JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Reflection Efficiencies of a Periodic Absorbing Surface

JACK C. MILLER

Pomona College, Claremont, California 91713(Received 30 September 1963)

The reflection efficiencies for the various orders of diffraction produced by a grating are calculated underthe following assumptions. The grating is for the most part flat, but with a fraction E of the periodicity in-terval a described by a profile function z~ f(x). The reflection and transmission efficiencies are calculatedas a series expansion in e. Numerical calculations have been performed for the case of ultrasoft x rays incidentat near grazing angles, using a complex index of refraction s = 1-,oX 2 ±+i0oX3 25 or tabulated values. No effectsof shadowing by the grating profile have been considered, and the numerical work has been done for asymmetrical triangular trench incised in the flat surface.

I. INTRODUCTION

THE problem of calculating the reflection efficiencyof a periodic surface for incident electromagnetic

or acoustic radiation is a relatively old one.1 In most ofthe treatments of this problem, however, the boundaryconditions appropriate to a perfect reflector are em-ployed.2 It is the purpose of this paper to describe amethod of solution for the more realistic case in whichboth reflection and refraction occur.

We begin by considering solutions to the wavefunction

(02u/aX2)+ (a2J/az 2)+k2u=O, (1)

which describes the following process. A wave, specifiedby the function u~ M(x,z), is incident upon the surfacez=f(x)=f(x+a) which forms the boundary betweentwo media; for z>f(x), k=o,/c while for z<f(x),k=rq(c,/c) with q the complex index of refraction; a isthe periodicity interval. We take the maximum valueof f (x) to define the x axis and so have the situation de-picted in Fig. 1. The angle of incidence z is measuredfrom the x axis as shown and we take the spatial partof the incident wavefunction to be

" M (x,z) =exp[i (co/c) (x cost - z sin6)]. (2)

The time dependent solution is then u(i)(x,z)e-i t. Thequantity e-its appears as a constant multiplicative fac-tor throughout and is, in consequence, omitted. In orderto specify the physical situations covered by the fol-lowing theory, we remark that uM (x,z) can be construedas either (1) the y component of an electric field polar-ized perpendicular to the plane of incidence (TE case),or (2) the y component of a magnetic field polarizedperpendicular to the plane of incidence (TH case), orfinally (3) the velocity potential of an acoustic wave,with v= -grad u W being the velocity of a vibratingparticle at the field point (x,z). Of course, the incidentfield implies the existence of a reflected field u(r) (x,z)and a transmitted field u(t) (x,z). The boundary condi-

'J. W. Strutt (Lord Rayleigh), The Theory of Sound (DoverPublications, Inc., New York, 1896), Vol. II, Sec. 272a.

2See, for example: W. C. Meecham, J. Appl. Phys. 27, 361(1956), or I. Abubakar, Proc. Cambridge Phil. Soc. 59, 231(1963). Compare also: G. Sprague, D. H. Tomboulian, and D. E.Bedo, J. Opt. Soc. Am. 45, 756 (1955). The results of this lastpaper, when applied to a situation with the parameters of our sur-face, are at considerable variance with the results obtained herein.

tions which we employ are continuity of (1) total fieldand (2) normal derivative of the total field across theboundary z= f(x). Our purpose is to calculate the de-tailed structure of U(r) (x,z) and U(t) (x,z). These solu-tions are found for a particular form of f (x), namely onefor which f(x) = 0 in a region of width w, and with therestriction that E= 1- (wia) is a small number.

II. REFLECTED AND TRANSMITTED FIELDS

Following the line of reasoning employed by Eckart3

we assume that the reflected and transmitted fields are

8(r) (xz) = E B., exp[i(w/c) (x cos;t'+z sin~ik')], (3)n

, (t) (xky) =E Cnexp~i (Xcwc) (x costs,-z sin.) ]. (4)

That some doubt attaches to these assumed forms is dis-cussed by Lippman.4 The question is whether the re-flected field can in fact be represented by a superposi-tion of plane waves whose propagation direction is avector having exclusively a positive z component. Forz>f(x) this is clearly a satisfiable condition, while forz<maxf(x), but such that k= w/c, it may not be. Weneglect the deviations in the structure of u(r) and u(t)introduced by this argument and assume the forms asgiven by Eqs. (3) and (4). For a complete representa-tion of the reflected and transmitted fields, a Green'sfunction analysis is probably necessary and we refer thereader to the discussion of Meecham's paper2 leading tohis Eq. (11).

III. BOUNDARY CONDITIONS

As mentioned in Sec. I, the continuity of total fieldacross the boundary is taken as one boundary con-

FIG. 1. Grating profilein the general case show-ing the various param- aeters used. Material with \ acomplex index of refrac- xtion 77 occupies the regionZ <f (X). X

C. Eckart, Phys. Rev. 44, 12 (1933).B. A. Lippman, J. Opt. Soc. Am. 43, 408 (1953).

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VOLUME 54, NUMBER 3 MARCH 1964

3 JACKl C. lMILLER V 5

FIG. 2. The struc-ture of the variousdiffracted orders with

8 \ Aangles considerablyexaggerated for pur-

; poses of clarity.

dition, so

u(i)[Xf(X))]+u(r)[Xf(X)] =U(t)[Xf(X)]. (5)

Upon using the fact that the various fields must exhibita periodicity equal to that of f(x) we find by replacingx by x+a,

cos~&4 = q cos 4n = cosOt+nX/a, (6)

with n=0, ±1, ±2,--- which are recognized as theusual grating conditions responsible for the various or-ders of spectra. Thus the reflected and refracted fieldspropagate in orders as suggested by Fig. 2.

For electromagnetic radiation the tangential com-ponent of the magnetic field in the TE case is

Htan= (icjolw)(ME,/Ov), (7)

and for the TH case the tangential component of theelectric field is

Eta.= (iyc/1&n72) (aOH/ d9v), (8)

where a/la denotes the normal derivative and /u is thepermeability of the medium. The requirement of con-tinuity in the normal derivative is thus equivalent tothe usual requirement that the tangential componentsof electric and magnetic fields are continuous across theboundary.

We now define some functions which appear inthe boundary conditions, after multiplying them byexp[-i(co/c)Xcos0]:

'k") (x) = expE-i(c/c)f(x) sinl], (9)

E n (t) (x) = exp[i27rnx/a+i (w/c)f(x) sinPn], (10)

En ) (x) = exp[i2rnx/a-i(7Zw/c)f(x) sinn4 j. (11)

With these definitions the boundary conditions become

+() (x)+Z B4 E4 (r) (x) -E CELEnEt ) (x) = 0, (12)n it

which states continuity of the wavefunction, and

4(i)(x) sin(t7-P)-_ B,,E7,(r) (x) sin(4.,,+±)In

-E gC4En(t) (x) sin( 4 -) =0, (13)n

expressing the continuity of normal derivative acrossthe surface with g=n for TE and g= 1/n for THI, wherer is the angle between the surface normal and the zaxis so that tan = -f' (x). Implicit in these boundaryconditions is the assumption that the whole surface isilluminated and that no part is shadowed by another

part. What we seek are sets of numbers Bn and Cnsatisfying Eqs. (12) and (13).

IV. APPROXIMATION METHOD OF SOLUTION

Unfortunately, the functions En(r) (x) and En(t) (x)do not form an orthogonal set, so it is not possibleto isolate a given pair of coefficients Bn and C.. Onemethod of handling this lack of orthogonality would beto construct an orthogonal set from the set En(t)(x),by use of the familiar Gram-Schmidt orthogonalizationprocess. We could then find a given C. in terms of suit-able matrix elements involving the orthogonal functiongenerated from En(t) (x) and (i (x)+Zn BnE4 (r) (x),and so eliminate Cn between Eqs. (12) and (13). Arepetition of this process would yield Bn from which theexact forms of the reflected and transmitted fields couldbe found. The generality of this method is interestingand important, but involves considerably more of acalculation than is needed to obtain some practicalresults.

By making the surface of the grating largely flat, wehave a situation in which the orthogonality of En(r) (x)and En()t (x) is nearly achieved. In other words, we con-sider the effects of taking f (x) to be zero over most ofthe interval of periodicity a. As a further requirementof f (x) we assume that the grating profile function issymmetric, so that for -(a-w)/2<x< (a-w)/2,f (x) = f (- x). As a simple example of these requirementswe assume that

a-w)f (X=-X+-) tanky:

a-w-- <x<O,2

a-w a-wf(x)= X--- tan-y: 0<x<-,

2 ~ 2(14)

f(x)=O: (a-w)/2<x< (a+w)/2,

so that the profile of the grating is as shown in Fig. 3.We can now proceed to make a Fourier transfor-mation of Eqs. (12) and (13) by multiplying themboth by (1/a) exp(-i27rn'x/a) and integrating fromx - (a-w)/2 to x= (a+w)/2.

D4 o0(0, sink)+Z D4 tn(O, -sinb 4i)Bnn

= E Dn/wn(0?10 sintn,)C., (15)

FIG. 3. Gratinga profile for detailed

K I w -calculations of Bt(1)I with silicon dioxide

as the grating mate-rial.

354 Vol. 54

March1964 REFLECTION EFFICIENCIES OF ABSORBING GRATINGS

where

Dn' n (YP) = int n+ EL,,ntn (1) COS-En n2] (16)

with1

and7 sin(n'-n)e

Lntn(r) = j dz expE-i(27rrT/X)f(aez/2)] cos7r-(n'- n)ez.

Equation (13) becomes

Dnto(,y, sintk) sintA-F Dntn(Y,- sinkn)Bn Sinlkn

=E gD.,n(-y,7 sintn)Cn sintn. (17)n

For each of the three terms of this last equation, anexpression involving

eM1n (r)=ie sinyf dz

XexpE-i(27rr/X)f(aEz/2)] sin7r(n'-n) Ez

has been omitted on the grounds that, for n and n' nottoo dissimilar and the imaginary part of (27rr/X)Xf(aezj2)<0, the terms in question are of order el.The three terms omitted are those formed by choosingr=sint#, n=O; r =-sint'.; and F= sing,, in Mnn (P)

This last remark suggests that the solutions to beobtained from Eqs. (15) and (17) can be expanded as apower series in e. Accordingly, to first order in E, wewrite,

sinz7-g singo

sin6+g singo

2 sin6 (18)Cn= 6n0+fzCn( )-

sin6+g sinto

The first terms on the right are easily recognized as theappropriate Fresnel equations for reflection from a flatsurface. They correspond to the zero-order approxima-tion e=O with g=,q for the TE case, and g= 1/ for theTH case. Upon substituting Eqs. (18) into the set con-sisting of Eqs. (15) and (17) and eliminating the Cnwe obtain an expression for Bn (1) in the form

Bn,(')[sinV'n+g sintn]

= (sint7 cos-y-g sintn)Lno(sinti)

sing~-g sing0siz- i- (sing cos~y-g sintn,)LnO (-sink)sina+g sinko

2 sinO

sin- +g sino (g singo cosy-g singn)

XLno(n sinQo). (19)

It is perhaps useful at this point to summarize thevarious expressions appearing in Eq. (19). We have

sing&e= (1- [cosi0+n (X/a)] 2} a

with the understanding that we are to write

sini,&n=i{[cos0+n(X/a)] 2 - 11

(20)

(21)

for those cases where either n or a combine to produce(cosO+nX/a) > 1

g sing.= {g2-[cosz9+n(X/a)]2 } 1.

For our choice of profile LnO(r) is now explicitly

ILno,(r) = dzeiaz r cos7rne (1- z),

(22)

(23)

where a = (7rae/X) tany. Since we want to consider valuesof n which are small, the further approximation can bemade that cos7rne(1 -z) = 1. From these approximationswe obtain

(24)

The amplitudes Dn(1) and (1) are obtained for TE andTH by substitution of R and k, respectively. For n#0O,the reflection efficiency for the unpolarized case isgiven by

r.(n)= (e2/2)[I l (i) 12+ 1Bn(1) 12](sin,6n/sinz), (25)

which is derived in the usual way from the polarizedefficiencies

rTE =E2 | P() | 2 (sin ,6./sintO);

rTH ( r: 2i| Bn(1) |2(sin ,n/sint) .

For n= 0 the reflection efficiency is found by taking thesquare of the absolute value of B. given by the first ofEqs. (18). The factor sin4tn/sinzY is the usual geometricfactor which arises out of calculating our reflectionefficiencies as the ratio of outgoing to incoming energyper unit area and per unit time.

V. NUMERICAL RESULTS

In order to see what the foregoing theory will predictin the ultrasoft x-ray region (wavelengths in the interval10 A<X< 100 A), calculations have been performed fora grating surface ruled on glass with 600 lines per mm.The optical properties of the glass are taken to be essen-tially those of silicon dioxide with the values of 6 and 3in the complex index of refraction -j= 1 -a+i3 given bya=5oX2 and A=loV3*25, where o0 = 2.340X10-3 A-2 andfo =1.568X 10-8 A-325. The values for 6 thus obtainedreproduce the tabular values for silicon dioxide given inTable V of Ref. 5. The values for i3 are, however, largerby a factor of 2.5 than given in Ref. 5. This latter dis-crepancy has been necessary in order to obtain a reason-

B B. L. Henke and J. C. Miller, "Ultrasoft X-Ray InteractionCoefficients," Tech. Rept. No. 3, Ultrasoft X-ray Physics Con-tract No. AF 49(638)-394 AFOSR, August 1959.

355

LnO(r) =(lfiar) (eil F - 1).

JACK C. MILLER

2 -e =--2'

10 20 30 40 50 60X IN ANGSTROMS

FIG. 4. First- and second-order reflection efficiencies in percentversus wavelength in angstroms for the silicon dioxide grating withprofile as shown in Fig. 3. The experimental points shown are takenfrom Ref. 6.

ably good fit to the reflection efficiency in first order froma glass grating as reported by Lukiriskii and Savinov.ACalculations have been performed for an incidence angleof i7 =20 with the grating parameters of trench angle,y=O.l rad and with a ratio of flat area to periodicityinterval of 0.773 making E-=0.22 7. These values are wellwithin the limits of actual gratings as shown byphotomicrographs.

Total external reflection should be responsible for theenhancement of reflection at a certain critical wave-length given by the condition ,n= 0 or sing = 0.Since

-q sine = { (1-6)2-32- [cosz7+n(X/a)]2+i23(1-5)}1, (26)

2

ANGLE IN DEGREES

FIG. 5. First- and second-order reflection efficiencies in percentversus incidence angle in degrees for the SiO2 grating with profileas shown in Fig. 3. The values shown are for fixed wavelength of40 .

' A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147(1963) [Opt. i Spektroskopiya 14, 285 (1963)].

7 See, for example, the gratings reported upon by H. A.Kirkpatrick, J. Quant. Spectry. Radiative Transfer 2, 715-724(1962). A private communication from this author also suggeststhat the values quoted are within the bounds of actual gratingparameters.

we have as an approximation to the total external re-flection condition

(1-5)2-3 2 - [cos+n(X/a)]2= 0. (27)

For the parameters given, this condition results incritical wavelengths for order a=-1 and n=-2 ofX-, 1=33.41 A and X,- 2 =55.73 A, respectively. Thesevalues can be located on a plot of reflection efficiency vswavelength for n= -1 and n= -2 (Fig. 4), and are ameasure of the main rise in the reflection efficiencies.The experimental values for the corresponding situationtaken from Ref. 6 are also shown. In view of the approxi-mations made in the foregoing theory, less agreementbetween theory and experiment should be expected forsuccessively higher order of diffraction. In both ordersthe effects of the oxygen K absorption edge are seen atthe equivalent wavelength Xk=

2 3 .3 A. The "turning-on"effect of the grating in evidence near X= 13 A for bothorders seen in Fig. 4 is in agreement with the reflectioncalculations given in Ref. 5, p. 31, and is obtained in theusual manner by the requirement that the real part ofXq sint 0 be zero, or in a first-order approximation that23=72, It is not difficult to show from the structure ofB (') that near t7= 0 all order have their reflection effi-ciencies proportional to sinkY. This is to be expected fromthe fact that with decreasing incidence angles the in-coming radiation sees a progressively larger percentageof the grating as a flat surface. Results for X= 40 A withunpolarized reflection efficiency in percent plottedagainst incidence angle in degrees are shown inFig. 5.

In general, we can see that on the basis of this grat-ing model the reflection efficiencies are quite sensitiveto the angle of the incised trench (called y in the textand Fig. 3) as well as the ratio of flat area to periodicityinterval. In the ultrasoft x-ray region of the spectrumthe calculated reflection efficiencies give the qualitativefeatures of observed grating performance correctly.Some preliminary calculations of reflection efficiency asa function of order, for fixed angle of incidence t7 andwavelength X, suggest that the distribution of energy isfairly equal among the first few orders. This is, qualita-tively at least, similar to the results reported byKirkpatrick.7 Further work is being undertaken to de-termine the effects of grating profile as well as theshadowing of a certain fraction of the grating surfacedue to nearly grazing incidence of radiation.

ACKNOWLEDGMENTS

The author wishes to thank Professor B. L. Henke forsuggesting this problem. That this work was supportedin part by the Air Force Office of Scientific Research isalso gratefully acknowledged.

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