1788 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986

Nontrivial grating that possesses only specularcharacteristics: normal incidence

Akhlesh Lakhtakia, Vasundara V. Varadan, and Vijay K. Varadan

Laboratory for Electromagnetic and Acoustic Research, Department of Engineering Science and Mechanics,The Pennsylvania State University, University Park, Pennsylvania 16802

Received July 11, 1985; accepted July 14, 1986

It is well known that gratings with horizontally pronounced periodicities produce nonspecular reflection (andtransmission) modes depending on the frequency and the periodicity. Do gratings with vertically pronouncedperiodicities always do similarly well? This question is resolved by considering a grating made of an infinitely longslab of constant width in which perfectly conducting gratings are periodically inserted. Only normal incidence isconsidered, and an elementary modal theory has been used in which diffraction from knife edges has been neglected.From the computed results it is shown that such a grating always reflects and transmits specularly perfectly overwide frequency ranges irrespective of the periodicity of the grating, the dielectric constant of the slab (real), thewidth of the slab, or the polarization of the incident electromagnetic plane wave.

INTRODUCTION

It is commonly observed that, when a plane wave impingesupon an infinite, periodically rough, bimaterial interface,the reflected and the transmitted fields consist of both thespecular and the nonspecular plane-wave modes,1' 2 in accor-dance with the Floquet theorem.3 Whereas the specularmodes always propagate with a finite energy content (unlesstotal internal reflection occurs, in which case the specularlytransmitted mode shuts off), the nonspecular modes may bepropagating or evanescent, depending on the frequency, theconstitutive properties of the two media, and the period ofthe interface. Nevertheless, over an appropriately large fre-quency range it is never seen that all nonspecular modes arealways evanescent. These properties of such interfacesmake them useful as diffraction gratings.

Observations of the same general nature have also beenmade for some other periodic systems, e.g., perfectly con-ducting screens perforated periodically by identical aper-tures, 4' 5 wire meshes, 6 arrays of parallel-oriented perfectlyconducting cylinders,7 and plane metallic bigratings of regu-larly spaced hemispherical cavities. 8 Indeed, one may beforced to ask the question: Do all periodic scattering sys-tems have the typical grating behavior?

In order to answer this question, the common threadamong these various observationsl- 8 must be discovered.We made the ansatz that in order to observe the typicalgrating behavior, the periodic scattering surfaces must havehorizontally pronounced periodicities. This ansatz is wellsatisfied by the examples mentioned thus far, some of whichare illustrated in Fig. 1. But what if the nature of theperiodicity is vertically pronounced? Some authors 9-11

have examined the response of such gratings, such as theones shown in Fig. 2. However, the answer to the questionposed here may not be found in these papers 9-11 because theauthors had other objectives.

The reflection and transmission characteristics of a simplegeometry are examined in this paper. The grating to beconsidered, shown in Fig. 3, consists of an infinitely long

dielectric slab of finite width in which thin, perfectly con-ducting plates have been periodically inserted. The prob-lem is assumed to be two dimensional for the sake of simplic-ity so that a scalar analysis becomes possible. An elemen-tary modal theory has been used here with the diffraction bythe perfectly conducting knife edges neglected. It is shownthat regardless of the media on either side of this grating, themedium of the grating, its period, or its height and frequen-cy, this grating behaves very differently from conventionalones. Although nonspecular modes are necessary to de-scribe the fields existing in such a situation, it is seen thatonly the specular reflection and transmission modes carryenergy. Only normal incidence, however, has been consid-ered in this treatment.

THEORY

Figure 3 is a schematic of the grating considered here. Me-dium 1 with constitutive parameters cl, p fills the space z >2d, while medium 3 (E3, 3) extends from z = 0 downward. Inthe 0 < z < 2d space lies the grating medium 2 (e2, ,u2) withthe exception that the surfaces

x=ha,d3a,+5a,±7a,..., 0<z<2dare perfectly conducting. All the three media are assumedto be homogeneous, linear, and isotropic.

Since only normal incidence is being considered here, thetotal field in medium 1 is given by

2N

J = E [An exp(-ifl3z) + A,, exp(ii3nz)]n=O

X exp[i(n - N)7rx/a], z > 2d, (1)

where the ensemble {An+ represents plane waves movingtoward z = +O, respectively. The Floquet conditions forperiodicity3 have been satisfied by this expansion, and

An- = 6nN

0740-3232/86/111788-06$02.00 © 1986 Optical Society of America

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Lakhtakia et al.

Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1789

2N

iP2(r) = E [B. exp(-3 2nEz) + B"E+n=O

X exp (02 Ez)]sin[(n + 1)7r(x + a)/2a],

where

02n = =k2 2 -(n + 1)7r/2a V/

2 ;

and for the TM-polarization case,2N

{ 2(r) = E [B '_ exp(-i0 2nMz) + B M+n=O

X exp(i32nMz)]cos[n7r(x + a)/2a],

I

II

III-I

1 ,

Fig. 1. Examples of periodic structures in which the periodicity ishorizontally pronounced. From top to bottom: an array of cylin-ders; a wire mesh; a perfectly conducting screen periodically perfo-rated with circular holes; and a bidimensional, periodically rough,bimaterial interface.

with bnm being the Kronecker delta function, represents theincident plane wave. Furthermore,

/3n = 1k12

- [(n - N)r/a1 2 11/2 (3)

where 0in is either positive real (propagating plane waves) orpositive imaginary (evanescent plane waves). The parame-ter N denotes a truncation of an otherwise infinite series.

Likewise, in medium 3, the total field can be representedin the form

2N

03(r) = 1 [Cn exp(-ii3nz) + Cnj exp(i03nz)]n=O

X exp[i(n - N)irx/a], z < 0,

where

f33n = {k32 -[(n -N)7r/a]21/2

and

Cn+O=0 V n.

Finally, the field representation in medium 2 has tdecided on. In keeping with the usual way of solving peiic boundary value problems, one need consider only a

cell, shown in Fig. 3 also, for this purpose. For thepolarization case,

(4a)

Fig. 2. Examples of periodic structures in which the periodicity isvertically pronounced. The black areas are metallic, and the areasshaded with slanted lines are filled with dielectrics.

UNIT CELL

(4b) > I

(4c) + Y \\ &

to be PERFECTLY CONDUCTING

_inol PLATES *.2a+IFig. 3. Schematic of the two-dimensional grating considered inthis paper. The unit cell of the grating is also shown. Note that theperiodicity is vertically pronounced.

(5a)

(5b)

(6a)

z

LLJ

II

II

II

II

Lakhtakia et al.

tI

)

I

1790 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986 Lakhtakia et al.

where

2nM = fk 22 _ (n7r/2a) 2

11 /2. (6b)

These expansions (5) and (6) have adequately taken intoaccount the conditions to be satisfied by 42 on the perfectlyconducting walls at x = +a, 0 < z < 2d.

Expansions (1) and (4)-(6) contain the unknowns thathave to be evaluated as a solution to this problem. First,satisfying the conditions

i,(x, 2d) = b2(x, 2d), Ia/OzM1j(x, 2d) = x12 1a/azP/q'2(x, 2d),

(7)

where { ilj, TE polarizationXij =Ei/ej; TM polarization

leads to the (4N + 2) X (4N + 2) matrix equations

(8)

It is interesting to examine the reflected and the transmit-ted fields when media 1 and 3 are identical. In that case, thematrices

[J1] - [J3], [L12] - [L32]

in Eqs. (12a) and (12b); hence, Eq. (12c) reduces to

C} = [D]{AI,

whence it is clear that

A,+= OVn

because

Cn+ = Vn,

and

Cn- = exp[i2d(k 2 - k)]bnN,

(14a)

(14b)

(14c)

(14d)

[Jr 1 I - 1 r 1( A- [LI = [( L(B-1[-[M1 ][J11 M ][J] JL 0 :[D]*J ___2J [ -x2 [M2 ][LJX12 [M [ IL ; (9)

then the conditions

x3(x, 0) = iP2(x, 0), 1a/azIV 3(x, 0) = X321a/Ozhl'2 (x, 0)

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lead to

[ [~] 4~'][ 1fCL [ I [L] I [L] PIB-[-[Al 3 ] [JI[Ms]3 J]Aj C+J L-x32[M 2 1 [LI.xa2[M2 [L] JB+j

(11)

The various matrices involved in Eqs. (9) and (11) are given inAppendix A.

If Eqs. (9) and (11) are symbolically rewritten as

[J,] [D]{AI = [L12]IBI, (12a)

[J3]ACQ = [L32]IB), (12b)

then a solution for the fields in media 1 and 3 can be ob-tained from

ICI = [J3 J ' [L3 2 1[L 12 1-'[J 11[DIIAI (12c)

after the coefficients of the field expansion in medium 2 havebeen eliminated.

The solution obtained from Eq. (12c) must satisfy theunitarity constraints, i.e.,

2N

Y RRnIN + Tn/NI = 1 (13a)

as the truncation parameter N is increased. The diffractioncoefficients

RfIN = Rel3lfl/#llNl IAn I2 (13b)

and

Tn/N = Re43sn//#1,N ICn I2 X31 (13c)

represent the modal power reflected into medium 1 andtransmitted into medium 3, respectively. Also, at the sametime, the coefficient vectors IA+I and IC-I must also convergesatisfactorily.

It is obvious that no reflected modes exist, not even thespecular one; the transmitted field is then purely specular.Provided that k2 is real, all energy incident upon the inter-face z = 2d from medium 1 is transferred into medium 3 atthe z = 0 interface. This is so regardless of the polarizationof the incident plane wave, the frequency, and the periodand width of the grating and does not even depend on thewave numbers k1 , k 2 and k3 = kj. Such behavior of thepresent grating is completely different from that of a simpleslab.12 In addition, this grating also responds differentlyfrom a grating studied by Cain 3 that was formed by twoparallel, periodically rough, bimaterial interfaces: nonspec-ular modes did, in general, exist in Cain's case. It wouldappear, therefore, that the refraction by the two interfaceshas been offset completely by the adherence of the field to arigidly set, infinitely repetitive set of parallel-spaced, per-fectly reflecting planes that are parallel to the incident wavepropagation vector. In order to examine this contention,consider the case of a perfectly reflecting, infinite plane onwhich a plane wave is incident in a direction perpendicularto the normal to the surface: there is no meaningful reflec-tion then, the incident plane wave continuing to glide alongthe reflector surface. In other words, the perfectly conduct-ing planes act as guides for the incident field that is trans-mitted directly from medium 1 into medium 3, having suf-fered, however, a phase shift; vide Eq. (14d).

RESULTS AND DISCUSSION

Presented in Tables 1-4 and in Figs. 4 and 5 are samplecomputed values of the specularly reflected power RN/N forvarious cases. In all the computations made, all the threemedia were assumed to be nonmagnetic and to possess realdielectric constants ei; in particular, medium 1 was assumedto be free space.

It was observed that, regardless of the polarization of theincident field as well as for any values of E2 and e3 , thereflected field in the far zone (z - -) possessed only thespecular mode. Furthermore, the transmitted field in the

Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1791Lakhtakia et al.

Table 1. Specularly Reflected and TransmittedPower When a Plane Wave of Either Polarization IsNormally Incident upon the Grating and upon a Slab

of Same Dimensionsa

Grating Slab

d/a 2k1a Reflected Transmitted Reflected Transmitted

0.1 6.0-20.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

0.5 6.0-20.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

1.0 6.0-20.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

5.0 6.0-20.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

a The grating period is' 2a, while 2d is the grating height. All media arenonmagnetic; El = Eso -2 = 5co, and e3 = 10c. Note that while reflection andtransmission for the grating are constant at all frequencies and d/a valuesconsidered, they record the extrema shown for the slab.

Table 2. Same as Table 1 Except That E3 = 2.5co

Grating Slabd/a 2k1a Reflected Transmitted Reflected Transmitted

0.1 6.0-20.0 5.069% 94.931% 5.07-26.99% 94.93-73.01%

0.5 6.0-20.0 5.069% 94.931% 5.07-26.99% 94.93-73.01%

1.0 6.0-20.0 5.069% 94.931% 5.07-26.99% 94.93-73.01%

5.0 6.0-20.0 5.07% 94.931% 5.07-26.99% 94.93-73.01%

far zone (z - -- ) also had only the specular component.

This is quite uncharacteristic of conventional gratings thatdo give rise to nonspecular propagating components. Uni-

tarity was always satisfied to within 0.01%, and the coeffi-

cient vectors {A+} and IC-I converged satisfactorily within aprespecific tolerance of 0.01%. Hence the results shown

here may not be in doubt.In Tables 1 and 2, the specularly reflected and transmitted

diffraction coefficients for the grating are shown for a nor-

malized frequency range of 6.0 < 2 kla < 20.0, while the

aspect ratio d/a = 0.1,0.5, 1.0, and 5.0. It is to be noted fromthese two tables that RN/N and TN/N remain constant for all

frequencies and aspect ratios considered. On the other

hand, if the perfectly conducting vertical plates were to beremoved, then the grating would become a slab. As is shown

in these tables, RN/N and TN/N for the slab would record clear

extrema as the frequency is changed but dia is kept con-stant. 12

In Table 3, data are presented for varying aspect ratios 0.2< d/a < 10.0 with 2k1a = 10.0 and 20.0. While e2 = 5c0, E3 has

two values, e3 = 2(2 and 0.5e2. In either case it is apparentthat the grating considered appears to behave independent-ly of d/a as well as of the frequency, but RN/N and TN/N seem

to be dependent on e3 and E2.

This dependence of RN/N (and TN/N) is explored further inFigs. 4 (e2 = Sco) and 5 (e2 = 3eo) while Q3 is varied, at d/a =

1.0, 2kla = 10.0 and 20.0, over the range 1.0 S e3/e0 < 10.0.

On comparing these figures it seems that RN/N is indepen-dent also of c2 (provided that it is real) but depends only onQ3. Further confirmation of this conclusion comes from Ta-ble 4, where Q3 = 5E0 but 1.0 < 62/EO < 10.0.

Some insight into the behavior of the present grating isafforded by reviewing Eq. (12c). The presence of medium 2

is wholly contained in the factor [L32] IL12 -1, apart from thematrix [D], which merely represents a phase shift. Somealgebra now leads to the simplification

[! = 0 o 10[L2 L =2 -t --…f1 O X3~'

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where [1] is the identity matrix. The net result is that

{2C_1 =

[[II + [M3]'[MlA] [II - [1M3P'[4 [Ml1X [-_ -['[Ml__ ] j

[I] - M31] 1l1] . III + [M31 iMl

XF[eJ-]__0 i ° 0IA 1l° [JI L 0 ' [D*I] A 31

(16)

Table 3. Same as Table 2 but the Height d/a of the Grating Is Now Varied Keeping the Normalized Frequency2kla Constanta

Grating Slab

2k1a E3/f2 d/a Reflected Transmitted Reflected Transmitted

10.0 2.0 0.2-10.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

0.5 0.2-10.0 5.069% 94.931% 5.07-26.99% 94.93-73.01%

2.0 0.2-10.0 26.987% 73.013% 5.07-26.99% 94.93-73.01%

20.0 0.5 0.2-10.0 5.069% 94.931% 5.07-26.99% 94.93-73.01%

a Note that while the specular reflection and transmission are constant for the grating, they record the extrema shown for the slab as d/a is varied.

Table 4. Specularly Reflected and Transmitted Power When a Plane Wave of Either Polarization Is NormallyIncident upon the Grating and upon a Slab of Same Dimensionsa

Grating Slab

2k1a e2Eb Reflected Transmitted Reflected Transmitted

10.0 1.0-10.0 14.589% 85.411% 2.097-35.851% 97.903-64.149%

20.0 1.0-10.0 14.589% 85.411% 0.553-36.286% 99.447-63.714%

a The grating period is 2a and the grating height is 2d. All media are nonmagnetic; El = (O and E3 =. 5e0.

b Sampling done at A(E 2 /o) 0.1.

1792 J. Opt. Soc. Am. A/Vol. 3, No. 11/November 1986 Lakhtakia et al.

25

cRa-

-J

IL

-JLL

QWa-

C?-J

20

15

10

5

0 L I I I I I I I I I1 2 3 4 5 6 7 8 9 10

Fig. 4. For the grating of Fig. 3, the specularly reflected power RN/N= 1.0 - TN/N is shown. This quantity is independent of the fre-quency as well as the aspect ratio d/a. For comparison, the re-sponse of a slab of identical dimensions at 2kja = 10.0 and 2k1a =20.0 is also provided.

25

0e

-J

J

-J

ir0wa.co

20

15

10

0 W9 I I I 1 1-1-k4 .j i I1 2 3 4 5 6 7 8 9 10

f 3 /GOFig. 5. Same as Fig. 4, except that the relative permittivity of thefiller medium [Eq. (2)] has been altered.

whence the special case of identical media 1 and 3 can beeasily deduced. Because [Ml] and [M3] are diagonal matri-ces, it appears from Eq. (16) that Eq. (12c) represents astructure like a planar interface between media 1 and 3, withmedium 2 providing a phase shift through [1D].

In summary, it appears that for the present grating, (a)only specular reflection and transmission exist, (b) the dif-fraction coefficients are independent of both frequency andthe width 2d, and (c) the diffraction coefficients depend onlyon E

3 and not on the filler material 2.The latter two conclusions are in contrast with the behav-

ior of a slab of identical dimensions of medium 2, where thediffraction coefficients depend critically on e2, E3 , the width2d, and the frequency. The first conclusion, on the otherhand, definitely sets the present grating apart from the moreconventional gratings made of periodically rough bimaterialinterfaces.

It must, however, be borne in mind that these conclusionsmay not extend to the higher frequencies, where diffractionby the perfectly conducting knife edges will certainly affectthe performance of the grating. In addition, the presentanalysis holds only for normal incidence.

APPENDIX A

The elements of the (2N + 1) X (2N + 1) matrices commonto both polarization cases are given by

(Mj)mn = 3jmbmn, i = 1, 3,

(M2)m n = 32 m.EMmn,

(D)mn = exp[-i2d(fln - 0 2n M)I 6 mn,

(D)mn = exp[i2d(3ln - f22nEM)6..mn.

(Al)

(A2)

(A3)

(A4)

The matrix elements (J)m,n and (L)m,n, on the other hand,are given for each polarization by

(J)m, nTE =Ea dx exp[i(n - N)]7rx/a]

X sin[(m + l)ir(x + a)/2a], (A5)

(J)mn TM = Ja dx exp[i(n - N)rx/a]

X cos[m7r(x + a)/2a],

(L)mnTE = a6mn,

(A6)

(A7)

and

(L)mn nTM = 2a0=6 + abmn(1 bnd- (A8)

The integrals involved in Eqs. (A5) and (A6) can be evaluat-ed in simple closed forms by using any standard table ofintegrals (e.g., Ref. 13).

REFERENCES

1. S. L. Chuang and J. A. Kong, "Scattering of waves from periodicsurfaces," Proc. IEEE 69, 1132-1144 (1981).

2. W. N. Cain, "Electromagnetic and acoustic scattering by pen-

5

Lakhtakia et al. Vol. 3, No. 11/November 1986/J. Opt. Soc. Am. A 1793

odic bimaterial interfaces," M.S. thesis (Pennsylvania StateUniversity, University Park, Pennsylvania, 1985).

3. P. C. Waterman, "Scattering by periodic surfaces," J. Acoust.Soc. Am. 57, 791-802 (1975).

4. C. C. Chen, "Diffraction of electromagnetic waves by a conduct-ing screen perforated periodically with circular holes," IEEETrans. Microwave Theor. Tech. MTT-19, 475-481 (1971).

5. C. H. Tsao and R. Mittra, "Spectral-domain analysis of frequen-cy selective surfaces comprised of periodic arrays of cross di-

poles and Jerusalem crosses," IEEE Trans. Antennas Propag.AP-32, 478-486 (1984).

6. M. I. Astrakhan, "Reflection and screening properties of planewire grids," Radio Eng. (USSR) 23, 76-83 (1968).

7. J. Y. Surrateau, M. Cadilhac, and R. Petit, "The perfectly con-ducting wire grating: computation of the diffracted field fromMaxwell's equations and Hamilton's canonical system," IEEETrans. Antennas Propag. AP-33, 404-408 (1985).

8. E. Toro and R. Deleuil, "Application of a rigorous modal theoryto electromagnetic diffraction from a biperiodic rough surface,"IEEE Trans. Antennas Propag. AP-33, 540-548 (1985).

9. J. A. DeSanto, "Coherent scattering from rough surfaces," inMathematical Methods and Applications of Scattering The-ory, J. A. DeSanto, A. W. Sdenz, and W. W. Zachary, eds.(Springer-Verlag, Berlin, 1980).

10. E. V. Jull, J. W. Heath, and G. R. Ebeson, "Gratings thatdiffract all incident energy," J. Opt. Soc. Am. 67, 557-560(1977).

11. J. W. Heath and E. V. Jull, "Perfectly blazed reflection gratingswith rectangular grooves," J. Opt. Soc. Am. 68, 1211-1217(1978).

12. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1975).

13. W. Grobner and N. Hofreiter, Integraltafel zweter teil Bes-timmte Integrale (Springer-Verlag, Vienna, 1973), pp. 106-107.