWlfo(O), f2Wlh(0), and f4(~)lf4(0)versus ~ are plotted. From these curves, we see that as N increasesthe number of oscillations increases. These spatial oscillations cause

650

oa

-0.50

-1.00

Fig. 3.6 Current distribution along antenna for N = 4. Length ofantenna is quarter wave.

55


the far-zone waves to interfere destructively in every direction exceptthe equatorial one, where they add constructively and thus produce asharp omnidirectional beam straddling the equatorial plane.

3.6 General Remarks on LinearArrays

A great variety of radiation patterns can be realized by arranging inspace a set of antennas operating at the same frequency. The fieldsradiated by the separate antennas interfere constructively in certaindirections and destructively in others, and thus produce a directionalradiation pattern. A knowledge of each antenna's location, orienta-tion, and current distribution, being tantamount to a complete descrip-tion of the monochromatic source currents, uniquely determines theresultant radiation pattern. Once the vector currents are known, theradiation pattern can be calculated in a straightforward manner by themethods described in Chap. 2. On the other hand, the converse prob-lem of finding a set of antennas that would produce a specified radiationpattern has no unique solution. For this reason, the problem of syn-thesizing a set of antennas to achieve a prescribed radiation pattern isconsiderably more challenging than the one of analyzing a prescribedset of antennas for its resultant radiation pattern. Actually, the inde-terminacy of the synthesis problem is circumvented by imposing at thestart certain constraints on the set which reduce sufficiently its gener-ality and then by specifying the desired radiation pattern with thatdegree of completeness which would make the problem determinate.l

Although any arrangement of antennas can be analyzed for its radia-tion pattern when the vector current distribution along each of theantennas is known, a synthesis procedure is possible only for certainsets. An important example of such a set is the configuration calledthe array, which by definition is composed of a finite number of identicalantennas, identically oriented, and excited in such a manner that thecurrent distributions on the separate antennas are the same in form butmay differ in phase and amplitude. It follows from this definition that

1See, for example, Claus Muller, Electromagnetic Radiation Patterns andSources, IRE Trans. Antennas Propagation, AP-4 (3): 224-232 (1956).

56

Radiation from wire antenna8

the radiation pattern of an array is always the product of two functions,one representing the radiation pattern of a single antenna in the arrayand the other, called the array factor or space factor, being interpretableas the radiation pattern of a similar array of nondirective (isotropic)antennas. This separability simplifies the problems of analysis andsynthesis to the extent that it permits the actual array to be replacedby a similar array of isotropic antennas.!Of all possible arrays, the linear array is the simplest to handle

mathematically and hence constitutes a natural basis for a discussionof antenna arrays. Here we shall limit our attention to linear arrays. 2

Let us consider then a linear array which for definiteness is assumed toconsist of n center-driven half-wave dipoles oriented parallel to the zaxis with centers at the points xp(p = 0, 1, ... , n - 1) on the xaxis (see Fig. 3.7). Each dipole is independently fed, has a length 2l,and is resonant (kl = 11'/2). Under the simplifying approximation thatthe proximity of the dipoles does not modify the dipole currents or,equivalently, that the dipoles do not interact with each other, 3 the cur-

! An isotropic antenna is no more than a conceptual convenience. Actuallya system of coherent currents radiating isotropic ally in all directions of freespace is a physical impossibility. This was proved by Mathis using a theoremdue to L. E. .J. Brouwer concerning continuous vector distributions on surfaces.See H. F. Mathis, A Short Proof that an Isotropic Antenna is Impossible,Proc. IRE, 39: 970 (1951). For another proof see C. J. Bouwkamp andH. B. G. Casimir, On Multipole Expansions in the Theory of ElectromagneticRadiation, Physica, 20: 539 (1954).

2 For comprehensive accounts of antenna arrays we refer the reader to theexcellent literature on the subject. See, for example, G. A. Campbell,"Collected Papers," American Tel. and Tel. Co., New York, 1937; RonoldKing, "Theory of Linear Antennas," Harvard University Press, Cambridge,Mass., 1956; S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Prac-tice," John Wiley & Sons, Inc., New York, 1952; H. Bruckmann, "Antennenihre Theorie und Technik," S. Hirzel Verlag KG, Stuttgart, 1939; J. D.Kraus, "Antennas," McGraw-Hill Book Company, New York, 1950; H. L.Knudsen, "Bidrag til teorien fjilrantennesystemer med hel eller delvis rota-tionssymmetri," I Kommission hos Teknick Forlag, Copenhagen, 1952.

3 In practice, one would take into account this mutual interaction or couplingby invoking the concept of mutual impedance. See, for example, P. S. Carter,Circuit Relations in Radiating Systems and Application to Antenna Problems,Proc. IRE, 20: 1004 (1932); G. H. Brown, Directional Antennas, Proc. IRE,25: 78 (1937); A. A. Pistolkors, The Radiation Resistance of Beam Antennas.Proc. IRE, 17: 562 (1929); F. H. Murray, Mutual Impedance of Two SkewAntenna Wires, Proc. IRE, 21: 154 (1933).

157


rent density along thepth dipole is taken to be that of an isolated dipole:

(-1 ~ z ~ 1) (64)

where Ap denotes the complex magnitude of the current. Hence theresulting current density for the entire array is the sum

n-l n-!

J = .r J(p) = e,o(y) cos kz L Apo(x - xp) (65)p=o p=o

This current density gives rise to the following expression for the radia-

z

Q

Fig. 3.7 Linear array of half-wave dipoles at points xo, Xl, • , Xn-l

along X axis. Each dipole is parallel to z axis. OQ is line ofobservation. Q' is projection of Q on xy plane.

58


tion vector:

N = J e-ike,'r' J (r')dV' = e. J e-ikY'sin 6 sin <PO(y')dy'

n-l

X J~le-ikz'COS6COSkz'dz' J l Ape-ikz'sin6cos<PO(X' - xp)dx'p=o

which upon integration reduces to

(7r )2 cos 2 cos 8 n -1N = e -. ~ A e-ikzp sin 6co••

• k sm28 1.. pp=o(66)

Substituting this result into Eq. (86) of Chap. 2 and recalling the vectorrelation e, X e. = -e", sin 8, one finds that the far-zone Poyntingvector has only a radial component given by

where

(67)

F(8)cos (~cos 8)

sin 8(68)

is the radiation pattern of each dipole, and

n-lA(8,4» = L Ape-ik"p.in6co ••

p=o

is the array factor. The radiation pattern of the entire array is

U(8,4» = 1F(8)A(8,4» I = F(8) IA (8,4»1

(69)

(70)

If we let if; denote the angle between the x axis and the line of observa-

59


tion (cos if! = sin e cos tP), the array factor (69) takes the form

n-lA(if!) = r Ape-ikxpcos{l

p=o(71)

which is recognized as the canonical expression for the complex radia-tion pattern of a similar array of isotropic antennas. Thus the radia-tion pattern U(e,tP) of the actual array is equal to the radiation patternF(e) of a dipole multiplied by the radiation pattern A (if!) of the similararray of isotropic radiators. More generally, expression (71) is validfor any linear array irrespective of the type of its member antennas.For example, if each half-wave dipole of the array were replaced by anantenna having a radiation pattern G«(J,tP), then the resulting radiationpattern would be given by U«(J,tP) = IG«(J,</l)A«(J,</l)I.The linear array considered above includes certain special cases which

are distinguished by the restrictions one imposes on the complex mag-nitudes Ap of the input currents and on the positions Xp of the antennas.One such case is that of an equidistantly spaced linear array for which

xp = pd (72)

where d is the uniform spacing. Imposing this spatial restriction (72)on the array factor (71) and expressing Ap as the product

(73)

which explicitly exhibits through the factor exp (-ip'Y) the progressivephasing 'Y of the currents, we get

n-l n-l

A(if!) = r ape-ip(kdcosH1) = r apeipa (74)p=o p=o

where the shorthand a = - kd cos if! - 'Y has been used. Then, if weintroduce the complex variable ~defined by

~ = eia (75)

the array factor (74) takes the form of a polynomial of degree n - 1 in

60


the complex variable ~:

n-l

A(lf) = 2: ap~P

p~o(76)

Since the coefficients ap are arbitrary, some of them may be zero.When this occurs, the antennas which correspond to the vanishingcoefficients are absent from the array and the remaining antennas donot necessarily constitute an equidistantly spaced array. Neverthe-less, an incomplete array of this sort can be considered equidistantlyspaced by regarding d as the "apparent spacing" and n as the "apparentnumber" of antennas. Thus we see that the polynomial (76) can beidentified with any linear array having commensurable separations.The importance of this one-to-one correspondence between polynomialand array stems from the fact that it permits application of the highlydeveloped algebraic theory of polynomials to the synthesis problem.A case in point is Schelkunoff's well-known synthesis procedure,! whichingeneously exploits certain algebraic properties of the polynomial (76).

When the coefficients ap of the polynomial (76) are equal to a con-stant, which for the present may be taken as unity, the array is said tobe "uniform." The array factor of such a uniform linear array withcommensurable separations has the closed form

n-l

I ~n - 1A(lf) = P =--~- 1

p=o

which, with the aid of ~ = exp (ia), becomes

A (If) = ei(n-l)a/2 si~ (na/2)sm (a/2)

(77)

(78)

Consequently the radiation pattern of a uniform linear array of equi-distantly spaced isotropic sources is given by

IA (If) I = I sin (na/2) I = I si~ [n(kd cos If + 1')/2] Isin (a/2) sm [(kd cos If + 1')/2]

(79)

It is sometimes convenient to divide A(f) by n and thus normalize its

1 S. A. Schelkunoff, A Mathematical Theory of Linear Arrays, Bell SystemTech. J., 22: 80 (1943).

61

(80)


maximum value to unity. The resulting function K(y.,) is the "nor-malized radiation characteristic" of the array and is given by

K(y.,) = !I sin (n(kd cos y., + 1')/2] In sin (kd cos y., + 1')/2]

If the sources are in phase with each other (I' = 0) and if the spacingis less than a wavelength (kd < 211"), the radiation characteristic K(f)consists of a single major lobe straddling the plane y., = 11"/2 and a num-ber of secondary lobes or "side lobes." As long as the spacing remainsless than a wavelength, the spacing has only a secondary effect uponthe radiation pattern. Hence, if I' = 0 and if kd < 211", the radiationis cast principally in the broadside direction and the array operates as a"broadside array." However, if the spacing becomes greater tha'n awavelength (kd > 211"), the radiation characteristic K(y.,) changes mark-edly; it develops a multilobe structure consisting of "grating lobes"which collectively resemble the diffraction pattern of a linear opticalgrating. I On the other hand, if the sources are phased progressivelysuch that kd = -I' or kd = 1',the radiation is cast principally in thedirection of the line of sources and the array operates as an "end-firearray." If the spacing is less than a half wavelength (kd < 7T), there is asingle end-fire lobe in the direction l/J = 7T when kd = 'Y. But if thespacing is equal to a half wavelength (kd = 7T), two end-fire lobes exist

simultaneously, one along y., = 0 and the other along y., = 11". Hence,when kd < 11"the array is a "unilateral end-fire array" and when kd = 11"it is a "bilateral end-fire array." An increase in the directivity of aunilateral end-fire array is realized when the condition of Hansen andWoodyard is satisfied, viz., I' = -(kd + 1I"/n) or I' = (kd + 1I"/n).2If one desires the major lobe to point in' some arbitrary directiony., = y.,1, then the phase I' and the spacing d must be chosen such thatkd cos y.,1 + I' = o.When the coefficients ap of the polynomial (76) are smoothly tapered

in accord with the binomial coefficients

(n - 1) (n - I)!

ap = p = (n - 1_ p)!p! (81)

1See, for example, A. Sommerfeld, "Optics," pp. 180-185, Academic PressInc., New York, 1954.

2 W. W. Hansen and J. R. Woodyard, A New Principle in DirectionalAntenna Design, Proc. IRE, 26: 333 (1938).

62


kd=lr kd= 511'.4

kd= 311'4

kd=31r. 2Fig.3.8 Radiation characteristic K(t/t) of a uniform linear array for

various spacings. Calculated from Eq. (80) with 'Y = 0 andn = 12. Broadside array. Grating lobes.

63


kd= 71r kd=211"4

kd= 511" kd=311"2

Fig. 3.8 Continued.

64


kd=4'l1'

kd=81T

Fig. 3.8 Continued.

65


kd='Y=JI..4

kd='Y=!!:.2

Fig.3.9 Radiation characteristic K(~) of auniform linear array cal.culated from Eq. (80) with n = 12 for various values ojkd = 'Y Unilateral end-fire array. Bilateral end-fire array.

66


kd= 'Y= 3 'IT'4

kd='Y~1r

Fig. 3.9 Continued.

67


the array factor becomes

(82)

Hence, the radiation pattern of such a "binomial array" is given by

IA(~-)I = 2n-1lcosn-1 (a/2) I = 2n-1Icosn-1 [(kd cos 1/1+ 1')/2]1 (83)

For I' = 0 and kd = 11", the binomial array yields the following broad-side pattern, which is distinguished by the fact that it is free of sidelobes:!

(84)

Comparing a uniform broadside array with a binomial broadside arrayhaving the same number of radiators, we see from the above examplesthat the broadside lobe of the former is narrower than the broadsidelobe of the latter. Thus by tapering the strengths of the radiators wereduce the side lobes, but in so doing we broaden the broadside lobe.However, it is possible to choose the coefficients ap such that the widthof the broadside lobe is minimized for a fixed side-lobe level, or con-versely, the side-lobe level is minimized for a fixed width of the broad-side lobe. Indeed, Dolph2 demonstrated that for the case in which thenumber of sources in the array is even and d ~ X/2, such an optimumpattern can be achieved by matching the antenna polynomial (76) to aChebyshev polynomial. Then Riblet3 extended the discussion to thecase in which the number of sources is odd and d < X/2. And finallyPokrovskii,4 through the use of the so-called Chebyshev-Akhiezer poly-nomials, which constitute a natural extension of the Chebyshev poly-nomials, succeeded in handling the general case where d ~ X/2 ord < )0../2. Certain simplifications in the practical calculation of such

1 J. S. Stone, U.S. Patents 1,643,323 and 1,715,433.2 C. L. Dolph, Current Distribution for Broadside Arrays which Optimize

the Relationship between Beam Width and Side-lobe Level, Proc. IRE, 34:335 (1946).

3 H. J. Riblet, Discussion on Dolph's Paper, Proc. IRE, 35: 489 (1947).4 V. L. Pokrovskii, On Optimum Linear Antennas, Radiotekhn. i ElektrQrt"

I: 593 (1956).

68

Radiation fronl wire antennas

Chebyshev arrays were made by Barbiere1 and by Van der Maas.2

For a continuous distribution of isotropic radiators along a straightline, i.e., for a line source, the problem of an optimum broadside pattern(narrow beam width and low side lobes) was solved by T. T. Taylor. 3

When the sources are incommensurably spaced, the point of depar-ture is no longer the polynomial (76) but the more general expression(71). Clearly expression (71) is considerably more difficult to handlethan expression (76), especially when the number of sources becomeslarge; but with the use of a computer, numerical results can be obtainedin a straightforward manner. An unequally spaced array is generallymore "broadband" than an equally spaced array, in the sense that itsradiation pattern remains essentially unaltered over a broader band ofoperating frequencies. King, Packard, and Thomas4 studied thisattribute of unequally spaced arrays by numerically evaluating theradiation pattern for Ap = 1 and Xp chosen according to various spacingschemes. A general discussion of unequally spaced linear arrays hasbeen reported by Unz,. and certain equivalences between equally andunequally spaced arrays have been noted by Sandler. 6

Returning to the case of a uniform array whose radiation character-istic is given by expression (80), we see that if nand kd( <1l') are fixedand 'Yis varied from a to kd, the major lobe rotates from the broadsidedirection to the end-fire direction. This suggests that by continuouslyvarying the phase 'Y the beam can be made to sweep continuously overan entire sector. It is on this principle that electrical scanningantennas operate. 7 The phases of the antennas are controlled elec-

1D. Barbiere, A l\lethod for Calculating the Current Distribution ofTchebyscheff Arrays, Proc. IRE, 40: 78 (1952).

2 G. J. van der Maas, A Simplified Calculation for Dolph-TchebyscheffArrays, J. Appl. Phys., 25: 121 (1954).

3 T. T. Taylor, Design of Line-source Antennas for Narrow Beamwidth andLow Side Lobes, IRE Trans. Antennas Propagation, AP-3 (1): 16 (1955).

4 D. D. King, R. F. Packard, and R. K. Thomas, Unequally-spaced Broad-band Arrays, IRE Trans. Antennas Propagation, AP-8 (4): 380 (1960).• H. Unz, Linear Arrays with Arbitrarily Distributed Elements, Electronics

Research Lab., series 60, issue 168, University of California, Berkeley, Nov. 2,1956.

6 S. S. Sandler, Some Equivalences between Equally and Unequally SpacedArrays, IRE Trans. Antennas Propagation, AP-8 (5): 496 (1960).

7 For a review of the scanning properties of such arrays see, for example,W. H. von Aulock, Properties of Phased Arrays, Proc. IRE, 4.8: 1715 (1960).

69


trically by phase shifters which form an integral part of the feedsystem. Although in many operational radars the scanning is donemechanically; electrical scanning is used in the case of large arrayantennas because it provides scanning patterns and scanning rates thatcannot be obtained by mechanical means.Without further calculation, we can deduce the radiation pattern of

a rectangular array of dipoles. We do this by compounding the radia-tion pattern of a parallel arrayl with that of a collinear array. Anexpression for the radiation pattern of a collinear array of half-wavedipoles can be constructed from expressions (68) and (71). We notethat the parallel array of Fig. 3.7 is transformed into a collinear arraywhen the dipoles are rotated until their axes are aligned with the x axis.Clearly then, in view of expression (68), the radiation pattern of eachrotated dipole is given by

F(I{;)cos (; cos I{;)

sin I{;

and the array factor remains the same as it was before the rotation, viz.,

n-lA(I{;) = 2: Ape-ikx• CDS'"

p=o

Hence the radiation pattern of the collinear array turns out to be

(7r )cos 2 cos I{; n-l

U(I{;) = 1F(I{;)A(I{;)j = . '\' A e-ikx.CDS'"sml{; ~ p

p=o(85)

It follows from this expression (by replacing I{; with 0) that the radia-tion pattern of a collinear array of dipoles lying along the z axis withdipole centers at the points Zp is given by

(-rr )cos 2 cos 0 n-l

U(O) = . '\' A e-ikz.CDS8sm 0 ,~p

p=o(86)

1The linear array shown in Fig. 3.7 is called a "parallel array" whenever itbecomes necessary to distinguish it from a collinear array.

70


By substituting this expression for F(8) in Eq. (70) we get the radiationpattern of a rectangular array of half-wave dipoles which are parallel tothe Z axis and have centers at the points x = xl" Z = Zq (p = 0, 1, ... ,n - 1; q = 0, 1, ... , m - 1) in the xz plane. We can regardexpression (86) as the radiation pattern of each element of the parallelarray, i.e., we can replace F(8) of expression (70) with U(8) of expres-sion (86), and thus obtain the following expression for the radiationpattern of a rectangular array of dipoles:1

cos (,: cos 8) n-1 m-l

U(8,</1) = s~n 8 ILL Apqe-ih. sin 6 cos ~e-ikz' cos 61 (87)1'=0 q=O

where Apq denotes the complex magnitude of the current in the dipoleat x = xl" Z = Zq. If the magnitudes of the dipole currents are equalto a constant, say 10, and if the array constitutes a two-dimensionalperiodic lattice with uniform spacings dx and d. in the x and Z directions(Xl' = pdx, Zq = qd.), expression (87) reduces to

~(':~8) -.U(8 </1)= I 2 I sin [n(kdx sin 8 cos </1)/2] sin [m(kd. cos 8)/2]\

' 0 sin 8 sin [(kdx sin 8 cos </1)/2] sin [(lcd. cos 8)/2]

(88)

We see that such a rectangular array can cast a narrow beam in thedirection (8 = 7r/2, </1 = 7r/2) normal to the plane of the array. Alongthe axis of this beam at a distance r = ro from the array, the radialcomponent of Poynting's vector is given by

(89)

or by

s - ~R.(LxL.) 2 (90)r - '\J;- 87r2r02 dx2d.2

1Although this expression was derived by considering a parallel array ofsimilar collinear arrays, it is valid also for the more general case where thecomplex amplitudes Apq of the dipole currents are arbitrarily chosen.

71


where Lz( = ndz) and L.( = md.) are by definition the effective dimen-sions of the array.lIn view of expression (90), it appears that Sr increases quadratically

with the area LzL. of the array. However, expression (89) is validonly for "small" or "moderately sized" arrays because as the array isenlarged the field at the fixed observation point (T = TO, 8 = 71"/2,q, = 71"/2) changes in nature from a far-zone, or Fraunhofer, field to anear-zone, or Fresnel, field. If we take L(= VLz2 + Ly2) as thetypical dimension of the array, the condition2 that the array be con-tained well within the first Fresnel zone is

From this it follows that

L2 < X2 + XTO- 16 2

Since X/TO « 1, expression (92) reduces to

(91)

(92)

(93)

Thus we see that the critical value of L is Le = VXTo/2. If L < Le,

the observation point is in the far zone and the previously derivedformulas are valid. On the other hand, if L > Le, the observationpoint is in the near zone and to find the radiation one must take intoaccount the fact that the field is now of the Fresnel type. Tetelbaum3

1The effective dimensions so defined are the limiting values of the actualdimensions (L.)actual = (n - l)dz and (L.)actual = (m - l)d. + 2l as n,m~ 00.

2 Let To-I cos wt be the field at observation point due to the dipole at origin.Then (To2 + £2)-~2 cos (wt + q,) is the field at observation point due to thefarthest dipole. Assuming that YTo2 + L2 ~ To in the denominator, we seethat the resulting field is cos (wt) + cos (wt + q,) = A cos (wt + a). It fol-lows that the modulus A is given by A 2 = 2 + 2 cos q,. The second term ispositive as long as q, ~ 71"/2, with q, == (271"/X)(YTo2 + L2 - To). Hence wehave the condition (271"/X)(YTo2 + L2 - To) ~ 71"/2, or YTo2 + L2 - To ~ ~.

3 S. Tetelbaum, On Some Problems of the Theory of Highly-directiveArrays, J. Phys., Acad. Sci. U.S.S.R., 10: 285 (1946).

72


has performed such a calculation for the case of a square array; hisresults show that as the array is made larger, S. at first increases inaccord with Eq. (90)and then behaves in a manner dictated by Cornu'sspiral of Fresnel diffraction theory. A similar calculation has beenmade by Polkl for the case of it uniformly illuminated rectangularaperture antenna.

3.7 Directivity Gain

The directivity gain g of a directional antenna can be calculated fromthe relation

(r ~ 00) (94)

where S.(r,8,t/» denotes the radial component of the far-zone Poyntingvector, (S.) max the major-lobemaximumof S.(r,8,t/», dn( = sin 8 d8 dt/»the element of solid angle, and r the radius of a far-zone sphere. Thisrelation directly yields g = 1 for an isotropic antenna and g > 1 for allother antennas.Unless the antenna happens to be a short dipole or some other

equally simple antenna, the problem of calculating directivity gain iscomplicated by the fact that the integral representing the time-averagepower P radiated by the antenna, viz.,

(4.. 2P = 10 ST(r,8,t/»r dn (r ~ 00) (95)

cannot be evaluated by elementary means. The same difficulty arisesin connection with the calculation of the radiation resistance R of anantenna,2 because to find R from the definition R = 2P/12, where I is

1C. Polk, Optical Fresnel-zone Gain of a Rectangular Aperture, IRE Trans.Antennas Propagation, AP-4 (1): 65-69 (1956).

2 M. A. Bontsch-Bruewitsch, Die Strahlung cler komplizierten recht-winkeligen Antennen mit gleichbeschaffenen Vibratoren, Ann. Phys., 81: 425(1926).

73


an arbitrary reference current, one is again faced with the task ofcalculating P. As an alternative, it is always possible to calculate Pby Brillouin's emf method, 1but the integral to which this method leadsis generally as difficult to evaluate as the integral (95) posed byPoynting's vector method. The situation is eased considerably whenthe antenna is highly directional, for then ST(r,8,q,) may be approxi-mated by a function that simplifies the evaluation of the integral (95).Let us first consider the simple case of a short wire antenna. From

Eq. (19) we see that for kl «1, the far-zone radial component of thePoynting vector has the form

ST(r,8,q,) = ~ sin2 8r

(96)

where K is a constant that will drop out of the calculation due to thehomogeneity of relation (94). The maximum of ST(r,8,q,) occurs at8 = 7r/2 and has the value

(97)

Substituting expressions (96) and (97) into definition (94), we find thatthe gain of a short dipole is given by

2 3g = =-10" sina 8 d8 2

(98)

As the antenna is lengthened, its gain increases moderately. To showthis, we recall from Eq. (19) that for a center-driven antenna of arbi-trary length the far-zone radial component of the Poynting vector hasthe form

ST(r 8 q,) = !i [cos (kl co~ 8) - cos kl]2, , r2 SIn 8

(99)

where, now, K = Vp./Elo2/87r2. Substituting this expression intoEq. (95), we obtain the following integral representation for the time-

1A. A. Pistolkors, The Radiation Resistance of Beam Antennas, Proc.IRE, 17: 562 (1929).

74


average power radiated by the antenna:

P = 27rK (1r [cos (kl COS.8) - cos klJ2 d810 sm 8 (100)

To evaluate this integral we introduce the new variables u( = kl cos 8)and v( = kl - u). Thus

( r [cos (kl co~ 8) - cos klJ2 d810 sm 8

_IJkl 2( 1 1)- 2" -kl (cos U - cos kl) kl _ u + kl + u du

= J_klkl (cos ~l-=- c~s kl)2 du = J02kl [(1 + cos 2kl)(1 - cos v)

. 2kl ( . 1 . 2) cos 2kl ( )] dv- sm sm v - 2" sm v - -2- 1 - cos 2v vand hence

P = 27rK [ C + In 2kl - Ci 2kl + sin22kl (Si 4kl - 2Si 2kl)

+ COS22kl (C + In kl + Ci 4kl - 2Ci 2kl)] (101)

where

Si x = (z sin ~d~10 ~is the sine integral

Ci x = - f'" cos ~d~ = C + In x _ (z 1 - cos ~d~z ~ 10 ~

is the cosine integral, and C( = 0.5722 ... ) is Euler's constant.With the aid of a table of sine arid cosine integrals, I P can be easily

I See, for example, E. Jahnke and F. Emde, "Tables of Functions," DoverPublications, Inc., New York, 1943.

75


computed from expression (101). In the case of a half-wave dipole(kl = 7r/2) it follows from Eqs. (99) and (101) that (Sr)max = K/r2 andP = 27r(1.22)K. Inserting these results into Eq. (94), we find thatthe gain of a half-wave dipolel is g = 1.64. Similarly, in the case of afull-wave dipole we would find that g = 2.53. These examplesillustrate that the gain of a linear antenna increases rather slowly withlength, and to get really high gains from thin-wire antennas one mustoperate them in multielement arrays.As an antenna of high-gain capabilities, let us now consider a uniform

parallel array whose far-zone Poynting vector, in accord with Eqs.(67) and (79), has the radial component

S _ ~ _1_ cos (~cos ()) sin (n(kd sin ()cos cP + ")')/2] 2r - "\J; 87r2r2 sin () sin ((kd sin ()cos cP + ")')/2] (102)

By virtue of approximation (25) we can write this expression in thesimpler form

S _ I~(0.945)2 \ . ()sin (n(kd sin ()cos cP + ")')/2] 12 (103)

r -"\JE 87r2r2 sm sin ((kd sin ()cos cP + ")')/2]

It is clear that the maximum of Sr occurs at ()= 7r/2 and cP = cPo,where cPo is fixed by kd cos cPo + ")'= O. Thus

Substituting (103) and (104) into definition (94), we get

47rn2

g = {2" { •.. 3 ()sin2 (n(kd sin ()cos cP + ")')/2]d()dcPJo Jo sm sin2 ((kd sin ()cos cP + ")')/2]

(104)

(105)

The integral in this expression can be evaluated exactly2 through the1The relative gain gr of an antenna is its gain over a half-wave dipole.

That is, gr = g/1.64.2 C. H. Papas and R. King, The Radiation Resistance of End-fire and

Collinear Arrays, Proc. IRE, 36: 736 (1948).

76


use of Bonine's first integral theorem 1

(2" f" . 3 1:1sin2 [n(kd sin 1:1cos cP + 'Y)/2j dl:1dcPJo Jo sm sin2 [(kd sin 1:1cos cP + 'Y)/2j

8~n + 8 nI-l ( ) () (sin u sin u + cos u)=- ~ n-q cos q'Y ----- --3 U u3 u2

q=l

(106)

where u = qM. Hence the gain (105) of the array can be expressed interms of the finite series (106):

4~2(J = n-l - (107)

8~n + 8 \' ( ) () (sin u sin u + cos u)3 ~L n - q cos q'Y --u - US U2q=l

This expression is convenient for numerical calculation, especiallywhen the number n of dipoles is small. When n is very large and thearray is operating as a broadside array ('Y = 0, d ::;; X), we have thesimple limiting form

4nd(J =- X

(n ~ 00) (108)

which may be obtained2 by comparing the denominator of expression(107) with the Fourier expansions of the functions x, x2, and x3 for theinterval (0,2~).Let us now calculate the gain of a large uniform rectangular array.

We recall from Eq. (88) that its radiation pattern is given by

cos (~ cos 1:1). .U(1:1 ) = I ~ ISl~l na Sl~ mf31,cP 0 3m 1:1 sm a sm f3 (109)

where a = (kdx/2) sin 1:1cos cP and f3 = (kd./2) cos 1:1.The spacings dx

and dz are assumed to be less than a wavelength (dx < x, d. < X) and

1N. J. Sonine, Recherches sur les fonctions cylindriques et Ie developpementdes fonctions continues en series, llfath. Ann., 16: 1 (1880).

2 See, for example, K. Franz and H. Lassen, "Antennen und Ausbreitung,"p. 255, Springer-Verlag OHG, Berlin, 1956.

77


hence the radiation pattern consists of two broadside beams, one in thedirection 0 = 1r/2, q, = 1r/2 and the other on the opposite side of thearray in the direction 0 = 1r/2, q, = 31r/2, Each of these beams hasthe maximum value

Umax = Ionm (110)

With the aid of expressions (109) and (110) and the fact that Sr isproportional to U2/r2, definition (94) leads to the following expressionfor the gain: '

g=47rn2m2

cos2 (~cos 0) . .r 2•. r" ? Sl~2 nex Sl~2 m{3 dOdq,J 0 J 0 sm 0 sm2 ex sm2/3

(111)

Since the array is large, most of its radiation is concentrated in the twonarrow broadside beams. Because of the symmetry of the radiationpattern, the q, integration may be restricted to the beam lying in theinterval (0,1r), and because of the sharpness of the beam, the followingapproximations obtain:

cos q, "'" 1r/2 - q,dex = (kdx/2) (cos 0 cos q, dO - sin 0 sin q, dq,) "'" - (kdx/2)dq,

d/3 = - (kdz/2) sin 0 dO "'" - (kdz/2)dO

Applying these approximations to the integral in Eq. (111), we get

(112)

Here the actual limits have been replaced by infinite ones on the groundthat the two factors in the integrand rapidly decrease as ex and /3 departfrom zero. Since the ex integration yields n1r and the {3integration m1r,expression (112) yields the following limiting value for the gain of the

78


rectangular array:1

(n, m~ 00) (113)

If the array were backed by a reflector, which eliminates one of thebeams and concentrates all the energy in the other, the limiting valueof the gain would be twice as large, viz.,

(n, m~ 00) (114)

In terms of the effective dimensions of the array Lz( = ndz) andLz( = mdz) and the effective area of the array A (= LzLz), the limitingvalues (113) and (114) respectively become

(115)

1 If the limits are chosen to include only the broadside beam, then the inte-gral in Eq. (112) must be replaced by

Integral =

f"lm f"ln sin2ndsin2m{3 f" f" sin2xsin2y--2 - -{32 dOld{3 = nm -2- --2- dxdy-,,1m -"In 01 -" -.. X Y

Since

then

f" sin2 ~ - sin2 ~ I" f" sin 2~_"~d~ = ~ _"+ _,,-~-d~

The first term on the right vanishes and the second term is equal to 2Si (211").Using this result, we get Integral = 4nm[Si (211"))2.Hence the corresponding

. f h .. 211"(ndz)(mdz) [ 11" J2 Th. ItexpressIOn or t e gam IS g = -y2 2Si (211")' IS resu agreeswith Eq. (113) since [1I"/2Si(211"))2is approximately equal to 1.

79


for the array without a reflector, and

(116)

for the array with a reflector.It is clear from the above results that the gain of an array can be

increased by increasing its size. However, it is also possible inprinciple to achieve very high gain, i.e., supergain, with an array oflimited dimensions.l Since the elements of such superdirective arraysare closely spaced, their mutual interactions playa determining role.These interactions have the effect of storing reactive energy in theneighborhood of the elements and of thus making narrow the band-width of the array. 2 Moreover, the large currents that superdirectivitydemands lead to high ohmic losses and consequently to reductions inoperating efficiency. 3 In addition to narrow bandwidths and lowefficiencies, superdirective antennas are burdened with the require-ment that the amplitudes and phases of the currents be maintainedwith a relatively high degree of precision. Superdirective arrays areuseful in those cases where a very sharp narrow beam is desired regard-less of the cost in bandwidth, efficiency, and critical tolerances. Someaspects of the supergain phenomenon are closely related to the problemof optical resolving power. 4

I S. A. Schelkunoff, A Mathematical Theory of Linear Arrays, Bell SystemTech. J., 22: 80 (1943).

2 L. J. Chu, Physical Limitations of Omni-directional Antennas, J. Appl.Phys., 19: 1163 (1948).

3 R. M. Wilmotte, Note on Practical Limitations in the Directivity ofAntennas, Proc. IRE, 36: 878 (1948); T. T. Taylor, A Discussion on theMaximum Directivity of an Antenna, Proc. IRE, 36: 1135 (1948); H. J.Riblet, Note on Maximum Directivity of an Antenna, Proc. IRE, 36: 620(1948).

4 G. Toraldo di Francia, Directivity, Super-gain and Information Theory,IRE Trans. Antennas Propagation, AP-4 (3): 473 (1956).

80

Multipoleexpansion

of theradiation field 4

One method of expanding a radiation field in multipoles is todevelop in Taylor series the Helmholtz integral representa-tions of the scalar and vector potentials and then to identifythe terms of the series with formal generalizations of theconventional multi poles of electrostatics and magnetostatics.Another method of expansion consists in developing theradiation field in spherical E and H waves and defining theE waves as electric multipole fields and the H waves as mag-netic multi pole fields. In this chapter a brief account isgiven of these two methods.

4.1 Dipole and QuadrupoleMoments

We assume that a monochromatic current density J(r') isdistributed throughout some bounded region of space.Then by virtue of the conservation of charge there also existsin the region a monochromatic charge density p(r') given by

\7' • J(r') = iwp(r') (1)

To deduce a relation which we will use for defining themoments of the charge in terms of the current and for fram-ing the gauge of the potentials produced by the charge and

81


current, we multiply this equation of continuity by an arbitraryfunction f(r,r') and then integrate with respect to the primed coordi-nates. Thus from Eq. (1) we obtain

J p(r')f(r,r')dV' = L jf(r,r')V' • J(r')dV' (2)

Here the region of integration includes the entire space occupied bythe current and charge, and the normal component of the current iszero on the surface which bounds the region. Using the identityv' . (Jf) = f"V' • J + J . V'f, and noting that the term fV'. (Jf)dV'disappears because by the divergence theorem it equals the surfaceintegral ffn' J dS' whose integrand disappears, we see that Eq. (2)leads to the desired relation

J p(r')f(r,r')dV' = £ J J (r') . V'f(r,r')dV' (3)

On the proper selection of f(r,r'), the left side of this relation becomes amoment of the charge and the right side becomes an equivalent repre-sentation of the moment in terms of the current.When f = 1, relation (3) reduces to

f p(r')dV' = 0 (4)

and we thus see that the total charge is zero. Moreover, when wedenote the cartesian components of r' and J by x~ and Ja (ex = 1,2,3),and when we successively assume thatf = x~andf == x~x~(ex, (3 = 1,2,3),relation (3) gives rise to the first and second moments of the ,charge,which define respectively the cartesian components pa and Qafj of theelectric dipole moment p and the electric quadrupole moment Q:

pa = J p(r')x~ dV' = ~ J J",(r')dV'

Q",fj= J p(r')x~x/l dV'

= ~ J [J a(r')x~ + J fj(r')x~ dV' ]

82

(ex = 1, 2, 3)

(ex, (3= 1, 2, 3)

(5)

(6)

Multipole expansion of the radiation field

It is clear from these expressions that p is a vector and Q is a dyadicwhose components constitute a symmetrical matrix, QafJ = QfJa. Invector form, Eqs. (5) and (6) are

p = J p(r')r' dV' = ~ J J (r')dV'Q = J p(r')r'r' dV' = ~ J (J(r')r' + r'J(r')JdV'

(7)

(8)

These relations show how p and Q can be calculated from a knowledgeof either the charge or the current.Associated with the electric charge and electric current are their

magnetic counterparts, the magnetic charge density Pm (1") andmagnetic current density Jm(r'). These conceptual entities servethe purpose of establishing a formal duality between electric andmagnetic quantities. The magnetic current density is defined byJm(r') = (w/2i)r' X J(r') and the magnetic charge density is deducedin turn from Jm(r') by the conservation law 'V'. Jm(r') = iWPm(r').Since Jm(r') and Pm(r') obey the conservation law, it follows that theyalso obey a relation which is formally the same as Eq. (3), viz.,

J Pm(r')f(r,r')dV' = ~ J Jm(r') . 'V'f(r,r')dV' (9)

Choosing f to be successively the cartesian components of 1", andrecalling the definition of Jm(r'), we get the following expression for thefirst moment of the magnetic charge density:

J Pm(r')r' dV' = ~ J Jm(r')dV' = ~ J 1" X J(r')dV' (10)

Since the first moment of Pm(r') is by definition the magnetic dipolemoment m, this equation gives the following expression for m in termsof the electric current:

m = ~Jr' X J(r')dV' (11)

Clearly m can be regarded as a pseudo vector whose cartesian com-

83


ponents arel

3

m-r = ~ I Eali-r J x~J B(r')dV'a,B=l

('Y = 1, 2, 3) (12)

or as an antisymmetrical dyadic m having the cartesian components

(a, (3= 1, 2, 3) (13)

The simplest current configuration that possesses an electric dipolemoment is the short filament of current

J = e.Ioo(x')o(y') (-l ~ z' ~ l) (14)

Substituting this expression into Eqs. (7), (8), (11) we find that theelectric dipole moment is given by

ip = e. - 21Io

w(Hi)

and the electric quadrupole and magnetic dipole moments are zero.The dual of this configuration is the small filamentary loop of current

J = eq,Jq, = eq,Ioo(p' - a)o(z') (16)

where a is the radius of the loop. Substituting into Eqs. (7), (8), (11)we find in this case that the only nonzero moment is the magneticdipole moment given by

10 Jm = 2 ep X e<p p'o(p' - a)o(z')p'dp'det/dz' = ez'Tra2Io (17)

As an example of a configuration having an electric quadrupole mom.ent,

1 The three-index symbol Eap-r has the following meaning: Eali-r = 0 whenany two of the subscripts are the same, EaB-r = 1 when a, {3,'Yare all differentand occur in the order 12312 . . . (even permutations of 123), and EaB-r = -1when a, {3,'Yare all different and occur in the order 21321 ... (odd permuta-tions of 123). That is, E123 = Em = E3l2 = 1 and Em = E132 = Em = -1.

84


let us take two antiparallel short filaments of current separated by a.distance d:

J = ezlo[/l(x' - d/2)/l(y') - /lex' + d/2)/l(y')]

Substituting in Eq. (8), we get

(-l ::;z' ::; l) (18)

Q = ~ 21Io(eze. + e.ez) J [x'/l(x' - d/2) - x'/l(x' + d/2)]dx'

or

Q = (~ 21I0) (eze. + c.ez)d (19)

Since the dipole moment of each filament is given by Eq. (15) andd = de. is the vector separation of the two filaments, we can write theelectric quadrupole moment as

Q = pd + dp (20)

If we choose the function f(r,r') in relation (3) to be the free-spaceGreen's function, i.e., if we let

eikjr-r'l

f(r,r') = 41Tlr- r'l

then by virtue of the identity

eiklr-r'l eiklr-r'l'il'~--- -'il---Ir - r'l - Ir - r'l

relation (3) yields

J eikjr-r'l i J eikjr-r'lper') 4;[r _ r'l dV' = -: -;;;'il. J(r') 41Tlr_ r'l dV'

(21)

(22)

(23)

The left side of this equation is the Helmholtz integral representationof Eq,(r); the integral on the right is the Helmholtz integral representa-

85


tion of (1/ jL)A(r). Hence, Eq. (23) expresses the Lorentz conditioncoupling the vector potential A(r) with the scalar potential !fJ(r), viz.,

~ • A(r) = iWEjL!fJ(r) (24)

Thus we see that the Lorentz gauge is the one that is consistent withthe conservation of charge.

4.2 Taylor Expansion of Potentials

As in the previous section, let us start by assuming that in a boundedregion of space we have an arbitrary distribution of monochromaticcurrent density J(r') and a distribution of charge density p(r') derivedfrom it by the equation of continuity. Then the scalar and vectorpotentials of the electromagnetic field produced by such a source aregiven by the Helmholtz integrals

1 J eiklr-r'l!fJ(r) = -4 p(r') I '1dV'

~E r -- r

jL J eik!r-r'lA(r) = 4~ J(r') Ir __ r'l dV'

(25)

. (26)

To expand these potentials in Taylor series we need the three-dimen-sional generalization of the familiar one-dimensional Taylor series. Werecall that the one-dimensional Taylor series expansion of a functionf(x) is given by

(27)

When x is replaced by r, h by some vector a, and hd/dx by a'~, theTaylor series (27) heuristically takes the ~hree-dimensional form

L'"1f(r + a) = , (a . ~)nf(r)n.

n~O

86

(28)


By letting a = - r' we see that for any function of r - r' the expansionis

f(r - r') = ~ ~ (-r' . v)nf(r)1., n!n=O

and hence we have

eiklr-r'! I'" 1 eikr__ = -(-r'.v)n-Ir - r'l n! r

n=O

(29)

(30)

With the aid of expansion (30) we develop the Helmholtz integralrepresentation (25) of the scalar potential in the Taylor series

If>(r) = ~ J p(r')[l - r'. V + 72(r'. V)2 -41rE

Keeping only the first three terms, we get

eikr']-dV'r (31)

[ J ] eikr [ J ] eikr

41rElf>(r) = p(r')dV' r - p(r')r'dV'. Vr+ 72 [J p(r')r/r/ dV' ] :VV e:r

(32)

where in the third term (r' . V)2 has been written as the double scalarproduct r/r/: VV of the dyadics r'r' and VV. The first term is zero byvirtue of Eq. (4). The second and third terms involve respectivelythe electric dipole and quadrupole moments, as defined by Eqs. (7)and (8). Thus the leading terms of the Taylor expansion of If>(r) canbe written in the following concise form:

1 (eikr

eikr

If>(r) = - - P' V- - 72Q:VV- +41rE r r .. -) (33)

where p is the electric dipole moment and Q the electric quadrupolemoment.Similarly, the Taylor series development of the HelmholtZ integral

87


representation (26) for the vector potential is

41l-A(r) = I-lfJ(r')[l - r'. V + 72(r'. V)2 - .

Considering only the first two terms, we get

ikr

'J~dV'r(34)

(471-/p)A(r) = [J J(r')dV' ] e;' - [J J(r')r' dll' ] . V e;' (35)

The first integral by Eq. (7) is equal to -iwp. To express the secondintegral in terms of Q and m, we decompose the dyadic J(r')r' into itssymmetric and antisymmetric parts:

J(r')r' = %[r'J(r') + J(r')r'J - %[r'J(r') - J(r')r'J

When integrated, the first symmetric part yields by Eq. (8) the sym~metric dyadic (-iw/2)Q and the second anti symmetric part yields byEq. (13) the antisymmetric dyadic m. Thus the expansion for thevector potential up to the second term turns out to be

. eikr iWI-l eikr eikr

41l'A(r) = -tWI-lP- + - Q' V- - I-lm . V-. r 2 r r(36)

If one prefers to think of m as a pseudo vector, then the operatorm . V, where m is an antisymmetric dyadic, has to be replaced bym X V, where m is a pseudo vector. Accordingly, an alternative formof expression (36) is

(37)

Hence, for a source that can be described as a superposition of anelectric dipole, a magnetic dipole, and an electric quadrupole, the scalarand vector potentials are given by Eqs. (33) and (37). If a source issuch that poles of higher multiplicity are required, it becomes moreconvenient to calculate q,(r) and A(r) by evaluating directly the Helm-holtz integrals than to use the method of multipole expansion.,

88


4.3 Dipole and Quadrupole Radiation

The electromagnetic field of a monochromatic source can be found bysubstituting the Taylor expansions of the scalar and vector potentials,viz.,

et>(r) = _! (p' VG - ;YzQ:VVG+ ... )E

A(r) = -iwJL(pG - ;YzQ • VG - !:. m X VG + ... )W

where G = eikr/47rr, into the relations

(38)

(39)

E = -Vet> + iwA (40)

which yield the electromagnetic field E, H. By virtue of the linearityof the system, the resulting electromagnetic field may be thought of asthe vector sum of the individual electromagnetic fields of the variouspoles. Since each multipole radiates a spherical wave and the mostnatural coordinate system for a mathematical description of the radia-tion is a spherical one centered on the multipoles, we assume that themultipoles are located at the origin of a spherical coordinate system(r,8,et» defined in terms of the cartesian system (x,y,z) by x = r sin 8cos et>, y = r sin 8 sin et>,z = r cos 8. A consequence of this assumptionis that the free-space Green's function G = eikr / 47rr which appears inexpressions (38) and (39) is a function of the radial coordinate r only.From expressions (38) and (39) we see that the potentials of the

electric~dipole part of the source are

1et>elec.dip. = - - p . VG

E

Aelec.dip. = -iwJLpG

(41)

(42)

Applying relations (40) to these potentials and using the identitiesV(p' VG) = (p • V)VG and V.X (pG) = -p X VG, which follow fromvector a.-nalysisand the constancy of p, we obtain the electric and mag-

89


netic field of the electric dipole:

1Eelee.dip. = - [(p' V)Va + k2pG]

E

Helee.dip. = iwp X va

Since the gradient operator in spherical coordinates is

a la 1 av = er-+er--+e<l>-'--. ar r a9 r sm 9 at/>

the vector va which appears in Eqs. (43) and (44) is given by

eikr

( 1)va = v - = er ik - - a41l'r r

Moreover, in spherical coordinates, p has the form

(43)

(44)

(45)

(46)

(47)

where P = VP' P = VPr2 + ps2 + p<I>2is the strength of the electricdipole. Hence the scalar product of p and V yields

a la 1 aP • v = P - + ps - - + p .•.---, ar r a9. 'I' r sin 9 at/>

(48)

When this operator acts on the vector (46) and it is recalled that(ajar)e, = 0, (aja9)er = es, and (ajat/»er = e<l>sin 9, we get

( 2ik 2) (ik 1)(p' V)Va = erpr -k2 - - .. + - a + esps - - - ar r2 r r2

('k 1)+ e<l>p",7 - 1=2 a (49)

Using this result and expression (47), we easily obtain from Eq. (43)the spherical component of Eelee.dip. in terms of the spherical components

90


of p. Thus

1 ( 2ik 2)(Er)elec.dip. = - pr - - + 2 GErr

(ES)clcc.dip. = !Ps (ik - ~ + k2) GErr

1 (ik 1 )(E4»clcc.dip. = - P4> - - 2 + k2 GErr

In cartesian coordinates, p has the form

p = e",p", + eypy + e.p.

(50)

(51)

(52)

(53)

Scalarly multiplying this expression by Cr, es, e4>in succession, notingthat

pr = P' er PS = P' es P4> = p'e4>

and recalling that

er ' e", = sin e cos q, er ' ey = sin e sin q, er ' e. = cos eeS ' e", = cos e cos q, es ' Cy = cos e sin q, es ' e. = ~ sin e

e4>'e",= -sinq, e4>'ey = cosq,

we get the following connection between the cartesian and sphericalcomponents of p:

~=~~e~q,+~~e~q,+~~eps = p", cos e cos q,+ py cos e sin q, - p. sin eP4> = -p", sin q, + py cos q,

(54)

(55)

(56)

Substituting these expressions into Eqs. (50) through (52), we find thatthe spherical components of E.lcc.dip. in terms of the cartesian compo-

91

Tlteory of electromagnetic wave propagation

nents of p are given by

(Er)elee.dip. = !(pz sin e cos q, + Pu sin e sin q, + p. cos e)f

(Ee)elee.dip. = ! (Pz cos e cos q, + Pu cos e sin q, - p. sin e)f

x (~- ! + k2)Gl' 1'2

(E ) - 1 ( . + ) (ik 1 k'2) G'" elec.dip. - - - pz SIll q, Pu cos q, - - -2 +' :rf l' l'

(.58)

(.59)

To find the spherical components of Helee.dip. in terms of the sphericalcomponents of p, we substitute expressions (46) and (47) into Eq. (44)and thus obtain

(H e)elee.dip. = iwp", (ik - ~) G

(H"')elec.dip. = -iwpe (ik - ~) G

(60)

(61)

(62)

With the aid of Eqs. (55) and (56) we can express these spherical com-ponents of (H)eleo.dip. in terms of the cartesian components of p:

(H e)elee.dip. = iw( -pz sin q, + Pu cos q,) (ik - 0G

(H",)elee.dip. = -iw(pz cos e cos q, + Pucos e sin q, - p. sin e)

(63)

X (ik - ~) G (64)

Since (H r)elec.dip. is identically zero and (Er)elee.dip. is not, the radiationfield of the electric dipole is an E wave or, equivalently, a TM wave.But as kr is increased, (Er)elee.dip. becomes negligibly small compared to(Ee)elee.dip. and (E",)elee.dip., and therefore in the far zone the radiation

92


field has the structure of a TEM wave, and the simple relation

er X (E)elec.dip. = ~ (H)elec.dip. is valid there. As kr is decreased, the

magnetic field becomes negligible compared to the electric field, andthis electric field approaches the electric field of an electrostatic dipole.

As can be seen from expressions (38) and (39), the potentials of themagnetic dipole are

cPmag.dip. = 0

Amag.dip. = - I'm X 'VG

(6.5)

(66)

When substituted into the second of relations (40), this vector potentialyieldl:l the magnetic field of the magnetic dipole. That is,

Hmag.dip. = - 'V X (m X 'VG) (67)

Since m is a constant vector, it follows from vector analysis that'VX (m x'VG) = m'V2G - (m. 'V)'VG. But 'V2G = -k2G for r > O.Hence the magnetic field (67) of the magnetic dipole may be writtenalternatively as

Hmag.dip. = -[(m. 'V)'VG + k2mG] (68)

The electric field of the magnetic dipole is found by substituting expres-sions (65) and (66) into the first of relations (40). Thus

(69)

:~ On comparing Eq. (43) with Eq. (68) and Eq. (44) with Eq. (69), wesee that the electromagnetic field of an electric dipole, except for certainmultiplicative factors, is formally equivalent to the electromagneticfield of a magnetic dipole, with electric and magnetic fields interchanged.Hence, we can obtain the components of the magnetic dipole by simplyapplying a duality transformation to the already obtained field com-ponents of the electric dipole. Since the radiation field of an electricdipole is an E wave and the dual of an E wave is an H wave, the radia-tion field of a magnetic dipole thus must be an H wave or, equivalently,a TE wave.

93

Them..y of electromagnetic wave propagation

According to expressions (38) and (39), the potentials of the electricquadrupole are

IcPelec.quad. = 2E Q : 'V'VG

iwp,Aelec.quad. = 2 Q . "VG

(70)

(71)

With the aid of relations (40), these potentials yield the followingexpressions for the electric and magnetic fields of the electric quadrupole:

IEelec.quad. = - ,,['V(Q:'V'VG) + k2(Q. 'VG)] (72)

;<;E

iwHeleo.quad, = 2' 'V X (Q . 'VG) (73)

First let us find the components of Helec.quad.. In spherical coordinates,Q can be written as

Q = ererQrr + ereBQrB + Cre~Qr~ + eBerQBr + eBeBQBB + eBe~QB~

+ e~erQ~r + e~eBQ~B + e~e~Q~~ (74)

Scalarly postmultiplying this expression by 'VG = erf, where f is a short-hand for (ik - l/r)G, we get the vector

which, when substituted into Eq. (73), yields

(Hr)elec,quad. = i;[;0 (Q~r sin 0) - af)cP QBr] r stu 0iw[ f f) If)]

(H B)elec.quad. = 2' r sin 0 acP Qrr - Q~r r ar (rf)

iw[ If) fa](H ~)elec.quad. = 2' QBr r ar (rf) - r f)O Qrr

94

(75)

(76)

(77)

(78)


Since

1 a (f) 1 a [( 'k 1) eikr] k2 eikr _ k2G.rar r =,ar ~r- 471'r -t- 471'r= (79)

as r -t 00, the only parts of the magnetic field components that survivein the far zone are

iwk2(H B)elec.quad, = ""2 Q~rG

iwk2(H~)elec,quad, = - 2 QB,G

(80)

(81)

To represent these far-zone field components in terms of the cartesiancomponents of Q, we note that in cartesian coordinates Q has the form

Q = exexQxx + exeuQXY + exe.Qx. + eyexQyX + eyeyQyy + eye.Qy.

+ e.exQ.x + e.eyQ.y + e.e.Q.. (82)

which when premultiplied by eB, e~ and postmultiplied by er yields

QOr = eo. Q . e, = Qxx cos (J sin (J cos2 cJ>+ Qyy cos (J sin (J sin2 cJ>

- Q •• sin (J cos (J + (QXY + Qyx) cos (J sin (J cos cJ>sin cJ>

+ Qx. cos2 (J cos cJ>- Qzx sin2 (J cos cJ>+ Qy. cos2 (J sin cJ>- Q.y sin2 (J sin cJ>

and

Q~r = e~ . Q • e, = (Qyy - Qxx) sin (J sin cJ>cos cJ>- QXY sin (J sin2 cJ>

.+ QyX sin (J eos2 cJ>- Qx. sin cJ>cos (J + Qy. cos cJ>cos (J

Invoking the symmetry of Q and using some simple trigonometricidentities, we reduce these results to

QBr = ~ sin 2(J(Qxx cos2 cJ>+ Quv sin2 cJ>- Q •• + QXY sin 2cJ»

+ cos 2(J(Qx. cos cJ>+ Qy. sin cJ» (83)

Q~r = ~(Qyy - Qxx) sin (J sin 2cJ> + Qyx sin (J cos 2cJ>- Qx. sin <p cos (J + Qy. cos cJ>cos (J (84)

95


Substituting Eqs. (84) and (83) into Eqs. (80) and (81), we obtain thefollowing expressions for the spherical components of the far-zone mag-netic field of an electric quadrupole in terms of the cartesian compo-nents of the quadrupole moment:

iwk2 eikr ..(H8)elec.quad. =2 41l"r [~(Qyu - Qzz) sm (J sm 2et>

+ QyZ sin (J cos 2et> - Qzz sin et>cos (J + QyZ cos et>cos (J] (85)

_ iwk2 eikr 1 • 2 • 2(H 4»elec.quad. - - 2 41l"r [~ sm 2(J( Qzz cos et>+ Qyy sm et>

- Qzz + QZY sin 2et» + cos 2(J(Qzz cos et>+ QyZ sin et»] (86)

In the far zone the relation er X Eelec.quad. = f!!: Hele •.quad. is valid.. ~;

Consequently the spherical components of the far-zone electric field ofthe electric quadrupole are derivable from expressions (85) and (86) byuse of the following simple connections:

(87)

(88)

Alternatively one may calculate the far-zone <;omponents of Eelec.quad.

directly from Eq. (72). We have already calculated the quantityQ •VG which appears in the second term of Eq. (72). The result ofthis calculation is shown by Eq. (75). Hence, the only quantity wenow must calculate is V(Q: VVG) for r -+ 00. By definition of thedouble scalar product, we have

Q:VVG = r Qij(ei' V)(ej' V)G (89)i,;

where ej denotes the unit vectors in spherical coordinates. Since G is afunction of r only, this definition yields the expression

i)2Q :VVG = Qrr cr2 G (90)

96


which, in the far zone, reduces to

(r ~ <Xl) (91)

Taking the gradient of this quantity and keeping only its far-zone term,we get

(92)

Substituting Eqs. (75) and (92) into Eq. (72), we see that in the farzone (Er)elec.qu.d. disappears an_dthe other two components of Eelec.qu.d.are given by

(Ee)elec.qu.d. = ik3(93)- 2; QerG

(E.p)elec.qu.d. = ik3(94)- 2;Q.prG

With the aid of Eqs. (80) and (81) it is clear that this result agrees withEqs. (87) and (88).

4.4 Expansion ofRadiation Fieldin Spherical Waves

There is an alternative type of multipole expansion which in certaininstances is more natural than the one based on the Taylor series expan-sion. In this section we shall construct such an expansion by firstdeveloping the radiation in spherical E and H waves, then defining theE waves as electric multipoles and the H waves as magnetic multipoles,and finally calculating the expansion coefficients through Bouwkampand Casimir's method.!Outside the bounded region V 0, which completely contains the mono-

chromatic source currents, the electric and magnetic fields can be con-

Ie. J. Bouwkamp and H. B. G. Casimir, On Multipole Expansions in theTheory of Electromagnetic Radiation, Physica, 20: 539 (1954).

97


veniently derived from two scalar functions by use of the expressions

E = V X V X (rv) + iWILV X (ru)

H = V X V X (ru) - iWEV X (rv)

(95)

(96)

The two scalar functions u and v are the Debye potentialsl which sat-isfy the scalar Helmholtz equation

(97)

and obey the Sommerfeld radiation condition. Such a representationof an electromagnetic field in terms of the Debye potentials is quitegeneral. Indeed, it has been proved2 that every electromagnetic fieldin a source-free region between two concentric spheres can be repre-sented by the Debye potentials; the proof rests on Hodge's decomposi-tion theorem for vector fields defined on a sphere. 3

We choose a spherical coordinate system (r,e,q,) with center some-where ~ithin Vo• Then the acceptable solutions of Eq. (97) are thespherical wave functions

(n ;:::0,m = 0, II, ... , In)

(98)

The radial functions hn(kr) are the spherical Hankel functions of thefirst kind, which satisfy the differential equation

r2 ~;2n + 2r a:lrn+ [k2r2 - n(n + l)jhn = 0 (99)

and obey the radiation condition. They are related to the fractional-order cylindrical Hankel functions of the first kind by

I P. Debye, Dissertation, Munich, 1908; also, Der Lichtdruck auf Kugelnvon beliebigen Material, Ann. Phys., 30: 57 (1909).

2 C. H. Wilcox, Debye Potentials, J. Math. Meeh., 6: 167 (1957).3 P. Bidal and G. de Rham, Les formes differentielles harmoniques, Com-

mentarii Mathematiei Helvetiei, 19: 1 (1956).

98


The fact that hn can be expressed in terms of the exponential functionis sometimes useful; for examplel

i eikrho(kr) = - kT

(100)

etc.

The angular functions Y nm(O,q,) are the surface spherical harmonics ofdegree n and order m. They constitute a complete set of orthogonalfunctions on the surface of a sphere. Displaying explicitly the normal-ization constant, we write

(-n ~ m ~ n)

(101)

where Pnm(cos 0) are the associated Legendre polynomials of degree nand order m j then in view of

[ 'If' [m( ]2' _ 2 (n + m)!Jo Pn cosO) smOdO-2n+l(n_m)! (-n ~ m ~ n)

(102)

we see that this choice of normalization constant leads to the orthog-onality relations

(103)

where 5ij = 1 for i = j and 0 for i ~j. Here t,he quantity Ynm* is theconjugate complex of Ynm and is simply related to Yn-m as follows:

(104)

1See P. M. Morse, "Vibration and Sound," 2d ed., pp. 316-317, McGraw-Hill Book Company, New York, 1948.

99


This relation clearly follows from definition (101) when we recall

(n - m)'p,,-m(cos 8) = (_1)m (n + m) i Pnm(cos 8) (105)

The Debye potentials are linear superpositions of these spherical wavefunctions (98). That is,

00 m=n

v(",8,cp) = L L anmlfnmn=Om=-n

(106)

'"u(r,O,cP) = L

11=0

m=nL bnm!/lnm

m=-ll.

(107)

The expansion coefficients anm, bnm could be calculated from a knowl-edge of u and v on the surface of a sphere of radius r = To by using theproperty that the functions !/Inm are orthogonal over the surface of thesphere, viz.,

However, we shall not determine them in this way. Rather, we shalldetermine them from the radial components of the electric and mag-netic fields, in accord with the method of Bouwkamp and Casimir.!Substituting expressions (106) and (107) for the Debye potentials

into representations (95) and (96), we get the following expansion of theelectromagnetic field in spherical wave functions:

(109)n.m

(110)n,m

We ~re free to consider this electromagnetic field as a superposition oftwo electromagnetic fields, one being an E type field (E, ~ 0, Hr = 0)and the other an H type field (Hr ~ 0, Er = 0). Accordingly we

! C. J. Bouwkamp and H. B. G. Casimir, On Multipole Expansions in theTheory of Electromagnetic Radiation, Physica, 20: 539 (1954). Also,H. B. G. Casimir, A Note on Multipole Radiation, Helv. Phys. Acta, 33: 849(1960).

100


decompose the electromagnetic field E, H as follows:

E = E' + E"H = H" + H'

(111)(112)

where E/, H' denote the E type field and E", H" the H type field.Comparing Eqs. (109) and (110) with Eqs. (111) and (112) respectively,we see that the E type field is given by

n,m

H' = -iWf L anm'V X (rif;nm)n,m

and the H type field by

E" = iwJ.l L bnm'V X (rif;nm)n,m

H" = L bnm'V X 'VX (rif;nm)n,m

Moreover, if we let

E~m = 'VX 'VX (rif;nm) } ~lectric multipoles ofH~m = -iWf'V X (rif;nm) degree n and order m

and

E~m = iwJ.l'V X (rif;nm) } magnetic multipo.les ofH:.'m = 'VX 'VX (rif;nm) degree n and order m

then Eqs. (113) through (116) become

H' = L anmH~mn,m

E" = L bnmE:.'mn,m

H" = ~ b H"L., nm nmn,m

(113)

(114)

(115)

(116)

(117)

(118)

(119)(120)

(121)

(122)

(123)

(124)

101


We define E~m, H~m to be the electromagnetic field of the electricmultipole of degree n and order m, and E;,'m, H;,'m to be the electro-magnetic field of the magnetic multi pole of degree n and order m, sothat expressions (121) through (124) constitute the multipole expan-sion of the electromagnetic field. Thus a superposition of the termsn = 1, m = 0, :!: 1 yields a dipolar field, and a superposition of theterms n = 2, m = 0, :!: 1, :!: 2 yields a quadrupolar field, and soforth.As yet the expansion coefficients have not been fixed; before we start

to calculate them, let us deduce the spherical components of themultipole fields. To find the spherical components of E~m, H~m andE;,'m, H;,'m, we make use of the following relations. We note that

= n(n + 1)J/;nm (125)

where the second equality follows from Eq. (99). We also note that

(126)

Denoting the unit vectors in the 0 and q, directions by eg and e.p

respectively, we obtain by vector analysis the angular components ofV X (rJ/;nm) and V.X V X (rJ/;nm):

1 a aeg' V X V X (rJ/;nm) = ;: ar ao (rJ/;nm)

e.p • V X V X (rJ/;nm) = -!-o aa a: (rJ/;nm)r SIn r 'I'

1 aeg • V X (rJ/;nm) = -. -0 a- J/;nmsm q,

ae.p • V X (rJ/;nm) = - ao J/;nm

(127)

(128)

(129)

(130)

With the aid of relations (125) through (130) we see that the sphericalcomponents of the multipole fields (117) through (120) are given by

102


(E~m)r = n(n + 1) h"Y"mr

(E~mh = ~ ; (rh,,) :() Y"m

(E~m)4> = i:n () dd (rh")Y,,mr sm r(H~m)r = 0

(H~m)e = ~E() h"Y"msm

(H~m)4> = iWEh" :() Y"m

and

(HI!) = n(n + 1) h Y mnm , r n- n

(H'.:m)e = ~ ; (rh,,) :() Y"m

(H'.:m)4>= . i:n () dd (rh,,) Y"mr sm r(E~:m)r = 0

(E'.:m)e = - ~WM()h"Y"msm

(E'.:m)4>= -iWMh" :() Y"m

spherical components ofelectric multipole fieldof degree n and order m(n ~ 0, -n 5 m 5 n)

spherical components ofmagnetic multi pole fieldof degree n and order m(n ~ 0, -n 5 m 5 n)

(131)

(132)

With the aid of relations (125) and (126) it follows from Eqs. (109)and (110) that r . E and r . H can be written as1

r' E = ~ a"mn(n+ l)lf"mn,m

n,'n

(133)

(134)

By virtue of the orthogonality of the functions If,,m over the surface of asphere, it is obvious from expansions (133) and (134) that the coeffi-

1Comparing these expressions for r' E and r . H with expressions (106)and (107) for v and u we see that apart from the factor n(n + 1) they arerespectively the same.

103


cients anm and bnm can be determined from a knowledge of the scalarfunctions r . E and r . H over the surface of a sphere. Accordingly wenow shall find the expansion coefficients anm, bnm from the currentswithin V 0 by first calculating r . E and r . H in terms of the currentsand then incorporating these results with expansions (133) and (134).The validity of this procedure is assured by the theorem 1 that anyelectromagnetic field in the empty space between two concentricspheres is completely determined by the radial components Er, Hr'Our task now is to obtain r . E and r . H in terms of the currents

within Vo. We recall (see Sec. 1.1) that the E and H produced by amonochromatic J must satisfy the Helmholtz equations

v X V X H - k2H = V X J

V X V X E - k2E = iwp.J

(135)

(136)

When these equations are scalarly multiplied by the position vector r,we get the relations

r . V X V X H - k2r . H = r. V X J

r •V X V X E - k2r •E = iwp.r • J

which by vector analysis reduce2 to

(V2 + k2) (r . H) = - r . V X J

(V2+ k2) (r . E + i.- r . J) = .;...r . V X V X JWE "'WE

In terms of the free-space Green's function

, eiklr-r'lG(r,r) = 4 I '11rr-r

(137)

(138)

(139)

(140)

(141)

1Bouwkamp and Casimir, loco cit.2 We use the vector identity (V2+ k2)(r' C) = 2V. C + r' V(V. C) -

r . V X V X C + k2r •C, where C is an arbitrary vector field. To obtainEq. (139) we let C = H. To obtain Eq. (140) we let C = E and C = Jsuccessively.

104


the solutions of the scalar Helmholtz equations (139) and (140) whiohsatisfy the Sommerfeld radiation condition are

r' H = J G(r,r')r'. v' X J(r')dV' (142)Vo

r' E = J:- r' J - J:- J G(r,r')r'. V' X V' X J(r')dV' (143)1-WE 1-WE Vo

These relations are valid for r inside and outside Vo• For r outside V 0,the first term on the right side of Eq. (143) is identically zero and wehave

r •E = .i J G(r,r')r'. v' X v' X J(r')dV'WE Vo

It is known that for r' < r

n=", eiklr-r'l ik '\' .,G(r,r) = 41rlr _ r'l = 411" 1.. (2n + 1)Jn(kr )hn(kr)Pn(cos 'Y)

n=O

(144)

(145)

where in(kr') = (1I"/2kr')'tiJn+",,(kr') and cOS'Y= cos (Jcos (J' + sin (Jsin (J' cos (I/J - I/J'). It is also known that

m=n

Pn(cos 'Y)= L (-1)mPnm(cos (J)Pn-m(COS(J')eim(4>-4>') (146)m=-n

Recalling

Ytnm = hn(kr) Y nm«(J,I/J)

= [(2n + 1) ~: ~ :~: r hn(kr)P nm(COS(J)eimq, (147)

and introducing the functions Xnm, which are defined by

(148)

105


we find from expansions (145) and (146) that the free-space Green'sfunction can be expressed as follows:

(149)

Substituting this expression into Eq. (142) we get

'kr' H = ~ L (-I)"iYnm(r) J Vo Xn-m(r')r' . \7' X J(r')dV' (150)

n,m

An integration by parts yields

Jv. Xn-m(r')r'. \7' X J(r')dV' = Jv. J(r') . \7' X (r'xn-m)dV' (151)

and hence the expansion for r . H becomes

r' H = 1:L (-I)"'lfnm(r) Jvo J(r') • \7' X (r'xn-m)dV' (152)n,m

From Eq. (144) it similarly follows that

r. E = - ~ &. ~ (-I)"iYnm(r) f J(r/). \7'471" '\j~L Vo

n,mX \7' X (r/Xn-m)dV' (153)

Comparing Eq. (152) with Eq. (134) and Eq. (153) with Eq. (133), wefinally obtain the desired formulas

1 {j; (_l)m f J( ')..." ...,1 (' -m)dV'anm = - 471" '\j;- n(n + 1) Vo r . v X v X r Xn

b - ik (_l)m f J(')"'" (' -m)dV'nm - 471" n(n + 1) v. r • v X r Xn

(154)

(155)

which give anm and bnm in terms of the current.From the above analysis we see that a multipole expansi9n of the

electromagnetic field E, H radiated by a monochromatic current J is

106


obtained by developing E and H in basic multipole fields E~m, H~m andE;.'m,H;.'m, that is, by writing E and H in the following form:

E = L an",E~m + L bnmE;.'mn,m n,m

H = L anmH~m + L bnmH~mn,m n.m

(156)

(157)

Here the basic multipole fields are given explicitly in terms of sphericalwave functions by the definitions (117) through (120), and the expan-sion coefficients anm, bnm (which constitute a decomposition of theknown current J into electric and magnetic multipoles superposed atthe origin of coordinates) are deduced from J by evaluating theintegrals (154) and (155).In the far zone (kr -7 00), the basic multipole fields can be expressed

most conveniently in terms of the operator

1L=-:-rXV~ (158)

which in wave mechanics is known as the angular-momentum operator.To show this, we note that the asymptotic form of the spherical Hankelfunction is

(kr -7 00) (159)

From this form it follows that

and

(160)

(161)

107


Hence in terms of the operator L we have

(162)

(163)

Using expressions (162) and (163), we thus see that the basic multipolefields (117) through (120) in the far zone are

eikrE:.'m = -(-i)niVJ.l/E-LYnm r

eikrH" = -(-i)ni - e X LY mnm r T n

(164)

(165)

(166)

(167)

Substituting these multipole fields into expansions (156) and (157),we find that the far-zone electromagnetic field is given by

eikr _H = r [-LanmVE/J.l (-i)n+II .•Ynm

+ L bnm(_i)n+le, X Lynm] (169)

108

Radio-astronomical 5

antennas

Observational radio astronomy is concerned with the meas-urement of the radio waves that are emitted by cosmic radiosources.! With the apparently single exception of the mono-chromatic radiation at A= 21 centimeters, i.e., the "hydro-gen line" emitted by interstellar hydrogen, cosmic radiowaves are rapidly and irregularly varying functions of time,resembling noise. The measurable properties of cosmicradio waves are their direction of arrival, state of polariza-tion, spectrum, and strength. For ground-level observa-tions, radio astronomy is limited essentially to the band rang-ing approximately from 1 centimeter to 10 meters, becausewaves of wavelength greater than about 10meters are unableto penetrate the earth's ionosphere and those of wavelengthless than about 1 centimeter are absorbed by the earth'satmospheric gases. However, radio-astronomical observa-

! For a popular exposition on radio astronomy, see the delightfuland informative monograph by F. G. Smith, "Radio Astronomy,"Penguin Books, Inc., Baltimore, 1960. For a comprehensivetreatment of the subject, see J. L. Pawsey and R. N. Bracewell,"Radio Astronomy," Oxford University Press, Fair Lawn, N.J.,1955; also 1. S. Shklovsky, "Cosmic Radio Waves," Harvard Uni-versity Press, Cambridge, Mass., 1960. See also F. T. Haddock,Introduction to Radio Astronomy, Proc. IRE, 46: 3 (1958); andR. N. Bracewell, Radio Astronomy Techniques, Handbuch derPhysik, LIV, Springer-Verlag OHG, Berlin. See also J. L. Stein-berg and J. Lequeux, "Radio Astronomy," McGraw-HilI BookCompany, New York, 1963.

109


tions have also been made in the band ranging from about 3 milli-meters to 1 centimeter.The instrument that is used to measure cosmic radio waves is the

"radio telescope." It consists of three basic components operatingin tandem, viz., a receiving antenna, a sensitive receiver, and arecording device. Functionally, the antenna collects the incidentradiation and transmits it by means of a wave guide or coaxial lineto the input terminals of a receiver; the receiver in turn amplifies andrectifies the input signal; and then the recording device, which is drivenby the rectified output of the receiver, presents the data for analysis.The rectified output of the receiver is a measure of the power fed tothe receiver by the antenna.Since the cosmic signals arriving at the input terminals of the

receiver are noiselike and similar to the unwanted noise signals whichare unavoidably generated by the receiver itself, the receiver must beable to distinguish the desired noise signal from the undesired one.This is a difficult requirement and is met by a "radiometer," whichconsists of a high-quality receiver and special noise-reducing circuitry.To reduce even further the effects of the receiver noise, radiometerssometimes are supplemented with a low-noise amplifier such as a maser!or a parametric amplifier operating in front of the receiver.2

The part of the radio telescope that we shall consider in this chapteris the antenna, and our presentation will cover only the radiationtheory of such radio-astronomical antennas. The reader interested inthe more practical and operational aspects of the subject is referred tothe literature. 3

1The first application of a maser (X band) to radio astronomy was madeby Giordmaine, Alsop, Mayer, and Townes [J. A. Giordmaine, L. E. Alsop,C. H. Mayer, and C. H. Townes, Proc. IRE, 47: 1062 (1959)]. See alsoJ. V. Jelley and B. F. C. Cooper, An Operational Ruby Maser for Observa-tions at 21 Centimeters with a 60-Foot Radio Telescope, Rev. Sci. Instr., 32:166 (1961).

2 See, for example, F. D. Drake, Radio-astronomy Radiometers and TheirCalibration, chap. 12 in G. P. Kuiper and B. M. Middlehurst (eds.), "Tele-scopes," The University of Chicago Press, Chicago, 1960.

3 See, for example, J. G. Bolton, Radio Telescopes,chap. 11 in G. P. Kuiperand B. M. Middlehurst (eds.), "Telescopes," The University of ChicagoPress, Chicago, 1960.

110

Radio-astronomical antennas

5.1 Spectral Flux Density

Since an incoming cosmic radio wave is a plane transverse electromag-netic (TEM) wave, its field vectors E(r,t) and H(r,t) are perpendicularto each other and to the direction of propagation. Consequently thePoynting vector of the wave, viz., S(r,t) = E(r,t) X H(r,t), is parallelto the direction of propagation, and its magnitude is given by thequadratic quantity

S(r,t) = VfO/ILO E(r,t) • E(r,t) (1)

It is an observed fact that each component of the field vectors, insofaras its time dependence is concerned, has the character of "noise."That is, at any fixed position r = ro the field vectors are rapidly andirregularly varying functions of time, yet in their gross behavior theyare essentially independent of the time and in particular do not vanishat t = :!: 00. They constitute what is known as a stationary random(or stochastic) process.l By virtue of this noiselike behavior of thewave, it is the spectral density of the time-average value of S(r,t), andnot the instantaneous value of S(r,t) itself, that constitutes a meaning-ful measure of the strength of the incoming wave.In order to resolve the incoming signal into its Fourier components,

we must introduce the truncated function ET(ro,t) defined by

ET(ro,t) = E(ro,t)

ET(ro,t) = 0

for It I ~ T

for It I > T(2)

where 2T is a long interval of time. Since ET(ro,t) vanishes at t = :!: 00,its Fourier transform

1 JooAT(w) = 271" _ 00 ET(ro,t)eiwt dt (3)

1For general theory of stochastic (random) processes see, for example,S. O. Rice, Mathematical Analysis of Random Noise, Bell System Tech. J.,23: 282 (1944); 25: 46 (1945); S. Chandrasekhar, Stochastic Problems inPhysics and Astronomy, Rev. Mod. Phys., 15 (1): 1 (1943).

111


and its Fourier integral representation

(4)

always exist as long as T is finite. We know from the theory ofstochastic processes that the transform AT(w) increases without boundas T -> 00, whereas the quadratic quantity IAT(w)12/T tends to adefinite limit,l i.e.,

lim -T1 IAT(w)!2 = finite limit

T-••,(5)

This fact suggests that a quadratic quantity such as the time-averagePoynting vector be considered. According to Eqs. (1) and (4), themagnitude of Poynting's vector is

and its time-average value, defined as

(S(ro,t» = lim 21TI~ST(ro,t)dt

T-+oo, T

is given by

(S(ro,t» = r; lim 21T1 T [100

AT(w')e-iw't dw''\J""io T--.oo -T-oo

(7)

Since

lim ..!-.. 1T e-i(w'+w")t dt = lim.!!:.o(w' + w")T--.oo 2T -T T--.oo T

(9)

1 For rigorous mathematical theory see N. Wiener, Generalized HarmonicAnalysis, Acta Math., 55: 117 (1930).

112


where 5 is the Dirac delta function, Eq. (8) reduces to

(S(ro,t) = G lim -T1r II'" Ar(w'). AT(w")5(w' + w")dw'dw''\JJ;o T...•oo-'"

= ~ lim -T1rI 00 AT(w). Ar( -w)dw (10)'\JJ;o T...•oo -00

Using the relation Ar(w) = A~( -w), which is a consequence of the factthat Er(ro,t) in Eq. (3) is real, we see that Eq. (10) may be written asthe one-sided integral

(S(ro,t» = r '" [ G lim 2T1rAr(w) • A~(w)] dwJo '\JJ;o T...•'"

(11)

This expression gives the time-average power density of the incomingwave in watts meter-2; hence the quantity

S.,(ro) == ~ lim [2T1rAT(w) • A~(w)]'\JJ;o T...• '"

(12)

which is known to be finite by virtue of Eq. (5), gives its spectral fluxdensity I in watts meter-2 (cycles per second)-I.

I An alternative definition of the spectral flux density S.,(ro) in terms ofthe electric field E(ro,t) is based on constructing the autocorrelation function

q,(q) = I~ lim 21TI~E(t). E(t + q)dt ~ ~ (E(t) . E(t + q»'\J}.Lo T ...• '" T '\J ;,

and then identifying its Fourier transform with the spectral flux density.That is,

If'" 2fcooS., = - q,(q)e''''' dq = - q,(q) cos wq dq1r -'" 1r 0

If S., is to be the spectral density, the integral of S., over all frequencies mustyield the time-average power. To show that this requirement is met by theabove definition, we note that

hO

O 2h'" hoo hoos'" dw = - dq q,(q) cos wq dw = 2 dq <P(q)f,(q) = <P(O)o 1r 0 0 0

and recognize that q,(O) is indeed the time-average power.

113


From a strictly mathematical viewpoint, to obtain the spectral fluxdensity 8", one would have to observe the incoming signal for aninfinitely long period of time. In practice, obviously, this is neitherpossible nor desirable. As a practical expediency, the definition isrelaxed by choosing T large enough and yet not so large as to iron outsignificant temporal variations of 8",. The best that can be done is toobserve the incoming signal for successive periods of time equal to thetime constant T of the receiving system. What is observed then is thesignal smoothed by successive averaging over finite periods of durationT. The finiteness of T produces fluctuations in the record, as does thefiniteness of the bandwidth Aw of the receiving system. The period 1

of the fluctuations is approximately 1/ Aw; hence in a time interval T,

the incoming signal effectively consists of n( = TAw) independentpulses, whose standard deviation is 1/ y"n = 1/~. In view ofthis we can write

(13)

where oR is the standard deviation of the readings R, and K is a dimen-sionless constant whose value depends on the detailed structure of thereceiving system. Thus we see that the finiteness of T and Aw producesan uncertainty, or spread, in the readings. Consequently, in orderthat an incoming signal be detectable, the deflection produced by itmust be greater than the deflection oR produced by the inherentfluctuations of the receiving system.Since 8", is the power per unit area per unit bandwidth, the power P

1To see this, we consider the Fourier integral representation (4) for thefrequency band w - Aw/2 to w + Aw/2 over which AT(W) is assumed con-stant. Then

Hence the period of the envelope is 411'/Aw.

114

Radio-astron.omical antennas

flowing normally through an area A in the frequency range", - !:.",/2to '" + !:.",/2 is given by

P = AS",!:.", (14)

5.2 Spectral Intensity, Brightness,Brightness Temperature, Apparent

Disk TemperatureIn the previous section we defined the spectral flux density S", of cosmicradiation. In this section we shall define in terms of S", some otheruseful measures of cosmic radiation.One of these measures of cosmic radiation is the spectral intensity,

defined by

dP", = I",(n')du nl . n' dn(n') (15)

where dP", is the radiant power per unit bandwidth flowing through anelement of area du into a solid angle dQ(n'), nl the unit vector normalto du, and n' the unit vector along the axis of the solid angle (see Fig.5.1). The quantity I",(n') is the spectral intensity of the radiation

n'

dO (n')""""

Fig. 5.1 Geometric construction for definition of spectral intensity.Radiation is emitted by source and passes radially outwardthrough solid angle an.

115


traveling in the direction n'. Another such measure is the spectralbrightness, which is defined in the same way as the spectral intensityexcept that n"( = -n') is now the direction from which the radiationis corning (see Fig. 5.2). Accordingly, the power per unit bandwidthfalling on the area drT from the solid angle dn(n") is given by

dP., = b.,(n")drT n2 . n" dn(n") (16)

where b.,(n") is the spectral brightness of the incoming radiation andn2 = -nl. Comparing expressions (15) and (16), we get the relation

b.,(-n')dn( -n') = I.,(n')dn(n') (17)

which places in evidence the fact that brightness refers to radiationtraveling toward drT and intensity refers to radiation traveling awayfrom drT.The quantity dP.,j(drT nl' n') is the power per unit bandwidth per

unit area normal to the direction of travel of the radiation and henceit is equal to dS.,. Thus we see that the spectral intensity of the

D'

Fig.5.2 Geometric construction for definition of brightness. Radi-ation is emitted by distributed source in sky and passesradially inward through solid angle dn.

116


emitted radiation is the spectral flux density per unit solid angle, i.e.,

I = dB",'" dn

(18)

Similarly the spectral brightness of the received radiation is the spec-tral flux density per unit solid angle, i.e.,

b = dB",'" dn

(19)

If the source of radiation is distributed over the sky, then a convenientmeasure of the amount of radiation that falls on a receiving antennafrom a given direction is the spectral brightness in that direction. Asshown in Figs. 5.1 and 5.2, it is most convenient to choose the origin ofcoordinates at the source for L, and at the receiver for b.,. The spec-tral brightness b"" like the spectral intensity I w, is a function of 8, q, butnot of r.The units of I", and b", are the same since they are defined in the same

way, except that in the former the radiation is traveling outward fromthe vertex of the solid angle where the source is located, and in thelatter it is traveling in toward the vertex where the receiver is located.Specifically, the units of I", and b", are watts meter-2 (cycles per second)-lsteradian-I.It is sometimes convenient to specify the radiation in terms of the

temperature that a blackbody would require in order to produce themeasured spectral brightness. According to Planck's law for black-body radiation in free space, the spectral brightness B", of a blackbodyat temperature T (in degrees Kelvin) is given by

B", = 2hc 1X3 exp (he/kXT) - 1

whereh = Planck's constantk = Boltzmann's constantc = velocity of lightX = wavelength

(20)

117


But in radio-astronomical applications he «kAT and hence expression(20) may be replaced by the Rayleigh-Jeans approximation to Planck'slaw, viz.,

(21)

The spectral brightness temperature T.,b of the radiation coming towardthe receiving antenna along a direction 8, c/J is obtained by equatingexpression (21) to b.,. Thus the spectral brightness temperature T.,bis related to the spectral brightness by

(22)

Like b." the quantity T.,b is a function of 8, c/J only. In case the sourcesubtends a solid angle flo at the receiver, it follows from Eqs. (19) and(22) that

Noting that the "apparent p.isk temperature" T.,d is defined by

2k rS., = };2 } o. T.,d dO

we see from Eq. (23) that T.,d is related to T.,b by

(23)

(24)

(25)

and in this sense constitutes a measure of the average value of thespectral brightness temperature.

5.3 Poincare Sphere, Stokes.ParametersBy its very nature a monochromatic electromagnetic wave must beelliptically polarized, i.e., the end point of its electric vector at each

118


point of space must trace out periodically an ellipse or one of its specialforms, viz., a circle or straight line. On the other hand, a polychro-matic electromagnetic wave can be in any state of polarization, rangingfrom the elliptically polarized state to the unpolarized state, whereinthe end point of the electric vector moves quite irregularly. Cosmicradio waves are generally in neither of these two extreme states, butrather in an intermediate state containing both elliptically polarizedand unpolarized parts. A wave in such an intermediate state is saidto be "partially polarized" and is describable by four parametersintroduced by Sir George Gabriel Stokes in 1852 in connection with hisinvestigations of partially polarized light.! In this section we definethese Stokes parameters and show that they serve as a completemeasure of the state of polarization.As an exemplar, we consider the case of a plane monochromatic TEM

wave. The electric vector E(r,t) of such a wave traveling in the direc-tion of the unit vector n has the form

E(r,t) = Re {Eoei(k.r-wt)I (26)

where k( = n2'n-jX) is the wave vector and Eo the complex vectoramplitude. Because the wave is plane and TEM, vector Eo is aconstant and lies in a plane perpendicular to n, that is, n' Eo = O.Since the polarization of the wave is governed by Eo, and since Eo is aconstant, the state of polarization is the same everywhere. Thisconstancy of polarization is peculiar to plane, homogeneous waves, thepolarization of more general types of electromagnetic fields possiblybeing different at different points of space. For example, if the fieldwere a wave generated by a source of finite spatial extent, the polari-zation would vary with radial distance from the source as well as withpolar and azimuthal angles.Without sacrificing generality, we choose a cartesian coordinate

system x, y, z such that the z axis is parallel to n. With respect tothis system Eo can be written as

(27)

1G. G. Stokes, On the Composition and Resolution of Streams of PolarizedLight from Different Sources, Trans. Cambridge Phil. Soc., 9: 399 (1852).Reprinted in his "Mathematical and Physical Papers," vol. III, pp. 233-258,Cambridge University Press, London.

119


where ez, eu are unit vectors along the x and y axes respectively, andwhere the amplitudes az, au as well as the phases oz, Oy are real con-stants. Thus from Eqs. (26) and (27) it follows that the cartesiancomponents of E(z,t) are given by the real expressions

Ez = az cos (t/> + oz) E, = 0 (28)

where for brevity the shorthand t/> = wt - kz has been used.nating t/> from these expressions, we get

(E)2 (E)2 E E-.!: + -!!. - 2 -.!: -!!. cos 0 = sin 2 0az au az au

where

y

2c)'

-------2c .•------ ....•

Elimi-

(29)

(30)

"

Fig. 5.3 Polarization ellipse for right-handed polarized wave havingorientation angle if;.

120


Taking E", and Ey as coordinate axes, we see that Eq. (29) representsan ellipse whose center is at the origin E", = Ey = O. Geometricallythis means that at each point of space the vector E rotates in a planeperpendicular to n and in so doing traces out an ellipse. As is evidentfrom expressions (28), the rotation of E and the direction of propaga-tion n form either a right-handed screw or a left-handed screw,depending on whether sin 0 < 0 or sin 0 > 0 respectively. Accord-ingly, in conformity to standard radio terminology, the polarization ofa wave receding from the observer is called right-handed if the electricvector appears to be rotating clockwise and left-handed if it appears tobe rotating counterclockwise. See Fig. 5.3.To determine the polarization ellipse of a monochromatic wave, a set

of three independent quantities is needed. One such set obviouslyconsists of the amplitudes a"" ay and the phase difference O. Anotherset is made up of the semimajor and semiminor axes of the ellipse,denoted by a and b respectively, and the orientation angle 1/; betweenthe major axis of the ellipse and the x axis of the coordinate system.These two sets are related such that a, b, 1/; can be found from a"" ay, 0and vice versa. The well-known connection relations are!

a2 + b2 = a",2 + a/

2 2a",aytan 1/;= 2 2COSOa", - ay

Moreover, we have

(0 :::; 1/; < 11")

(31)

(32)

(33)

where X is an auxiliary angle defined by

btanx=+--a (34)

The numerical value of tan x yields the reciprocal of the axial ratio albof the ellipse, and the sign of x differentiates the two senses of polariza-

1See, for example, M. Born and E. Wolf, "Principles of Optics," pp. 24-31,Pergamon Press, New York, 1959.

121


tion, e.g., for left-handed polarization 0 < X ~ 71"/4and for right-handedpolarization -71"/4 ~ X < o.The Stokes parameters for the monochromatic plane TEM wave (28)

are the four quantities

(35)

But since the quantities are related by the identity

(36)

only three of the four parameters are independent. Alternatively, theStokes parameters can be written in terms of the orientation angle Vtand the ellipticity angle X as follows:

81 = 80 cos 2x cos 2if; 82 = 80 cos 2x sin 2if; 83 = 80 sin 2x (37)

where 80 is proportional to the intensity of the wave. From theseexpressions we see that if 81, 82, 83 are interpreted as the cartesiancoordinates of a point on a sphere of radius 80, known as the Poincaresphere, 1 the longitude and latitude of the point are 2if; and 2x respec-tively (see Fig. 5.4). Thus, there is a one-to-one correspondencebetween the points on the sphere and the states of polarization of thewave. In order that the wave be linearly polarized, the phase differ-ence 0 must be zero or an integral multiple of 71", and consequently,according to Eq. (33), X must be zero. Thus we see that the points onthe equator of the Poincare sphere correspond to linearly polarizedwaves. In order that the wave be circularly polarized, the amplitudesaz and au must be equal and the phase difference 0 must be either 71"/2or-71"/2, depending on whether the sense of polarization is left-handed orright-handed respectively. Hence, from Eq. (33) it follows that for aleft-handed circularly polarized wave 2x = 71"/2and for a right-handedcircularly polarized wave 2x = -71"/2; that is, the north and south polesof the Poincare sphere correspond respectively to left-handed and right-handed circular polarization. The other points on the Poincare sphere

1H. Poincare, "Theorie mathematique de la lumiere," vol. 2, chap. 12,Paris, 1892.

122


Fig. 5.4 Poincare sphere is a sphere of radius 80 •.• A point on spherehas latitude 2X and longitude 2if;.

represent elliptic polarization, right-handed in the southern hemisphereand left-handed in the northern hemisphere.Since E(z,t) has only two components Ex, Ey it can be represented,

for any fixed value of z( = zo), as a vector in the complex plane whosereal and imaginary axes are Ex and Ey respectively. That is, as a func-tion of t, to each value of E(zo,t) there corresponds, a point Ex + iEy inthe Argand diagram. I With the aid of this representation an ellip-tically polarized wave may be decomposed into a right-handed and aleft-handed circularly polarized wave. We note that in the complexplane circularly polarized waves of opposite senses are given by thecomplex vectors PI exp (iwt) and P2 exp (-iwt + i'Y), the former being

1 See, for example, K. C. Westfold, New Analysis of the Polarization ofRadiation and the Faraday Effect in Terms of Complex Vectors, J. Opt. Soc.Am., 49: 717 (1959).

123


ri/!:ht-handed and the latter left-handed. Thus in terms of the moduliPI, P2 the semimajor and semiminor axes of the polarization ellipse ofthe wave consisting of the superposition of these two circularly polar-ized waves are P2 + PI and P2 - PI. The orientation angle I/; of theellipse is given by 21/; = 1', where l' is the phase angle between the com-plex vectors at t = 0 (see Fig. 5.5). Since the axial ratio of thepolarization ellipse is (P2 - PI)/(P2 + PI), the angle X is given bytan x = (P2 - PI)/(P2 + PI). From this it follows that

• P22 - PI2sm 2x = 2 + 2P2 PI

2P2PIcos 2x = 2 + 2P2 PI

(38)

Substituting Eqs. (38) into expressions (37), recalling that 21/; = 1', andnoting that so( = P22 + P12) is the intensity of the wave, we get the fol-lowing expressions for the Stokes parameters in terms of the moduliP2, PI and the phase difference 1':

80 = 2(P22 + P12)

83 = 2(.P22 - P12)

iEy

82 = 4 P2PI sin l'(39)

Fig. 5.5 Complex plane. Splitting of elliptical polarization into twooppositely polarized circular components.

124


The state of polarization can be measured in a number of differentways. For example, as is suggested by expressions (35), the state ofpolarization can be measured by using two linearly polarized receivingantennas in such a way that one yields ax, the other yields ay, and thephase difference between their responses yields o. Alternatively, fromexpressions (39) it is seen that the state of polarization also can bemeasured by using two circularly polarized antennas of opposite senses,one of the antennas yielding Pz, the other yielding Pi, and the phasedifference of their responses yielding 'Y. The accuracy of these methodsof measurement depends largely on how purely linear is the linearlypolarized antenna and how purely circular is the circularly polarizedantenna. The techniques of measuring the polarization of monochro-matic waves are well known and will not be discussed here.lUsing the monochromatic wave (26) as a prototype, we now examine

the case of a plane polychromatic TEM wave, which by virtue of itspolychromatic character can be elliptically polarized, or unpolarized,or partially polarized. We assume that the frequency spectrum of thewave is confined to a relatively narrow band of width dw so that theelectric vector of the wave, in analogy with expression (26), may havethe simple analytic representation

E(z,t) = Re IEo(t)ei(.wlc-wo I (40)

where w now denotes sOll1eaverage value of the frequency. Becausethe bandwidth is narrow, Eo(t) may change by only a relatively smallamount in the time interval 1/dw and in this sense is a slowly varyingfunction of time. If the bandwidth were unrestricted, the moot ques-tion of representing a broadband signal in analytical form would ariseand the problem would have to be reformulated.2 In practice, how-

l See, for example, H. G. Booker, V. H. Rumsey, G. A. Deschamps, M. L.Kales, and J. 1. Bohnert, Techniques for Handling Elliptically PolarizedWaves with Special Reference to Antennas, Proe. IRE, 39: 533 (1951);D. D. King, "Measurements at Centimeter Wavelength," pp. 298-309,D. Van Nostrand Company, Inc., Princeton, N.J., 1950; J. D. Kraus, "An-tennas," pp. 479-484, McGraw-Hill Book Company, New York, 1950.

2 See A. D. Jacobson, Theory of Noise-like Electromagnetic Fields of Arbi-trary Spectral Width, Calteeh Antenna Lab. Report, No. 32, June, 1964.Also,Robert M. Lerner, Representation of Signals, chap. 10in E. J. Baghdady(ed.), "Lectures on Communication System Theory," McGraw-Hill BookCompany, 1961.

126


ever, this difficulty is compulsorily bypassed inasmuch as the instru-ments used in measuring polarization are inherently narrowbanddevices.Writing Eo(t) in the form

Eo(t) = e"a,,(t)e-wz(t) + eyauCt)e-i6.(t)'.~ (41)

where the amplitudes a,,(t), ay(t) and the phases o,,(t), Oy(t) are slowlyvarying functions of time, we see from Eq. (40) that the cartesian com-ponents of E(z,t) are given by

E" = a,,(t) cos [ep + o~(t)]

Ey = ay(t) cos [ep + o,,(t) + o(t)]

E. = 0

(42)

where ep = wt - zw/c, o(t) = Oy(t) - o,,(t). Although the amplitudesand phases are irregularly varying functions of time, certain correlationsmay exist among them. .It is these correlations that determine theStokes parameters and consequently the polarization of the wave. Bydefinition, the Stokes parameters of the polychromatic wave (42) arethe time-averaged quantities

80 = (a,,2(t) + (ay2(t)

82 = 2(a,,(t)auCt) cos o(t)

81 = (a,,2(t) - (a/(t)

83 = 2(a,,(t)ay(t) sin o(t)(43)

which are generalizations of the monochromatic Stokes parameters (35).It can be shown 1 that the polychromatic Stokes parameters satisfy therelation

..802 ;::: 812 + 822 + 832 (44)

where the equality sign holds only when the polychromatic wave iselliptically polarized.The polychromatic wave (42) is 'elliptically polarized when the ratio

q of the amplitudes (q = ay/a,,) and the phase differences 0 are absolute

1See, for example, S. Chandrasekhar, "Radiative Transfer," pp. 24-34,Dover Publications, IhC., New York, 1960.

126


constants. That is, when q and IJ are time-independent, the electricvector of the wave traces out an ellipse whose size continually varies ata rate controlled by the bandwidth ~w but whose shape, orientation,and sense of polarization do not change. To demonstrate this, we notethat for an elliptically polarized wave the Stokes parameters (43) become

80 = (1 + q2)(az2(t»

82 = 2q(az2(t» cos IJ

81 = (1 - q2)(az2(t»

83 = 2q(az2(t» sin IJ(45)

Since these parameters satisfy the identity 802 = 812 + 822 + 83

2, only

three of them are independent. In analogy with Eqs. (37) we can writethe Stokes parameters of an elliptically polarized polychromatic wavein the form

81 = 80 cos 2x cos 2if; 82 = 80 cos 2x sin 2if; 83 = 80 sin 2x (46)

Consequently the orientation angle if;of the polarization ellipse is givenby

82 2qtan2if; = - = --cos IJ81 1 - q2

and its ellipticity angle X by

. 2 83 2q.SIn X = - = -- sm IJ80 1+ q2

(47)

(48)

Since q and IJ are time-independent, it is clear from Eqs. (47) and (48)that if;and X are time-independent, in confirmation of the fact that theshape, orientation, and sense of polarization do not change.

We return to the polychromatic wave (42) and now assume that thephase of Ey is shifted with respect to the phase of Ez by an arbitraryconstant amount E. The cartesian components of such a polychro-matic wave are given by

Ez = az(t) cos [q, + IJz(t)J

Ey = ait) cos [q, + IJz(t) + lJ(t) + EJ

E. = 0

(49)

127

Theory of electrOlpagnetic wave propagation

As is clear from Fig. 5.6, the component of the electric field along the x'axis making an angle ()with the x axis is Ez'«(),E) = Ez cos ()+ Ell sin ()and its square is Ez,2«(),e) = Ez2 cos2 () + E/ sin2 () + 2EzEII cos ()sin ().Substituting expressions (49) into this quadratic form, we find that tMinstantaneous value of Ez,2«(),e) is

Ez,2«(),e) = az2(t) cos2 T cos2 () + aIl2(t) cos2 [T + B(t) + EJ sin2 ()

+ 2az(t)ay(t) cos T cos [T + B(t) + EJ cos fJ sin ()

where T = 4>+ Bz(t). Recalling that az(t), all(t), B(t) are slowly varyingfunctions of time and that

cos T = cos [4>+ Bz(t)J = cos [wt - zwjc + Bz(t)J

is a rapidly varying function of time, we find that the mean value of2Ez,2«(),e), which we denote by I«(),e), has the following'representation:

I«(),e) = 2(Ez,2«(),e» = (az2(t» cos2 () + (aIl2(t» sin2 ()

+ [(az(t)all(t) cos B(t» cos e - (az(t)ay(t) sin B(t» sin EJ sin 2() (50)

y

Fig. 5.6 Linearly polarized antenna that picks up the component Ez:~~', of electric field along x' axis. Its response is proportiQnal

to the mean-square value of Ez'.

128


With the aid of definitions (43) this representation leads directly to therelation

[(O,E) = ~[80 + 81 cos 20 + (82 cos E - 8a sin E) sin 20] (51)

which shows that [(O,E) is linearly related to the Stokes parameters. Itis evident from relation (51) that the Stokes parameters can be deter-mined by measuring [(O,E) for various values of 0 and E.If [(O,E) happens to be independent of 0 and E, the wave is said to be

"unpolarized." In other words, an unpolarized wave is one thatsatisfies

[(O,E) = ~8o (52)

independently of 0 and E, or, equivalently, the necessary and sufficientcondition that the wave be unpolarized is

81 = 82 = 8a = 0 (53)

If the polychromatic wave consists of a superposition of several phys-ically independent waves, the intensity of the resulting wave is the sumof the intensities of the independent waves. That is, if [(n) denotesthe intensity of the nth independent wave, the intensity [ of the com-posite wave is given by

(54)

Moreover, since each of the independent waves satisfies relation (51),we have for the nth independent wave

where 80(n), 81 (n), 82(n), 8a(n) are the corresponding Stokes parameters.Hence, from Eqs. (54) and (55) we get the expression

(56)

129

Theory of elec!romagnetic wave propagation

which, when compared with expression (51), shows that each of theStokes parameters of the composite wave is the sum of the respectiveStokes parameters of the independent waves. That is, the Stokesparameters are additive in the sense that

(57)

where 80, 81, 82, 83 are the Stokes parameters of the composite wave and80(n>, 81(n), 82(n), 83(n) (n = 1,2,3, ... )aretheStokesparametersoftheindependent waves into which the composite wave can be decomposed.With the aid of this additivity of the Stokes parameters we can show

that a polychromatic wave is decomposable uniquely into an unpolar-ized part and an elliptically polarized part, the two parts being mutuallyindependent. To do this, we denote the Stokes parameters of the com-posite wave by (80,81,82,83), those of the unpolarized part by (80(1) ,0,0,0),and those of the polarized part by (80(2),81(2),82(2),83(2». Then by theadditivity relation (57) we have

(58)

The degree of polarization m is defined as the ratio of the intensity ofthe polarized part to the intensity of the composite wave, i.e., bydefinition

80(2)m=-

80(59)

From relation (44) we know that the Stokes parameters of the polarizedpart are connected by the relation

(60)

which, with the aid of the last three equalities of Eq. (58), can bewritten as

(61)

It follows from definition (59) and relation (61) that in terms of theStokes parameters of the composite wave the degree of polarization is

130


given by

(62)

Furthermore, the orientation of the polarization ellipse is given by

(63)

and its ellipticity by

(64)

where use has been made of Eqs. (58) and (61). Thus we see thatwhen the Stokes parameters 80, 81, 82, 83 of a partially polarized waveare known we can calculate the degree of polarization from Eq. (62),and the properties of the polarization ellipse of the polarized part ofthe wave from Eqs. (63) and (64). Since X is restricted to the interval-7r/4 ~ X ~ 11'/4, Eq. (64) unambiguously yields a single value for x.However, Eq. (63) can be satisfied by two values of t/I differing by 11'/2,the restriction that t/llie in the interval 0 :S t/I ~ 11' not being sufficientto fix t/I unambiguously. But from the first two of Eqs. (46) we seethat t/I must be chosen such that 81 and 82 have the proper signs. Con-sequently, t/I is determined by Eq. (63) and by the requirement that theappropriate part of the Poincare sphere be used.Another way of decomposing a polychromatic wave is to express it

as the superposition of two oppositely polarized independent waves.Two waves are said to be "oppositely polarized" if the orientation andellipticity angles t/l1, XI of one of the waves are related as follows to theorientation and ellipticity angles t/l2, X2 of the other wave:

(6.5)

This means that the major axes of the polarization ellipses of oppositelypolarized waves are perpendicular to each other, that the axial ratios ofthe ellipses are equal, and that the senses of polarization are opposite.

131


Let (80,81,82,83) denote the Stokes parameters of the polychromaticwaves, and let (80(1) ,81(1) ,82(1) ,83(1» and (80(2),81(2),82(2) ,83(2» denote theStokes parameters of the two independent and oppositely polarizedwaves. By the additivity of the Stokes parameters we have

This relation is satisfied if we choose

80(1) = ~80 - ex

80(2) = ~80 + ex

(66)

(67)

where ex is an unknown quantity. Then the Stokes parameters of theoppositely polarized waves are

(~80 - ex) cos 2Xl cos 21/11(~80 - ex)

(~80 - ex) cos 2Xl sin 21/11

and

O/z80 - ex) sin 2Xl(68)

(~80 + ex) (~80. + ex) cos 2X2 cos 21/12

(~80 + ex) sin 2X2(69)

Since these waves are oppositely polarized, we have

cos 2Xl = cos 2X2 sin 2Xl = - sin 2X2(70)

In view of these relations we see that if the additivity theorem isapplied to the Stokes parameters of the original wave and to the Stokesparameters (68) and (69) of the two oppositely polarized waves, thefollowing relations are obtained:

- 2ex cos 2Xl cos 21/11 = 81

- 2ex cos 2Xl sin 21/11 = 82

-2ex sin 2Xl = 83

132

(71)


Squaring and adding Eqs. (71), we obtain

(72)

From Eqs. (67) and (72) it followsthat the intensities of the two oppo-sitely polarized waves are

80(1) = 7280 - 72 v'812 + 822 + 832

80(2) = 7280 + 72 v'812 + 822 + 832

From Eqs. (71) we see that 1/;1 and Xl are given by

(73)

(74)

Thus we see that a polychromatic wave whose Stokes parameters are(80,81,82,83) can be decomposed into two polarized waves having theintensities (72)80 :!: (72)(812 + 822 + 832)~ and being in the opposite

states of polarization (x,1/;) and ( -x, 1/; + ;) where X and 1/; are given

by Eqs. (74).Since we have used a fixed cartesian system of coordinates (x,y,z) to

describe the Stokes parameters, the question of how these parameterschange under a rotation of the axes naturally arises. To find the lawof transformation, we need to consider only an elliptically polarizedwave. This follows from the fact that a partially polarized wavealways can be decomposed into two oppositely polarized independentwaves. Let (80,81,82,83) denote the Stokes parameters of one of theelliptically polarized waves when referred to the original system, andlet (8~,8i,8~,8;) denote these parameters when referred to the rotatedsystem. The rotation consistsof a clockwisetwisting of the coordinatesabout the z axis and through an angle cf>. By virtue of the fact thatthe wave is elliptically polarized, we can write the Stokes parameters(80,81,82,83) in terms of the ellipticity angle x and orientation angle 1/;,as follows:

80 80 cos 2x cos 21/; 80 cos 2x sin 21/; 80 sin 2x

133


Obviously, when referred to the rotated system these parametersbecome

So So cos 2x cos 2(if; - f/J) So cos 2x sin 2(if; - f/J) So sin 2x

Clearly then, the Stokes parameters referred to rotated coordinates aregiven by

S~ = So

S~ = So cos 2x cos 2(if; - f/J) = SI cos 2f/J + S2 sin 2f/J

s~ = So cos 2x sin 2("" - f/J) = S2 cos 2f/J - SI sin 2f/J

S; = S3

(n))

where (SO,SI,S2,S3) and (s~,s~,s~,s;) are the Stokes parameters in, respec-tively, the original and rotated coordinates. The parameters So and S3

are invariant under the rotation, i.e., the intensity and the ellipticity ofthe wave do not change when the axes are rotated. On the other hand,SI and S2 do not remain the same and hence the orientation angle if;changes when the axes are rotated.

5.4 Coherency Matrices

In the previous section it was demonstrated that the state of polariza-tion of a narrowband polychromatic wave is specified completely bythe four Stokes parameters So, SI, S2, S3. In this section we shall showthat the state of polarization can be specified alternatively by means ofa 2 X 2 matrix whose elements characterize the state of coherencybetween the transverse components of the wave.Let us again consider a plane TEM narrowband (quasi-monochro-

matic) polychromatic wave traveling in the z direction. In accordwith Eqs. (42) the cartesian components of such a wave are

E", = Re {a",(t)eikze-io'e-iwt}

Ell = Re {all(t)eikze-io.e-iwt}

Ez = 0

134

(76)

Radio-astronOlnical antennas

In terms of the complex vector A, whose components are given by

A. = 0 (77)

the electric field components (76) may be written in the form

E. = 0 (78)

which shows that A is the phasor of the electric vector of the wave.Unlike the phasor of a monochromatic wave, A is time-dependent.The elements J pq of the coherency matrix J are defined by

(p,q = x,y) (79)

If Ap and Aq are physically independent then (ApA:) = O. It isobvious from definition (79) that

(80)

and hence the coherency matrix

(81)

is hermitian.To find the connection between the Stokes parameters and the coher-

ency matrix, we note that when expressions (77) are substituted intodefinition (79) we get

Jxx = (ax2(t» J yy = (ay2(t»

J xy = (ax(t)ay(t)ei<l(t) = (ax(t)ay(t) cos o(t» + i(a.(t)ay(t) sin o(t» (82)

JyX = (ax(t)ayc-i<l(t) = (ax(t)ay(t) cos oCt»~ - i(ax(t)ay(t) sin o(t»

where oCt) == Oy(t) - ox(t). Comparing expressions (82) with expres-sions (43), we find that the Stokes parameters are related to the ele-

135


ments of the coherency matrix as follows:

82 = JZy + JyZ 83 = i(Jyz - JZY)

Jzz = 72(80 + 81) J1I1I = 72(80 - 81)(83)

These relations show that the Stokes parameters and the elements ofthe coherency matrix are linearly related and that a specification of thewave in terms of the latter is in all respects equivalent to its specifica-tion in terms of the former.1Since the additivity theorem applies to the Stokes parameters, it

must, in view of the linear relations (83), also apply to the coherencymatrix, in the sense that if JW, J(2), • • • , J(N) are the coherencymatrices of N independent waves traveling in the same direction, thenthe coherency matrix J of the resulting wave is the sum of the coherencymatrices of the independent waves, viz.,

N

J = L J(n)

n=1(84)

To show this, we let Az(n), Ay(n) be the cartesian components of thephasor of the nth independent wave. Then by superposition thecartesian components of the phasor of the resulting wave are

N

Az = L A",(n)

n=1

N

Ay = L Ali(n)n=1

(85)

The elements of the coherency matrix of the resulting wave are

N NJpq = (A~:) = L L (Ap(n)Aq(m)*)

n=1 m=1N

= L (Ap(n)Aq(n)*) + L (Ap(n)Aq(m)*) (86)n=l n"'m

1Compare with E. Wolf, Coherence Properties of Partially PolarizedElectromagnetic Radiation, Nuovo Cirrumto, 13: 1165 (1959).

136


Each term of the last summation is zero since Ap(n) and Aq(m) forn ~ m are independent. Hence we have

N

J pq = L J pq (n)n=l

(87)

where J pq(n) denotes the elements of the coherency matrix of the nthindependent wave. Thus the additivity theorem (84) is verified.From Schwarz's inequality, which is expressed by

.and fWll;l definition (79) it followstllp,t

(88)

or, because of Eq. (80), that

(89)

'The equality sign in these expressions obtains only when Api Aq isconstant, which in turn means that the determinant of the coherencymatrix vanishes only if the wave is elliptically pola~ized. If thedeterminant does not vanish, then the wave is partially polarized.Thatis,

det J =;= 0

det J > 0

for elliptic polarization

for partial polarization(90)

We know from our study of the Stokes parameters that for anunpolarized wave 80 ~ o and 81 = 82 = 83 = O. Casting this intothe l!1nguageof the coherency matrix, we see from Eqs. (83) thatJ zz = J yy = (72)80. Thus we find that the coherency matrix of anunpolarized wave has the form

(91)

137


Moreover, from expressions (46) and Eqs. (83) we see that the coher-ency matrix of an elliptically polarized wave has the form

J = ~ [ (1 + cos 2x cos 2if;)2 (cos 2x sin 2if; - i sin 2x)

(cos 2x sin 2if; + i sin 2x)] (92)(1 - cos 2x cos 2if;)

where if; is the orientation angle of the polarization ellipse and X is itsellipticity angle. To see what this matrix looks like for certain simplestates of polarization, we recall that for linear polarization X = 0, forright-handed circular polarization x = -71"/4, and for left-handedcircular polarization X = 71"/4. Hence from expression (92) we findthat

J = ~ [1 + cos 2if;2 sin 2if;

sin"2if; ]1 - cos 2if; (93)

is the coherency matrix of a linearly polarized wave making an angle if;with the x axis;

(94)

is the coherency matrix for right-circular polarization; and

(95)

is the coherency matrix for left-circular polarization.It follows from relations (83) that the coherency matrix can be

expanded in terms of the Stokes parameters and certain elementarymatrices which in wave mechanics are called the Pauli spin matrices.l

That is,

1 3

J = 2" l spOpp=o

(96)

where 8p (p = 0, 1, 2, 3) are the Stokes parameters, do is the unit

I See, for example, U. Fano, A Stokes-parameter Technique for the Treat-ment of Polarization in Quantum Mechanics, Phys. Rev., 93: 121 (1954).

138


matrix

(97)

and dl, d2, da are the Pauli spin matrices

(98)

From Eq. (91) we see that do represents an unpolarized wave. Further-more, using the decompositions

dl = [~ ~IJ = [~ ~J- [~ ~Jd2 = [~ ~J = 72 [~ ~J - 72 [ ~ 1 ~

1]

da = [~i ~J= 72 [~i ;J - ~~n ~iJ(99)

and recalling the states of polarization that the matrices (93), (94), and(95) express, we see that dl characterizes the excess of a linearly polar-ized wave making an angle"" = 0 over a linearly polarized wave makingan angle"" = 7r/2; d2 the excess of a linearly polarized wave making anangle"" = 7r/ 4 over a linearly polarized wave making an angle"" = 37r / 4;and da the excess of a wave polarized circularly to the left over onepolarized circularly to the right.If we decompose the wave into an unpolarized part and an elliptically

polarized part, then the ratio of the intensity of the polarized part tothe intensity of the original wave is the degree of polarization m of thewave. The quantity

(100)

is the trace (or spur) 'of the matrix and represents the intensity of theoriginal wave. The degree of polarization is given by the expression

m = VI - 4 det J/(Tr J)2 (101)

139


which can be derived from Eqs. (62) and (83). Since this expressioninvolves only the rotational invariants det J and Tr J, the degree ofpolarization does not change with a rotation of the coordinate axis.From Eqs. (63) and (83) it follows that the orientation of the polari-zation ellipse of the polarized part of the wave is given by

tan 21/1 = JXY + JyX

Jxx - Jyy

and from Eqs. (64) and (83) that its ellipticity is given by

. 2 . JyX - JXYsm X = ~-=============y(Tr J)2 - 4 det J

(102)

(103)

Under rotation X does not change because the denominator of Eq. (103)is a rotational invariant, as is the numerator i(Jyx - JXY). However,1/1 does change under rotation, as might have been expected. Thus wesee that m and X are independent of the choice of orientation of thecoordinate axes, while 1/1 is not.The quantities Tr J and det J do not change when the coherency

matrix is transposed; on the other hand, the quantity J yx - J xy simplychanges in sign. Therefore, from Eq. (103) we see that X simplychanges in sign when the coherency matrix is transposed. Since thesign of X determines the sense of polarization, this means that if acoherency matrix describes a wave with a certain sense of polarization,then the transpose of the matrix describes a wave traveling in the samedirection but with the opposite sense of polarization; or if a coherencymatrix describes a wave traveling in a certain direction, the same matrixalso describes a wave traveling in the opposite direction with oppositepolarization.

5.5 Reception of PartiallyPolarized Waves

In this section we shall calculate how much power a given antenna canextract from an incident polychromatic wave. We shall carry out thecalculation by recalling the results of the conventional case, where the

140


incoming wave is monochromatic, and then generalizing these resultsto the case where the incoming wave is polychromatic. This methodof analyzing the problem, which uses the monochromatic theory ofantennas as a point of departure, appears to be the most tractable,because it takes advantage of the fact that the receiving properties ofan antenna are most conveniently expressed in terms of its mono-chromatic behavior as a transmitter.

Hence, for the present we confine our attention to the conventionalmonochromatic theory of receiving antennas. According to thistheory an antenna, actually or effectively, has two circuit terminalsand with respect to these terminals its behavior is as follows: When theantenna is driven by a monochromatic voltage source applied to itsterminals and no radiation is incident, the source "sees" an impedance,namely, the input impedance Zi of the antenna; on the other hand,when a monochromatic wave of the same frequency is incident on theantenna and the terminals are open-circuited, a voltage appears acrossthe terminals, namely, the open-circuit voltage Vo• Then, in accordwith Thevenin's theorem of circuit theory, when the antenna operatesas a receiving antenna having a load impedance Zz connected to itsterminals, the equivalent circuit of the antenna consists of the voltageVo in series with Zi and Zz. From this equivalent circuit it is clearthat the power absorbed by the load is a maximum when Zi and Zz areconjugate-matched, i.e., Zi = Zi. Under this condition of optimumpower transfer, the power generated by Vo is divided equally betweenthe power absorbed by Zi and the power absorbed by Zz. Physically,the power absorbed by Zi consists of the (reversible) power that iscarried away from the antenna by the scattered, or reradiated, portionof the incident power and the (irreversible) power that goes into ohmiclosses, i.e., into the heating of the antenna structure. In the hypo-thetical case where the conjugate-matched antenna is free of ohmiclosses, one-half of the applied incident power is scattered into spaceand the other half is absorbed by the load.

The power that an incident monochromatic wave delivers to theconjugate-matched load of a receiving antenna is related to thebehavior of the antenna as a transmitter. To present this relation, letus suppose that the antenna in question is driven as a transmitter by amonochromatic voltage applied to its terminals. Let us also supposethat the antenna is located at the origin of a spherical coordinate system

141


(r,8,e/». Then if the electric vector (actually phasor) of the far-zonefield radiated by the antenna is Erad, the radial component of thePoynting vector of this field is

the field polarization vector is

EradPr.d(8 -1..) - ---,-,===

,'I' - vErad. Er.d'

and the gain function is

(104)

(105)

g(8,e/»47rr2Srad(r,8,e/»

fo 41r Srad(r,8,e/»r2 dn(106)

where dn( = sin 8 d8 de/» is an element of solid angle. Alternatively,let us now suppose that the antenna is operated as a receiving antennawith a conjugate-matched load attached to its terminals, and that aplane monochromatic wave is incident on it from a direction 8 = 80,

e/> = e/>o. If the electric vector of the incident wave is Einc, the radialcomponent of the Poynting vector of the incident wave is

(107)

and the field polarization vector is

(108)

Then, in compliance with the reciprocity theorem, 1the power absorbedby the load is given by the relation

(109)

1S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Practice,"pp. 390-394, John Wiley & Sons, Inc., New York, 1952.

142


where the quantities g(B,q,) and prad(B,q,) describe the behavior of theantenna in transmission and the quantities Sinc(B,q,) and pinc(B,q,)describe the incident wave in reception.The polarization loss factor

(110)

which appears in relation (109) can take on any value in the rangeo ::; K ::; 1, depending on how closely the polarization of the waveradiated in a direction (O,q,) is matched to the polarization of theincident wave falling on the antenna from the same direction. When

(111)

the radiated wave and the incident wave are matched completely andK = 1. If the field polarization vector of the incident wave is con-jugate-matched in this sense to the field polarization vector of theradiated wave, the power absorbed by the conjugate-matched load is amaximum and, according to Eq. (109), has the value!

(112)

By definition the ratio (Pabs)max/Sinc(B,q,) is the effective area A (B,q,) ofthe receiving antenna,2 and consequently the effective area of theantenna in reception is proportional to the gain function of the antennain transmission, i.e.,

A2A (B,q,) = 47r g(O,q,) (113)

With the aid of this result and definition (110) we can write Eq. (109)in the alternative form

Pab• = A(B,q,)Sinc(B,q,)K(B,q,) (114)

!Y.-C. Yeh, The Received Power of a Receiving Antenna and the Criteriafor Its Design, Proc. IRE, 37: 155 (1949).

2 Compare C. T. Tai, On the Definition of the Effective Aperture of Anten-nas, IRE Trans. Antennas Propagation, AP-9: 224-225 (March, 1961).

143


which explicitly displays the dependence of the absorbed power on theeffective area of the antenna and on the polarization loss factor.To generalize the above discussion to the case where the incident

wave is partially polarized and polychromatic, we write Eq. (114) inthe equivalent form

(115)

where pradprad. is the dyadic associated with the wave radiated in a----direction (8,1/», and pincpinc. denotes the transpose of the dyadicpincpinc. associated with the wave incident from the same direction(8,1/». Moreover, we can in turn write Eq. (115) as

(116)

Now if the incident wave happens to be a polychromatic wave and ifover the entire spectrum of the wave the antenna is conjugate-matchedto the load, then Eq. (116) remains valid for each frequency of thespectrum. Assuming that the antenna and load are so matched, weget the total absorbed power by integrating Eq. (116) over all fre~quencies or, as mentioned in Sec. 5.1, by averaging with respect totime. Such an integration would require a knowledge of the frequencydependence of prad and A. However, we shall assume that prad, andhence A, is independent of frequency over the spectrum of the poly-chromatic wave and shall thus obtain the following expression for thetotal absorbed power

(117)

Let us now consider the case where the incident polychromatic waveis narrowband and has the form

(118)

Here the complex components Eg(t) and E",(t) are slowly varying func-tions of time, w is a mean frequency, and k = wlc. For such an inci-

144


dent wave the matrix of the components of the time-average value ofthe dyadic EincEinc* is

(i = 1,2; j = 1,2) (119)

The coherency matrix of the incident wave in Eq. (117)is the transpose[4] of [h]. The matrix of the components of the dyadic A (pradprad*) ,that is,

(i = 1,2; j = 1,2) (120)

is the effective-area matrix of the antenna. In terms of the effective-area matrix [Ai;] of the receiving antenna and the coherency matrix[l;] of the incident wave, the power absorbed in the conjugate-matched load of the receiving antenna is given by the compact relation

(i = 1,2; j = 1,2) (121)

which follows directly from Eq. (116) and definitions (119), (120).We can divide the incident wave into two mutually independent

parts, viz., an unpolarized part and a polarized part. We do this bysplitting [I;;] into

(122)

and noting that the first matrix on the right represents the unpolarizedpart and the second matrix on the right represents the polarized part.Taking the trace of this matrix equation, we obtain the expression

(123)

whose left side represents the average value (Sinc) of the incident powerdensity, and whose right side represents the power density 2a of itsunpolarized part plus the power density (3(qll + q22) of its polarizedpart. By definition the degree of polarization m is the ratio of thepower density of the polarized part to the total power density; hence in

145


terms of m Eq. (123) can be written as

(124)

From this it follows that

(125)

Since we are free to choose {j, we make the choice

(126)

on the grounds of convenience. In view of expressions (125) and (126)we see that Eq. (122) can be written in terms of the time-average powerdensity (Sine) of the incident wave and its degree m of polarization:

As a consequence of choice (126) we have

(128)

and because [qij] represents a completely polarized wave, we have

(129)

Moreover, by virtue of the fact that [[;;] is hermitian, we also have

(130)

From conditions (128), (129), and (130) we see that the components ofqij may be written in the following way in terms of the orientation angle1f/ and the ellipticity X of the polarization ellipse of the polarized part ofthe incident wave falling on the antenna from a direction (8,cP):

[qU q12] _ ~ [ 1 + cos 2x cos 21f/q21 q22 - 2 cos 2x sin 21f/- i sin 2x

146

cos 2x sin 21f/+ i sin 2x]1 - cos 2X cos 21f/

(131)


Similarly, since [Aij] represents a completely polarized wave, viz., thewave the antenna would radiate if it were used as a transmitter, wecan write it in terms of the orientation angle 1/;' and ellipticity angle x'of the wave radiated in a direction (8,4»:

A-. - 1 A 8 [ 1+ cos 2x' cos 21/;'[ ,,] - % (,4» cos 2x' sin 21/;' - i sin 2x'

cos 2x' sin 21/;'+ i sin 2x' ]1 - cos 2x' cos 21/;'

(132)

Substituting Eq. (127) into Eq. (121), we get

Pab• = Tr [Aij][i-;;] = ~'2(1 - m)(All + A22)(Sinc)

+ m(Allqll + A12q12 + A21q21 + A22q22)(Sinc) (133)

and then, using Eqs. (131) and (132), we find that the time-averagepower absorbed by the conjugate-matched load is given by)

Pab• = H(1 - m)A(8,4»(Sinc(8,4>)) + mA(8,4»(Sinc(8,4>)) cos2 ~ (134)

where

cos'Y == cos 2x' cos 2x cos (21/;' - 21/;) - sin 2x' sin 2x (135)

On the Poincare sphere, 'Y is the angle between the point (21/;,-2x)describing the polarization ellipse of the incident wave and the point(21/;',2x') describing the polarization ellipse of the radiated wave.When 1/;' = 1/; and x' = - x, that is, the two points coincide and'Y= 0, the polarizations of the radiated and incident waves are con-jugate-matched and there is no polarization loss. This, of course,means that the two polarization ellipses have the same orientation inspace and the same axial ratio. It also means that the sense of rotationof the incident wave is the same as the sense of rotation of the radiatedwave if the former is viewed from infinity and the latter from theantenna. If viewed from some fixed position, the senses of rotationwould appear to be opposite.

) H. C. Ko, Theoretical Techniques for Handling Partially Polarized RadioWaves with Special Reference to Antennas, Proc. IRE, 49: 1446 (1961).

147


The first term on the right of Eq. (134) represents the contribution toPab. of the unpolarized part of the incident wave, whereas the secondrepresents the contribution of the polarized part. If the polarizationof the antenna in a direction (O,e/» is conjugate-matched to the incidentwave coming from the same direction, then l' = 0 and the powerabsorbed in the conjugate-matched load resistance is a maximum, i.e.,

(136)

Moreover, if the incident wave is completely polarized, we have m = 1and hence

(137)

On the other hand, if the incident wave is completelyunpolarized, wehave m = 0 and hence

(138)

In this case there is no question of matching.

5.6 Antenna Temperatureand Integral Equation forBrightness Temperature

From the discussion in the previous section we know that if a planeunpolarized polychromatic wave is incident from a direction 0, cfJ on alossless receiving antenna located at the origin of a spherical coordinatesystem (r,O,cfJ), the power absorbed by the matched load (the receiver)is given by

(139)

where Sine is the spectral flux density of the incident wave, A.w thebandwidth of the receiver, and A the effective area of the antenna.The validity of this expression rests on the assumption that A and Sineare independent of frequency within the relatively narrow bandwidth

148


~w. To find the absorbed power for the case where the source is dis-tributed over the sky, we note that the elemental contribution to theabsorbed power of the radiation falling within a cone of solid angle dnand within a bandwidth ~w can be expressed as

(140)

where, in view of Eq. (23), dSinc is related to the brightness temperatureTb of the sky by

(141)

The subscript w has been dropped from Tb for simplicity. Then weassume that the radiation falling on the antenna from any direction isincoherent with respect to the radiation from the other directions. Byvirtue of this assumption, the total absorbed power is the sum of theelemental powers delivered to the antenna by various incident rays.In other words, if the incident rays are physically independent, thetotal absorbed noiselike power may be calculated by integratingexpression (140) over the solid angle subtended by the distributedsource:

(142)

The quantity in the brackets has the dimension of temperature and isknown as the antenna temperature. It provides a convenient measurefor the noiselike power picked up by the antenna in a bandwidth ~w.Thus antenna temperature T" is defined by

(143)

or, in terms of the gain function g(8,q,), by

(144)

149


According to this definition, one possible physical interpretation ofTa is as follows: If the antenna is completely enclosed by a surfacewhich radiates as a blackbody at temperature Ta, then the antenna willabsorb in its load resistance the power kTa ~w. Alternatively, Ta maybe regarded as the temperature to which the effective input resistanceof the receiver (which, if matched, equals the radiation resistance of theantenna) must be raised so that the noise power, produced by the ther-mal motion of the electrons and delivered to the receiver through a loss-less line, would equal P <lb. in accordance with the relation P <lb. = kTa ~w.

Antenna temperature as defined by Eq. (144) is a measure of the inci-dent radiation only; it is not a measure of the temperature of thematerial in the antenna structure.So far we have tacitly assumed that the direction of the main lobe

of the receiving antenna is fixed and lies along the axis 8 = 0 of thespherical coordinate system. However, this is an unnecessary restric-tion and can be removed easily. For example, if we let n be a unit vec-tor pointing in the direction of the main lobe and n' be a unit vector inthe direction of the solid angle dfl(n'), then Eq. (144) can be formallywritten in the following more general form:

(145)

This is the integral equation for the brightness temperature Tb(n').By changing the orientation n of the antenna so that its radiationpattern effectively scans the sky, we can measure Ta as a function of n.Moreover, by measuring the radiation pattern or by predicting ittheoretically, we can deduce the gain function. Accordingly, weregard Ta(n) and g(n,n') as known quantities, and find the brightnesstemperature Tb(n') of the sky in terms of Ta(n) and g(n,n') by solvingthe integral equation. A practical way of solving the integral equationis by successive approximations.! To show what the scheme of themethod is, let us write Eq. (145) in operator form

(146)

1See J. G. Bolton and K. C. Westfold, Galactic Radiation at Radio Fre-quencies, Au;tralian J. Sci. Res., 3: 19 (1950).

150


where K(n,n') is the integral operator defined by

K(n,n')f(n') == 4~ J g(n,n')f(n')dn(n') (147)

the quantity fen') being a typical function of n'. Also, for simplicity,we do not bother to write explicitly the arguments n, n'. Thus inoperator form Eq. (145) becomes

or equivalently

(148)

Suppose as a zero-order approximation to '1'b we choose the known func-tion T a and then take

'1'1 = Ta + (1 - K)Ta (149)

as the first-order approximation to Tb• By applying the same proce-dure to '1'1, we obtain the second-order approximation to Tb:

Clearly for the nth approximation to Tb we have

Tn = Ta + (1 - K)Tn-1

or in terms of Ta

(150)

(Hil)

Tn = Ta + (1 - K)Ta + (1 - K)2Ta + ... (1 - K)nTa (152)

5.7 Elementary Theory of the Two-element Radio Interferometer

To attain high resolving power, antenna arrays having multilobe receiv-ing patterns are used. The high resolving power of such arrays stems

151


Direction of incidentradiation

Antenna

It-- Axial planeIIIIIIII

Transmission line

Antenna

Fig. 5.7 Two-element interferometer. Receiver is connected to twoidentical and similarly oriented antennas. Direction ofincident radiation makes angle'" with base line and angle awith axial plane. Separation of antennas is l. Receiver isat electrical center of transmission line.

from the fact that each lobe of the multilobe pattern becomes narrowerand hence more resolvent as the spacing between adjacent antennas isincreased.The simplest array that exhibits a multilobe receiving pattern is the

two-element radio interferometer,! consisting of two identical and sim-ilarly oriented receiving antennas separated by a distance l and con-nected to a single tuned2 receiver by a transmission line (Fig ..5.7). Tofind the receiving pattern of such an interferometer, we note that by

lOne of the first applications of the two-element radio interferometer,which we recognize as the radio analog of Michelson's optical interferometer,was made by L. L. McReady,J. L. Pawsey, and R. Payne-Scott, Solar Radia-tion at Radio Frequencies and Its Relation to Sunspots, Proc. Roy. Soc.,(A) 190: 357 (1947).

2 Because the receiver is sharply tuned we can use a monochromatic theoryin most of the analysis.

152


the reciprocity theorem its receiving and radiation patterns must bethe same and we recall from Sec. 3.5 that its radiation pattern mustbe the product of the radiation pattern F of one antenna and the arrayfactor A of the two antennas. Thus it follows from the reciprocity andmultiplication theorems that the receiving pattern of the interferometeris IFAI and that the power fed to the receiver is proportional to IFA12.In a typical two-element interferometer F has one main lobe and thefactor A has numerous lobes; these are called "grating" lobes. Con-sequently, the array factor A is responsible for the multilobe structure(fringes) of the receiving pattern and F gives the pattern's slowly vary-.iog envelope (Fig. 5.8).Since we are interested in the resolving properties of the interfer-

ometer and since they depend chiefly on A, we may, insofar as radia-tion falling within the central portion of the main lobe of F is concerned,set the factor F equal to unity and thus assume that the receiving pat-tern of the interferometer is given by [AI alone. Accordingly, from

Gratinglobes

Fig: 5.8 Pola,. plot of typical receiving pattern of two-element inter-.jerome,ter.

153


Eq. (79) of Sec. 3.6 we see that the receiving pattern of the two-elementinterferometer is given byl

IA(lf)1 = 2 cos ("72kl cos 11') (153)

where 11' is the angle between the direction of the incident wave and thebase line, i.e., the straight line joining the two antennas. In terms ofthe complementary angle a( = 7rj2 - 11'), i.e., the angle the direction ofthe source makes with the plane perpendicular to the base line (axialplane), the radiation pattern is

IA (a) I = 2 cos ("72kl sin a) (V54)

The power P fed to the receiver of the interferometer is proportional toIA(a)12 and hence

pea) = 2Po cos2 (>~kl sin a) = Po!1 + cos (kl sin a)] (155)

where Po denotes the power fed to the receiver by a single antenna.As a point source of radiation sweeps across the sky, the angle a changesand P oscillates between the limits 0 and 2Po• This is strictly true forsmall a only. Actually, when a becomes large, the power fed to thereceiver is no longer given by expression (155) alone, but by the productof expressio)1 (155) and IFj2. The factor IFI2 has the effect of taperingoff the oscillations (Fig. 5.9). The nulls of the receiving pattern occurwhere

kl sin a = (2n + 1)7r (n = 0, 1,2, ... ) (156)

and the maxima occur where

kl sin a = 2n7r (n = 0, 1, 2, ... ) (157)

For small values of a, i.e., for values of a such that sin a = a, the width

1We obtain this expression from Eq. (79) of Chap. 3 by setting n = 2 andl' = O. The fact that the receiver of the interferometer is located at theelectrical center of the transmission line connecting the two antennas requiresthat'Y = O.

154


of each grating lobe is given by the simple relation

211" XAa = kf = r (158)

which shows that as the spacing l is increased the width of each gratinglobe is decreased. It also shows that for a fixed spacing the width ofeach grating lobe is decreased as the wavelength Xto which the receiveris tuned is decreased.In the derivation of formula (155) it was tacitly assumed that the

incident radiation comes from a point source. We now shed thisrestriction and consider the more realistic case where the source hasangular extent. In this case the received power is given by

P(ao) = I[1 + cos (kl sin a)Jf(a - ao)da (159)

where f(a - ao) is the distribution across the incoherent source and aois the angle that the mean direction of the source makes with the axialplane. If the width of the source is 2w, the limits of int~gration area = ao - wand a = ao + w. We assume that ao and 2w are small,i.e., we assume that the source is narrow and near the axial plane.Expression (159) is a generalization of expression (155) and reduces toit when f(a - ao) is the Dirac delta function Il(a - ao).

cx __

Fig. 5.9 Rectangular plot of receiving pattern of two-element inter-ferometer for point source. The minima are zero. Themaxima are tapered, by virtue of the fact that IFI2 is notequal to one for all values of a. Actually, IFI2 behaves in amanner indicated by the envelope.

155


ao---Fig. 5.10 Rectangular plot of receiving pattern for a narrow source

having uniform distribution.

However, if f(a - ao) has a narrow rectangular shape, i.e., iff(a - ao) = Po/2w for ao - w ::::;;a ::::;;ao + wand f(a - ao) = 0 forall other values of a, expression (159) leads to

P(ao) = f(1 + cos kla)f(a - ao)da = Po(l + V cos klao) (160)

where the quantity V defined by

V = sin klwklw

(161)

is the "visibility factor," a term borrowed from optics.l As the rec-tangular distribution sweeps across the sky, ao changes and P(ao)oscillates sinusoidally between Po(l - V) and Po(1 + V). The ratioof the minimum value to the maximum value is the modulation indexM given by

1- VM=I+V (162)

From this we see that if the distribution function is rectangular thewidth of the source can be determined by measuring M and then com-puting w from Eqs. (161) and (162). See Fig. 5.10.

1See, for example, M. Born and E. Wolf, "Principles of Optics," pp.264-267, Pergamon Press, New York, 1959.

156


More generally, if the source is narrow but otherwise arbitrary, itfollows from Eq. (159) that for small values of ao the received power isgiven by

P(ao) = J[l + cos (kl sin a)J.f(a - ao)da

= f[1 + cos (kla)]f(a - ao)da

= fj(a - ao)da + f cos (kla)f(a - ao)da (163)

The first term on the right is Po, the power fed to the receiver by oneantenna; the second term on the right we denote by Pl. Accordingly,we write

where

PI = f cos (kla)f(a - ao)da

(164)

(165)

If we let u = a - aoand note that f(u) == 0 for lui> w, then Pi canbe cast in a form that explicitly displays its amplitu4e and phase, viz.,

Pi = J cos [kl( u + ao) ]f( u )du = Re eiklao J_"'",eik1uf( u )du= Re eiklaoQ(kl)ei~(kl) = Q(kl) cos [klao + cf>(kl)] (166)

Here the amplitude Q(kl) and the phase cf>(kl) are defined by

The inverse Fourier transform of Eq. (167) yields the relation

1J:'"f(u) = - Q(kl) cos [klu - cf>(kl)]d(kl)11" 0

. (167)

(168)

which expressesj(u) in terms of the amplitude and phase of the observedquantity Pi, viz., Q(kl) and cf>(kl). Relation (168) shows that it is pos-sible, in principle, to find the distribution by measuring the amplitude

157


and phase with different base lines. However, the measurement ofphase sometimes presents difficulties. Unfortunately, it is not possibleto determine uniquely the distribution from a knowledge of the ampli-tude alone, unless some information is available beforehand about thegeneral shape of the distribution function.As an example of a two-element radio interferometer with a horizon-

tal base line we mention the one in Owens Valley, California, which isoperated by the California Institute of Technology~ Each element ofthe interferometer is a steerable parabolic reflector antenna, 90 feet indiameter, placed on the ground. It is used for the measurement ofangular diameters at centimeter and decimeter wavelengths, and forpositional work. 1A two-element interferometer having a vertical base line can be

effected by placing a single horizontally beamed antenna on a cliff ofheight l/2 overlooking the sea. The surface of the sea acts as an imageplane. Thus the elevated antenna and its image constitute a two-element interferometer. 2 The elevated antenna is horizontally polar-ized to take advantage of the fact that the surface of the sea approxi-mates a perfect reflector most closely for horizontal polarization. Theimage antenna is out of time-phase with respect to the elevated antennaand hence the power received from a point source is given by

P(OI) = 2Po[1 - cos (kl sin 01)] (169)

where 01 is the angle the direction of the source makes with the axialplane, i.e., the surface of the sea, and Po is the power the elevatedantenna would receive if it were not operating as an interferometer.In this case the nulls of the receiving pattern occur where

kl sin 01 = 2n1l' (n = 0, 1,2, ... ) (170)

and the maxima occur where

kl sin 01 = (2n + 1)11' (n = 0, 1, 2, ... ) (171)

1 For details the reader is referred to J. G. Bolton, Radio Telescopes, chap. 11in G. P. Kuiper and B. M. Middlehurst (eds.), "Telescopes," The Universityof Chicago Press, Chicago, 1960.

2 An interferometer of this type is called a "sea interferometer," a "cliffinterferometer," or a "Lloyd's mirror" after its optical analog.

158


From Eq. (169) we see that as the source rises above the horizon andcuts through the grating lobes of the interferometer, the received powerincreases from zero and oscillates in characteristic fashion. Then asthe source rises above and out of the beam, the received power graduallytapers to zero, an effect which would have been displayed by Eq. (169)had it been multiplied by IFj2.

5.8 Correlation InterferometerThe two-element interferometer discussed in the previous sectionbehaves as though the incident radiation were monochromatic becausethe receiver of the interferometer is sharply tuned and accepts only avery narrow band of the incident radiation's broad spectrum. Sincethe energy residing outside this band is rejected and thus wasted, thesensitivity of the interferometer is limited by the bandwidth of thereceiver. Increasing the bandwidth would increase the sensitivity butwould also deteriorate the multilobe pattern and hence decrease theprecision of the system. This means that in a phase-comparison typeof interferometer the bandwidth is necessarily narrow and the sensi-tivity is limited by the bandwidth. In addition to this inherent limita-tion on the sensitivity there is a practical limitation on the resolvingpower. As the antennas are moved farther apart for the purpose ofincreasing the resolving power, it becomes more difficult to compareaccurately the phases of the antenna outputs. The awkwardness ofmeasuring the phases of two widely separated signals places a practicallimitation on the antenna separation and this in turn places a limitationon the resolving power.Because of these and other limitations, the two-element phase-com-

parison interferometer has been superseded in certain applications bymore sophisticated systems. In this section we shall discuss one suchsystem, namely, the correlation interferometer of Brown and Twiss.l

But before we do this, let us discuss the concept of degree of coherence2

upon which it is based.

1R. H. Brown and R. Q. Twiss, A New Type of Interferometer for Use inRadio Astronomy, Phil. Mag., 45: 663 (1954).

2 F. Zernike, The Concept of Degree of Coherence and Its Application toOptical Problems, Physica, 5: 785 (1938).

159


To measure the degree of coherence of the polychromatic radiationfrom an extended source we use two identical and similarly orientedantennas. These antennas receive the incident radiation and conse-quently develop at their respective output terminals the voltages VI(t)and V2(t), which for mathematical convenience are assumed to have theform of an analytical signal. The resulting voltages are fed into areceiver whose output is the time-average power given by

P = ([VI(t) + V2(t)][Vt(t) + VW)))

= (VI(t) Vi (t» + (V2(t)VW» + 2 Re (VI(t)vt(t» (172)

The first term on the right is the time-average power output of oneantenna operating singly, and the second term is the time-averagepower of the other antenna operating singly. Hence, the third term isthe only one that involves the mutual effects or mutual coherence ofthe incident radiation. Accordingly, as a quantitative measure of themutual coherence of the incident radiation, we choose the complexquantity l' defined by

(V I(t) Viet»~l' = v(VI(t)ViCt»(V2(t)vi(t)

(173)

and referred to as the complex degree of coherence. The modulus h'lof l' is known as the degree of coherence. By the Schwau inequa.lityit can be shown that

11'1s:; 1 (174)

When 11'1= 0 the incident radiation is incoherent; when 11'1= 1 theincident radiation is coherent; and when 0 < 11'1< 1the incident radia-tion is partially coherent. In terms of 1', expression (172) for the poweroutput of the receiver becomes

P = (VI(t)Vi(t» + (V2(t)Vi(t»

+ 2 V(VI(t) Vi(t»(V2(t) Vi(t» 11'1cos (arg'Y) (175)

where arg'Y is the phase of 1', that is, l' = 11'1exp (i arg 1'). Forsimplicity we assume that the power outputs of the antennas when

160


operated separately are equal; that is, we assume that

(176)

With the aid of this assumption expression (175) reduces to the relation

P = 2Po[1 + h'l cos (arg'Y)] (177)

which clearly indicates that the degree of coherence 11'1 of the incidentradiation is measured by the visibility. Expression (177) provides anoperational definition o(the degree of coherence.To show how I' is related to data that specify the source and to the

spacing of the antennas we proceed as follows. We choose a cartesiancoordinate system with origin 0 in order that the antennas be locatedalong the x axis at the points x = =+= 1/2. For simplicity the source isassumed to be a line source lying along the 1; axis of a parallel cartesiansystem with origin 0'. The distance between 0 and 0' is R. See Fig.5.11. We think of the source as being divided into elements of lengthdh, db dl;a, . . . , and we denote the respective antenna output volt-ages due to the radiation from the mth element by the analytic signalstrml(t) and Vm2(t). The respective antenna output voltages due to theradiation from the entire source are given by the sums

(178)

We assume that each element of the source is an isotropic radiator.Consequently the radiation from the mth element produces the voltage

( Rml) e-iw(t-Rmdc)Vml(t) = Am t - - R

C ml

in one antenna and the voltage

(R 2) e-iw(t-Rm,/c)

Vm2(t) = Am t - ~ R .C . m2

(179)

(180)

in the other. Here Rm1 and Rm2 are the distances from the mth ele-ment to the antennas, C is the velocity of light, w is the mean frequency

161


Line source

Anten na No. 1\

x--1/2

Rm2

Antenna NO.2/

Fig. 5.11 Arrangementfor the measurement of degree of coherence.Two identical and similarly oriented antennas are exposedto the polychromatic radiation from line source. R is dis-tance from 0 to 0'. The angle that the line connecting 0with the center of the line source makes with the axial line00' is ao. The angle that the line from 0 to the elementd~m makes with the line 00' is am. The distances from theelement d~m to the antennas are Rm1 and Rm2.

of the incident radiation, and Am is the complex amplitude function.It follows from expressions (178) that

(V1(t)vt(t» = ~ (Vml(t)V;::l(t»+ ~~ (Vml(t)V:1(t»m fn;Jlin

(181)

However, the isotropic radiators that make up the source are assumedto be statistically independent and to have a mean value of zero, Le.,

162

when m :;6 n (182)


and consequently the cross-product terms of Eq. (181) vanish. Thuswe get

(VI(t)vt(t» = L (Vml(t)V~I(t»m

Similarly, we obtain

(V 2(t)VW» = L (Vm2(t)V~2(t»m

and

(VI(t) VW» = L (V ml(t) V~2(t»m

(183)

(184)

(185)

Substituting expressions (179) and (180) into Eqs. (183) and (184)respectively, and noting that Am is stationary, we see that

(VI(t)Vt(t» = ~ _1_ / A (t _ Rml) A * (t _ Rml)\';;:RmI2 \ m C m C /

= l R~12 (Am(t)A~(t» (186)m

and

(V2(t)VW» = f R~22 (Am (t - R;2) A~ (t ~ R;2))

= l R~22 (Am(t)A~(t» (187)m

Since Rmi and Rm2 are approximately equal, these two expressions inthis approximation are equal to each other and to Po. That is, inagreement with assumption (176) we have

(188)

Substituting expressions (179) and (180) into Eq. (185), we obtain

163


where k = wle. Since Rm1 and Rm2 are approximately equal, and sinceAm is stationary, Eq. (189) reduces to

(190)

To cast this expression into the form of an integral, we introduce thefollowing geometric considerations. From Fig. 5.11 it is clear that

(191)

Since R» (~m+ J) and R» (~m- J) it follows from Eqs. (191)

that

Rm2 - Rm1 = _ ~l and (192)

Using these approximations, we see that Eq. (190) becomes

(V1(t)VW» = L ~2 (Am(t)A;::(t»eiklt ••!Rm

(193)

Moreover, from Fig. 5.11 it is clear that tan am = ~mIR, but since am issmall we have the simpler relation

(194)

With the aid of relation (194) we may cast Eq. (193) in the followingform:

(V1(t)VW» = l ~2 (Am(t)A;::(t»eikla.. (195)m

which suggests that the sum may be written as an integral. If we let

164

Radio-astroIiomical antennas

then Eq. (195) in the limit becomes

(196)

where ao is the angle that the line connecting 0 with the center of thesource makes with the axial plane.

Substituting Eqs. (188) and (196) into expression (173), we find thatthe complex degree of coherence is related to the source distributionfunction f(a - ao) and to the antenna spacing kl by the relation

'Y = -l J f(a - ao)eik1a daPo

Since

Po = ff(a - ao)da

we may also write 'Y in the homogeneous form

ff(a - ao)eik1a da'Y=-~----ff(a - ao)da

(197)

(198)

(199)

If in accord with the notation of the previous section we denote theamplitude and phase of the integral appearing in Eq. (196) by Q(kl)and q,(kl) respectively, then we may write

ff(a - ao)eik1a da = Q(kl)eikla'ei4>(kl) (200)

and from Eqs. (199) and (200) note that the degree of coherence isgiven by

I I = Q(kl)'Y Q(O)

and the phase of 'Y by

arg 'Y = klao + q,(kl)

(201)

(202)

Substituting expressions (201) and (202) into Eq. (177), we obtain the

165


expression

{Q(kl) }P = 2Po 1 + Q(O) cos [klao + q,(kl)] (203)

which places in evidence the equivalence of visibility and degree ofcoherence.In the special case where f(a - ao) is a rectangular function of

width 2w, that is, f(a - ao) = Po/2w for ao - w ~ a ~ ao + wandf(a - ao) = 0 for all other values of a, we have

J . Po fa.+w . sin klw .f(a - ao)e'kla da = - e,kla da = Po -- e,kla.2w a.-w klw (204)

Hence for such a rectangular distribution the degree of coherence isgiven by

I I = Q(kl) = sin klw'Y Q(O) klw

and the phase of 'Yby

arg'Y = klao

(205)

(206)

Thus we see that for a uniform source the magnitude 'Yis related byEq. (20.5) to the width of the source and the phase arg'Y is related byEq. (206) to the angle between the axial plane and the line runningfrom the origin to the center of the source.The correlation coefficient p(V1,V2) of VI(t) and V2(t) by definition is

where

U2(V1) = «VI - (V1»(vt - (Vi»)

U2(V2) = «V2 - (V2»(V: - (v:m

(207)

(208)

are the variances. Since VI(t) and V2(t) have zero mean value, i.e.,

(209)

166


the expression for p reduces to

(210)

Comparing expressions (173) and (210), we see that the complex degreeof coherence 'Y is equal to the correlation coefficient p(V 1,V 2)' Hencewhat is actually measured in the above arrangement is the amplitudeand phase of the correlation coefficient p( V 1,V2).

Now let us suppose that the circuits are changed (to a Brown andTwiss system) so that we can measure the correlation coefficient of thesquare of the moduli Ml(t) and M2(t) of Vl(t) and V2(t) respectively.The correlation coefficient p(M 12 ,M 22) by definition is

«M12 - (MI2»(M22 - (M22»)p(M 1

2,M 22) = u(M !2)u(M 22)where

u2(M 12) = «M 12 - (M 12»2)

u2(M22) = «M22 - (M22»2)

(211)

(212)

Under the assumption that the receiver noise is negligible compared tothe desired signal, it can be shown by statistical calculations! that

(213)

But

and hence

(214)

1 E. N. Bramley, Diversity Effects in Spaced-aerial Reception of IonosphericWaves, Proc. Inst. Elec. Engrs., 98 (3): 9-25 (1951); also, J. A. Ratcliffe,Some Aspects of Diffraction Theory and their Application to the Ionosphere,Rept. Prog. Phys., 19: 188-267 (1956).

167


This means that the correlation coefficient of the squares of the moduliof the antenna voltages is equal to the square of the degree of coherenceof the incident radiation. In the case where the source is a rectangulardistribution of width 2w, we see from Eqs. (205) and (214) that

(M 2 M 2) = I sin klw 12

p . 1, 2 klw (215)

With the aid of this result w can easily be computed from a knowledgeof the correlation coefficient p(M 12,M 22).

Although p(M 12,M 22) yields information about I'YI only, and p(V1, V2)

yields information about I'YI as well as arg 'Y, the former is easier tomeasure, as no phase-preserving link between the antennas is required.The correlation interferometer of Brown and Twiss may be defined

as an interferometer that measures p(M 12,M 22). It differs from a con-ventional interferometer, which measures p(V1, V2). Since no radio~frequency phase-preserving link is necessary in the measurement ofp(M 12,M 22), the antennas can be separated greatly and thus high resolv-ing powers can be realized.

168

Electromagneticwaves in a

plasma 6

In recent yea-Faconsiqerable attention has beeil focused onthe theory of electl'Omagneticwave propagation in a plasmamedium. In large measure this interest in the theory hasbeen stimulated by its applicability to current problems inradio communications, radio' astronomy, and controlledthermonuclear fusion. For example, the theory has beeninvoked to m'plain such phenomena as the propagation ofradio Waves in the ionosphere,! the propagation of cosmicradio waves in the flolar atmosphere, in nebulae, and iilinterstellar and interplanetary space, 2 the reflection ofradio waveS frpm meteor tra.ils3 and from the envelope ofionized g!ts that s\lrrOl,l,m:Js a !SPacecraft as it pen~trates

1 K. G.Budden, "Radio Waves in the Ionosphere," CambridgeUniversity Press, New York, 1961; also, J. A. Ratcliffe, "TheMagneto-ionic Theory," Cambridge University Press, New York,1961.

2 V. L. Ginzburg, "Propagation of Electromagnetic Waves inPlasma," Gord{)n l!todBreach, Science Publishers, Inc., New York,1961; also, I. S.Shldovsky, "Cosmic Radio Wlleyes," Harvard Uni~versity Press, Cambridge, Mass., 1960.

3 N. Herlofson, Plasma Resonance in Ionospheric Irregularities,ArkjlJ Fysik, 3: 247 (1951); also, J. L. Heritage, S. Weisbrod, andW. J. Fay, "Experimental Studies of Meteor Echoes at 200M~gll.<;ycles ill Electromagnetic Wave Propagation," in.M. Desi~rant and ,T. 4 Mi<;hiel~ (eds.), Apade.mi(l Pre~s Inc., New Yor~,1960.

1(;19


the atmosphere,l and the propagation of microwaves in laboratoryplasmas.2In these applications the medium through which the electromag-

netic wave must travel is formally the same: it is a plasma, or moredescriptively, a macroscopically neutral ionized gas consisting prin-cipally of free electrons, free ions, and neutral atoms or molecules.This means that from one application to another the nature of theproblem does not change essentially, despite the large variations themedium may undergo in, say, its degree of ionization and its tempera-ture. However, in the presence of a beam of charged particles inter-acting with the plasma, an electromagnetic wave does acquire char-acteristics which differ qualitatively from those in a beam-free plasma.One such characteristic is, for example, wave amplification by beam-generated plasma instabilities. 3 Accordingly, phenomena of this kindhave to be treated separately and for this reason are excluded fromthe present discussion.In this chapter we shall analyze the problem of electromagnetic wave

propagation in a plasma medium by calculating the constitutive param-eters of the plasma and then treating the problem as a conventionalproblem in the theory of electromagnetic wave propagation in a con-tinuous medium.

6.1 Alternative Descriptions ofContinuous Media

We recall from electromagnetic theory that for a continuous mediumat rest Maxwell's equations can be written in the following elementary

1Proc. Symp. Plasma Sheath, vol. 1, U.S. Air Force, Cambridge ResearchCenter, December, 1959.

2 V. E. Goland, Microwave Plasma Diagnostic Techniques, J. Tech. Phys.,U.S.S.R., 30: 1265 (1960).

3 R. A. Demirkhanov, A. K. Gevorkov, and A. F. Popov, The Interactionof a Beam of Charged Particles with a Plasma, Proc. Fourth Intern. Conf. onIonization Phen. in Gases,vol. 2, p. 665, North Holland Publishing Company,Amsterdam, August, 1959.

170

Electromagnetic waves in a plasma

form,l

1 {JIJo V' X B = Jt + EO at E

V' X E = - :t BEOV" E = Pt

V'.B=O

(1)

(2)

(3)

(4)

which describes the macroscopic electromagnetic field in the mediumby the two vector fields E and B and characterizes the medium by thetotal macroscopic charge density Pt and total macroscopic current den-sity Jt. The constants IJo and EO denote respectively the permeabilityand dielectric constant of the vacuum.The total charge density Pt consists of the free charge density P and

bound charge density Pb; similarly the total current density Jt consistsof the free current density J and bound current density Jb, that is,

Pt = P + Pb (5)

(6)

The free charge is that part of the total charge which exists independ-ently of the field. On the other hand, the bound charge is an attributeof the multipoles that are induced in the medium by the electromag-netic field. Indeed, Pb and Jb are given by the series2

Pb = - V'• P + 7~V'V':Q + ...{J L a

Jb = at P - 2 at V'•Q + v X M +

(7)

(8)

1See, for example, R. W. P. King, "Electromagnetic Engineering," McGraw-Hill Book Company, New York, 1945; also, L. Rosenfeld, "Theory of Elec-trons," North Holland Publishing Company, Amsterdam, 1951.

2 We keep only the leading terms. When the series are terminated at acertain degree of approximation, the number of electric multipoles exceedsthe number of magnetic multi poles by one. In compliance with ihis rule wehave kept two electric multi poles P and Q and one magnetic multi pole M.

171


where P, M, Q denote respectively the volume densities of the electricdipoles, magnetic dipoles, and electric quadrupoles that are producedby the action of the electromagnetic field on the neutral molecules ofthe medium. In other words, P, M, Q are functionals of E and B.In view of the series (7) and (8), Maxwell's equations (1), (2), (3),

(4) become

1 a a Ia;;; v x B = J + EOat E + at P - 2 at v . Q + v XM + (9)

V X E = - ft B (10)

EoV.E=p-V,P+~VV:Q+ (11)

V. B = 0 (12)

If we define the electric displacement D by

V.D = p (13)

then on comparing this relation with Eq. (11) we see that this definitionleads to

D = EoE+ P - ~V . Q + ...

Moreover, if we define the vector H by

1H=-B-M/Joo

(14)

(15)

(16)

then Eq. (9) leads to

VXH=J+ftD

Hence, when D is defined by Eq. (13) and H by Eq. (15), the Maxwellequations (9), (10), (11), (12) assume their conventional form:

aVxH=J+-D ataVxE=--B at

V.D = p

V.B = 0

172

(17)

(18)

(19)

(20)


To apply these considerations to the case of an electromagnetic wavepassing through a plasma medium, we note that the wave, in principle,interacts with all three components of the plasma, viz., the free elec-trons, the free ions, and the neutral molecules. However, the inter-action of the wave with the neutral particles is so feeble in comparisonto the interaction between the wave and the charged particles that itcan be neglected. This means that P, M, Q, which constitute ameasure of the interaction between the wave and the neutral particles,can be set equal to zero. Moreover, since the ions are much moremassive than the electrons, the velocity imparted to the ions by thewave is negligibly small compared to the velocity given to the electrons.That is, when an electromagnetic wave passes through a sufficientlyionized plasma only the free electrons of the plasma influence appre-ciably the transmission of the wave.The interaction between the wave and the electrons is introduced

into Maxwell's equations through the current density term J. As willbe shown subsequently (see Eq. 43), the electronic current density Jproduced in the plasma by the wave is related in the steady state to theelectric vector E of the wave by a linear relation of the form

J = aE + iwbE (a, b = positive real) (21)

unless E exceeds a value at which nonlinearities come into play. Ittherefore follows that when an electromagnetic wave whose electricvector E lies within the bound of linearity passes through a sufficientlyionized plasma, the Maxwell equations for the phenomenon in thesteady state become

v X H = aE + iwbE - iwEoE

V X E = iWJLoH

Let us write Eq. (22) in the form

V X H = (a + iwb)E - iWEOE

(22)

(23)

(24)

where (a + iwb)E appears as a conduction current and -iWEOE as avacuum displacement current. This form suggests that we think ofthe complex factor (a + iwb) as a complex conductivity given by

(To = (T, + iui = a + iwb (25)

173


and thus describe the plasma as a conductor having a permeability p'o,

a dielectric constant EO, and a complex conductivity qc. However, weshall not use this mode of description here. Instead, we shall interpretthe term iwbE of Eq. (22) as a polarization current and thus considerthe plasma as a lossy dielectric.To do this, we recall that for a lossy dielectric by definition we have

v X H = qE - iwP - iWEoE (26)

where q is the conductivity of the dielectric and P is the polarization ofthe neutral molecules of the dielectric. Also for a dielectric we have

P = xcE (27)

where Xc, the electric susceptibility of the dielectric, is always positive.Since the relation

D = EoE + P = EE (28)

defines the dielectric constant E of the dielectric, it follows that thedielectric constant of the dielectric is given by

E = EO+ Xc (29)

Clearly, for a true dielectric E is always greater than EObecause Xc ~ O.If we are to describe the plasma as a lossy dielectric, we must identify

Eq. (22) with Eq. (26), setting aE = qE and iwbE = -iwP = -iwx.E.This means that the conductivity q of the dielectric must equal a andits electric susceptibility X. must equal -b, that is, q = a and Xc = -b.Since b is positive, x. must be negative. Thus, if the effect of themotion of the electrons is to be accounted for by a conductivity and apolarization, then we must think of the plasma as a lossy dielectricwhose electric susceptibility is negative. The constitutive parametersof the dielectric are then given by

q=a J.l = J.lo E = EO - b (30)

Here we note that in contrast to an actual dielectric E is less than EO.Also we may combine the conductivity with the dielectric constant

174


and thus obtain a complex dielectric constant Ec• If this is done, theplasma is described by the constitutive parameters

Jl = jlO Ec =a- •....+ EO - bUAI

(31)

6.2 Constitutive Parametersof a Plasma

When a high-frequency electromagnetic wave passes through a plasma,only the interaction between the wave and the free electrons need beconsidered. Therefore, from a statistical point of view the macroscopicstate of the plasma can be described in terms of a single distributionfunction f(r,w,t), which determines the probable number of electronsthat at the time t lie within the spatial volume dx dy dz centered at rand have velocities within the intervals dwx, dwu, dw. centered at w.This function of the position vector r, the velocity vector w, and the

time t must satisfy the Boltzmann (or kinetic) equation

df af (d )- == - + w. "ilf + - w . "il f = Cdt at dt"

(32)

where "il••f is the gradient of f in velocity space, "ilf is the gradient of fin coordinate space, and C is the temporal rate of change inf caused bycollisions. The acceleration dw/dt is related to E and B of the wave inaccord with the Lorentz force equation

m ~ w = q(E + w X B) (33)

where q and m denote respectively the charge and mass of the electron.Substituting expression (33) into Eq. (32), we obtain

aj qat + w. "ilf + m (E + w X B) . "il••f = C (34)

which shows explicitly that the driving force is the macroscopic elec-tromagnetic field E, B. Multiplying this equation by mw and

175

ihtegrating over all ve16cltYe-s, we 6btMn1 themalcro~Cbpi'ceql1s,tibnofmotion

(35)

tfi this equation the particle density n(r,l) and the macroscopic velocityv(r,t) are defined respectively by

"n(r,t) = 11f j(r,w,t)dwJi,wtliw.

v(r,t) == ~ Iii wj(r, w,t)dWxdwudw.

The dyadic S is the stress, defined by

"S == mil I (w .•...v)(w - v)j(r,w,t)dwi:dwydW.

(36)

(37)

and the vector G is the net gain of momentum due to collisions.In the present case all the nonlinear terms as well as the v X .a term

are dropped from Eq. (35), and thus the equation of motion is reduced,in the steady state, to the following simple form:

-'- iwnmv = nqE + G (3\)

Moreov~r, since G is the net gain in momentum per unit Volume perunit time, we may write

G = -nmvWefl (40)

where the proportionality constant We!! is the collision frequency andmeasures the numoor of effective collisions an electron makes per unittime. Furthermore, the density of electronic current J and the plasma

1 See, for example, L. Spit~er, Jt., "Physics of Fully Ionized Gases," Inter~science Publishers, Inc., New York, 1956.

176

---~--"._ •.- ~~~._ ..'~---~


frequency Wp are defined by

J=nqv

and by

(41)

(42)

Hence, from the equation of motion (39) and the expressions (40), (41),and (42) we find that the electronic current density J is related to Eas follows:

2 2 2J = EOWp E = EOWelfWp E + iw EOWp E-iw + Welf W2 + Wefl2 W2 + Wel12

(43)

Comparing expression (43) with Eq. (21) of the previous section, wedetermine the coefficients a and b; and then by using relations (30) ofthe previous section, we find the constitutive parameters of the plasma.Accordingly, if we think of the plasma as a lossy dielectric, its con.ductivity is given by

its dielectric constant by

E = EO (1 - 2 ~p2 2)W Well.

and its permeability by

I/o = 1/00

(44)

(45)

(46)

The elementary derivation of the constitutive parameters givenabove makes use of the collision frequency merely as an unknownparameter, without providing any information about its value. Toevaluate WeI/' the microprocesses which the plasma particles undergomust be taken into account explicitly. This has been done elsewhere

177


by kinetic theory and the results show that Well is not constant at all.Nevertheless, expressions (44) and (45) with Well taken to be constantadequately describe the plasma for our present purposes.

6.3 Energy Density inDispersive Media

Using Maxwell's equations for a lossless medium, we can write

a aV' • Set) = - E(t) • - D(t) - H(t) • - B(t)at at

where

Set) = E(t) X H(t)

(47)

(48)

is the Poynting vector. The quantity V'. Set) represents the rate ofchange of the -electromagnetic energy density wet), that is,

a .V' • set) = - - wet)at

From Eqs. (47) and (49) we see that

aw a a- = E(t) • - D(t) + H(t) • - B(t)at at at

(49)

(.50)

For a simple, nondispersive, lossless dielectric E is a real constant and f.L

is equal to f.LO; hence

D(t) = EE(t) B(t) = f.LoH(t) (51)

and relation (50) reduces to

a a- wet) = !I [7~EE(t) • E(t) + ~f.LoH(t) • H(t»)at vt(52)

which shows that the electromagnetic energy density for a simple,

178


nondispersive, lossless dielectric is given by

w(t) = %eE(t) • E(t) + %/loH(t) • H(t) (53)

The first term on the right is the electric energy density w. and thesecond term is the magnetic energy density wm:

w.(t) = %eE(t) • E(t)

wm(t) = %/loH(t) • H(t)

(54)

(55)

In the case of harmonic time dependence, where E(t) = Re {Ee-iwt}and H(t) = Re {He-iwt}, the time-average energy densities may bewritten in terms of the phasors E, H as follows:

Wm = 7.:t:/loH. H*'/1). = 7.:t:eE. E*

(.56)

(.57)

To define the electric and magnetic energy densities of an electro-magnetic wave in a plasma, we must assume that the plasma is lossless,because it is only for a lossless medium that electromagnetic energycan be rationally defined as a thermodynamic quantity. For thisreason we must limit our consideration to situations where the collisionfrequency w.!! is so small that we may set it equal to zero, In keepingwith this restriction, we consider a plasma whose collision frequency iszero and note that its constitutive parameters are

/l = /lO 0'=0 (58)

as can be seen by setting w.!! equal to zero in Eqs. (44) and (45).Since /l is a constant, the magnetic energy density can be evaluated bymeans of relation (54) or (.56). However, since e is a function of fre-quency, the medium is dispersive and relations (.'55)and (57) no longercan be used to evaluate the electric energy density. For example, ifwe use relation (.57)we obtain the expression

IV. = 7.:t:eo(1 - :;22) E . E* (59)

179


which predicts that w. < 0 when w < Wp, in contradiction to the factthat w. must always be positive-definite.Since the plasma is dispersive we cannot compute the electric energy

density on a monochromatic basis. The reason for this is that sinceow./at = E(t) • aD(t)/at, the expression for the electric energy density,viz., w.(t) = JE(t) • aD(t)/at dt + C, contains the integration constantC, whose value depends on how the field is established. To determineC, we assume that the wave is quasi-monochromatic; then for t -Y - 00

we have E( - 00) = 0, w.( - 00) = 0, and hence C = O. That is, for aquasi-monochromatic wave that starts in the remote past from valuezero and builds up gradually, the integration constant is zero and w.(t)is fully determined.A high-frequency wave whose amplitude is slowly modulated is a

simple type of wave that builds up gradually in time and thus serveswell in calculating electric energy density. Accordingly, we assumethat the time dependence of the electric vector in the lossless plasmahas the form

E(t) = 72Eo[COS (w + Ilw)t - cos (w - Ilw)t]

- Eo sin Ilwt sin wt (60)

where Eo is a constant vector and Ilw is small compared to w. SinceD = E(w)E, the resulting displacement vector is

D(t) = 72EO[E(W + Ilw) cos (w + Ilw)t - E(W - Ilw) cos (w - Ilw)t] (61)

and the resulting displacement current is

aat D(t) -72Eo[(W + IlW)E(W + Ilw) sin (w + Ilw)t

- (w - Ilw)E(w - Ilw) sin (w - Ilw)t] (62)

Expanding (w + IlW)E(W + Ilw) and (w - IlW)E(W - Ilw) in a Taylorseries and retaining only the first two terms, we get the approximateexpressions

a(w + Ilw)E(w + Ilw) = WE + Ilw aw (EW) +

a(w - IlW)E(W - Ilw) = WE - Ilw aw (EW) +

180

(63)

(64)


which when substituted into expression (62) lead to the followingexpression for the displacement current:

ft D(t) = - Eo [ WE sin t:.wt cos wt + t:.w:w (WE) cos t:.wt sin wtJ (65)

We see from Eq. (50) that the rate of change of the electric energydensity is

a aat w. = E(t) • at D(t) (66)

and hence the energy gained during the time interval tt - to is given by

(II aW.(tl) - w.(to) = J to E(t) • at D(t)dt (67)

From expression (60) it is evident that E(t) is zero when t = 0 and hasthe form of a high-frequency carrier sin wt whose modulation envelopesin t:.wt increases slowly with time. The time required for E(t) to buildup from zero to its maximum value is t:.wt = 71"/2 or t = 71"/2t:.w. Theenergy gained during the time interval to = 0 to tt = 71" /2t:.w is given by

(,,/2/),I» aw. = J 0 E(t) • at D(t)dt

Substituting expressions (60) and (65) into Eq. (68), we get

E E("/2/),,,,. .

W. = O' OWE Jo sm2 t:.wtsm wt cos wt dt

+ E E a ( ) ("/2/),,,, ., do • 0 t:.waw WE J 0 sm2 wt sm t:.wt cos t:.wt t

(68)

(69)

The first integral on the right is negligibly small compared to thesecond. In the second integral we may replace sin2 wt by ~ and thusapproximate the integral by

("/2/)',,, • 1~ Jo sm t:.wt cos t:.wtdt = 4t:.w (70)

181


It follows that the time-average electric energy density is given by

(71)

If instead of the form (60) for E(t) we take E(t) = Re {Eo(t)e-U.l},where Eo(t) is a slowly varying function, we would get again

(72)

Since E = EO(l - w1} / w2), expression (72) leads to

(73)

which shows that We is the sum of two terms, the first representing theenergy in the vacuum and the second representing the kinetic energyof the electrons.l To demonstrate that the second term does equal thetime-average kinetic energy of the electrons, we recall from Eq. (39)of the previous section that for a lossless plasma

-iwnmv = nqE

The time-average kinetic energy density is, therefore, given by

_ I nq2K = %nmv'v* = --E.E*

4 w2m

Using definition (42) of the plasma frequency, we get

(74)

(75)

(76)

which is identical with the second term of expression (73).Thus we see that' for a lossless plasma the time-average electro-

1 Formally, this result can also be obtained from the energy theorem ofChap. 1; see F. Borgnis, Zur e!ektromagnetischen Energiedichte in Medienmit Dispersion, Z. Physik, 159: 1-6 (1960).

182


magnetic energy density is given byl

tV = ~loIoH • H* + !i. (w~)E • E*4 ow

(77)

6.4 Propagation of Transverse Wavesin Homogeneous Isotropic Plasma

To determine the propagation properties of transverse electromagneticwaves in a homogeneous isotropic plasma, we consider a linearlypolarized plane transverse wave whose electric vector E(l) has the form

(78)

where Eo(t) is a slowly varying function of time, w is the real meanangular frequency, and k is the propagation constant or mean wavenumber, which may be complex. In a medium whose constitutiveparameters are ~, 1010, u, the electric vector must satisfy

(79)

Since in the present case E(t) ~s transverse, Le., perpendicular to thedirection of propagation, the quantity V X V X E may be replaced by- V2E. Moreover, since Eo(t) is a slowly varying function in compari-sion to e-u..l, we may replace iJ/ot by -iw and 02/ot2 by -w2. Thuswhen expression (78) is substituted in Eq. (79) we find that thepropagation constant is given by

(80)

Since w is assumed real, it is clear from Eq. (80) that k is generally

1See, for example, L. Brillouin, Congr. intern. elee./ Paris, 1932, vol. 2,pp. 739-788, Gauthier-Villars, Paris, 1933.

183


complex. Accordingly, we write k in the form

k = {:3 + ia = ~ 71 + iac (a, (:3, 71 = positive-definite) (81)

which displays as real quantities the phase factor {:3,the attenuationfactor a, and the index of refraction 71. To obtain explicit expressionsfor these factors in terms of the constitutive parameters, we substituterelations (81) into Eq. (80). Thus we find that

{:3=WV~[~+~GY + (2SYTa = W V~ [- ~+ ~GY+ (;wyr_ 1 [€ ~(t)2 (U)2]~'71-- -+ - + -V;;; 2 2 2w

(82)

(83)

(84)

Applying expressions (82), (83), and (84) to a lossless (nonabsorp-tive) plasma whose constitutive parameters are € = €o(l - wp2jw2),

J.I. = J.l.o, u = 0, we get

a=O

for w > Wp (85)

{:3=O 71=0 for w < Wp (86)

{:3=0 a=O 71 = 0 for w = Wp (87)

These expressions show the marked difference in behavior between awave whose operating freq~ency is greater than the plasma frequencyand a wave whose operating frequency is less than the plasma fre-quency. When w > Wp, the wave travels without attenuation at aphase velocity greater than that of light in vacuum. On the otherhand, when w < Wp the wave is evanescent (nonabsorptively damped)and carries no power. At w = Wp the wave is cut off; the magnetic

184

Elect.romagnetic waves in a plasma

field is zero and the electric field must satisfy V'X E(t) = o. Hence, atcutoff a transverse electromagnetic wave cannot exist. However, alongitudinal electrical wave, sometimes called a~'plasma wave" or"electrostatic wave," can exist. To examine the properties of such awave, spatial dispersion must be taken into account.

There are three types of velocity that pertain to the transverse wave:the phase velocity Vph, whose value can be found from a knowledge of71 by using the relation Vph = c/'T/; the group velocity Vg, which bydefinition is iJw/iJ{3; and the velocity of energy transport Ven, which isdefined by the ratio S./7IJ. Again restricting the discussion to a losslessplasma, we see from expressions (85) that the phase and group velocitiesare given by

(88)

(89)

Since an increase of wavelength (or, equivalently, a decrease of fre-quency) results in an increase in phase velocity, the dispersion is s~idto be "normal."

To find Ven, we note that the time-average value of the Poyntingvector of the wave is z directed and has the value

~

-- ES. = 72 Re e.' (E X H*) = 72 Re - Eo' Et

JJ.o(90)

Moreover, we note that the time-average energy density (77) in thiscase reduces to

Therefore, the velocity of energy transport assumes the form

(91)

S.V~n = -=- =w

(7~) Re V~(72)E + (7::i)W ilE/ilw

(92)

185


Substituting E = Eo(l - wp2 / w2) into this form, we find that the velocityof energy transport in a lossless plasma is given by

~

W2Ven = C 1 - ..2...

w2(W ~ Wp) (93)

which is identical to expression (89) for the group velocity.Let us consider now a plasma with small losses. In the limiting case

where lEI» u/w, the losses are incidental and expressions (82), (83),and (84) reduce to

{3=wV;; (W ~ wp) (94)

Using relations (44) and (45), i.e.,

E = Eo (1 - 2 ~2 2)W Well

(95)

we see that expressions (94) yield

and the corresponding phase and group velocities are given by

C

Vph = ~ 21 .Wp

- w2 + Weu2

(96)

(97)

(98)

(99)

(100)

Comparing expression (88) with expression (99), we see that the phasevelocity is decreased by the presence of loss. On the other hand, com-paring expression (89) with expression (100), we see that the group

186


velocity is increased by the presence of loss. The interpretation ofgroup velocity as the velocity of energy transport breaks down whenthe medium is dissipative.

6.5 Dielectric Tensor of MagneticallyBiased Plasma

When a magnetostatic field Bo is applied to a plasma, the plasmabecomes electrically anisotropic for electromagnetic waves. That is,the permeability of the plasma remains equal to the vacuum permea-bility p.o, whereas the dielectric constant of the plasma is transformedinto a tensor! (or dyadic) quantity £.

To derive the dielectric tensor of a magnetically biased plasma, whichfor simplicity is assumed for the present to be lossless, we use themacroscopic equation of motion (35). In the present instance thisequation reduces to

- inmwv = nq(E + v X Bo) (101)

and yields the following expression for the macroscopic velocity of theplasma electrons:

v=-w2(q/m)E - iW(q2/m2)E X Bo + (q3/m3)(E. Bo)Bo

-iw [(~ Bo) .(~ Bo) - w2](102)

Since the density of the electronic convection current J by definition isequal to nqv, it follows from expression (102) that J is given by

where Wp is the plasma frequency (wp2 = nq2/mEo) and where the1See, for example, C. H. Papas, A Note Concerning a Gyroelectric Medium,

Calteeh Tech. Rept. 4, prepared for the Officeof Naval Research, May, 1954.

187


amplitude1 WQ of the vector

(,)Q == !I Bom (104)

represents the gyrofrequency of the electrons. From a knowledge of Jwe can find the dielectric constant of the plasma by noting that thetotal current density is the sum of the convection current density J andthe vacuum displacement current density -iWEOE, and then by regard-ing this total current density as a displacement current in a dielectricmedium whose dielectric constant I: is fixed by the relation

J - iWEoE = -iwl:' E (105)

According to expression (103), it appears that J is generally not parallelto E; the quantity I:must be a tensor or dyadic to take this into account.Since, by definition, the displacement vector D is calculated from

D = I:.E (106)

the tensor character of I:also means that D is not generally parallel to E.Although a tensor is independent of coordinates, its components are

not. If we are given the components of a tensor with respect to onecoordinate system, we can find its components with respect to any othercoordinate system by applying the transformation law connecting thecoordinates of one system with those of the other. Therefore we arefree to choose any coordinate system without risking loss of generality.In the present instance, for simplicity, we choose a cartesian system ofcoordinates (x,y,z) whose z axis is parallel to Bo, that is, Bo = ezBo;Cz is the z-directed unit vector. When Bo > 0, the vector Bo isparallel to the z axis; and when Bo < 0, the vector Bo is antiparallel tothe z axis. The components of I: in this cartesian system are denotedby Eik, with i, k = x, y, z.Substituting expression (103) into Eq. (105) leads to the following

expressions for the components Eik of I: in the cartesian system whose

1 This means that WQ = (q/m)Bo. For electrons q is negative and henceWg = (-lql/m)Bo.

ISS


z axis is parallel to Do:

Ezz = EO (1 - 2 Wp2 2) = EuuW - Wg

(107)

(108)

(109)

The remaining components Ezz, Ezz, Euz, Ezu are identically zero. We notethat when the magnetostatic field Bo vanishes, Wg vanishes and thediagonal terms become equal to each other, i.e.,

(110)

and the off-diagonal terms disappear. That is, when Bo = 0, theplasma becomes isotropic as it should. Also we note that when Bo isreplaced by - Bo, the gyrofrequency Wg changes sign and, consequently,the components satisfy the generalized symmetry relation

(111)

as must the components of the dielectric constant of any mediumwhose anisotropy is due to an externally applied magnetostatic field.lIn addition we see that the components constitute a hermitian matrix,i.e.,

(112)

The hermitian nature of the dielectric tensor results from the assump-tion that the plasma is lossless.Expressions (107), (108), and (109) for the components of the dielec-

tric tensor may be easily generalized to take into account collisionlosses. For the case where the collision losses are appreciable, we mustadd to the right side of Eq. (101) a collision term. Thus for the equa-

l See, for example, A. Sommerfeld, "Lectures on Theoretical Physics," vol.5, "Thermodynamics and Statistical Mechanics," p. 163, Academic Press Inc.,New York, 1956.

189


tion of motion of the electrons we get

-inmwv = nq(E + v'x Bo) ~ nmVWefl (113)

where Well is the collision frequency. Rewriting this equation in theform

-inm(w + iWell)V = nq(E + v X Bo) (114)

and comparing with Eq. (101), we see that the resulting expression forJ is the same as expression (103), with W replaced by W + iWell' Itthen follows from Eq. (105) that the cartesian components of thedielectric tensor of a lossy dielectric are given by

, (1 Wp2(W + iWell) ) ,Exx = EO - [( +')2 2] = EwW W ~Well - Wg

• Wp2Wg

-~EO W(W + iWel1 + Wg)(W + iWel1 - Wg)

E~. = Eo [1 _ ( ~2. )]ww ~Well

,-£1/%

(115)

(116)

(117)

where the prime is used to distinguish the lossy components from theloss-free ones. As in the lossless case, we again have

(118)

but, unlike the lossless case, the components <k do not constitute ahermitian matrix. We can, however, decompose E:k uniquely as follows,

, +iEik = Eik - rTik

W(119)

so that Eik and rTik are hermitian.When the frequency of the electromagnetic waves that are passing

through a magnetically biased plasma is very low, the motion of theplasma ions must be included in the analysis. We can find the dielec-tric constant in this low-frequency case by calculating the convectioncurrent as the sum of the ionic current and the previously determined

190


electronic current, and by finding t from a knowledge of J through theuse of relation (105).To proceed with the calculation, we note that the equation of motion

for the ions is formally the same as the equation of motion for theelectrons. Accordingly, since we have (in the loss-free case)

-inmwv = nq(E + v X Bo) (120)

as the equation of motion for the electrons, then for the ions the equa-tion of motion must be

-inimiWVi = niqi(E + Vi X Bo) (121)

Here mi denotes the ionic mass, qi the ionic charge, ni the ionic popula-tion density, and Vi the macroscopic velocity of the ions. We knowfrom previous calculation that the electronic convection current nqv isgiven by expression (103). Hence, it follows from the similarity ofEqs. (120) and (121) that the ionic convection current niqivi is given bythe same expression (103) but with Wp replaced by the ionic plasma fre-quency Wpi and Wg replaced by the ionic gyrofrequency, where

qiWgi = - Bomi

(122)

Superposing nqv and niq,v" we get J, that is,

J = nqv + niqivi (123)

and then substituting this J into relation (105), we find that thenonzero components of t for a loss-free magnetically biased plasma aregiven byl

(124)

(125)

(126)

1See, for example, E. Astrom, On Waves in an Ionized Gas, Arkiv Fysik, 2:443 (1950).

191


These components are in accord with the generalized symmetry rela-tion (111)~nd with the hermiticity condition (112).The hermitian property of the dielectric tensor is a consequence of

the assuinpt~on that the plasma is loss-free. To show that thehermiticity of the tensor is preserved under a rotation of the coordinatesystem, we introduce another cartesian system x', y', z', which isobtained from the original cartesian system x, y, z by a pure rotation.Let ai, with i = x', y', z', denote the unit vectors along the axes of theprimed system; and as before let Ci, with i = x, y, z, denote the unitvectors along the axes of the unprimed system. In the unprimedsystem the dielectric tensor is given by

(i, k = x, y, z) (127)

and in the primed system it must have the form

t = ~aiakEik (a) (i, k = x', y', z') (128)

where Eik(a) denote the components of t with respect to the primedsystem. Since ai • ak = Oik, it follows from expression (128) that

Emn (a) = am. £ • an (129)

Substituting expression (127) In expression (129), we obtain therelation

which, by means of the shorthand

'Yik == ai' ek

can be written as.

Similarly we obtain

192

(130)

(131)

(132)

(133)


Since Eik is hermitian, it follows from Eqs. (132) and (133) that Enm(l» isalso hermitian, i.e.,

(134)

Thus we see that hermiticity is preserved under a rotation of the axes.Although we have found it convenient to express the constitutive

relation of a magnetically biased plasma by means of a single tensor, itis simpler in certain considerations to deal instead with the elementaryvector operations that carry E into D. To determine these operations,we assume for simplicity that the plasma is loss-free. Consequently,its dielectric tensor has the form

-ig 0)a 0o b

(135)

where a, b, and g are real quantities. Splitting this matrix as follows,

(a 0 0) (0 0

E= 0 a 0 + 0 0OOa 00

~)+(~b - a 0

(136)

and then substituting it into the constitutive relation D = I: • E, weobtain

D = aE + (b - a)e.(e.' E) + ige. X E (137)

as an alternative statement of the constitutive relation. In the casewhere the motion of the ions can be neglected, i.e., in the case wherea, b, and ig are given respectively by expressions (107), (109), and (108),we have

When the biasing field is weak or when the frequency is high, the ratiowg/w is small compared to unity and relation (138) to first order in

193


Wo/ w becomes

W 2D = EE + iEO -T (,)0 X E

w (139)

where E = Eo(l - Wp2/(2) is the dielectric constant of an isotropicplasma.Returning to the dielectric tensor (135), we ask whether there is a

special coordinate system with respect to which the dielectric tensor isdiagonal. The ansy;C! to this is that since the tensor is hermitian itsmatrix can be diagonalized by a unitary transformation which amountsto a complex rotation in Hilbert space.l More simply, however, weobserve that when the dielectric tensor (135) is substituted into theconstitutive relation D = t. E, we obtain

Dz = aEz - igEy

Dy = igEz + aEy

Dz = bEz

(140)

(141)

(142)

With the aid of the following combinations o( expressions (140) and(141),

Dz + iDy = (a - g)(Ez + iEy)

Dz - iDII = (a + g)(Ez - iEy)

we get the matrix equation

(Dz + iDy) (a - g 0Dz - iDy. = 0 a + g

Dz 0 0

(143)

(144)

(145)

which displays the dieiectric constant as a diagonal matrix. SinceD . (cz ::!: icy) = (czDz + eyDy + ezDz) • (ez ::!: icy) = Dz ::!: iDy, thecomponent Dz + iDy is the projection of D on the vector Cz + icy and

1See, for example, Hermann Weyl, "The Theory of Groups and QuantumMechanics," chap. 1, Dover Publications, Inc., 1931. Translated from theGerman by H. P. Robertson.

194


Dx, - iDy is the projection of D on the vector eX, - iey• Thus we seethat the elements of the matrices in Eq. (145) are referred to a coordi-nate system x' = I/V2 (x + iy), y' = 1/0 (x - iy), z' = z, whoseunit vectors are eX,'= 1/0 (ex,+ iey), ey' = 1/0 (ex, - ie)y, ez' = ez•

The vectors eX,',ey', ez', which are unit orthogonal vectors in the hermitiansense, that is, ei 'ek* = Oik where i, k = x', y', z', constitute the prin-cipal axes of the dielectric tensor.l

6.6 Plane Wave in MagneticallyBiased Plasma

In this section we shall study the propagation and polarization prop-erties of a plane monochromatic wave in a magnetically biased homo-geneous plasma which for simplicity is assumed to be lossless. Weregard the plasma as a continuous medium whose conductivity is zero,whose permeability is equal to the vacuum permeability /-'0, and whosedielectric constant is the tensor £ given by Eqs. (107), (108), and (109)of the previous section.By definition, the electric vector of a plane monochromatic wave has

the form

E(r) = Eoeik•r (146)

where Eo is a constant vector, k is the vector wave number, and r isthe position vector. We may write k as

k = n~v

(147)

where n is the unit vector in the direction of propagation and v is thephase velocity of the wave. The problem is to determine the vector k,which describes the propagation of the wave, and the vector Eo, whichdescribes the polarization of the wave.

1 For an exhaustive discussion, see G. Lange-Hesse, Vergleich der Doppel-brechung in Kristall und in der Ionosphare, Archiv der Elektrischen Vber-tragung, 6: 149-158 (1952).

195


The vector E must satisfy the Helmholtz equation

v X V X E = w2}J.o£ • E

as can be seen from the Maxwell equations

(148)

V X E = iW}J.oH V X H = -i~£. E (149)

by taking the curl of the first and then using the second to eliminate H.Substituting expression (146) into Eq. (148), and using relation (147),we obtain

1v2Eo - n(n • Eo) = - - £. Eo

EO c2 (150)

wherec = I/V }J.OEOis the vacuum velocity of light. Without loss ofgenerality we choose a cartesian system of coordinates so oriented thatthe z axis is parallel to Bo and the yz plane contains n. As shown inFig. 6.1, the angle between nand Bo is denoted by 8. Accordingly, thex, y, z components of the vector equation (150) are given by

Eox (1 - ~ EXX) - EOy (~ EXY) + 0 = 0c2 Eo c2 EO

Eox (- ~ EYX) + EOy (cos2 8 - ~ EYY) + Eo.( - cos (J sin 8) = 0 (151)C EO C EO

o + EOy( - cos 8 sin 8) + Eo. (sin2 8 - ~~) = 0C EO

where Eox, Eoy, Eo. are the cartesian components of Eo. Since thesethree simultaneous equations are homogeneous, they yield a nontrivialsolution only when

o

196

-sin8cos8

o

-sin8cos(J =0

• 2 (J V2Ensm ---

C2 EO

(152)


%

Fig. 6.1 Arbitrary di-rection n ofwave propa-gation inplasma withapplied mag-netostaticfield Bo•

D

y

With the aid of the quantities EI, E2, Ea, which are defined by

Exz • fzuE} = - - t-

Eo EOE2 = E",,,, + i E","

Eo Eo

Ezzfa = -Eo

(153)

we find that Eq. (152) can be written

(154)

This equation determines two values of v2/e2 for each value of 9.In the case where the propagation is parallel to Bo, we have 9 = 0;

accordingly, Eq. (154) yields the two solutions

v2 1 1 1 (155)- = - =e2 EI Exx • Exy X

--t- 1 ----EO Eo 1+ Y

and

v2 1 1 1 (156)- = - =e2 E2 E",,,,+ i E",u X1---

EO EO 1 - Y

197


where X = (Wp/W)2 and Y = -wo/w.! From these expressions it fol-lows that the propagation constants of the two waves that travel par-allel to Bo are given by

(157)

and

(158)

Moreover, when the propagation is along the y axis, i.e., perpendicularto Bo, f) is equal to 7r/2 and in this case the two solutions of Eq. (154) are

and

(159)

1X

1 - 1 - P/(1 - X)

(160)

For the propagation constants of the corresponding two waves, we have

w --- w~ W 2k' /2 = - VI - X = - 1 - -.!!-"c C w2

and

(161)

(162)

! Since q in the case of electrons is a negative quantity, then wo, which isgiven by (q/m)Bo, is also a negative quantity. We wish Y to be a positivequantity and therefore we include a minus sign in the definition.

198


In general, when 8 is arbitrary we have the two solutions.

~ = [1- --1 -Y-T2----,X 1=1=Y=T=4===2]-11 - 2 1 _ X :!: \}"4 (1 _ X)2 + Y L

and hence

(163)

(164)

(165)

where YT = Y sin 8 and YL = Y cos 8.Thus we see that there are two waves traveling in any arbitrary

direction 8, and that one of them has a propagation constant k~ givenby expression (164) while the other has a propagation constant k':given by expression (165). Since as a function of X the propagationconstant k~ resembles the propagation constant of a wave in an isotropicplasma more closely than k': does, the wave whose propagation constantis k~ is sometimes referred to as the ordinary wave and the wave whosepropagation constant is k': as the extraordinary wave. Indeed k~/2'the value k~ has when 8 = 71'/2, is identically equal to the propagationconstant of a wave in an isotropic plasma.Each of the field vectors of a wave is proportional to exp Uk. r).

Therefore the Maxwell equations yo X E = iw~H, yo X H = -iwDreduce to the relations

2k X E = iw~oH 2k X H = -iwD (166)

which clearly indicate that the vectors k, E, D lie in a plane perpendic-ular to H (Fig. 6.2). Since H is necessarily perpendicular to k, thewave cannot be an H wave (also known as a TE wave). In general thewave must be an E wave (also known as a TM wave), but in certainspecial directions the wave is a TEM wave. The Poynting vector

199


E

s

Fig. 6.2 The vectors k,E, D, S lie inthe plane of thepaper, and H isperpendicularto it. D and Hare perpendicu-lar to k. S isperpendicularto E and H. Sis generally notparallel to k.

(167)

S(= ~ Ex H*)of the wave is not parallel to k except in those directionsof travel where the wave is TEM.lLet us again consider the special case where the propagation is par-

allel to Bo• In this case, (J = 0 and Eqs. (151) reduce to

E Oz (1 - ~ Ezz) - E 0 (~EZY) = 0e2 Eo Y e2 Eo

Eoz (_ ~ Eyz) + Eoy (1 _ ~ EW) = 0e2 EO e2 EO

Eo. (- ~ EZZ) = 0e2 EO

with v2/e2 given by Eq. (155) and by Eq. (156). From the third ofthese equations, we see that Eo. is zero. Consequently, the two wavesthat travel parallel to Bo are TEM waves. When v2/e2 is given by Eq.(155), the first or second of Eqs. (167) yields

Eoz .-=tEOy

and when v2/e2 is given by Eq. (156), we find that

Eoz .- =-tEOy

(168)

(169)

1However, it has been shown by S. M. Rytov, J. ExpU. Theoret. Phys.,U.S.S.R., 17: 930 (1947), that the time-average Poynting vector is parallel tothe group velocity.

200


Therefore, the electric vectors of the two waves traveling parallel toBo can be written as

and

E" = (ex + ieu)Ceiko"x

(170)

(171)

where A and C are arbitrary amplitudes. Clearly E' is a left-handedcircularly polarized wave, whereas E" is a right-handed circularly polar-ized wave.l The sum of these two waves yields the composite wave

To study the polarization of this composite wave, we consider the-ratio Ex/ Ey• From (172) we obtain

Ex .1 + (C/A) exp [i(k~' - k~)z]- = 1, --~--~~----Ey 1- (C/A)exp[i(k~' - k~)z]

(173)

If the waves E' and E" are chosen to have equal amplitudes, then theconstants A and C become equal. As a consequence of this choice, Eq.(17:3)reduces to

Ex _ (k~ - k~' )Ey

- cot 2 z (174)

Since this relation is real, the composite wave at any position z is lin-early polarized; however, the orientation angle of its plane of polariza-tion (the plane containing E and k) depends on z and rotates as z

1A geometric interpretation may be obtained by considering the realvectors He E'e-iwt and Re E"e-iwt. Setting A = C = 1, we obtain from Eqs.(170) and (171) the expressions

Re E'e-iwt = e. cos (k~z - wt) + ey sin (k~z - wt)

Re E"e-iwt = e. cos (k~'z - wt) - ey sin (k~'z - wt)

Clearly, at any fixed time the locus of the tip of the vector Re E'e-iwt is aright-handed helix. As time increases this helix rotates counter-clockwise.On the other hand, the locus of the tip of the vectorRe E"e-iwt is a left-handedhelix, which rotates clockwise.

201


increases or decreases. In other words, the composite wave undergoesFaraday rotation. The angle T through which the resultant vector Erotates as the wave travels a unit distance is given by

k~ - k~'T = 2 (175)

The rotation is clockwise because k~ > k~' always. With the aid ofexpressions (157) and (158), we see that T can be written in the form1

(176)

which displays the dependence of the Faraday rotation T on frequency.We note that if a wave travels parallel to Bo it undergoes a clockwise

Faraday rotation. On the other hand, if a wave travels antiparallel toBo it undergoes a Faraday rotation of the opposite sense. That is, onreversing the direction of propagation, a clockwise wave becomes coun-terclockwise, and vice versa. This means that if the plane of polariza-tion of a wave traveling parallel to Bo is rotated through a certainangle, then upon reflection it will be rotated still further, the rotationfor the round trip being double the rotation for a single crossing.For weak biasing fields the Faraday rotation depends linearly on Bo•

To deduce this fact from expression (176), which in terms of the param-eters X = (wp/w)2 and Y = -wo/w can be written as

T = ~~ (~1 - 1 : Y - ~1 - 1 ~ Y) (177)

we expand the square roots and retain only the first two terms in accordwith the assumption that X« 1 and Y« 1. Thus we obtain therelation

lwT = -- XY =2c

_ 1.. (wp)2 W2c w 0

(178)

which shows that the Faraday rotation T for weak biasing fields (Y «1)and high frequencies (X «1) is linearly proportional to Wo and hence

1Recall that Wo is a negative quantity.

202


linearly proportional to Bo• Since Wo is negative for electrons, weagain see that T is positive (clockwise rotation) in the case of parallelpropagation.In the other special case, propagation being perpendicular to Bo,

that is, along the y axis, we have () = 7rj2, and Eqs. (151) reduce to

Eox (1 - ~ EX.:) - Eoy (~EXll) = 0c2 EO c2 EO

Eox (_ ~ EYX) + Eoy (_ ~ EYY) = 0c2 EO c2 EO

Eo. (1 - ~E •• ) = 0c2 EO

When in accord with Eq. (159) we choose

(179)

(180)

(181)

(182)

then from Eqs. (179), (180), and (181) it follows that Eox and Eoy areidentically zero, and the only surviving component of the electric vectoris Eo.. Thus we see that one of the two waves traveling in the y direc-tion is a linearly polarized TEM wave whose electric vector is parallelto Bo and has the form

E' A ik',,'211= e. e (183)

where A is an arbitrary constant. Since the propagation constantk~/2 as given by Eq. (161) is independent of Bo and equal to the prop-agation constant of a wave in an isotropic plasma, this TEM wave (theordinary wave) is independent of Bo in its propagation properties andbehaves as though it were a TEM wave in an isotropic plasma.To obtain the extraordinary wave propagating perpendicular to Bo,

the other possible value of v2/ c2 as given by Eq. (160) is used. That is,

v2 Exxj EO

C2 = (Exx/EO)2 + (EXy/EO)2(184)

is substituted into Eqs. (179), (180), and (181). ,Thus it is found that

203


Eo. vanishes identically and that

Eo"EOy =

E .1-X-P_..!!!!.=~-----Ell" XY

(185)

Therefore the electric vector of this extraordinary wave has the form

(186)

where C is an arbitrary constant. The magnetic vector H" is obtainedby substituting E" into the first of Eqs. (166). Thus

H" . k':'21 - X - Y2 C ik'.;,'lI= -~e.------ eWfJoo XY

(187)

From expressions (186) and (187), we see that the extraordinary wavetraveling perpendicular to Bo is an E wave (TM wave) with its mag~netic vector parallel to Bo•For propagation in an arbitrary direction (J, it followsfrom Eqs. (151)

that the ratio p of the electric vector components perpendicular to n isgiven byl

for the ordinary wave whose propagation constant is k~ and by

" _ E'; _ i [1 Y T2 + ~1 Y T

4 + Y 2JP - E~ - - YL 2 1 - X 4 (1 - X)2 L

(188)

(189)

for the extraordinary wave whose propagation constant is k~. HereE, is the component of E in the direction of the unit vector ee, which isdefined bye" X e, = n. That is, E, = -E. sin (J + Ell cos (J. Theratio E"jE, is a measure of the polarization of the part of E that istransverse to the direction of propagation n and is sometimes referredto as the polarization factor. The projection of the tip of E on a plane

1Without loss of generality, we still take n to lie in the zy plane.

204


Fig. 6.3 Polarization ellipses of ordinary and extraordinary wavestraveling into the plane of the paper. Ordinary wave iscounterclockwise. Extraordinary wave is clockwise.

transverse to n sweeps out an ellipse and, accordingly, the wave is saidto he elliptically polarized. We note that p' p" = 1 and consequentlythe ordinary and extraordinary waves are oppositely polarized. Inthe case of the ordinary wave the sense of polarization is counterclock-wise and in the case of the extraordinary wave it is clockwise. SeeFig. 6.3.

6.7 Antenna Radiation inIsotropic Plasma

So far we have been concerned with only the plane wave solutions ofMaxwell's equations for a homogeneous plasma medium. Now, as ageneralization to a case that involves spherical waves, we consider the

205


far-zone radiation field of a primary source in an unbounded plasma.For simplicity, the primary source is taken to be a thin, center-driven,straight-wire antenna of length 2l, and the ambient plasma is assumedto be homogeneous and isotropic. The antenna is driven monochro-matically at an angular frequency wand the time-average power fedinto its input terminals is Pi. The problem is to find for fixed Pi and wthe far-zone radiation field of the antenna as a function of X( = Wp2/(2).

Actually the basic part of the calculation has already been made inChap. 3. Indeed, all we are required to do is to replace E by Eo(1 - X)and k by (w/e) Vi - X in expressions (17), (18), and (19) of Sec. 3.2.However, since these expressions are valid only in the far zone, we mustbe careful not to violate the condition (w/e) Vi - X r »1. Clearlythis condition can be met for the range 0 :::;X < 1 by making r, thedistance from the center of the antenna to the observation point, suffi-ciently large; but for X = 1 (plasma resonance) the condition is vio-lated. Moreover, at X = 1 we have cutoff, i.e., no wave propagationcan occur, and the power fed into the antenna goes into heating theplasma.As in Sec. 3.2, we place the antenna along the z axis of a cartesian

coordinate system, with one end of the antenna at z = -l and theother end at z = l. With respect to the concentric spherical coordinatesystem (r,O,cf» shown in Fig. 3.1, we see that the far-zone field compo-nents of the antenna immersed in a homogeneous isotropic plasmamedium are

Ee = 1 . ~H~Vi - x'1~

iei(w/c) yl- XrH~ = - 2 Io(X)F(O,X)

rrr

(190)

(191)

and the radial component of the time-average Poynting vector in thefar zone is

(192)

This follows from Eqs. (17), (18), and (19) of Chap. 3 when E is replaced

206


by Eo(1- X) and k is replaced by (wle) V'l=X. The radiation pat-tern F«(),X) of the antenna is given by

F«() X) == cos [(wle) y1=X l cos.()j - cos [(wle) y1=X lj (193), SIn ()

To find how 10, the magnitude of the current at the driving point,depends on the time-average real power Pi fed into the antenna's inputterminals and on the parameter X( = wp2lw2) which completelydescribes the plasma medium into which the antenna radiates, we notethat since the plasma is assumed to be lossless, the time-average powerP radiated by the antenna must be equal to Pi. Substituting expres-sion (192) into the definition

P. (2" (" S .= 10 10 rr2 sm ()d() dq,

and equating P to Pi, we find that lois related to Pi as follows:

Pi =. _1__ I~ I02(X) (" F2«(),X) sin ()d()yl- xVEo 411" 10

More conveniently, we write this relation in the form

(194)

(195)

(196)

where the new parameter Rrad, the so-called radiation resistance of theantenna, has the representation

Rrad(X) = y_l__ ~ 21 (" F2«(),X) sin ()d()l-X'\J~ 11"10

(197)

By substituting expression (193) into the integral and performing theoperations that led to Eq. (101) of Chap. 3,we obtain

Rrad(X) = y 1 I~ 21 [c + In 2a - Ci 2a1 - X '\J EO 11"

+ sin22a (Si 4a - 2Si 2a) + co~2a (C + In a + Ci 4a - 2Ci 2a) ]

(198)

207


whereC(= 0.5722) is Euler's constant and a == (wle) VI - Xl. Thuswe see from Eq. (196) that 10 depends on Pi and Rrad as follows,

(199)

and from expression (198) that Rrad(X) can be calculated for any X inthe range 0 S X < 1.In view of relation (199) the far-zone field expressions (190) and (191)

can be written as

(200)

(201)

These are the desired forms because they show how the far-zone fieldsEo, Hq, depend on X. In the special case where X = 0 they reduce, asthey should, to the conventional expressions for the far-zone fields of astraight-wire antenna in vacuum. In the other special case where Pi,w, and 1 are fixed and X is made to approach unity, we find that

1 (w)2F(O,X) -+ 2 C [2(1 - X) sin 0

Rrad(X) -+ 1.- ~ (~)4 [4(1_ X)%67r '\j;;; e

Consequently, as X -+ 1, expressions (200) and (201) reduce to

~ ei(w/c)Vl-Xr _ /- sin 0Eo "'J '\j;;; r V Pi (1 _ X)l4

ei(w/c)Vl-Xr _Hq,"'J----vP;(I- X)l4sinOr

(202)

(203)

(204)

(205)

This shows that as X -+ 1, the antenna's radiation pattern approachesthe radiation pattern of a Hertzian dipole. It also shows that the

208


wave impedance Z, which is given by

(206)

increases without bound as X ~ 1.

6.8 Dipole Radiation inAnisotropic Plasma

As was shown in Chap. 2, the radiation field of a monochromatic sourcein an unbounded homogeneous isotropic medium can be calculated byeither the method of potentials or the method of the dyadic Green'sfunction. As long as the medium is homogeneous and isotropic thesetwo methods are equally convenient. However, in the case where thesurrounding medium is anisotropic, the method of potentials1 leads todifficulties in the early stages of the calculation and the Green's func-tion method becomes the more fruitful of the two. Indeed, Bunkin,2Kogelnik,3 and Kuehl4 used the Green's function method with consider-able success to analyze various aspects of the problem of a primarysource in an anisotropic medium. Recalling some of their results, weshall now show how one proceeds in the Green's function method tofind the radiation field of a dipole immersed in an unbounded homo-geneous anisotropic plasma.The electric field E of a monochromatic source J immersed in an

1A. Nisbet, Electromagnetic Potentials in a Heterogeneous Non-ConductingMedium, Proc. Royal Soc. (London), (4) 240: 375-381 (1957).

2 F. V. Bunkin, On Radiation in Anisotropic Media, J. Exptl. Theoret.Phys., U.S.S.R., 32: 338-346 (1957); also Soviet Physics JETP, 5: 277-283(1957).

8 H. Kogelnik, The Radiation Resistance of an Elementary Dipole inAnisotropic Plasmas, Proc. Fourth Intern. Conf. on Ionization Phen. in Gases(Uppsala, 1959), pp. 721-725, North Holland Publishing Company, Amster-dam, 1960. Also J. Res. Natl. Bur. Std., 64D (5): 515-523 (1960).

4 H. Kuehl, Radiation from an Electric Dipole in an Anisotropic ColdPlasma, Caltech Antenna Lab. Rept. 24, October, 1960; also Phys. Fluids, 5:1095--1103(1962).

209


unbounded anisotropic plasma medium must satisfy

(207)

Moreover, E must have the form of a wave traveling away from thesource. Hence, we are required to find the particular integral of Eq.(207) that satisfies the radiation condition.By virtue of the linearity of Eq. (207), the desired solution may be

expressed in the form

E(r) = iW/loJr(r,r') . J(r')dV' (208)

where the integration extends throughout the region of finite extentoccupied by the current. If this form is to be the solution of Eq. (207),the dyadic Green's function r(r,r') must satisfy

v X V X r(r,r') - W2/lo£' r(r,r') = u8(r - r')

or

(209)

vv. r(r,r') - V2r(r,r') - W2/lo£ • r(r,r') = u8(r - r') (210)

where u is the unit dyadic and 8(r - r') is the three-dimensionalDirac delta function.To facilitate the construction of the dyadic Green's function, we

express it as a Fourier integral. That is, we write

r(r r') = ~ J 00 A(k)eik.(r-r'l dk, 811"8 - 00

(211)

and by so doing transform the problem of finding r into one of firstfinding the dyadic function A(k) and then evaluating the integral ink space. Substituting expression (211) into Eq. (210) and recallingthe integral representation

8(r - r') = _1_ J 00 e'ik'(r-r'l dk811"8 - 00

we see that A(k) is determined by

V(k) •A(k) = u

210

(212)

(213)

(214)


where

V(k) = -kk + k2u - W2~O£

With the aid of the theory of matrices, Eq. (213) yields for A(k) theexpression

A(k) = adj V(k)det V(k)

(215)

Here det V(k) stands for the determinant of the matrix of V(k) and- adj V(k) represents the dyadic whose matrix is the adjoint of thematrix of V(k).l It therefore follows from Eqs. (211) and (215) thatthe integral form of the dyadic Green's function is

r(r r') = ~ f 00 adj V(k) eik'(r-r') dk, 811"3 - 00 det V(k) (216)

This form obeys the radiation condition and hence constitutes the onlysolution of Eq. (211) that leads to a physically acceptable result.Since the source of radiation in the present instance is an oscillating

electric dipole, we write the current distribution as

J(r') = -iwplJ(r') (217)

where p denotes the electric dipole moment. Substituting the current(217) and the Green's function (216) into the form (208), we obtain theintegral representation

E(r) = W2f..lo f 00 [adj V(I{)] . P eik'r dk811"3 -00 det V(k) (218)

which is the desired expression for the electric field E of the dipole p.Thus we see that in the Green's function method the problem ofcalculating the field of a dipole in a homogeneous anisotropic mediumsplits into an algebraic part, which consists in finding the adjoint andthe determinant of the matrix components of the dyadic V(k), and into

1See, for example, H. Margenau and G. M. l\1urphy, "The Mathematics ofPhysics and Chemistry," p. 295, D. Van Nostrand Company, Inc., Princeton,N.J., 1943.

211


an analytic part, which requires the evaluation of the integral inexpression (218).According to Kuehl, when the dipole oscillates at a high frequency,

i.e., when X = Wp2/W2« 1 and Y2 = w//w2« 1, the dipole's far~zoneelectric field in the spherical coordinates r, 8, t!> is given by

(

W)2 ei(w/c)(I-X/2)rE = - - p. sin 8 4 (eg cos {3r - e", sin (3r)

C 1I"Eor(219)

for a z-directed dipole of moment p. parallel to the biasing field Bo, andby

(

W)2 ei(w/c)(I-X/2)rE = - p.Vi - sin2 8 cos2 t!> 471" [ee cos ({3r + a)

C Eor

- e", sin ({3r + a)] (220)

for an x-directed dipole of moment p. perpendicular to the biasing fieldBo• Here {3= k (~)XY cos 8 and a = tan-I (tan t!>/cos 8). Compar-ing these expressions with the corresponding ones for a dipole in anisotropic plasma, we see that in the case of high frequencies theanisotropy does not change the amplitude VE . E* of the radiatedfield E but does change its state of polarization: it causes the field toundergo Faraday rotation.

6.9 Reciprocity

Let EI, HI be the electromagnetic field radiated by a current Jloccupying a finite volume V I and let E2, H2 be the electromagnetic fieldradiated by a current J 2 occupying another finite volume V 2. The twosource currents oscillate monochromatically at the same frequency andthe medium occupying the space V3 outside of VI and V2 is anisotropicand may be inhomogeneous.Clearly EI, HI are related to JI and E2, H2 are related to J2 by the

equations

v X HI = JI - iwt. EI

212

(221)


Multiplying the first one by E2 and the second one by El, and thensubtracting the resulting equations, we get

E2• yo X HI - El• yo X H2 = E2• Jl - E1• J2

- iwE2 • t •El + iwEl • t •E2 (222)

With the aid of yo X El = iw~oHl and yo X E2 = iw~oH2 we write theleft side of Eq. (222) as a divergence and thus obtain

yo • (El X H2 - E2 X HI) = E2 • J 1 - E1 • J 2

- iwE2 • t •E1 + iwEI • t •E2 (223)

Integrating this relation throughout all space and converting the leftside of the resulting equation to a surface integral which vanishes byvirtue of the behavior of the fields over the sphere at infinity, we areled to the expression

(224)

where

(225)

When U is zero, Eq. (224) yields the relation

(226)

which defines what we usually mean by reciprocity. 1 That is, twomonochromatic sources are said to be reciprocal when the source cur-

1The reciprocity theorem for electromagnetic waves is a generalization ofRayleigh's reciprocity theorem for sound waves (see Lord Rayleigh, "Theoryof Sound," 2d ed., vol. II, pp. 145-148, Dover Publications, Inc., New York,1945) and stems from the work of Lorentz [seeH. A. Lorentz, AmsterdammerAkademie van Wetenschappen, 4: 176 (1895-1896)]. For a detailed discussion.see P. Poincelot, "Pr6cis d'6lectromagn6tisme th6orique," chap. 18, Dunod,Paris, 1963.

213


rents and their radiated electric fields satidy relation (226) or, equiva-lently, when the quantity U vanishes. Clearly, U vanishes when thedielectric constant of the medium is symmetric (Elk = EM). However,for a magnetically biased plasma the dielectric constant is hermitianand hence U does not necessarily vanish. This means that in the caseof an anisotropic plasma reciprocity does not necessarily hold.Nevertheless, the concept of reciprocity can be generalized, at least

formally, to include the case of an anisotropic plasma.! Such ageneralization is based on the fact that the dielectric tensor of amagnetically biased plasma is symmetrical under a reversal of the bias-ing magnetostatic field, i.e.,

or

t(Be) = t( - Be)

(227)

(228)

where the tilde indicates the transposed dyadic. When the biasingfield is Be, we have for the fields produced by Jl the Maxwell equation

(229)

Moreover, when the biasing field is -Be, we have for the fields pro-duced by J2 the Maxwell equation

V'X H2( - Be) = J2 - iwt(- Be) . E2( - Be) (230)

which, in view of the symmetry relation (228), assumes the form

V'X H2( -Be) = J2 - iwt(Be) • E2( -Be) (231)

Proceeding as before, we find from Eqs. (229) and (231) the relation

(232)

! For application to ionospheric propagation see K. G. Budden, A Reciproc-ity Theorem on the Propagation of Radio Waves via the Ionosphere, Proc.Cambridge Phil. Soc., 50: 604 (1954).

214


which is the desired generalization of the reciprocity theorem to thecase of an anisotropic plasma.1 If J2 is such that E2( - Bo) = E2(Bo),or if J1 is such that E1(Bo) = E1( -Bo), this relation reduces to theusual reciprocal relation (226).1Reciprocity and reversibility are not unrelated properties. If the current

density J transforms into -J' when t is replaced by -t', the Maxwell equa-tions can be made invariant under time reversal by replacing D by D', Hby -H", Bby -B', and E by E'. However, in a lossy medium the presenceof a conduction current term qE makes it impossible for the Maxwell equationsto be invariant under time reversal.

215

TheDoppler 7

effect

If a source of monochromatic radiation is in motion relativeto an observer, the observed frequency of radiation willincrease as the source and observer approach each other andwill decrease as they get farther apart. This principle,enunciated by Christian Dopplerl in 1843, is called the"Doppler principle" or the "Doppler effect."

Basically the Doppler effect is a consequence of thecovariance of Maxwell's equations under the Lorentz trans-formation. For the usual case where .the source andobserver are in free space, the exact relativistic formulationof the Doppler effect is well known. But in the presence ofmaterial media the Doppler effect is more intricate andinvolves questions which as yet have not been completelysettled.

In this chapter the problem of calculating the Dopplereffect in material media is discussed. It is shown that forhomogeneous media the calculation can be made by usingthe principle of phase invariance, whereas for inhomo-geneous media. a more elementary point of departure isrequired.

1 Ch. Doppler, nber das farbige Licht der Doppelsterne,Abhandlungen der Koniglichen Bohmischen Gesellschaft der W issen-schaften, 1843. See also E. N. Da C. Andrade, Doppler and theDoppler Effect, Endeavor, vol. 18, no. 69, January, 1959.

217


7.1 Covariance of Maxwell'sEquations

According to the theory of relativity, the Maxwell equations must havethe same form in all inertial frames of reference, i.e., they must becovariant under the Lorentz transformation.l This means that if wewrite the Maxwell equations in an inertial frame K and then by aproper Lorentz transformation pass from the coordinates x, y, z, t ofK to the coordinates x', V', Zl, t' of another inertial frame K' which ismoving at a uniform velocity with respect to K, the dependent func-tions, i.e., the four field vectors, the current density vector, and thecharge density, must transform in such a way that the transformedequations have the same formal appearance as the original equations.The Lorentz transformations can be considered a consequence of the

postulate that the velocity of light in vacuum has the same value c in allframes of reference. To show this, we make the spatial origins of Kand K' coincident at t = t' = 0 and introduce the convenient notationXl = x, X2 = y, Xa = Z, X4 = ict, x~ = x', x~ = V', x~ = x', x~ = ict'.Then, in this notation, the postulate demands that the condition

(1)

be satisfied. Here and in analogous cases we suppress the summationsign and use the convention that repeated indices are summed from 1to 4. This condition in turn leads to the requirement that the coordi-

1The covariance of the Maxwell equations under the Lorentz transformationwas proved by Lorentz and Poincare, and physically interpreted by Einstein.Their work, however, was intentionally restricted to the Maxwell equationsof electron theory, Le., to the so-called microscopic Maxwell-Lorentz equations,and said nothing of material media. The required generalization of the theoryto the case of material media was finally worked out by Minkowski from thepostulate that the macroscopic Maxwell equations are covariant under theLorentz transformation. See, for example, W. Pauli, "Theory of Relativity,"Pergamon Press, New York, 1958; A. Sommerfeld, "Electrodynamics,"Academic Press Inc., New York, 1952; V. Fock, "The Theory of Space Timeand Gravitation," Pergamon Press, NewYork, 1952; E. Whittaker, "A Historyof the Theories of Aether and Electricity," vol. II, Harper & Row, Publishers,Incorporated, New York, 1953; C. M~ller, "The Theory of Relativity," OxfordUniversity Press, Fair Lawn, N.J., 1952.

218

The Doppler effect

nates x; and xl' be related by the linear transformations

(2)

whose coefficients aI" obey the side conditions

forfor

JI = AJI~A

(3)

These linear transformations constitute the complete Lorentz group oftransformations. Since the determinant lal',1 may equal +1or -1,this complete group splits naturally into the positive transformationsfor which lal',1 = 1 and the negative transformations for whichlal',1 = -1. From these the positive transformations are selectedbecause they include the identity transformation

x; = XI' (J.L = 1, 2, 3, 4) (4)

The positive transformations, which can be thought of as a rotation infour-dimensional space or, equivalently, as six rotations in the XIX2,

XIX3, XIX4, X2X3, X2X4, X3X4 planes, contain not only the proper Lorentztransformations but also extraneous transformations involving thereversal of two or four axes. Therefore, when these extraneous trans-formations are excluded, those that remain of the positive transforma-tions constitute the proper Lorentz transformations.Assuming that the coordinates undergo a proper Lorentz transforma-

tion, we define a 4-vector as a set of four quantities AI' (J.L = 1, 2, 3,4)that transform like the coordinates:

(5)

Moreover, we define a 4-tensor AI" of rank 2 as a set of 42 quantitiesthat obey the transformation law

(6)

and a 4-tensor AI">" of rank 3 as a set of 43 quantities that obey thetransformation law

(7)

219


In terms of the quantities FafJ, GafJ, J'a (a, (3 = 1,2,3,4), whose valuesare given by

0 B. -B"i--Ec 2O

-B. 0 B20 i--EFafJ = c " (8)

B" -B2O 0 -~Ec •

~E", iE i0eE.c c "

[ 0H. -H" -icDo]

-H. 0 H2O -icD" (9)GafJ = H" -H", 0 -icD.icD", icD" icD. 0

J.~[n (10)

wp

the two Maxwell equations

V.B = 0

become

aVxE=--B at (ll)

(a, f3, ~ = 1,2,3,4) (12)

and the other two Maxwell equations

avxH--D=Jat

become

V.D = p (13)

(a = 1, 2, 3, 4) (14)

220

The Doppler effect

From the postulate that Maxwell's equations are covariant under aproper Lorentz transformation of the coordinates, viz., that the four-dimensional forms (12) and (14) are covariant, it follows that F a{3 andGa{3 are 4-tensors of rank 2 and J a is a 4-vector. This means thatwhen the coordinates undergo a proper Lorentz transformation

(/.I = 1, 2, 3, 4) (15)

the quantities J a (the 4-current) transform like the coordinates:

(/.I = 1, 2, 3, 4) (16)

and the field tensors Fa{3, Ga{3 transform like the product of thecoordinates:

F;. = apaa.{3F a{3

G;. = apaa.{3Ga{3

(J,l, II = 1, 2, 3, 4)

(J,l, II = 1, 2, 3, 4)

(17)

(18)

So far the only restrictions we have placed on the reference framesare that their spatial origins be coincident at t = t' = 0 and that theirrelative velocity v be uniform. Now we shall place an additionalrestriction on the reference frames, namely, that they have the sameorientation. With the velocity and orientation specified, the coeffi-cients ap• can be uniquely determined from Eqs. (2) and (3) and thecondition lap.1 = 1. One can show that if the two inertial frames Kand K' have the same orientation, and if their relative velocity is v,then the coefficients ap• are given by

V 21+ (oy -I)-=-

v2

(oy _ 1) VyV.v2

(oy _ 1) v,vx02

(oy _ 1) vxvyv2

V 21 + (oy - 1)-.!'-

v2

(oy _ 1) V.Vyv2

(oy _ 1) V"'V,v2

(oy _ 1) vI/v,v2

V 21 + (oy - 1) -!-

v2

'Y

(19)

Using these values of the coefficients and expressing the results inthree-dimensional form, we find that the transformation law (15) for

221


the position 4-vector XI" which can be written as (r,iet), becomes

, ~.~r = r - -yvt + ('Y - 1) -- Vv2

, ( r. v)t =-y t-7

where

(20)

(21)

1-y = VI - {32

{3=~e

and that the transformation law (16) for the 4-vector (J,iep) assumesthe form

, J.vJ = J - 'YVP + ('Y - 1) - Vv2

Also, we find that the transformation law (17) leads to

E.vE' = 'Y(E + v X B) + (1 - 'Y) -2 v

V

( 1) B.vB' = -y B - C2v X E + (1 - -y) V2v

and that the transformation law (18) yields

( 1) D.vD' = 'Y D + C2v X H + (1 - 'Y) V2 v

H.vH' = -y(H - v X D) + (1 - 'Y) -2- V

V

(22)

(23)

(24)

(25)

(26)

(27)

Clearly Eqs. (22) and (23) follow from Eqs. (20) and (21) by replacingr by J and iet by iep. Also Eqs. (26) and (27) follow from Eqs. (24)and (25) by replacing E by eD and B by Hie.

222

The Doppler effect

Thus we see that when the coordinates and time undergo the properLorentz transformations expressed by Eqs. (20) and (21), the Maxwellequations with respect to K, viz.,

aVxH=J+-D ata

vxE=--B atV.D=p V. B = 0 (28)

transform into the Maxwell equations with respect to K', viz.,

v' X H' = J' + ~,D' v' X E' = - ~ H'at'V'. D' = p' v' . B' = 0 (29)

provided the primed quantities are related to the unprimed quantitiesby relations (20) through (27).

7.2 Phase Invariance andWave 4-Vector

If a reference frame K is at rest with respect to a homogeneous medium,the Maxwell equations in K admit solutions of the form

E(r,t) = Re Eoei(k.r-",O

B(r,t) = Re Boei(k.r-..O

(30)

(31)

where Eo is a constant and Bo, which is related to Eo by Bo = (1/ w)k X Eo,is likewise a constant. Expressions (30) and (31) represent in K theelectric and magnetic vectors of a plane homogeneous wave of angularfrequency wand wave vector k.To see what form this plane wave takes in a reference frame K'

moving at uniform velocity v with respect to K, we first substituteexpressions (30) and (31) into the transformation law (24) and thusobtain the expression

E'(r,t) = Re E~ei(k.r-",t) (32)

223


where E~is a constant given by

E~ = -y(Eo + v X Bo) + (1 - -y) Eo' v v. v2 (33)

Then we transform the coordinates rand t into the coordinates r' and t'of K' by means of the proper Lorentz transformation

. r'. vr = r' + -yvt' + (-y - 1) -- Vv2

t = -y (ti + r'c: v)(34)

(35)

Applying this transformation to expression (32), we see that the electricvector of the wave in K' takes the form

E' (r' ,t') = Re E~ei(k'.r'-""t')

where

,w k. vk = k - -y - v + (-y - 1) - Vc2 v2

w' = -y(w - v. k)

(36)

(37)

(38)

This shows that in going from K to K' the plane wave (30) is trans-formed into the plane wave (36).By the mann~r in which k' and w' appear in expression (36), we are

led to the interpretation that k' is the wave vector of the wave in K'and w' is its frequency. Accordingly, we regard relation!! (37) and (38)as t~e transformation laws for the wave vector and the frequency.Comparing these relations with Eqs. (20) and (21), we see that

(k,i ~) transforms like the 4-vector (r,ict). Hence'

is a 4-vector. It is called the wave 4-vector.

224

The Doppler effect

The phase cP of the wave in K is defined by

cP = k. r - wt

and in terms of kl' and XI' it takes the form

(40)

(41)

Since kl' and XI' are 4-vectors, it follows from Eq. (41) that cP is invariant.What we have shown above is that the phase cP of a uniform plane

wave in a homogeneous medium remains invariant under a properLorentz transformation of the coordinates. This invariance of thephase, sometimes referred to as the principle of phase invariance,applies not only to waves in vacuum but also to waves in homo-geneous media, even if these homogeneous media be anisotropic anddispersive. However, in the case of inhomogeneous media the Maxwellequations do not admit uniform plane wave solutions and hence pre-clude the possibility of devising an invariant phase. 1

7.3 Doppler Effect and Aberration

As in the previous section, we consider a plane monochromatic wavetraveling in a homogeneous medium. We recall that if k and warerespectively the wave vector and angular frequency of the wave in thereference frame K, which is at rest with respect to the medium, thenthe wave vector k' and the angular frequency w' of the 'wave, asobserved in a reference frame K' moving with uniform velocity v withrespect to K, are given by

k' w ( k.v= k - 'Y - v + 'Y - 1) - Vc2 v2

w' = 'Y(w - v • k)

(42)

(43)

1K. S. H. Lee and C. H. Papas, Doppler Effects in Inhomogeneous Aniso-tropic Ionized Gases, J. Math. Phys., 42 (3): 189-199 (September, 1963).

225


where

1'Y = VI _ {32

From Eq. (42) we "~n calculate the angle between the directions of k'and k and thus obtain the aberration of the wave vector due to therelative motion of the reference frames. Also, from (43) we can calcu-late the difference between w' and w, which gives the correspondingDoppler shift in frequency.To derive the aberration formula, we note that the spatial axes of

K and K' are similarly oriented, i.e., the x', y', z' axes are parallelrespectively to the x, y, z axes, and we assume that v is parallel to thex axis and hence to the x' axis. Since v = ezv, it follows from thescalar multiplication of Eq. (42) by the unit vectors ez and ell that

wk' cos 8' = -yk cos 8 - 'Y 2 vc

k'sin8' = ksin8

(44)

(45)

where 8' is the angle between k' and v, and 8 is the angle between k andv. Dividing Eq. (45) by Eq. (44) and using the relations k = wlvph,where Vph is the phase velocity in K, n = clvph, where n is the index ofrefraction in K, and (3 = vic, we get the aberration formula

tan 8' =! sin 8 1 tan 8'Y cos 8 - fi = ~ 1 - fi sec 8

n n(46)

In vacuum, we have n = 1 and, accordingly, Eq. (46) reduces to thefamiliar relativistic formula for aberration.The formula (43) for the Doppler effect can be written as

w' = 'Y(w - vk cos 8) = 'Yw(I - (3n cos 8) (47)

where 8 is the angle between the wave vector k and the relative velocityv. From this equation we see that a wave of angular frequency w inreference frame K appears to have a different frequency w' when

226

The Doppler effect

observed from the moving frame K'. The Doppler shift in frequency,viz., the quantity w' - w, is a maximum when (J = 0 and is a minimumwhen (J = 7rj2. In the latter case we have the relation

w' = ')'w

which expresses the so-called "transverse Doppler effect."

(48)

7.4 Doppler Effect in HomogeneousDispersive Media

We shall now apply the Doppler formula to the situation in which amonochromatic source and an observer are in a homogeneous dispersivemedium. We shall limit the discussion to two cases: in one the sourceis fixed with respect to the medium and in the other the observer isfixed with respect to the medium. The observer is assumed to be inthe far field of the source so that, to a good approximation, the wavesincident upon the observer are plane.In the case where the source is fixed with respect to the medium, we

choose the reference frame K to be at rest with respect to the mediumand the source, and the reference frame K' to be moving with theobserver at velocity v with respect to K. Hence, from Eq. (47) we seethat

w' = ')'w[1 - (3n(w) cos (J] (49)

where w is the source frequency in K, and w' is the frequency observedin the moving frame K'. The index of refraction n(w) is evaluated inK. Since n(w) ~ 0, it follows from expression (49) that when theobserver is moving toward the source «(J = 7r), w' is greater than w, asin a vacuum. However, when the observer is moving away from thesource «(J = 0), w' is not necessarily less than w. Under special circum-stances (for example, when the medium is a nearly resonant plasma),n(w) could be so small that w' would be greater than w, in contradistinc-tion to the corresponding phenomenon in a vacuum, where w' wouldnecessarily have to be less than w.

227


In the case where the observer is fixed with respect to the medium,we choose K to be at rest with respect to the medium and the observer,and K' to be moving with the source at velocity v with respect to K.Accordingly we again have

W' = 'Yw[1 - [3n(w) cos OJ (50)

but now w' is the source frequency and w the observed frequency.When 0 = 1r the source is moving away from the observer, and wheno = 0 it is moving toward the observer. Due to the dispersive natureof n(w), expression (50) is not, in general, monotonic between w' and w.Therefore, a given value of w' may yield more than one value of w.This means that the radiation incident upon the observer may appearto have several spectral components even though the source is oscillat-ing at a single frequency. This splitting of the emitted monochromaticradiation into several modes is called the complex Doppler effect. Thiseffect has been studied by Frankl in connection with the problem ofdetermining the radiation of an oscillating dipole moving through arefractive medium. If the medium were nondispersive, expression (.50)would, of course, yield a monotonic relation between w' and w, andhence no complex Doppler modes would be generated.As an illustrative example, let us examine the complex Doppler effect

in the special instance where the medium is a homogeneous plasma.For such a medium, Eq. (50) becomes

W' = 'Y(w - [3yw2 - wp2 cos 0) (51)

where Wp is the plasma frequency. A plot of w' versus w is shown inFig. 7.1. The curve has two branches, one given by the solid line andthe other by the broken line. The broken line represents Eq. (51) foro = 1r (source receding from the observer), and the solid line representsEq. (51) for 0 = 0 (source approaching the observer). The twobranches join at point A, where w = w" and w' = 'YW". The solidbranch is a minimum at point B, where w = 'YW" and w' = Wp• Theasymptotes make with the axes an angle If which depends on the relative

11. M. Frank, Doppler Effect in a Refractive Medium, J. Phys. U.S.S.R.,7 (2): 49-67 (1943). See also, O. E. H. Rydbeck, Chalmers Res. Rept. 10,1960.

228

The Doppler effect

Fig. 7.1 A sketch of the source frequency w' versus the observedfre-quency w, in the case where the observer is at rest withrespect to a homogeneous isotropic plasma medium and thesource is moving through the medium at relative velocity {3c.

velocity v according to the relation tan if; = VI - {3/v'f+I3. Fromthe curve, we see that for a given value w~of w' greater than 'YWp, we geta single value w, of w when the source is receding, and a single value Wa

of w when the source is approaching. We also see that if w~ is less than'YWp but greater than Wp, the wave due to the receding source is beyondcutoff, and the wave due to the approaching source splits into two, thusyielding two values of Wa instead of only one. One of these two fre-quencies is always greater than w~, while the other may be greater orless than w~ depending on how close w~ is to -ywp' Finally, we note thatif w~ is less than Wp, even the wave due to the approaching source isbeyond cutoff.

229


7.5 Index of Refraction of a MovingHomogeneous Medium

To compute the index of refraction of a homogeneous medium movingat velocity v with respect to a reference frame K, we choose a frameK' that is at rest with respect to the medium, and we assume that inK' there is a monochromatic plane wave having wave vector k' andfrequency w'. In K the wave is perceived as a plane wave of wavevector k and frequency w. The index of refraction of the medium isdefined by n' = ck'/w' in K' and by n = ek/w in K.As a point of departure for the calculation, we use the transformations

w' k'.vk = k' + "I - v + ("I - 1)-- Ve2 v2

w = 'Y(w' + v. k')

From Eq. (52) we find that k is given by

(52)

(53)

(54)

Dividing Eq. (54) by Eq. (53) and noting that k' . v = k'v cos 0', weobtain

kw

(55)

Since by definition n = ek/w and n' = ek'/w', it follows from Eq. (55)that

ek vn'2 + 2"(2n'(3cos 0' + ("(2 - l)n'2 cos2 0' + "(2(32n = -;; = "((1 + n'(3 cos 0') (56)

Although this relation relates n to n', it is not yet the relation we want,because it involves the angle 0'. To obtain the desired relation, wemust eliminate 0' in favor of the angle 0 between k and v. Accordingly,

230

The Doppler effect

we invoke the aberration relations

-yen' cos (J' + fJ)cos (J = -:=. ==.============Vn'2 sm2 (J' + -y2(n' cos (J' + {j)2

(J' -y(n cos (J - (j)

cos = --:=============Vn2 sin2 (J + -y2(n cos (J - {j)2

(57)

(58)

which follow from Eq. (46).led to

n' cos (J' + {3n cos (J = 1 + n'{3cos (J'

Combining Eqs. (56) and (57), we are

(59)

which, with the aid of Eq. (58), yields the following quadratic equationfor n:

(60)

Solving this equation and choosing the root that yields n = n' forv = 0, we obtain the desired relation:

n = VI + -y2(n'2 - 1) (1 - {32cos2 (J) - (3-y2(n'2 - 1) cos (J (61)1 - -y2(n'2 - 1){32 cos2 (J

Here n' is the index of refraction of the medium in the K' frame, whichis at rest with respect to the medium, n is the index of refraction in theK frame;with respect to which the medium is moving at velocity v,and (J is the angle between v and the wave vector k.We see from Eq. (61) that the index of refraction n of a moving

medium depends on the velocity v( = (3c) of the medium and on theangle (J between k and v. When {32 « 1, Eq. (61) reduces to the follow-ing equation,

n = n' - (n'2 - 1){3cos (J (62)

which is valid for dispersive as well as nondispersive media. In the

231


case where the direction of k is parallel (8 = 0) or antiparallel (8 = 11")

to v and the medium is nondispersive, Eq. (62) yields

Vph = ~ i: v (1 - ~)n' n'2 (63)

where Vph( = c/n) is the phase velocity of the wave in K. This is thewell-known formula of Fresnel. The coefficient (1 - 1/n'2) is calledthe Fresnel drag coefficient. The Fresnel formula was verified exper-imentally by Fizeau who used streaming water as the moving medium.For a dispersive medium Eq. (63) has to be modified. To find what

this modification is, we note that in Eq. (62) the index of refraction n'is a function of w'. Since the Doppler formula (47) for low velocities«32 « 1) yields w' = w =+= (3nw, where the upper sign is for 8 = 0 andthe lower one is for 8 = 11", we see that

n'(w') = n'(w =+= (3nw) (64)

Expanding this relation about wand keeping only the first two terms,we get

'( ') '( ) an'(w)n w = n w =+= (3nw ~

Substituting this expansion into the equation

n = n' =+= (n'2 - 1)(3

(65)

(66)

which follows from Eq. (62) when 8 = 0 and 8 = 11", and neglectingterms in (32, we get

, an'(w)n = n'(w) =+= [n'2(w) - 1](3 =+= (3wn (w) ~

Since Vph = c/n, we then deduce from Eq. (67) that

c [ 1] w an'(w)Vph = n'(w) i: v 1 - n'2(w) i: v n'(w) ~

232

(67)

(68)

The Doppler effect

This is the form that Eq. (63) takes for a dispersive medium. We seethat the dispersive nature of the medium is accounted for by the lastterm on the right side. This term is sometimes referred to as the"Lorentz term". It was verified experimentally by Zeeman.

7.6 Wave Equation for MovingHomogeneous Isotropic Media

In a frame of reference K' which is at rest with respect to a homogeneousisotropic medium, the vector potential A' (r' ,t') and the scalar potentialcfJ'(r',t') due to a current density J'(r',t') and a charge density p'(r',t')clearly must obey the inhomogeneous wave equations

[\7'2 - n'2 ~] A'(r' t') = - 'J'(r' t')c2 i)t'2 ' /-l,

['2 n'2 i)2] , , , _ 1",\7 - C2 at'2 cfJ (r ,t) - - ;: p (r ,t )

(69)

(70)

where /-l' and e' are the permeability and the dielectric constant of themedium and n' is the index of refraction. With the aid of the 4-vectorsJ: and A:, whose values are given by

J' = [~t]a J~icp'

A: = [~~]-: cfJ'c

(71)

these equations can be combined to give

(n'2 i)2)(-1- KC2(4) \7'2 - - - A' = ,,'J'a C2 i)t'2 a ,.. a (72)

where K = /-l'e' - (1/c2) = (n'2 - 1)/c2 and 1l4a is the Kronecker delta.We wish to transform Eq. (72) to reference frame K, with respect to

which the medium is moving at velocity v. Since A: and J: are 4-vec-

233


tors, they transform as follows:

(73)

Here All and Jil are 4-vectors in K, and the a"l1 are the coefficients of theproper Lorentz transformation that carries K' into K. To transformthe differential operator that appears in Eq. (72), we write

(74)

The first two terms on the right side constitute an invariant operator,and hence

(75)

By means of the transformations

r'. vr = r' + 'Yvt' + ('Y - 1) -2- V

V

it can be shown that

n'2 - 1 iJ2 (a )2c2 at'2 = K'Y

2 at + v • V

(76)

(77)

(78)

Thus from relations (75) and (78) we see that the operator (74) trans-forms as follows:

n'2 a2 1 a2 (a )2V'2 - - - = V2 - - - - K'Y2 - + V • Vc2 at'2 c2 at2 at (79)

Now, with the aid of the transformations (73) and (79), it becomesevident that equation (72) in K' transforms into the following equation

234

The Doppler effect

in K:

where the operator L is defined by

1 iP (a )2L == V'2 - - - - K"'/ - + v • V'c2 at2 at

(80)

(81)

Multiplying Eq. (80) by aav, summing on a, and using the orthogonalityrelation (3), we find that

(82)

For a = 4, Eq. (80) yields

(83)

Therefore we can cast Eq. (82) in the form

(84)

Using Eq. (19), we see that

(85)

where Uv is the velocity 4-vector ("(v,i"(c). With the aid of this result,Eq. (84) yields

,LAv = -p.'Jv - ~,~ U".lfJUfJ (86)

This is the equation into which Eq. (72) is transformed when the frameof reference is changed from K' to K.

In three-dimensional form, Eq. (86) leads to the following equa-tions for the vector potential A(r,t) and the scalar potential </J(r,t) in

235


reference frame K:

[ 1 82 (8 )2]V'2 - "& at2 - K'Y2 at + v. V' A(r,t)

(87)

[ 1 82 (8. )2JV'2 - "& 8t2 - K'Y2 at + v • V' q,(r,t)

,= -J.l.'C2p - ~,~ 'YC2('YJ' v - 'YC2p) (88)

where, as before, K = (C2E'J.I.' - 1)jc2 = (n'2 - 1)jc2. With aknowl-edge of these equations, we can find the vector and scalar potentials ofa source surrounded by a homogeneous isotropic medium moving at avelocity v with respect to the source. Moreover, these equations enableone to calculate the electric vector E = -V'q, - (8j8t)A and the mag-netic vector B = V' X A of the source in the presence of a wind.The above discussion is based on the transformation of the inhomo-

geneous wave equation from the K' frame to the K frame. Actually,the same results can be achieved by using the tensor form of Maxwell'sequations as the point of departure. 1 To show this, we recall thatMaxwell's equations can be written as follows:

8Ga(3 = J8X(3 a

(89)

(90)

These tensor equations hold in all Lorentz frames, and in particularthey hold in K' and K. In K' the constitutive relations are

D' = E'E' and H' = .; H'J.I.

(91)

1K. S. H. Lee, On the Doppler Effect in a Medium, Antenna Lab. Rept.29, California Institute of Technology, December, 1963. See also, J. M.Jauch and K. M. Watson, Phenomenological Quantum-electrodynamics,Phys. Rev., 74: 950, 1485 (1948).

236

The Doppler effect

Expressing D', E', H', B' in terms of D, E, H, B of the reference frameK, we find with the aid of the Eqs. (24), (25), (26), and (27) that theconstitutive relations in K are

1D +"2 v X H = l(E + v X B)C

H - v X D = .!.(B _.!.v XE)JJ.' c2

When written in tensor form, these constitutive relations become

(92)

(93)

Ga{JV{J = c2lFapVp (94)

1Ga{JVp + G{JpVa + GpaV{J = ., (Fa{JVp + F{JpVa + FpaV{J) (95)

JJ.

where as before Vp denotes the velocity 4-vector ('Yv,i'Yc).To express the field tensor Ga{J explicitly in terms of the field tensor

Fa{J, we multiply Eq. (95) by VP' Noting that

VpVp = -c2

we thus find that

By virtue of the constitutive relations (94), we have

(97)

Hence, it follows from Eq. (96) that

(98)

Substituting expression (98) into the Maxwell equation (90), we findthat

oFa{J + u u of{Jp _ u U oFa• - 'JoX{J K a • OX~ K p {J OX{J - JJ. •• (99)

237


However, from Eqs. (90) and (94) we see that

(100)

Therefore, Eq. (99) becomes

(101)

Now we have two equations for the field tensor Fa/3, one being theMaxwell equation (89) and the other being equation (101). If we

write the field tensor Fa/3 in terms of the 4-potential A, = (A,i, ~}that is, if we write

(102)

then Eq. (89) is satisfied. Substituting expression (102) into Eq. (101),we obtain the following equation for the 4-potential:

Rearranging terms, we get

Since the 4-potential is not completely determined by Eq. (102), we arefree to impose on it the following additional condition,

(105)

238

The Doppler effect

which is called the "generalized Lorentz condition" for the 4-potential.When this condition is satisfied, Eq. (104) reduces to

(106)

This equation is identical to Eq. (86) and, in three-dimensional form,amounts to Eqs. (87) and (88).To show that Eq. (106) can be used to find the index ofrefraction of

a moving media, we assume that A(r,t) has the form of a plane wave:

(107)

Substituting this expression into Eq. (87), with the right side set equalto zero, we find

(108)

On solving this equation for n = ek/w, we are led to relation (61) forthe index of refraction.

239

Name IndexAlsop, L. E., 110Andrade, E. N. Da C.,217n.

Astrom, E., 191n.Aulock, W. H. von, 69n.

Baghdady, E. J., 125n.Barbiere, D., 69Beverage, H. H., 46n.Bidal, P., 98n.Bladel, J. van, 26n.Blumenthal, 0., 9n.Bohnert, J. I., 125n.Bolton, J. G., 110n., 150n.,158n.

Bontsch-Bruewitsch,M. A., 73n.

Booker, H. G., 125n.Bopp, F., 189n.Borgnis, F., 182n.Born, M., 121n., 156n.Bouwkamp, C., 9, 35-36,45n., 50n., 57n., 97, 100,104n.

Bracewell, R. N., 109n.Bramley, E. N., 167n.Brillouin, L., 34, 183n.Brouwer, L. E. J., 57n.Brown, G. H., 57n.Brown, R. H., 159Bruckmann, H., 57n.Bruijn, N. G. de, 50n.Budden, K. G., 169n.,214n.

Bunkin, F. V., 209

Campbell, G. A., 57n.Carter, P. S., 57n.Casimir, H. B. G., 57n.,97, 100, 104n.

Cerenkov, P. A., 47Chandrasekhar, S., 111n.,126n.

Chu, L. J., 80n.Cooper, B. F. C., 110n.Courant, R, 21n., 51n.

Debye, P., 98Demirkhanov, R. A., 170n.Deschamps, G. A., 125n.Desirant, M., 169n.Dirac, P. A. M., 21n.Dolph, C. L., 68Doppler, Ch., 217Drake, F. D., 110n.

Einstein, A., 218n.Emde, F., 16, 75n.

Fano, D., 138n.Fay, W. J., 169n.Feshbach, H., 11n.Fock, V., 218n.Frank, I., 47n., 228n.Franz, K., 77n.Friis, H. T., 42n., 57n.,142n.

Geverkov, A. K., 170n.Ginzburg, V. L., 169n.Giordmaine, J. A., 110n.Goland, V. E., 170n.

Haddock, F. T., 109n.Hansen, W. W., 62Harrison, C. W., Jr., 42n.Helmholtz, H. von, 9Heritage, J. L., 169n.Hedofson, N., 169n.Hilbert, D., 21n., 51n.Hodge, W. V. D., 98

Jacobson, A. D., 125n.Jahnke, E., 75n.Jauch, J. M., 236n.Jelley, J. V., 47n., 110n.

Kales, M. L., 125n.Kellogg, E. W., 46n.King, D. D., 69, 125n.King, R: W. P., 42n., 45,57n., 76n., 171n.

Knudsen, H. L., 57n.Ko, H. C., 147n.Kogelnik, H., 209Kraus, J. D., 57n., 125n.Kuehl, H., 209Kuiper, G. P., 110n., 158n.

Labus, J., 39n.Lange-Hesse, G., 195n.Lassen, H., 77n.Lawson, J. D., 50n.Lee, K. S. H., 225n., 236n.Lequeux, J., 109n.Lerner, R. M., 125n.Lorentz, H. A., 213n., 218.

Maas, G. J. van der, 69McReady, L. L., 152n.Margenau, H., 211n.Mathis, H. F., 57n.Mayer, C. H., 110n.Meixner, J., 9, 189Michiels, J. L., 169n.

Middlehurst, B. M., 110n.,158n.

Minkowski, H., 218n.Ml'Sller,C., 218n.Morse, P. M., 11n., 99n.Muller, C., 56n.Murphy, G. M., 211n.Murray, F. H., 57n.

Nisbet, A., 209n.

Packard, R F., 69Papas, C. H., 45n., 76n.,187n., 225n.

Pauli, W., 218n.Pawsey, J. L., 109n., 152n.Payne-Scott, R, 152n.Pistolkors, A. A., 57n., 74n.Pock1ington, H. C., 38n.Poincare, H., 122, 218n.Poincelot, P., 213n.Pokrovskii, V. L., 68Polk, C., 73Popov, A. F., 170n.

Ratcliffe, J. A., 167n.,169n.

Rayleigh, Lord, 213n.Rham, G. de, 98n.Riblet, H. J., 68, 80n.Rice, C. W., 46n.Rice, S. 0., 111n.Robertson, H. P., 194n.Rosenfeld, L., 171n.Rumsey, V. H., 125n.Rydbeck, O. E. H., 228Rytov, S. M., 200n.

Sandler, S. S., 69Schelkunoff, S. A., 7n.,42n., 43n., 57n., 61,80n., 142n.

Schwartz, L., 21n.Shk1ovsky, I. S., 109n.,169n.

Smith, F. G., 109n.Smythe, W. R, 11n.Sommerfeld, A., 20n., 62n.,189n., 218n.

Sonine, N. J., 77Spitzer, L., Jr., 176n.Steinberg, J. L., 109n.Stokes, G. R, 9n., 119Stone, J. S., 68n.Stratton, J. A., In.

Tai, C. T., 143n.Tamm, I., 47n.

241


Taylor, T. T., 69, SOn.Tetelbaum, S., 72Thomas, R. K., 69Toraldo, G. de Francia,80n.

Townes, C. R., 110n.Twiss, R. Q., 159

Unz, R., 69

242

Watson, G. N., 51n.Watson, K. M., 236n.Weisbrod, S., 169n.Westfold, K. C., 123n.,150n.

Weyl, R., 194n.Whittaker, E. T.; 5In.,21Sn.

Wiener, N., 112n.

Wilcox, C. R., 98n.Wilmotte, R. M., 80n.Wolf, E., 121n., 136n.,I56n.

Woodward, P. M., 50n.Woodyard, J. R.., 62

Yeh, Y.-C., 143n.

Zernike, :F., 159n.

Subject IndexAberration, 226Angular-momentum oper-

ator, 107Antenna, dipole, 44, 208isotropic, 57radio-astronomical, 109-

110scanning, 69-70straight wire, 37-56current in, 37-42integral relation for,48-50

pattern synthesis, 50-56

radiation from, 42-47Antenna temperature, 149-

151Antipotentials, 13-14, 23Apparent disk tempera-

ture, 118Area, effective, in matrix

form, 145of receiving antenna, 143

Argand diagram, 123Array factor, 57, 59, 60Arrays, binomial, 62-68broadside, 62, 68, 69Chebyshev, 68-69collinear, 70end-fire, 62, 69linear, 57-70parallel, 70rectangular, 71superdirective, 80uniform, 61

Attenuation factor, 184,186

Autocorrelation function,113

Axial ratio, 121

Binomial theorem, 53Blackbody spectral bright-

ness, 117Boltzmann equation, 175Boundary conditions, 8-9Brightness temperature,

150Brown and Twiss inter-

ferometer, 159, 167-168

Cerenkov radiation, 47Coherence, degree of, 160-

168Coherency matrix, 135-

140, 145Collision frequency, 176,

177,178

Complex dielectric con-stant, 175

Conjugate matching, 141,143

Constitutive parameters,of anisotropic plasma,189-191

of isotropic plasma, 174,177

of lossy dielectric, 173-174

of simple media, 2, 5-6transformation of, 192

Cornu spiral, 73Correlation coefficient,

166-167Correlation function, 113Correlation interferometer,

159-168Coulomb gauge, 11Covariance of Maxwell's

equations, 223Current 4-vector, 222

Debye potentials, 97-98Degree, of coherence, 159-

161, 165-166of polarization, 130-131,

139-140, 145-146Dipole (see Electric dipole;

Magnetic dipole)Dirac delta function, 21Directivity gain, defini-

tion, 73full-wave dipole, 76half-wave dipole, 76rectangular array, 78-80short dipole, 74uniform parallel array,

76-77Dispersion, 185Distribution function, 175Doppler effect, 217, 226-

227complex, 228-229

Duality, 6-8Dyadic Green's function,

19, 26-29, 210-211

E wave, 81, 92, 100-101,204

Electric dipole, 82, 89-93field of, 90-93, 102short filament of cur-

rent, 84Electric energy density, in

dispersive media, 178-183

instantaneous, 14

Electric energy density,time-average, 15":17

Electric potential, scalar,10

vector, 13, 14Electric quadrupole, 82, 83fields of, 94-97two antiparallel fila-

ments,85Electric wall, 9Electrostatic wave, 185EMF method, 34, 74Energy theorem, 17Evanescent wave, 184Extraordinary wave, 199,

204-205

Far zone, definition, 32of multipoles, 108of rectangular array,

71-72Faraday rotation, 202, 212Field tensors, 220, 238Four-potential, 238-239Four-tensor, 219Four-vector, 219Fraunhofer field, 72Fresnel drag formula, 232Fresnel field, 72

Gain (see Directivity gain)Gain function, 142Gauge, Coulomb, 11Lorentz, 11

Giorgi system of units, 1Grating lobes, 153Green's function, dyadic,

26-29, 32, 210-211scalar, 20-29, 89, 104-

106Gyrofrequency, 188, 191

Hwave, 81, 93,100-101Hankel function, spheri-

cal,98-99Helmholtz equation, sca-

lar, 11, 12, 19,21-23,38, 98

vector, 104, 196Helmholtz integral, 22, 23,

85,87Helmholtz's partition

theorem, 9Hermite polynomials, 51-

54Hermiticity, of coherency

matrix, 135of dielectric tensor, 189-

193

243


Hertzian dipole, 44, 208Hilbert space, 194Hodge's decomposition

theorem, 98Hydrogen line, 109

Impedance of antenna, 141Inertial frame of reference,

217-218Intensity, polychromatic

wave, 129spectral, 115-117

Interferometer, correla-tion, 159-168

two-element, 151-159Irrotational vector, 9Isotropic antenna, 57

Kronecker delta, 24

Legendre polynomials,associated, 99

Lorentz condition for four-potential, 238-239

Lorentz force, 2, 175Lorentz gauge, 11, 13, 19,

23, 86Lorentz transformation,

217-223

Magnetic dipole, 83, 93field of, 93loop of current, 84

Magnetic energy, 14-17Magnetic potential, scalar,

13vector, 10

Magnetic wall, 9Maxwell's equations, 1, 4,

171-174in tensor form, 220, 236

Modulation index, 156Multipolar fields, 101-108

Newton's law, 4Noise, 111

Ordinary wave, 199, 203-205

Orientation angle, 121-124, 127, 131, 133,138, 140, 146-147

Pauli spin matrices, 139Phase invariance, 223-225Phasor, 4, 135, 136Planck's law, 117Plasma, 170anisotropic, dielectric

tensor of, 187-195dipole radiation in,209-212

plane waves in, 195-205

reciprocity relationfor, 212-215

244

Plasma frequency, 177Poincare sphere, 122, 131,

147Poisson's equation, 12Polarization, 109, 118-134degree of, 130-131, 139-

140, 146measurement of, 125sense of, 121-123, 131

Polarization loss factor,143

Polarization vector, 171,172,174

Polarized wave, circularly,122-124, 138-139

elliptically, 118-134,137-139

linearly, 122, 123, 138-139

oppositely, 131-133partially, 119, 125, 133,

137, 140-148Potentials, 9-12, 19-24in spherical wave func-

tions, 100Taylor expansion of,

87-88Power, absorbed, 142radiated, 29-34

Poynting's vector, 15-16,29-34, 111-113, 142-143, 178, 185

of center-driven an-tenna, 44, 206

of linear array, 59of monochromatic

source, 33, 36of rectangular array, 71of traveling wave of

current, 46Poynting's vector theorem,

16, 178

QuadrupOle (8ee Electricquadrupole)

Radiation characteristic,normalized, 62

Radiation condition, 20,27-29, 98, 104

Radiation pattern, ofantenna in plasma,207, 208

of center-driven an-tenna, 44-50

of collinear array, 70-71of linear array, 59of monochromatic cur-

rent source, 48of rectangular array, 71of traveling wave of

current, 47of two-element inter-

ferometer, 153Radiation resistance, 37,

45, 207

Radio astronomy, 109Radio telescope, 109-110Radiometer, 110Random (stochastic) proc-

ess, 111-112Rayleigh-Jeans law, 118Reciprocity, 212-215Reciprocity theorem, 142Refraction, index of, 184-

186,226-227,230-233Reversibility, 215

Schelkunoff's synthesismethod,61

Schwarz's inequality, 137Sea interferometer, 158-

159Sommerfeld's radiation

condition, 20, 98Spectral brightness, 115-

118Spectral flux density, 113-

117Spectral intensity, 115-116Spherical wave expansion,

97Spur, 139Stationary random proc-

ess, 111Stochastic process, 111Stokes parameters, VS.

coherency matrix,135-136, 138-139

for monochromaticwave, 122-124

for polychromatic wave,126-133

under rotation, 133-134Stress dyadic, 176Superdirectivity,80Synthesis of radiation

patterns, 48-56

Taylor's series, 86-88, 180Thevinin's theorem, 141Trace of matrix, 139Truncated function, 111

Unilateral end-fire array,62

Unit dyadic, 24Unitary transformation,

194Unpolarized wave, 129

Variance, 166Velocity, energy transport,

185group, 185-187phase, 185-186, 226, 232

Visibility factor, 156, 161,166

Wave four-vector, 224-226Wave impedance, 209

A CATALOG OF SELECTED

DOVER BOOKSIN ALL FIELDS OF INTEREST

ED

A CATALOG OF SELECTED DOVER

BOOKS IN ALL FIELDS OF INTEREST

DRAWINGS OF REMBRANDT, edited by Seymour Slive. Updated Lippmann,Hofstede de Groot edition, with definitive scholarly apparatus. All portraits,biblical sketches, landscapes, nudes. Oriental figures, classical studies, togetherwith selection of work by followers. 550 illustrations. Total of 630pp. 9~ x 12\(

21485-0,21486-9 Pa., Two-vol. set $25.00

GHOST AND HORROR STORIES OF AMBROSE BIERCE, Ambrose Bierce. 24tales vividly imagined, strangely prophetic, and decades ahead of their time intechnical skill: "The Damned Thing," "An Inhabitant of Carcosa, " "The Eyes ofthe Panther," "Moxon's Master," and 20 more. 199pp. 5%x 8~. 20767.6 Pa. $3.95

ETHICAL WRITINGS OF MAIMONIDES, Maimonides. Most significant ethicalworks of great medieval sage, newly translated for utmost precision, readability.Laws Concerning Character Traits, Eight Chapters, more. 192pp. 5%x 8~.

24522-5 Pa. $4.50

THE EXPLORATION OF THE COLORADO RIVER AND ITS CANYONS,J. W. Powell. Full text of Powell's I ,OOO-mileexpedition down the£abled Coloradoin 1869. Superb account of terrain, geology, vegetation, Indians, famine, mutiny,treacherous rapids, mighty canyons, during exploration of last unknown part ofcontinental U.S. 400pp. 5%x 8~. 20094-9 Pa. $6.95

HISTOR Y OF PHILOSOPHY, Julian Marias. Clearest one-volume history on themarket. Every major philosopher and dozens of others, to Existentialism and later.505pp. 5%x 8~. 21739-6 Pa. $8.50

ALL ABOUT LIGHTNING, Martin A. Uman. Highly readable non-technicalsurvey of nature and causes of lightning, thunderstorms, ball lightning, St. Elmo'sFire, much more. Illustrated. 192pp. 5%x 8~. 25237-X Pa. $5.95

SAILING ALONE AROUND THE WORLD, Captain Joshua Slocum. First manto sail around the world, alone, in small boat. One of great feats of seamanship toldin delig-htful manner. 67 illustrations. 294pp. 5%x 8~. 20326-3 Pa. $4.95

LETTERS AND NOTES ON THE MANNERS, CUSTOMS AND CONDI.TIONS OF THE NORTH AMERICAN INDIANS, George Catlin. Classicaccount of life among Plains Indians: ceremonies, hunt, warfare, etc. 312 plates.572pp. of text. 6~ x 9\4. 22118-0,22119.9 Pa. Two-vol. set $15.90

ALASKA: The Harriman Expedition, 1899, John Burroughs, John Muir, et al.Informative, engrossing accounts of two-mon!!}, 9,000-mile expedition. Nativepeoples, wildlife, forests, geography, salmon industry, glaciers, more. Profuselyillustrated. 240 black-and-white line drawings. 124 black-and-white photographs. 3maps. Index. 576pp. 5%x 8~. 25109-8 Pa. $11.95

CATALOG OF DOVER BOOKS

THE BOOK OF BEASTS:Being a Translation from a Latin Bestiary of the TwelfthCentury, T. H. White. Wonderful catalog real and fanciful beasts: manticore,griffin, phoenix, amphivius, jaculus, many more. White's witty erudite commen-tary on scientific, historical aspects. Fascinating glimpse of medieval mind.Illustrated. 296pp. 5%x 8\4.(Available in U.S. only) 24609-4Pa. $5.95

FRANK LLOYD WRIGHT: ARCHITECTURE AND NATURE With 160Illustrations, Donald Hoffmann. Profusely illustrated study of influence ofnature-especially prairie-on Wright's designs for Fallingwater, Robie House,Guggenheim Museum, other masterpieces. 96pp. 9\4x 101<\. 25098-9Pa. $7.95

FRANK LLOYD WRIGHT'S FALLINGWATER, Donald Hoffmann. Wright'sfamous waterfall house: planning and construction of organic idea. History of site,owners, Wright's personal involvement. Photographs of various stages of building.Preface by Edgar Kaufmann, Jr. 100illustrations. 112pp. 9\4x 10.

23671-4Pa. $7.95

YEARS WITH FRANK LLOYD WRIGHT: Apprentice to Genius, Edgar Tafel.Insightful memoir by a former apprentice presents a revealing portrait of Wrightthe man, the inspired teacher, the greatest American architect. 372black-and-whiteillustrations. Preface. Index. vi + 228pp. 8\4x II. 24801-1Pa. $9.95

THE STORY OF KING ARTHUR AND HIS KNIGHTS, Howard Pyle.Enchanting version of King Arthur fable has delighted generations with imagina-tive narratives of exciting adventures and unforgettable illustrations by the author.41 illustrations. xviii + 313pp. 6%x 9\4. 21445-1Pa. $5.95

THE GODS OF THE EGYPTIANS, E. A. Wallis Budge. Thorough coverage ofnumerous gods of ancient Egypt by foremost Egyptologist. Information onevolution of cults, rites and gods; the cult of Osiris; the Book of the Dead and itsrites; the sacred animals and birds; Heaven and Hell; and more. 956pp. 6li x 9\4.

22055-9,22056-7Pa., Two-vol. set $21.90

A THEOLOGICO-POLITICAL TREATISE, Benedict Spinoza. Also containsunfinished Political Treatise. Great classic on religious liberty, theory of govern-ment on common consent. R. Elwes translation. Total of 421pp. 5%x 8~.

20249-6Pa. $6.95

INCIDENTS OF TRAVEL IN CENTRAL AMERICA, CHIAPAS, AND YU-CATAN, John L. Stephens. Almost single-handed discovery of Maya culture;exploration of ruined cities, monuments, temples; customs of Indians. 115drawings. 892pp. 5%x 8~. 22404-X,22405-8Pa., Two-vol. set $15.90

LOS CAPRICHOS, Francisco Goya. 80 plates of wild, grotesque monsters andcaricatures. Prado manuscript included. 183pp. 6%x 9%. 22384-1Pa. $4.95

AUTOBIOGRAPHY: The Story of My Experiments with Truth, Mohandas K.Gandhi. Not hagiography, but Gandhi in his own words. Boyhood, legal stu~ies,purification, the growth of the Satyagraha (nonviolent protest) movement. Cnucal,inspiring work of the man who freed India. 480pp. 5%x 8~. (AvaIlable In U.S. only)

24593-4Pa. $6.95


ILLUSTRATED DICTIONARY OF HISTORIC ARCHITECTURE, edited byCyril M. Harris. Extraordinary compendium of clear, concise definitions for over5,000 important architectural terms complemented by over 2,000 Ime drawmgs.Covers full spectrum of architecture from ancient ruins to 20th-century Modermsm.Preface. 592pp. 7Y, x 9%. 24444-X Pa. $14.95

THE NIGHT BEFORE CHRISTMAS, Clement Moore. Full text, and woodcutsfrom original 1848 book. Also critical, historical material. 19 illustrations. 40pp.4%x 6. 22797-9 Pa. $2.50

THE LESSON OF JAPANESE ARCHITECTURE: 165 Photographs, JiroHarada. Memorable gallery of 165 photographs taken in the 1930's of exquisiteJapanese homes of the well-to-do and historic buildings. 13 line diagrams. 192pp.8%x lilt 24778-3 Pa. $8.95

THE AUTOBIOGRAPHY OF CHARLES DARWIN AND SELECTED LET-TERS, edited by Francis Darwin. The fascinating life of eccentric genius composedof an intimate memoir by Darwin (intended for his children); commentary by hisson, Francis; hundreds of fragments from notebooks, journals, papers; and letters toand from Lyell, Hooker, Huxley, Wallace and Henslow. xi + 365pp. 5%x 8.

20479-0 Pa. $5.95

WONDERS OF THE SKY: Observing Rainbows, Comets, Eclipses, the Stars andOther Phenomena, Fred Schaaf. Charming, easy-to-read poetic guide to all mannerof celestial events visible to the naked eye. Mock suns, glories, Belt of Venus, more.Illustrated. 299pp. 51ix 81i. 24402-4 Pa. $7.95

BURNHAM'S CELESTIAL HANDBOOK, Robert Burnham, Jr. Thorough guideto the stars beyond our solar system. Exhaustive treatment. Alphabetical byconstellation: Andromeda to Cetus in Vol. I; Chamaeleon to Orion in Vol. 2; andPavo to Vulpecula in Vol. 3. Hundreds of illustrations. Index in Vol. 3. 2,OOOpp.6li x 9\4. 23567-X, 23568-8, 23673-0 Pa., Three-vol. set $37.85

STAR NAMES: Their Lore and Meaning, Richard Hinckley Allen. Fascinatinghistory of names various cultures have given to constellations and literary andfolkloristic uses that have been made of stars. Indexes to subjects. Arabic and Greeknames. Biblical references. Bibliography. 563pp. 5%x 8Y,. 21079-0 Pa. $7.95

THIRTY YEARS THAT SHOOK PHYSICS: The Story of Quantum Theory,George Gamow. Lucid, accessible introduction to influential theory of energy andmatter. Careful explanations of Dirac's anti-particles, Bohr's model of the atom,much more. 12 plates. Numerous drawings. 240pp. 5%x 8Y,. 24895-X Pa. $4.95

CHINESE DOMESTIC FURNITURE IN PHOTOGRAPHS AND MEASUREDDRAWINGS, Gustav Ecke. A rare volume, now affordably priced for antiquecollectors, furniture buffs and art historians. Detailed review of styles ranging fromearly Shang to late Ming. Unabridged republication. 161 black-and-white draw-ings, photos. Total of 224pp. 8%x II Ii. (Available in U.S. only) 25171-3 Pa. $12.95

VINCENT VAN GOGH: A Biography, Julius Meier-Graefe. Dynamic, penetrat-ing study of artist's life, relationship with brother, Thea, painting techniques,travels, more. Readable, engrossing. 160pp. 5%x 8Y,. (Available in U.S. only)

25253-1 Pa. $3.95


HOW TO WRITE, Gertrude Stein. Gertrude Stein claimed anyone could~nderst~nd her unconventional writing-here are clues to help. FascinatingImprovISatiOns, language expenments, explanations illuminate Stein's craft andthe art of writing. Total of 414pp. 4%x 6%. 23144.5 Pa. $5.95

ADVENTURES AT SEA IN THE GREAT AGE OF SAIL: Five FirsthandNarratives, edited by Elliot Snow. Rare true accounts of exploration, whaling,shipwreck, fiercenatives, trade, shipboard life,more. 33illustrations. Introduction.353pp. 5%x 8lj!. 25177.2Pa. $7.95

THE HERBAL OR GENERAL HISTORY OF PLANTS, John Gerard. Classicdescriptions of about 2,850 plants-with over 2,700 illustrations-includes Latinand English names, physical descriptions, varieties, time and place of growth,more. 2,706 illustrations. xlv + 1,678pp. 8lj!x 12'4. 23147.XCloth. $75.00

DOROTHY ANDTHE WIZARDIN OZ,L. Frank Baum. Dorothy and the Wizardvisit the center of the Earth, where people are vegetables, glass houses grow and Ozcharacters reappear. Classic sequel to Wizard of Oz. 256pp. 5%x 8.

24714.7Pa. $4.95

SONGS OF EXPERIENCE: Facsimile Reproduction with 26Plates in Full Color,William Blake. This facsimile of Blake's original "Illuminated Book" reproduces26 full.color plates from a rare 1826edition. Includes "The Tyger," "London,""Holy Thursday," and other immortal poems. 26 color plates. Printed text ofpoems. 48pp. 5'4x 7. 24636.1Pa. $3.50

SONGS OF INNOCENCE, William Blake. The first and most popular of Blake'sfamous "Illuminated Books," in a facsimile edition reproducing all 31 brightlycolored plates. Additional printed text of each poem. 64pp. 5'4x 7.

22764.2Pa. $3.50

PRECIOUS STONES, Max Bauer. Classic, thorough study of diamonds, rubies,emeralds, garnets, etc.: physical character, occurrence, properties, use, similartopics. 20 plates, 8 in color. 94 figures. 659pp. 6~x 9'4.

21910.0,21911.9 Pa., Two.vol. set $15.90

ENCYCLOPEDIA OF VICTORIAN NEEDLEWORK, S. F. A. Caulfeild andBlanche Saward. Full, precise descriptions of stitches, techniques for dozens ofneedlecrafts-most exhaustive reference of its kind. Over 800 figures. Total of679pp. 8~x 11.Two volumes. Vol. I 22800.2Pa. $11.95

Vol. 2 22801.0Pa. $11.95

THE MARVELOUS LAND OF OZ, L. Frank Baum. Second Oz book, theScarecrow and Tin Woodman are back with hero named Tip, Oz magic. 136illustrations. 287pp. 5%x 8lj!. 20692.0Pa. $5.95

WILD FOWL DECOYS, Joel Barber. Basic book on the subject, by foremostauthority and collector. Reveals history of decoy making and rigging, place inAmerican culture, different kinds of decoys, how to make them, and how to usethem. 140plates. 156pp. 7Y. x 10'4. 20011.6Pa. $8.95

HISTORY OF LACE, Mrs. Bury Palliser. Definitive, profusely illustrated chron.icle of lace from earliest times to late 19thcentury. Laces of Italy, Greece, England,France, Belgium, etc. Landmark of needlework scholarship. 266 illustrations.672pp. 6~x 9'4. 24742.2Pa. $14.95


ILLUSTRATED GUIDE TO SHAKER FURNITURE, Robert Meader. Allfurniture and appurtenances, with much on unknown local styles. 235 photos.146pp. 9 x 12. 22819-3Pa. $7.95

WHALE SHIPS AND WHALING: A Pictorial Survey, George Francis Dow. Over200 vintage engravings, drawings, photographs of barks, brigs, cutters, othervessels.Also harpoons, lances, whaling guns, many other artifacts. Comprehensivetext by foremost authority. 207black-and-white illustrations. 288pp. 6 x 9.

24808-9Pa. $8.95

THE BERTRAMS, Anthony Trollope. Powerful portrayal of blind self-will andthwarted ambition includes one of Trollope's most heartrending love stories.497pp. 5%x 8!-l. 25119-5Pa. $8.95

ADVENTURES WITH A HAND LENS, Richard Headstrom. Clearly writtenguide to observing and studying flowers and grasses, fish scales, moth and insectwings, egg cases, buds, feathers, seeds, leaf scars, moss, molds, ferns, commoncrystals, etc.-all with an ordinary, inexpensive magnifying glass. 209 exact linedrawings aid in your discoveries. 220pp. 5%x 8!-l. 23330-8Pa. $4.50

RODIN ON ART AND ARTISTS, Auguste Rodin. Great sculptor's candid, wide-ranging comments on meaning of art; great artists; relation of sculpture to poetry,painting, music; philosophy of life, more. 76superb black-and-white illustrationsof Rodin's sculpture, drawings and prints. 119pp. 8%x 11\4. 24487-3Pa. $6.95

FIFTY CLASSIC FRENCH FILMS, 1912-1982: A Pictorial Record, AnthonySlide. Memorable stills from Grand Illusion, Beauty and the Beast, Hiroshima,Mon Amour, many more. Credits, plot synopses, reviews, etc. 160pp. 8\4x 11.

25256-6Pa. $11.95

THE PRINCIPLES OF PSYCHOLOGY, William James. Famous long coursecomplete, unabridged. Stream of thought, time perception, memory, experimentalmethods; great work decades ahead of its time. 94 figures. 1,39Ipp. 5%x 8!-l.

20381-6,20382-4Pa., Two-vol. set $19.90

BODIES IN A BOOKSHOP, R. T. Campbell. Challenging mystery of blackmailand murder with ingenious plot and superbly drawn characters. In the besttradition of British suspense fiction. 192pp. 5%x 8!-l. 24720-1Pa. $3.95

CALLAS: PORTRAIT OF A PRIMA DONNA, George Jellinek. Renownedcommentator on the musical scene chronicles incredible career and life of the mostcontroversial, fascinating, influential operatic personality of our time. 64 black-and-white photographs. 416pp. 5%x 8\4. 25047-4Pa. $7.95

GEOMETRY, RELATIVITY AND THE FOURTH DIMENSION, RudolphRucker. Exposition of fourth dimension, concepts of relativity as Flatlandcharacters continue adventures. Popular, easily followed yet accurate, profound.141illustrations. 133pp. 5%x 8!-l. 23400-2Pa. $3.50

HOUSEHOLD STORIES BY THE BROTHERS GRIMM, with pictures byWalter Crane. 53 classic stories-Rumpelstiltskin, Rapunzel, Hansel and Gretel,the Fisherman and his Wife, Snow White, Tom Thumb, Sleeping Beauty,Cinderella, and so much more-lavishly illustrated with original 19th centurydrawings. 114illustrations. x + 269pp. 5%x 8!-l. 21080-4Pa. $4.50


~UNDIALS, AI~ert Waugh. Far and away the best, most thorough coverage ofIdeas, mathematIcs concerned, types, construction, adjusting anywhere. Over 100illustrations. 230pp. 5%x 8~. 22947-5Pa. $4.50

PICTURE HISTORY OF THE NORMANDIE: With 190Illustrations, Frank O.Braynard. Full story of legendary French ocean liner: Art Deco interiors, designinnovations, furnishings, celebrities, maiden voyage, tragic fire, much more.Extensive text. 144pp. 8%x lilt 25257-4Pa. $9.95

THE FIRST AMERICAN COOKBOOK: A Facsimile of "American Cookery,"1796, Amelia Simmons. Facsimile of the first American-written cookbook pub-lished in the United States contains authentic recipes for colonial favorites-pumpkin pudding, winter squash pudding, spruce beer, Indian slapjacks, andmore. Introductory Essay and Glossary of colonial cooking terms. 80pp. 5%x 8~.

24710-4Pa. $3.50

101PUZZLES IN THOUGHT AND LOGIC, C. R. Wylie, Jr. Solve murders androbberies, find out which fishermen are liars, how a blind man could possiblyidentify a color-purely by your own reasoning! 107pp. 5%x 8~.20367-0 Pa. $2.50

THE BOOK OF WORLD.FAMOUS MUSIC-CLASSICAL, POPULAR ANDFOLK, James J. Fuld. Revised and enlarged republication of landmark work inmusico-bibliography. Full information about nearly 1,000songs and compositionsincluding first lines of music and lyrics. New supplement. Index. 800pp. 5%x 8\(

24857-7Pa. $14.95

ANTHROPOLOGY AND MODERN LIFE, Franz Boas. Great anthropologist'sclassic treatise on race and culture. Introduction by Ruth Bunzel. Only inexpensivepaperback edition. 255pp. 5%x 8~. 25245-0Pa. $5.95

THE TALE OF PETER RABBIT, Beatrix Potter. The inimitable Peter's terrifyingadventure in Mr. McGregor's garden, with all 27 wonderful, full-color Potterillustrations. 55pp. 4\4x 5~. (Available in U.S. only) 22827-4Pa. $1.75

THREE PROPHETIC SCIENCE FICTION NOVELS, H. G. Wells. When theSleeper Wakes, A Story of the Days to Come and The Time Machine (full version).335pp. 5%x 8~. (Available in U.S. only) 20605-XPa. $5.95

APICIUS COOKERY ANDDINING IN IMPERIAL ROME, edited and translatedby Joseph Dommers Vehling. Oldest known cookbook in existence offers readers aclear picture of what foods Romans ate, how they prepared them, etc. 49illustrations. 301pp. 6li x 9\4. 23563-7Pa. $6.50

SHAKESPEARE LEXICON AND QUOTATION DICTIONARY, AlexanderSchmidt. Full definitions, locations, shades of meaning of everyword in plays andpoems. More than 50,000exact quotations. 1,485pp. 6~ x 9%.

22726-X,22727-8Pa., Two-vol. set $27.90

THE WORLD'S GREAT SPEECHES, edited by Lewis Copeland and LawrenceW. Lamm. Vast collection of 278 speeches from Greeks to 1970. Powerful andeffective models; unique look at history. 842pp. 5%x 8~. 20468-5Pa. $11.95


THE BLUE FAIRY BOOK, Andrew Lang. The first, most famous collection, withmany familiar tales: Little Red Riding Hood, Aladdin and th.eWonderful Lamp,Puss in Boots, Sleeping Beauty, Hansel and Gretel, Rumpelsultskm; 37 mall. 138illustrations. 390pp. 5%x 8!1. 21437-0 Pa. $5.95

THE STORY OF THE CHAMPIONS OF THE ROUND TABLE, Howard Pyle.Sir Launcelot, Sir Tristram and Sir Percival in spirited adventures of love andtriumph retold in Pyle's inimitable style. 50 drawings, 31 full-page. xviii + 329pp.6!1x 9'4. 21883-X Pa. $6.95

AUDUBON AND HIS JOURNALS, Maria I\udubon. Unmatched two-volumeportrait of the great artist, naturalist and author contains his journals, an excellentbiography by his granddaughter, expert annotations by the noted ornithologist, Dr.Elliott Coues, and 37 superb illustrations. Total of 1,200pp. 5%x 8.

Vol. 125143-8 Pa. $8.95Vol. II 25144-6 Pa. $8.95

GREAT DINOSAUR HUNTERS AND THEIR DISCOVERIES, Edwin H.Colbert. Fascinating, lavishly illustrated chronicle of dinosaur research, 1820's to1960. Achievements of Cope, Marsh, Brown, Buckland, Mantell, Huxley, manyothers. 384pp. 5'4 x 8'4. 24701-5 Pa. $6.95

THE TASTEMAKERS, Russell Lynes. Informal, illustrated social history ofAmerican taste 1850's-1950's. First popularized categories Highbrow, Lowbrow,Middlebrow. 129 illustrations. New (1979) afterword. 384pp. 6 x 9.

23993-4 Pa. $6.95

DOUBLE CROSS PURPOSES, Ronald A. Knox. A treasure hunt in the ScottishHighlands, an old map, unidentified corpse, surprise discoveries keep readerguessing in this cleverly intricate tale of financial skullduggery. 2 black-and-whitemaps. 320pp. 5%x 8!1.(Available in U.S. only) 25032-6 Pa. $5.95

AUTHENTIC VICTORIAN DECORATION AND ORNAMENTATION INFULL COLOR: 46 Plates from "Studies in Design," Christopher Dresser. Superbfull-color lithographs reproduced from rare original portfolio of a major Victoriandesigner. 48pp. 9~.x 12'4. 25083-0 Pa. $7.95

PRIMITIVE ART, Franz Boas. Remains the best text ever prepared on subject,thoroughly discussing Indian, African, Asian, Australian, and, especially, North-ern American primitive art. Over 950 illustrations show ceramics, masks, totempoles, weapons, textiles, paintings, much more. 376pp. 5%x 8. 20025-6 Pa. $6.95

SIDELIGHTS ON RELATIVITY, Albert Einstein. Unabridged republication oftwo lectures delivered by the great physicist in 1920-21. Ether and Relativity andGeometry and Experience. Elega •••••ideas in non-mathematical form, accessible tointelligent layman. vi + 56pp. 5%x 8!1. 24511-X Pa. $2.95

THE WIT AND HUMOR OF OSCAR WILDE, edited by Alvin Redman. Morethan 1,000ripostes, paradoxes, wisecracks: Work is the curse of the drinking classes,I can resist everything except temptation, etc. 258pp. 5%x 8!1. 20602-5 Pa. $4.50

ADVENTURES WITH A MICROSCOPE, Richard Headstrom. 59 adventureswith clothing fibers, protozoa, ferns and lichens, roots and leaves, much more. 142illustrations. 232pp. 5%x 8!1. 23471-1 Pa. $3.95


PLANTS OF THE BIBLE, Harold N. Moldenke and Alma L. Moldenke. Standardreference to all 230 plants mentioned in Scriptures. Latin name, biblical reference,uses, modern identity, much more. Unsurpassed encyclopedic resource for scholars,botanists, nature lovers, students of Bible. Bibliography. Indexes. 123black-and-white illustrations. 384pp. 6 x 9. 25069-5Pa. $8.95

FAMOUS AMERICAN WOMEN: A Biographical Dictionary from ColonialTimes to the Present, Robert McHenry, ed. From Pocahontas to Rosa Parks, 1,035distinguished American women documented in separate biographical entries.Accurate, up-to-date data, ••umerous categories, spans 400 years. Indices. 493pp.6!1x 9\4. 24523-3Pa. $9.95

THE FABULOUS INTERIORS OF THE GREAT OCEAN LINERS IN HIS-TORIC PHOTOGRAPHS, William H. Miller, Jr. Some 200superb photographs. capture exquisite interiors of world's great "floating palaces"-1890's to 1980's:Titanic, Ile de France, Queen Elizabeth, United States, Europa, more. Approx. 200black-and-white photographs. Captions. Text. Introduction. 160pp. 8li!x 11K

24756-2Pa. $9.95

THE GREAT LUXURY LINERS, 1927-1954:A Photographic Record, WilliamH. Miller, Jr. Nostalgic tribute to heyday of ocean liners. 186 photos of lie deFrance, Normandie, Leviathan, Queen Elizabeth, United States, many others.Interior and exterior views. Introduction. Captions. 160pp. 9 x 12.

24056-8Pa. $9.95

A NATURAL HISTORY OF THE DUCKS, John Charles Phillips. Greatlandmark of ornithology offers complete detailed coverageof nearly 200species andsubspecies of ducks: gadwall, sheldrake, merganser, pintail, many more. 74 full-color plates, 102black-and.white. Bibliography. Total of 1,920pp. 811x 11\4.

25141-1,25142-XCloth. Two-vol. set $100.00

THE SEAWEED HANDBOOK: An Illustrated Guide to Seaweeds from NorthCarolina to Canada, Thomas F. Lee. Concise reference covers 78species. Scientificand common names, habitat, distribution, more. Finding keys for easy identifica-tion. 224pp. 511x 8!1. 25215-9Pa. $5.95

THE TEN BOOKSOF ARCHITECTURE: The 1755Leoni Edition, Leon BattistaAlberti. Rare classic helped introduce the glories of ancient architecture to theRenaissance. 68 black-and-white plates. 336pp. 811x 11\4. 25239-6Pa. $14.95

MISS MACKENZIE, Anthony Trollope. Minor masterpieces by Victorian masterunmasks many truths about life in 19th-century England. First inexpensive editionin years. 392pp. 511x 8!1. 25201-9Pa. $7.95

THE RIME OF THE ANCIENT MARINER, Gustave Dore, Samuel TaylorColeridge. Dramatic engravings considered by many to be his greatest work. Theterrifying space of the open sea, the storms and whirlpools of an unknown ocean,the ice of Antarctica, more-all rendered in a powerful, chilling manner. Full text.38 plates. 77pp. 9\4x 12. 22305-1Pa. $4.95

THE EXPEDITIONS OF ZEBULON MONTGOMERY PIKE, Zebulon Mont-gomery Pike. Fascinating first-hand accounts (1805-6) of exploration of Missis.sippi River, Indian wars, capture by Spanish dragoons, much more. 1,088pp.5%x 8!1. 25254-X,25255-8Pa. Two-vol. set $23.90


A CONCISE HISTOR Y OF PHOTOGRAPHY: Third Revised Edition, HelmutGernsheim. Best one-volume history-camera obscura, photochemistry, daguer-reotypes, evolution of cameras, film, more. Also artistic aspects-landscape,portraits, fine art, etc. 281 black-and-white photographs. 26 in color. 176pp.8lp 11\i. 25128-4 Pa. $12.95

THE DORE BIBLE ILLUSTRATIONS, Gustave Dore. 241 detailed plates fromthe Bible: the Creation scenes, Adam and Eve, Flood, Babylon, battle sequences, lifeof Jesus, etc. Each plate is accompanied by the verses from the King James version ofthe Bible. 241pp. 9 x 12. 23004-X Pa. $8.95

HUGGER-MUGGER IN THE LOUVRE, Elliot Paul. Second Homer Evansmystery-comedy. Theft at the Louvre involves sleuth in hilarious, madcap caper."A knockout. "-Books. 336pp. 5%x 8~. 25185-3 Pa. $5.95

FLATLAND, E. A. Abbott. Intriguing and enormously popular science-fictionclassic explores the complexities of trying to survive as a two-dimensional being ina three-dimensional world. Amusingly illustrated by the author. 16 illustrations.103pp. 5%x 8~. 20001-9 Pa. $2.25

THE HISTORY OF THE LEWIS AND CLARK EXPEDITION"MeriwetherLewis and William Clark, edited by Elliott Coues. Classic edition of Lewis andClark's day-by-day journals that later became the basis for U.S. claims to Oregonand the West. Accurate and invaluable geographical, botanical, biological,meteorological and anthropological material. Total of 1,508pp. 5%x 8~.

21268-8,21269-6, 21270-X Pa. Three-vol. set $25.50

LANGUAGE, TRUTH AND LOGIC, Alfred J. Ayer. Famous, clear introductionto Vienna, Cambridge schools of Logical Positivism. Role of philosophy,elimination of metaphysics, nature of analysis, etc. 160pp. 5%x 8~. (Available inU.S. and Canada only) 20010-8 Pa. $2.95

MATHEMATICS FOR THE NON MATHEMATICIAN, Morris Kline. Detailed,college-level treatment of mathematics in cultural and historical context, w.ithnumerous exercises. For liberal arts students. Preface. Recommended ReadmgLists. Tables. Index. Numerous black-and-white figures. xvi + 641pp. 5%x 8~.

24823-2 Pa. $11.95

28 SCIENCE FICTION STORIES, H. G. Wells. Novels, Slar Begotten and MenLike Gods, plus 26 short stories: "Empire of the Ants," "A Story ofthe Stone Age,""The Stolen Bacillus," "In the Abyss," etc. 915pp. 5%x 8~. (Available in U.S. only)

20265-8 Cloth. $10.95

HANDBOOK OF PICTORIAL SYMBOLS, Rudolph Modley. 3,250 signs andsymbols, many systems in full; official or heavy commercial use. Arranged bysubject. Most in Pictorial Archive series. 143pp. 8~ x II. 23357-X Pa. $05.95

INCIDENTS OF TRAVEL IN YUCATAN, John L. Stephens. Classic (1843)exploration of jungles of Yucatan, looking for evidences of Maya civilization.Travel adventures, Mexican and Indian culture, etc. Total of 669pp. 5%x 8~.

20926-1, 20927-X Pa., Two-vol. set $9.90


DEGAS: An Intimate Portrait, Ambroise Vollard. Charming, anecdotal memoir byfamous art dealer of one of the greatest 19th-century French painters. 14black-and-white illustrations. Introduction by Harold L. Van Doren. 96pp. 5%x 8l-!.

25131-4 Pa. $3.95

PERSONAL NARRATIVE OF A PILGRIMAGE TO ALMANDINAH ANDMECCAH, Richard Burton. Great travel classic by remarkably colorful personality.Burton, disguised as a Moroccan, visited sacred shrines of Islam, narrowly escapingdeath. 47 illustrations. 959pp. 5%x 8l-!. 21217-3,21218-1 Pa., Two-vol. set $17.90

PHRASE AND WORD ORIGINS, A. H. Holt. Entertaining, reliable, modernstudy of more than 1,200 colorful words, phrases, origins and histories. Muchunexpected information. 254pp. 5%x 8l-!. 20758-7 Pa. $5.95

THE RED THUMB MARK, R. Austin Freeman. In this first Dr. Thorndyke case,the great scientific detective draws fascinating conclusions from the nature of asingle fingerprint. Exciting story, authentic science. 320pp. 5%x 8l-!.(Available inU.S. only) 25210-8 Pa. $5.95

AN EGYPTIAN HIEROGLYPHIC DICTIONARY, E. A. Wallis Budge. Monu-mental work containing about 25,000 words or terms that occur in texts rangingfrom 3000 B.C. to 600 A.D. Each entry consists of a transliteration of the word, the wordin hieroglyphs, and the meaning in English. 1,314pp. 6%x 10.

23615-3,23616-1 Pa., Two-vol. set $27.90

THE COMPLEAT STRATEGYST: Being a Primer on the Theory of Games ofStrategy, J. D. Williams. Highly entertaining classic describes, with manyillustrated examples, how to select best strategies in conflict situations. Prefaces.Appendices. xvi + 268pp. 5%x 8l-!. 25101-2 Pa. $5.95

THE ROAD TO OZ, L. Frank Baum. Dorothy meets the Shaggy Man, littleButton-Bright and the Rainbow's beautiful daughter in this delightful trip to themagical Land of Oz. 272pp. 5%x 8. 25208-6 Pa. $4.95

POINT AND LINE TO PLANE, Wassily Kandinsky. Seminal exposition of role ofpoint, line, other elements in non-objective painting. Essential to understanding20th-century art. 127 illustrations. 192pp. 6l-!x 9\4. 23808-3 Pa. $4.50

LADY ANNA, Anthony Trollope. Moving chronicle of Countess Lovel's bitterstruggle to win for herself and daughter Anna their rightful rank and fortune-perhaps at cost of sanity itself. 384pp. 5%x 8l-!. 24669-8 Pa. $6.95

EGYPTIAN MAGIC, 1<..A. Wallis Budge. Sums up all that is known about magicin Ancient Egypt: the role of magic in controlling the gods, powerful amulets thatwarded off evil spirits, scarabs of immortality, use of wax images, formulas andspells, the secret name, much more. 253pp. 5%x 8l-!. 22681-6 Pa. $4.50

THE DANCE OF SIVA, Ananda Coomaraswamy. Preeminent authority unfoldsthe vast metaphysic of India: the revelation of her art, conception of the universe,social organization, etc. 27 reproductions of art masterpieces. 192pp. 5%x 8l-!.

24817-8 Pa. $5.95


CHRISTMAS CUSTOMS AND TRADITIONS, Clement A. Miles. Origin,evolution, significance of religious, secular practices. Caroling, gifts, yule logs,much more. Full, scholarly yet fascinating; non-sectarian. 400pp .. ~% x 8l\.

23354-5 Pa. $6.50

THE HUMAN FIGURE IN MOTION, Eadweard Muybridge. More than 4,500stopped-action photos, in action series, showing undraped men, women, childrenjumping, lying down, throwing, sitting, wrestling, carrying, etc. 390pp. 7'1.x 10%.

20204-6 Cloth. $19.95

THE MAN WHO WAS THURSDAY, Gilbert Keith Chesterton. Witty, fast-pacednovel about a club of anarchists in turn-of-the-century London. Brilliant social,religious, philosophical speculations. 128pp. 5Jiix 8l\. 25121-7 Pa. $3.95

A CEZANNE SKETCHBOOK: Figures, Portraits, Landscapes and Still Lifes, PaulCezanne. Great artist experiments with tonal effects, light, mass, other qualities inover 100 drawings. A revealing view of developing master painter, precursor ofCubism. 102 black-and-white illustrations. 144pp. 8%x 6Jii. 24790-2 Pa. $5.95

AN ENCYCLOPEDIA OF BATTLES: Accounts of Over 1,560 Battles from1479 D.C. to the Present, David Eggenberger. Presents essential details of every majorbattle in recorded history, from the first battle of Megiddo in 1479 ilL to Grenada in1984. List of Battle Maps. New Appendix covering the years 1967-1984. Index. 99illustrations. 544pp. 6l\x 9\4. 24913-1 Pa. $14.95

AN ETYMOLOGICAL DICTIONARY OF MODERN ENGLISH, Ernest Week-ley. Richest, fullest work, by foremost British lexicographer. Detailed wordhistories. Inexhaustible. Total of 856pp. 6l\ x 9\4.

21873-2,21874-0 Pa., Two-vol. set $17.00

WEBSTER'S AMERICAN MILITARY BIOGRAPHIES, edited by RobertMcHenry. Over 1,000 figures who shaped 3 centuries of American military history.Detailed biographies of Nathan Hale, Douglas MacArthur, Mary Hallaren, others.Chronologies of engagements, more. Introduction. Addenda. 1,033 entries inalphabetical order. xi + 548pp. 6l\ x 9\4. (Available in U.S. only)

24758-9 Pa. $11.95

LIFE IN ANCIENT EG YPT, Adolf Erman. Detailed older account, with much notin more recent books: domestic life, religion, magic, medicine, commerce, andwhatever else needed for complete picture. Many illustrations. 597pp. 5Jiix 8l\.

22632-8 Pa. $8.95

HISTORIC COSTUME IN PICTURES, Braun & Schneider. Over 1,450costumedfigures shown, covering a wide variety of peoples: kings, emperors, nobles, priests,servants, soldiers, scholars, townsfolk, peasants, merchants, courtiers, cavaliers,and more. 256pp. 8Jiix 11\4. 231.~0-X Pa. $7.95

THE NOTEBOOKS OF LEONARDO DA VINCI, edited by J. P. Richter. Extractsfrom manuscripts reveal great genius; on painting, sculpture, anatomy, sciences,geography, etc. Both Italian and English. 186 ms. pages reproduced, plus 500additional drawings, including studies for Last Supper, Sforza monument, etc.860pp. 7Ysx 10%. (Available in U.S. only) 22572-0, 22573-9 Pa., Two-vol. set $25.90


THE ART NOUVEAU STYLE BOOK OF ALPHONSE MUCHA: All 72 Platesfrom "Documents Decoratifs" in Original Color, Alphonse Mucha. Rare copy-right-free design portfolio by high priest of Art Nouveau. Jewelry, wallpaper,stained glass, furniture, figure studies, plant and animal motifs, etc. Only completeone-volume edition. 80pp. 9%x 12\4. 24044-4 Pa. $8.95

ANIMALS: 1,419 COPYRIGHT-FREE ILLUSTRATIONS OF MAMMALS,BIRDS, FISH, INSECTS, ETC., edited by Jim Harter. Clear wood engravingspresent, in extremely lifelike poses, over 1,000 species of animals. One of the mostextensive pictorial sourcebooks of its kind. Captions. Index. 284pp. 9 x 12.

23766-4 Pa. $9.95

OBELISTS FL Y HIGH, C. Daly King. Masterpiece of American detective fiction,long out of print, involves murder on a 1935 transcontinental flight-"a verythrilling story"-NY Times. Unabridged and unaltered republication of theedition published by William Collins Sons & Co. Ltd., London, 1935. 288pp.5%x 8~. (Available in U.S. only) 25036-9 Pa. $4.95

VICTORIAN AND EDWARDIAN FASHION: A Photographic Survey, AlisonGernsheim. First fashion history completely illustrated by contemporary photo-graphs. Full text plus 235 photos, 1840-1914, in which many celebrities appear.240pp. 6l~x 9\4. 24205.6 Pa. $6.00

THE ART OF THE FRENCH ILLUSTRATED BOOK, 1700-1914, Gordon N.Ray. Over 630 superb book illustrations by Fragonard, Delacroix, Daumier, Dore,Grandville, Manet, Mucha, Steinlen, Toulouse-Lautrec and many others. Preface.Introduction. 633 halftones. Indices of artists, authors & tiLles, binders andprovenances. Appendices. Bibliography. 608pp. 8%x II \4. 25086-5 Pa. $24.95

THE WONDERFUL WIZARD OF OZ, L. Frank Baum. Facsimile in full color ofAmerica's finest children's classic. 143 illustrations by W. W. Denslow. 267pp.5%x 8~. 20691-2 Pa. $5.95

FRONTIERS OF MODERN PHYSICS: New Perspectives on Cosmology, Rela-tivity, Black Holes and Extraterrestrial Intelligence, Tony Rothman, et al. For theintelligent layman. Subjects include: cosmological models of the universe; blackholes; the neutrino; the search for extraterrestrial intelligence. Introduction. 46black-and-white illustrations. 192pp. 5%x 8~. 24587-X Pa. $6.95

THE FRIENDLY STARS, Martha Evans Martin & Donald Howard Menzel.Classic text marshalls the stars together in an engaging, non-technical survey,presenting them as sources of beauty in night sky. 23 illustrations. Foreword. 2 starcharts. Index. 147pp. 5%x 8~. 21099-5 Pa. $3.50

FADS AND FALLACIES IN THE NAME OF SCIENCE, Martin Gardner. Fair,willy appraisal of cranks, quacks, and quackeries of science and pseudoscience:hollow earth, Velikovsky, orgone energy, Dianetics, flying saucers, Bridey Murphy,food and medical fads, etc. Revised, expanded In the Name of Science. "A very ableand even-tempered presentation."- The New Yorker. 363pp. 5%x 8.

20394.8 Pa. $6.50

ANCIENT EG YPT: ITS CULTURE AND HISTOR Y, J. E Manchip White. Frompre-dynastics through Ptolemies: society, history, political structure, religion, dailylife, literature, cultural heritage. 48 plates. 217pp. 5%x 8~. 22548-8 Pa. $4.95


SIR HARRY HOTSPUR OF HUMBLETHWAITE, Anthony Trollope. Incisive,unconventional psychological study of a conflict between a wealthy baronet, hisidealistic daughter, and their scapegrace cousin. The 1870 novel in its firstinexpensive edition in years. 250pp. 5%x 8l>. 24953-0 Pa. $5.95

LASERS AND HOLOGRAPHY, Winston E. Kock. Sound introduction toburgeoning field, expanded (1981) for second edition. Wave patterns, coherence,lasers, diffraction, zone plates, properties of holograms, recent advances. 84illustrations. 160pp. 5* x 8\4.(Except in United Kingdom) 24041-X Pa. $3.50

INTRODUCTION TO ARTIFICIAL INTELLIGENCE: SECOND, EN-LARGED EDITION, Philip C. Jackson, Jr. Comprehensive survey of artificialintelligence-the study of how machines (computers) can be made to act intelli-gently. Includes introductory and advanced material. Extensive notes updating themain text. 132black-and-white illustrations. 512pp. 5%x 8l>. 24864-X Pa. $8.95

HISTORY OF INDIAN AND INDONESIAN ART, Ananda K. Coomaraswamy.Over 400 illustrations illuminate classic study of Indian art from earliest Hdrappafinds to early 20th century. Provides philosophical, religious and social insights.304pp. 6%x 9%. 25005-9 Pa. $8.95

THE GOLEM, Gustav Meyrink. Most famous supernatural novel in modernEuropean literature, set in Ghetto of Old Prague around 1890.Compelling story ofmystical experiences, strange transformations, profound terror. 13black-and-whiteillustrations. 224pp. 5%x 8l>.(Available in U.S. only) 25025-3 Pa. $5.95

ARMADALE, Wilkie Collins. Third great mystery novel by the author of TheWoman in White and The Moonstone. Original magazine version with 40illustrations. 597pp. 5%x 8l>. 23429-0 Pa. $9.95

PICTORIAL ENCYCLOPEDIA OF HISTORIC ARCHITECTURAL PLANS,DETAILS AND ELEMENTS: With 1,880 Line Drawings of Arches, Domes,Doorways, Facades, Gables, Windows, etc., John Theodore Haneman. Sourcebookof inspiration for architects, designers, others. Bibliography. Captions. 141pp.9 x 12. 24605-1 Pa. $6.95

BENCHLEY LOST AND FOUND, Robert Benchley. Finest humor from early30's, about pet peeves, child psychologists, post office and others. Mostlyunavailable elsewhere. 73 illustrations by Peter Arno and others. 183pp. 5%x 8l>.

22410-4 Pa. $3.95

ERTE GRAPHICS, Erte. Collection of striking color graphics: Seasons, Alphabet,Numerals, Aces and Precious Stones. 50 plates, including 4 on covers. 48pp.9%x 12\4. 23580-7 Pa. $6.95

THE JOURNAL OF HENRY D. THOREAU, edited by Bradford Torrey, F. H.Allen. Complete reprinting of 14 volumes, 1837-61, over two million words; thesourcebooks for Walden, etc. Definitive. All original sketches, plus 75photographs.1,804pp. 8l>x 12\4. 20312-3,20313-1 Cloth., Two-vol. set $80.00

CASTLES: THEIR CONSTRUCTION AND HISTORY, Sidney Toy. Tracescastle development from ancient roots. Nearly 200 photographs and drawingsillustrate moats, keeps, baileys, many other features. Caernarvon, Dover Castles,Hadrian's Wall, Tower of London, dozens more. 256pp. 5* x 8\4.

24898-4 Pa. $5.95


AMERICAN CLIPPER SHIPS: 1833-1858, Octavius T. Howe & Frederick C.Matthews. Fully-illustrated, encyclopedic review of 352 clipper ships from theperiod of America's greatest maritime supremacy. Introduction. 109 halftones. 5black-and-white line illustrations. Index. Total of 928pp. 5%x 8\>.

25115-2,25116-0 Pa., Two-vol. set $17.90

TOW ARDS A NEW ARCHITECTURE, Le Corbusier. Pioneering manifesto bygreat architect, near legendary founder of "International School." Technical andaesthetic theories, views on industry, economics, relation of form to function,"mass-production spirit," much more. Profusely illustrated. Unabridged transla-tion of 13th French edition. Introduction by Frederick Etchells. 320pp. 6ii x 9\4.(Available in U.S. only) 25023-7 Pa. $8.95

THE BOOK OF KELLS, edited by Blanche Cirker. Inexpensive collection of 32full-color, full-page plates from the greatest illuminated manuscript of the MiddleAges, painstakingly reproduced from rare facsimile edition. Publisher's Note.Captions. 32pp. 9%x 12\4. 24345-1 Pa. $4.95

BEST SCIENCE FICTION STORIES OF H. G. WELLS, H. G. Wells. Full novelThe Invisible Man, plus 17 short stories: "The Crystal Egg," "Aepyornis Island,""The Strange Orchid," etc. 303pp. 5%x 8\>.(Available in U.S. only)

21531-8 Pa. $4.95

AMERICAN SAILING SHIPS: Their Plans and History, Charles G. Davis.Photos, construction details of schooners, frigates, clippers, other sailcraft of 18thto early 20th centuries-plus entertaining discourse on design, rigging, nauticallore, much more. 137 black-and-white illustrations. 240pp. 6ii x 9\4.

246.~8-2 Pa. $5.95

ENTERTAINING MATHEMATICAL PUZZLES, Martin Gardner. Selection ofauthor's favorite conundrums involving arithmetic, money, speed, etc., with livelycommentary. Complete solutions. 112pp. 5%x 8\>. 25211-6 Pa. $2.95

THE WILL TO BELIEVE, HUMAN IMMORTALITY, William James. Twobooks bound together. Effect of irrational on logical, and arguments for humanimmortality. 402pp. 5%x 8\>. 20291-7 Pa. $7.50

THE HAUNTED MONASTERY and THE CHINESE MAZE MURDERS,Robert Van Gulik. 2 full novels by Van Gulik continue adventures of Judge Dee andhis companions. An evil Taoist monastery, seemingly supernatural events;overgrown topiary maze that hides strange crimes. Set in 7th-centurv China. 27illustrations. 328pp. 5%x 8\>. 23502-5 Pa. $5.95

CELEBRATED CASES OF JUDGE DEE (DEE GOONG AN), translated byRobert Van Gulik. Authentic 18th-century Chinese detective novel; Dee andassociates solve three interlocked cases. Led to Van Gulik's own stories with samecharacters. Extensive introduction. 9 illustrations. 237pp. 5%x 8\>.

23337-5 Pa. $4.95

Prices subject to change without notice.Available at your book dealer or write for free catalog to Dept. GI, DoverPublications, Inc., 31 East 2nd St., Mineola, N.Y. 11501. Dover publishes more than175 books each year on science, elementary and advanced mathematics, biology,music, art, literary history, social sciences and other areas.