An Accurate Absolute Gain Calibration of an Antenna for Radio Astronomy E. 1‘. W L , MEMBER, IEEE, AN- E. P. DELOLI

~ummary-The methods used to determine the absolute gain of an 80-square foot aperture horn-reflector antenna over a frequency range from 2.8 to 14.0 Gc, and in the absence of a high-level test range, are described. A “mirror” range, in which the size of the smooth reflecting surface required was relatively small, was used and the measurements were performed in the Fresnel zone of the antenna with corrections applied for the resulting gain reduction. These and other provisions reduced the probable error in the final values to about one per cent.

I. INTRODUCTION

MAJOR source of error in the measurement of the absolute flux of signals from radio stars is the uncertainty in the value of the antenna gain.

At microwave frequencies this gain has generally been calculated, as its accurate measurement normallq. re- quires uniform planewave illumination of the aperture area, an arrangement difficult to provide for the large aperture antennas used in radio astronomy. Horns have consequently been used as their gain may be predicted with reasonable accuracy; a common estimate is a probable error of two per cent.‘ However, these horns tend to be inconveniently long and the horn-reflector antenna,? with its rigid structure, low back-lobe levels and large aperture area for a short horn length is a de- sirable alternative. In this paper some unusual methods used to determine accurately the absolute gain of an 80- square foot aperture horn-reflector antenna over a wide range of microwave frequencies are described.

To a first approximation the aperture field of the horn reflector is a geometrical transformation of the accu- rately known field existing in the pyramidal horn, and hence the near-field gain reduction may be calculated with some precision. b‘hile there is appreciable litera- ture on the Fresnel-zone axial field intensity of certain simple aperture configurations, there has as yet been little indication of its application to the growing num- bers of large aperture antennas used for radio astronomy and space communications. More than the usual care and effort are generally required in determining the far- field characteristics of an antenna from near-field data, but for very large antennas there appears at present to be no accurate practical alternative to this approach,

1963. Manuscript received September 9, 1963; revised December 4,

Division, National Research Council, Ottawa, Canada. The authors are with the Radio and Electrical Engineering

3200 Mc/s,” Astrophys. J., vol. 132, pp. 279-285; September, 1960. N. LIT. Broten and LV. J. Nedd, “Absolute flux measurements a t

antenna for space communication,” Bell Sys. Tech. J., vol. 40, * A. B. Crawford, D. C. Hogg, and L. E. Hunt, “A horn-reflector

pp. 1095-1116; July, 1961.

and in view of the capital expenditures involved i t is rather surprising that i t seems to have been virtually neglected. Perhaps this is because the accuracy avail- able is limited if the range chosen is very short, while for a Fresnel-zone range of reasonable length the major problem, that of providing uniform illumination of the aperture area, often remains. This difficulty was over- come with the inexpensive simple arrangements dis- cussed in Section IV.

11. FRESNEL-ZONE GAIN REDUCTION FOR THE

HORN-REFLECTOR ANTENNA

Consider an aperture S in which a field of uniform phase exists and radiates into the region z > 0. According to scalar diffraction the or^,^ the field a t P ( x , y, 2;) in its Fresnel zone is

where

X = wavelength k=2n/X is the free-space propagation constant

F ( t , 7) =the aperture field distribution in terms of the

R= the distance from the origin of the coordi-

0 = the angular displacement of P from the z

aperture coordinates E , ?

nates to P

axis.

The arrangement of the coordinate system for integra- tion over the projected aperture of the horn-reflector antenna is shown in Fig. 1. The 2 axis coincides with the axis of the paraboloidal surface which is parallel to, and a distance y o from, the axis of the beam. On the beam axis x = 0, y =yo, z = R, and cos0 = 1 for R>>yo. Changing to polar coordinates, with 4 = r sin CY, 7 = r cos CY, the on-axis field is

[ 2R k

- j - (r’ - 2yor cos a) rdrda. 1 (2)

Rad. Lab. Ser., hlcGraw-Hill Book Company, Inc., New York, 3 S. Silver, “Microwave .-lntenna Theory and Design,” M.I.T.

N. Y . , vol. 12, p. 172; 1949.

439

440 IEEE TR.AhTSACTIONS O N ANTENNAS AND PROPAGATION J d Y

D APERTURE

+c Fig. 1-Geometry of the horn-reflector antenna.

ponent of aperture field is2

E(Q, 4 ) ~ = Eo. (1 - sin 4) cos - cos Q, ( 5 ) aa!

2 Q O

where EO is the field in the aperture at 01 = 4 = 0 .3 Eq. (5) takes account of both the space attenuation of the expanding spherical wave in the horn and the variation of the field across the horn. Using (5) in (4), the longi- tudinal component of the on-axis field in the Fresnel zone of the aperture is

The equation of the paraboloidal surface in Fig. 1 is with yo,^ the separation of the beam axis and the axis of the paraboloidal surface for longitudinal polarization

1 - sin 4 ( 3 ) The antenna gain may be obtained from the familiar 2f of the aperture field.

P = I

definition of directive gain,

where f is the local length of the paraboloidal surface and 4 is the angle measured off the horn axis. Clearly Y = p cos q5 and hence in terms of the horn coordinates a, 4, (2) is

where E(+, a) is the aperture field distribution and 2ao and 2q50 are the total flare angles of the horn.

The polarization of the on-axis field is that of the field in the projected aperture with longitudinal and transverse components parallel to the 7 and 4 axes of Fig. 1, respectively. Excitation of the aperture field is by the dominant waveguide mode in the horn antenna. A predominantly longitudinally or transversely polarized aperture field may be produced depending on the orien- tation of the exciting mode. The cross-polarized com- ponent in either case gives rise to additional sidelobes and reduces the gain; however, the magnitude of both effects can be readily calculated. The longitudinal com-

h R 2 ] E(0) I 2 ZOP,

G = I (7)

where Zo= 120n ohms is the characteristic impedance of free space and P, is the total power radiated. For longitudinal polarization, integration of the aperture field gives2

In the Fraunhofer zone the on-axis field may be ob- tained without resorting to numerical integration. As R+.o in (6) the exponential term in the integrand be- comes unity and

-cos Qo.".[(&)2 Q O - l ] l .

Hence the far-field gain of the antenna for longitudinal polarization is

1 + sin 4 0 coss (YO

1 - sin 4 0 > - - aosin&

1964 Ju11 and Deloli: Calibration o j Antenna for Rad.io dstronomy 441

The near-field gain G, may be obtained similarly, but a solution in closed form to the integral of ( 6 ) for E(O), has not been found, although approximations valid for limited ranges of RX may be derived. Of more immedi- ate use in this work is the gain reduction in the near- field, ;.e. I

In order to evaluate E(O),, y0.2 is required. For sym- metrical aperture field distributions the beam axis passes through the phase center of the aperture and its position may be calculated by the method of moments, but for asymmetrical distributions this approach is not generally val idJ According to ( j ) , the field distribution is asymmetrical about the geometric center of the horn- reflector aperture along the q axis of Fig. 1 ; however, a lengthy calculation outlined elsewhere5 shows that the beam axis and phase center are essentially coincident and the displacement of the beam axis from the axis of the paraboloidal surface is

L1 - sin +oJ

The antenna which was calibrated6 had flare angles c ro=+O=2Oo and focal length f = 1.855 meters so that from (12) y0,2=3.9w1 meters. I t readily follows that the aperture area is 7.444 m2, and from (10) the gain is

with X in meters. The near-field gain reduction can be calculated from (11) with IE(O)fl =0.31314 from (9), and I E(O),I evaluated numerically on a computer. The results of this calculation are shown in Fig. 2 where the near-field gain reduction is shown for the two aperture field polarizations and compared with that for a square aperture of the same area having uniform,’ and uniform and cosinusoidal field distribution^.^.^

unpublished notes. 4 G. C. McCormick, Sational Research Council, Ottawa, Canada,

6 E. V. Jull and E. P. Deloli, “The Precise Calibration of a Horn-

neering Division, National Research Council, Ottawa, Canada, Rept. Reflector Antenna for Radio Astronomy,’ Radio and Electrical Engi-

No. ERE-637; June, 1963.

richtfunkanlagen,” Frequena, vol. 10, pp. 33-44; February, 1956. 6 H. Laub and !A7. Stohr, “Hornparabolantenne f i i r breitband-

C. Polk, “Optical Fresnel-zone gain of a rectangular aperture,” IRE TRANS. ON ANTESSAS AND PROPAGATION, vol. AP-4, pp. 65-69; January, 1956.

8 “The Microware Engineers’ Handbook and Buyers’ Guide,” Horizon House-Microwave, Inc., Brookline, Mass., 2nd ed., p. 132; 1963.

2 3 1 5 6 7

L R W ‘ m-z

Fig. 2-Fresnel-zone gain of apertures with uniform phase.

I1 I. CALIBRATION OF THE GAIN STANDARD ANTENSAS

The on-axis gain of the horn-reflector antenna was required in the following frequency ranges:

S band, 2.8- 3.5 Gc J band, 6.0- 7.0 Gc K , band, 12.5-14.5 Gc

and since appreciable effort was required to achieve sufficient accuracy in each measured value their number was limited. As it is difficult to perform accurate abso- lute gain measurements on large antennas, the gain com- parison m e t h ~ d , ~ in which the signal received by the antenna is compared with that from an already cali- brated antenna, was adopted. Three pairs of identical optimum horns of convenient size were constructed from a previous designlo and calibrated absolutely on an out- door range. Fig. 3 shows schematically the arrangement used, in which an essential element was an accurately calibrated combination of a variable waveguide attenu- ator and a directional coupler arranged so that the initial difference between the compared signals was small. IVith the same biasing current in both bolometer detectors, their outputs were first adjusted to provide an identical response when successively attached to a source of the intensity used in the gain measurements. The gain meas- urements were then made a t horn separations of about 4L2;/X, where L is the largest aperture dimension, and

a survey of current techniques,” PROC. IRE, vol. 47, pp. 705-735; \VI A . Cumming, “Radiation measurements a t radio frequencies:

May, 1959. lo W. T. Slayton, “Design and Calibration of Nicroaave Antenna

Gain Standards,” U. S. Kava1 Research Lab., IVashington, D. C., Report No. 4; 1954.

448 IEEE TRANSACTIOSS O N ANTEhTNAS AND PROPAGATION J d y

CILIBRATED DIRECTIONAL M4TCHING

/- 1 ABSORBER

Fig. 3-Arrangement of the apparatus for calibration of the horns.

subsequently corrected for near-field effects.l’ In order to eliminate errors due to reradiation they were re- peated for different separations at intevals of X/16 over two cycles of the reradiation pattern and the average value taken. With these and other provision^,^ includ- ing controlling and correcting for the temperature of the bolometer detectors, an estimated probable error of & 1/2 per cent in the measured gain of the horns was obtained. Repetition of the measurements with a third horn revealed no difference in gain between the horns of each pair except at K , band, and there chiefly because of the different lengths of waveguide feeds. The meas- ured values a t S band, through which the somewhat speculative dashed line in Fig. 4 is drawn, are con- sidered more accurate than Slayton’s,lo but they show, as do his, oscillations about the calculated curve for the gain. These were also observed with the J-band horns but were less evident in the case of the higher gain ICu-

band horns, and appear to be a characteristic of small horns arising possibly from currents around the aperture boundary.

I V . CALIBRATION OF THE HORN-REFLECTOR ANTEXXA 4 . Dimensions of the Range

Since a conventional high-level test range was not available for the calibration of the horn-reflector an- tenna a “mirror” range was used. Suppose an omni- directional source a t a height hl above a perfectly con- ducting horizontal surface of infinite extent radiates horizontally polarized waves. Then the variation of the resultant field with height hs a t a distance R is as sin x where

2rhl(Jzl + h z )

X[R2 + (121 + 122)2]1/2 X = (14)

Let the source be a transmitting antenna with a radia- tion pattern broad in the vertical plane. If a receiving

vol. 41, pp. 109-115; January, 1953. l1 E. H. Braun, “Gain of electromagnetic horns,” PROC. IRE,

x (CM)

Fig. $-Gain of the S-band horns.

antenna, at h2>>h1 and R>>hz, is to be illuminated by the first lobe of the pattern above the horizon, the height required of the transmitting antenna is

The arrangement used is illustrated in Fig. 5. The source was an E-plane sectoral horn placed a few inches above a reflecting surface near ground level. The horn- reflector and gain standard antennas, mounted on a tower, were 55 feet above the ground and, for the meas- urements in the two lower frequency ranges, 493.3 feet, or about L2/2h at the shortest wavelength, from the 2.2X aperture transmitting antenna. In this way a broad symmetrical beam composed of horizontally polarized direct and reflected waves illuminated the apertures of the receiving antennas.

An attractive feature of this arrangement was the relatively small area of smooth reflecting surface re- quired; for, in decreasing hl, the Fresnel zones of the reflecting area contract and move towards the trans- mitting antenna.12 If only the first Fresnel zone lies on a smooth surface an essentially well proportioned beam results. I t was necessary, however, to lengthen the reflec- tor appreciably beyond the extremities of the first Fres- nel ellipse predicted for an infinite surface, as diffraction from the edge of the reflector produces an upward tilt- ing of the pattern lobes, calling for a subsequent in- crease in hl to properly center the beam. The reflecting surface consisted of a 12 X40-foot wooden platform covered with 0.001-inch thick aluminum foil and in- clined slight& to the horizontal so as to place the inter- vening ground of the range in the shadow of the trans- mitter. The tolerances required of the reflecting surface were easily met. Rayleigh’s optical criterion, which for small grazing angles $ limits surface variations to +X/16$ for a maximum pathlength deviation of X/8,

D. E. Kerr, “Propagation of Short Radio Waves,” M.I.T. Rad. Lab. Ser., McGraw-Hill Book Company, Inc., Sew York, N. Y. , vol. 13, pp. 414-413; 1951.

1964 Jull and Deloli: Calibration of Antenna for Radio Astronomy 443

R E F L E C T I N G SURFACE

Fig. 5-Arrangement of the equipment with the transmitting antenna height set for gain measurements at 3.2 Gc.

calls for & 1 inch in the most 'severe case. X tolerance of less than half this value was obtained without difficulty.

Gain measurements were made at J band and K , band on a longer range. Its dimensions were

1) Antenna separation, 989.4 feet 2) Reflecting surface, 1 2 X 56 feet 3) Transmitting antenna aperture length 4.4X.

B. Tolerable Modulation of the Illzmi.nating Field

The allowable length of the transmitting antenna aperture is governed chiefly by the separation of the re- ceiving antennas; with the dimensions in Section IV-A and the horn-reflector antenna aligned for maximum reception, the separation of its beam axis from that of the gain standard horn by 8 feet causes an increase in measured gain of 1/2 per cent. I t is also of value to investigate the effect of a modulation in the illuminat- ing field on the apparent gain of the horn-reflector an- tenna. This lengthy but routine calculation is outlined elsewhere5 and the results are shown in Fig. 6. Clearly the field variation in the longitudinal plane tends to cancel the asymmetry in the aperture distribution and increase the apparent gain while the variation in the transverse plane reduces the gain, but the change in both cases is notably small.

C. Investigation of the Illuminating Field

At the rear of the tower platform a small test antenna was arranged on a carriage so that by rotating the plat- form through 180°, the field in the area formerly oc- cupied by the receiving antenna apertures could be in- vestigated and automatically recorded. This field-prob- ing apparatus was first used to ascertain the correct height for the transmitting antenna and then to record variations in the illuminating field. Fig. 7 shows some results of the field probing. The variations of about 2-

LOHOITUDINAL VARIATION

+01

ii s P 2

FIELD AT APERTURE EDGE 0 ; -.a Y a

TRANSVERSE VARIATION z - g -.a L 4 I

- . 0 3

I

Fig. &Effect of a cosinusoidal variation in the illuminating field on the measured gain of the horn-reflector antenna.

inch period in these records indicate reflections arriving at angles of approximately 90" from the direction of maximum reception and roughly 40 d b below the il- luminating signal. IfTith the directivity of the gain standard included, the total discrimination against in- terfering signals was 65 d b in this instance and in all cases i t was a t least 50 db.

D. Arran.genzent of the Apparatus The apparatus used for the gain measurements is

shown schematically in Fig. 8, with the arrangement similar to that used in calibrating the horns and the same procedure adopted in matching and equalizing the response of the bolometer detectors. Absorbing ma- terial placed between the apertures of the receiving an- tennas to reduce their interaction did not measurably affect the respons'e of the larger antenna. In order to eliminate the error introduced by the remaining un- wanted reflections, the gain measurements were re- peated for several lateral positions of the gain com- parison horn.

In columns 2 and 6 of Table I are presented the gain of the horns and the longitudinally polarized gain of the horn reflector, respectively, both given in decibels above isotropic. The corrections for the near-field gain reduction were obtained from (11) and the corrections for field variations from the field-probing results used in conjunction with Fig. 6. The latter corrections, while small, make the largest contribution to the estimated probable errors in the final gain values, which are kO.04 d b for the S- and J-band values and 50.05 d b for the K,-band values. The aperture efficiency is evi- dently 62 per cent a t S band, 55 per cent at J band and 41 per cent a t K , band, with the variation caused by the aperture cover as discussed in the following section.

444 IEEE TRANSACTIONS O N AKTEhTNAS AND PROPAGATION July

Fig. 7-Illumination of the horn reflector and gain comparison antennas atf=6.8 Gc with R=989.4 feet, h ~ = 102 inches, &=55 feet.

TABLE I GAIN OF HORN-REFLECTOR ANTENNA (DECIBELS)

C A L I B R A T E D F IXED A N D V A R I A B L E A T T E N U A T O R

Fig. 8-Arrangement of the apparatus for gain measurements on the horn-reflector antenna.

Range (m)

150.4

301.6

10.71 9.52 9.37

4.61 8.82

4.41

4.84 4.61 4.41 2.31 2.22 3.14

Horn Gain

17.66 18.28 18.45 18.90 22.34 22.39

22.02 22.34 22.38 23.69 23.91 24.07

19.65 0.08 19.73 0.10 19.74 0.10 19.78 0.11 21.11 0.40 21.41 0.44

21.42 0.09 21.39 0.10 21.75 0.11 24.55 0.40 24.56 0.44 24.55 0.47

Illumi- nation

Zorrectiox

-0.07 $0.01 -0.04 -0.07

-0.03

-0.06 -0.02 -0.03 +O .03 +0.03 +0.01

-0.05

An- tenna Gain

37.32 38.12 38.25 38.72 43.80 44.21

43.47 43.81 44.21 48.67 48.94 49.10

1964 Ju11 and Deloli: Calibration of Antenna for Radio Astronomy

I I I I I I

J CALCULATE0 GAIN WITH NO APERTURE COVER

BASED ON A SQUARE APERTURE--- INCLUDING COVER LOSSES

MEASURED GAlN -----O-----O----

X ( C M )

Fig. 9--Gain of the horn-reflector antenna for longitudinal polarization of the aperture field.

v. DISCUSSION AND CONCLUDlNG REMARKS

To determine antenna gain with a probable error of only one per cent requires careful elimination or reduc- tion of errors in all the supporting measurements. In spite of the difficulties involved, the estimated accuracy does not appear to be unduly optimistic, for when the results obtained for the same frequencies from the two quite different ranges are examined, the difference is less than l / 2 per cent. With this agreement, not only is the near-field correction verified, but also the over-all accuracy of the procedure used to calibrate the horn- reflector antenna is indicated.

Fig. 9 shows the calculated antenna gain curve of (13) and the measured gain values which lie appreci- ably below it. This discrepancy is attributed almost wholly to reflection losses introduced by a glass fiber aperture cover, the approximate dimensions of which are shown in Fig. 10(a). Unfortunately this dielectric cover, which is of varying thickness, was not amenable to precise analysis, and the model shown in Fig. 10(b) and described in *Appendix I1 was used to estimate the cover losses. With this estimation included the discrep- ancy remaining is less than 1/2 db, but since the model chosen will underrate the loss for the actual cover, much of this may be further ascribed to the dielectric and the remainder to spillover, current losses and small differ- ences in the geometry of the aperture from that used in the analysis. In modifying the aperture field distribu- tion, the cover slightly reduces the near-field gain cor- rections, but this effect is of secondary importance. There is no evidence in Fig. 9 of the oscillations in the gain characteristics which limit the accuracy in the pre- dicted gain of small horns. Without its aperture cover, the gain of this antenna appears to behave as X-3, a desirable feature for accuracy in flux measurements carried out over a wide frequency band. The aperture cover of this antenna is necessary to protect the interior surfaces from the effects of precipitation. As a further

445

1 1 L 1 l - I 1 LI

(b) Fig. 1 0 4 a ) Approximate thickness of the aperture cover. Region 1:

tl=0.165 cm, 2 : t2=0.266 cm, 3: t3=0.470 cm. (b) klodel for esti- mation of the cover losses. t1=0.165 cm, k =0.324 cm, L1= L, =2.728 m, L1-L1’=L?-L2’=60.92 cm.

precaution against corrosion the antenna was filled with dry air at a pressure sufficient t o keep the aperture cover in a rigid’position, and while the antenna gain was con- sequently pressure-dependent at the higher frequencies, the required tolerances on the pressure were not severe.

Antenna test ranges which include a reflecting ground surface have been used b e f ~ r e , ~ b u t with the arrange- ments described in Section IV the ground is scarcely used and the expense of preparing and maintaining a graded surface is avoided. Further economy could be introduced without greatly affecting the available accu- racy by shortening the rather conservative near-field ranges used here; i f , for instance, they were halved, the height required of the receiving antennas would be halved. While reradiation was not detected on these ranges a t L2/2X, on shorter ranges the simple provision for its elimination mentioned in Section I11 could be used. Effort might also be spared in obtaining the near- field correction, for Fig. 2 shows that the Fresnel-zone behavior of this horn-reflector aperture closely resem- bles that of a square aperture of the same area with a uniform a cosinusoidal field distribution, for which the Fresnel-zone gain may be expressed in terms of the Fres- nel integrals and readily evaluated. In dealing with a horn-reflector antenna having a rectangular pyramidal horn, it should be possible to use a rectangular aperture of equal area.

446 IEEE TRANSACTIONS 0-V ANTENNAS AhTD PROPAGATION July

APPENDIX I Using oro=&= 20”, we get I1=0.44967 and with

FRESNEL-ZONE GAIN REDUCTION OF THE HORN- f = 1.855 meters,

REFLECTOR ANTENNA FOR TRANSVERSE POLAR- 67.689 IZATION OF THE APERTURE FIELD Gf = ->

X2 (21)

The transverse component of the aperture field is or 0.21 db below the gain for longitudinal polarization.

E(a, = Eo*(1 - sin 9) ‘Os - ‘Os CY* (16) what is essentially the phase center of the aperture and5 r9 The beam axis passes through the aperture plane at

290

Substituting this in (4), the expression for the transverse component of the on-axis field in the Fresnel zone is

2fI2

I1 y0.t = - J

Y0.C 1 + sin9 - - cos 4 cos x

1 - sin 4 f

where y o , t is the distance of the beam axis from the axis of the reflector for transverse polarization of the aper- ture field. Letting R+ 00, the on-axis field in the Fraun- hofer zone is

where

and the near-field gain reduction follows from (1 1). Since the total power radiated is2

the gain for transverse polarization is

where

With + a = 20” and I= 1.855 meters, 12=0.46522 and yo,$= 2.84054 meters.

APPENDIX I I

THE EFFECT OF A COVER OF NOYUNIFORM THICRNESS ON THE GAIN OF A RECTANGULAR APERTURE

Here the effect of a dielectric cover with the dimen- sions shown in Fig. 10(b) on the gain of a rectangular aperture of the same area as the horn-reflector aperture and with uniform and cosinusoidal illumination is con- sidered. The model, which is readily analyzed, was chosen because the two apertures differ in gain by only 7 per cent and have very similar gain reductions in the Fresnel zone. Tl and TZ are the complex transmission coefficients for regions 1 and 2 in Fig. 10(b), and the distribution of the incident field is Eo cos ( ~ g ) / L l . With the dielectric cover the on-axis field in the Fraunhofer zone of the aperture is

447

If T I = T2= 1, (23) reduces to TABLE I1

GAIS REDUCTION BY APERTCRE COVER (DECIBELS)

X (cm) 1 TI I j-2 1 G/Go

Eo 2L1L2 XR 7r

E(0)o = j - exp [ --jRR] - 7 (24)

Measurements a t X =4.61 cm on a sample of the cover indicated a relative dielectric constant of 3.67 and a loss tangent of 0.014. Using these and the dimensions of Fig. 10(b) in the expressions'3 for the transmission of a plane wave through a lossy dielectric sheet in air, with the electric field in the plane of incidence and the angle of incidence 20°, the values of TI and TZ in Table I1 were obtained. The gain reduction was then calcu- lated from (25).

l3 11:. PVI. Cady, kl. B. Karelitz, and L. A . Turner, "Radar Scanners and Radomes," R1.I.T. Rad. Lab. Ser., b'IcGraw-Hill Book Company, Inc.. New York, N. Y. , vol. 26, pp. 347-360; -1948.

ACKNOWLEDGMENT

The authors wish to thank the many staff members of the National Research Council Radio and Electrical Engineering Laboratories, Ottawa, Canada, whose assistance was essential to the completion of this work. Special mention should be made of W. A . Cumming, who gave many valuable suggestions, and W. W. Zuzak, who as a summer student assisted in the measurements. The National Research Council Computation Center provided the numerical evaluation of (6) and (17).

The Optimum Directivity of Uniformly Spaced Broadside Arrays of Dipoles

c. T. TN, FELLOW, IEEE

Summary-The optimum directivity of various types of uniformly spaced broadside arrays of dipoles is investigated theoretically in this paper. The formulation is processed with the aid of an array matrix. The expression for the optimum directivity and the corre- sponding excitation are expressed directly in terms of the elements of the array matrix. The computed values are assembled in several sets of curves, and the results are compared with the directivity of uniformly excited arrays.

Manuscript received September 16, 1963. The research reported here was supported in part by the U. S. l-aval Electronics Labora- tory, San Diego, Calif. under Contract hT123(953)-31663.-2 with The Ohio State University Research Foundation. An oral presentation of the material contained here and some of the extensions were given by the author a t the 1963 International Symposium of Antennas and Propagation held at Kational Bureau of Standards, Boulder, Colo.

The author is xt-ith the Antenna Laboratory, Department of Elec- trical Engineering, The Ohio State University, Columbus, Ohio.

INTRODUCTION HE PROBLERI dealing with the optimum direc- tivity of an array was first investigated by Uzkov' in the same year when Dolph? enunciated his

method of synthesis. By means of an orthogonal trans- formation in vector space, Uzkov obtained some impor- tant results concerning the optimum directivity of an array. In particular, he showed that the optimum

tive antenna design," Conzptes Rendus (Uoklady) de I 'dcademie de -4. I. Uzkov, approach to the problem of optimum direc-

Sciences des I'URSS, vol. 111, p. 35; 1946. C. L. Dolph, "A current distribution for broadside arrays whicl;:

optimizes the relationship between beamwidth and side-lobe level, PKOC. IRE, vol. 34, pp. 335-348; June, 1946.