214 J. Opt. Soc. Am. A/Vol. 5, No. 2/February 1988
Can the reflectivity of a nonlossy slab of arbitrary thicknessloaded with a strip grating be identical
to that of a plane interface over a wide range of frequencies?
Armand Wirgin
Laboratoire de Mecanique Theorique, Universit6 Paris VI, 4 Place Jussieu, 75005 Paris, France
Received March 27, 1987; accepted September 29, 1987
Can the reflectivity of a nonlossy slab of arbitrary thickness loaded with a strip grating be identical to that of a planeinterface over a wide range of frequencies? A negative response is provided to this question for broadsidetransverse magnetic polarized incident radiation. It is shown, however, that the strip-grating-loaded slab ofthickness h > 0 has the same far-field response as the bare slab of thickness h at all frequencies. These conclusionsare at odds with the recent results of Lakhtakia et al. [J. Opt. Soc. Am. A 3, 1788 (1986)] and also emphasize the non-uniqueness of the determination of a scattering structure from measurements, over a wide range of frequencies, ofits far-field response to a plane wave at a fixed angle of incidence.
INTRODUCTION
Several years ago, in the course of my studiesl 2 on the opti-cal properties of strip-grating-loaded nonlossy slabs, I foundthat the high-frequency response of these structures tobroadside radiation is identical to that of the bare slab of thesame thickness flanked by the same two media. In a recentpaper3 Lakhtakia et al. reported finding that the strip-grat-ing-loaded slab responds, in terms of reflected and transmit-ted power, in a manner identical to that of a plane interfaceflanked by the same two media as the slab for incidentbroadside radiation of widely ranging frequency polarized ineither the TE or the TM mode. If this finding were true, itwould have far-reaching consequences (e.g., glass plates withno reflection loss). It is therefore of some interest whetherany or both of the aforementioned findings are true.
In what follows, we arrive at a result that (1) substantiatesand extends to all frequencies our initial finding (for TMradiation), (2) invalidates what is found in Ref. 3, and (3)illustrates the nonuniqueness of the determination of a scat-tering structure from the sole knowledge of the far-fieldresponse, for a fixed angle of incidence and a varying fre-quency of illumination (see also the abstract of Ref. 3).
FORMULATION OF THE PROBLEM
Let space be divided, by two parallel planes L,(x 2 = h) andL2 (x2 = 0), into three regions Q1 = {x2 > h, -- < x, < -1, 02 =
0 < X2 <h, -- <x, < ol, and Q3 = {x2 <0, -- <X, <o(where xI, x2 , X3 are the three Cartesian coordinates). Q,and Q3 are assumed to be occupied by nonlossy, homoge-neous, isotropic dielectric media with (real, scalar) refractiveindices n, and n3 , respectively. For the sake of comparison,we shall treat two distinct problems. In problem 1 (P1), Q2is occupied by a nonlossy, homogeneous, isotropic dielectricmedium of refractive index 112 and is termed a bare slab. Inproblem 2 (P2), a periodic assembly of perfectly conductingstrips (strip grating of period d) is introduced into Q2, andthe material between the strips is considered to be the same
as the one that occupies all of Q 2 in P1. The mutuallyparallel strips are infinitely long in the X3 direction, of thick-ness (in the xl direction) a, of height h exactly equal to thethickness of the slab in P1, and of rectangular cross section(in the xl - x2 plane) and are placed within the slab in such amanner that one of the axes of each strip is perpendicular tothe planes Ll and L2 (see Fig. 1). In P2 the materials occu-pying Q, and Q3 are the same as in P1, and the region Q2 isnow termed a strip-grating-loaded slab. Since it is no moredifficult to do so, we assume that the angle of incidence Oi ofthe incoming plane-wave radiation (in Qj) is arbitrary. Welater specialize our results to broadside (normal) incidence(O. = 00). The plane of incidence is assumed to be the x -x2
plane, and the polarization of the incident radiation is takento be TM (i.e., with its magnetic field perpendicular to theplane of incidence). This restriction to the sole TM polar-ization mode is motivated by the fact that the latter is theonly one for which we have been able to obtain a closed-formsolution to P2 for the far-field scattering characteristics inthe special case 6i = 00, (d - a)/d = 1 (this being the casetreated in Ref. 3).
The geometry of the structure and the orientation of theplane of incidence are such that the total fields in all threesubspaces are TM polarized and independent of the X3 coor-dinate. Let the time dependence of the field be exp(-icot),and let uj(x) designate e3 Hj (with x = xiel + x2e2, el, and e2as the unit vectors along the xi and x2 axes, and Hj as thespatial component of the total magnetic field in Qj). The X3component of the incident field H 0 in Q, is expressed by
Recalling that n, n2, and n 3 are real scalars, we can easilyshow that the energy balance (in both P1 and P2) is ex-pressed by
p + - = 1,
(4b)
(10)
wherein
for 0 < x2 h,
ik2 x (± 2 'X2) = 2k2 OX1
where b = d - a is the spacer width and
Imn = m/kn,
it having been assumed that the three media have the semagnetic inductive capacities. In P1, conditions (4)absent, and the range of conditions (3) is -d/2 < x <In addition, u; satisfies the reduced wave (Helmholtz) eqtion
(aO2 + 2 + kj2) uj(x) = 0,
P = PnneW1
= -nne Y3
(the hemispheric reflectivity),
(the hemispheric transmissivity),
(5) W = set of indices n for which IknII(1) < k,
Lme W3 = set of indices n for which kn II(3) < k3;
Pn = IRnI2IKn(1)1/1Ki(1)1
(11)
(the intensity in the nthreflected spectral order),
Tn = Tn12Pr3IKn (3) /Ki(1)1 (the intensity in the nth
D = [(r2,r2 3 + 1)sin(k2h) + i(r2, + r23)cos(k2h)]/E,
(13)E = exp[iki,( 1 )h],
Rn = rn(h) [(r2r 2 3 - 1)sin(k2h)
+ i(r2,- 23)cos(k2 h)/D}Eno,0
Tn = tn(h) - [2ir/DF6n.
It follows that
p + T = po + To = IRO 12 + IT0 12I.13 = ro 12+ to 2r13 = 1, (15)
so that energy is conserved, as it must be. Equations (14)simply translate the fact that the slab far-field response isthat of specular reflection and transmission, since no energyis sent into the diffractive (n # 0) orders.
From Eqs. (14) it is obvious that Iro(O)l # Iro(h > 0)1 andIto(0)l X Ito(h > 0)1, except when sin(k 2 h) = 0, so that a slabdoes not, in general, respond (in terms of p and T) in the samemanner as a planar interface (between media of refractiveindices nj and n3). Lakhtakia et al.3 found that the intro-duction of a strip grating within the slab makes these tworesponses identical. We shall see if this is so by examiningP2.
IMPLICIT SOLUTION TO PROBLEM 2 FORARBITRARY INCIDENCE AND ARBITRARYPLATE THICKNESS
In Refs. 1 and 2 it was shown that the implicit solutions toEqs. (3)-(6) can be expressed by
The reflection and transmission coefficients are obtainedfrom the Am and Bm by means of
R = Ei(l)+no + (r 2 /Kn(l)) E [A.E.m=O
- BmEm(2)-]Km(2)Cnm,
Tn = (r 32/Kn(3)) > [Am - Bm]Km(2)Cnm-
m=O(18)
Equations (16), which result from what is termed the mode-matching technique,'-5 are often solved in a numerical man-ner by successive approximation: restriction of the numberof equations and unknowns to some finite value N, progres-sive increase of N, and cessation of the procedure whenstabilization of successive approximations is attained. An-other, more elegant, approach, which is applicable for thecase -y = 1, is also based, in the first stage, on the mode-matching technique but makes use, in the second stage, ofthe fact that for h - the equations can be solved explicitlyby the Wiener-Hopf (conventional residue calculus) meth-od. As h is, in reality, finite, we must modify the conven-tional residue method in a manner that was described indetail by Kent and Lee,6 who were specifically concernedwith the case n1 = n2 = n3. It turns out that we are led, bythe modified residue calculus technique, to another infiniteset of linear equations whose advantage over the former setis the speed by which successive approximations converge tothe solution, because of the explicit incorporation into theset of equations of information contained in the edge condi-tion (i.e., the condition whereby the field is specified to belocally square integrable everywhere outside the strips and,in particular, in the neighborhood of the strip edges). Themore general case, in which nj = n3 5# n2, was treated byKobayashi 7 by what he claimed to be a more rigorous variantof the modified residue calculus technique. Kent and Lee6
and Kobayashi7 obtained some numerical results for the casethat is of interest here (i = 0), albeit with the restrictionthat ni = n3 = n2, and found thatr = o = 1 for k 2h = 27r andk 2d = 3r/2. Kent and Lee also found that, under the samecircumstances, r = To = 1 for k 2h = 27r and k2d = 7r/2 and fork2h = r and k2d = 3ir/2. Thus, in all these examples, thepower transmitted by the structure is the same as when thestrips are absent.
By simple inspection of the coefficients of the linear equa-tions arising from the mode-matching technique, Lakhtakiaet al.3 claimed to have obtained a closed-form solution forthe case y = 1, Oi = 00, ni = 5-1 n2 (and numerical solutionsfor the more general situation nj 5z n3 0 n2). Their result isthat the strip grating produces the effect, as concerns re-
Armand Wirgin
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. A 217
flected and transmitted power, of wiping out the slab and,therefore, leaves only the interface between the media ofrefractive indices n, and n3. If the latter are equal, as isassumed by Kent and Lee and by Kobayashi, then the resultof Lakhtakia et al.3 is consistent with the previously men-tioned numerical results from Refs. 6 and 7. If, on the otherhand, n, #d n3 , neither the results of Ref. 6 nor those of Ref. 7can be used to test the conclusion of Ref. 3. It is the conten-tion of this paper that the conclusion is, in general, false. Toprove this, we now reexamine the solution of the mode-matching equations [Eqs. (16) and (18)] for the case of inter-est: Oi = ,y = , nl- n3 Fd n2 -
AN (ALMOST COMPLETE) CLOSED-FORMSOLUTION TO PROBLEM 2 FOR INFINITELYTHIN STRIPS, BROADSIDE INCIDENCE, ANDARBITRARY NONLOSSY MEDIA
It is possible to solve this problem by the method of Kobaya-shi,7 but it is much easier to proceed as follows, provided thatone can be content with an almost complete solution (i.e.,one that applies only to the far field). When = 1 and 0i =00, we find that
Cn2p = () p = 0, 1, 2,...,
Cn,2p+l + C-n,2p+l = P = 0, 1, 2,...,
n=0,I1, 2,...,so that
I2p,2q W = p,q/(EpKp(j)), P = 0, 1, 2....
I2p,2q+1) = 0 j = 1 3.
Consequently, for p, q = 0, 1, 2, . . .,
(19)
(20)
(21)
H2
p' = 2Ei(1)+6p0o' H2p2 O.
G2p,2q" = E 2p(2)+(1 -r12K2p(2)/Kp(1))6pq,
G2p,2q12 = E 2 p(2 )-(1 + P12K2p(2)/Kp(1))5pq,
G2p,2q21 = (1 - r3 2 K2 p(2 )/Kp(3 )) 6pq,
G2p,2q22 = (1 + T32 K2 p(2 )/Kp(3 )) bpq,
G2 p,2q+ljk = 0, j, k = 1, 2. (22)
This permits the extraction of explicit solutions from thesubset of Eqs. (16) for which is even:
A2p = af2p,
B2 p= 2 p, p=0,, 2 ,. (23)
where 2p and f2, are the solutions of the slab field coeffi-cients of P1 [Eqs. (13)]. This means that the even-indexcoefficients of the slab field in P2 are identical to the even-index slab field coefficients in P1. This does not mean thatthe slab fields of both of these problems are identical be-cause we have not demonstrated that A2p+l = a2p+1 = 0 andp = 0,1, 2,....
Inserting Eqs. (23) into Eqs. (18) gives
Rn = rn + Rn
Tn = tn + Tn', (24)
where rn and tn are the field coefficients of regions Q, and 23
for P1 [Eq. (14)] and Rn' and Tn' are the parts of Rn and Tncorresponding to summation over the odd indices of Am andBn in Eq. (18). From the fact that
O,2p+l o0 (25)
we obtain
Ro = 0,
To = 0, (26)
so that
Ro= ro,
To = to. (27)
This is the first fundamental result of this paper; it signifiesthat the specular reflection and transmission coefficients ofthe strip-grating-loaded slab are identical to those of thebare slab.
We have made no attempt to determine the Rn and Tnvalues for n id 0, so that Eqs. (24) constitute only a partiallycomplete closed-form solution to P2. It is, however, possi-ble to demonstrate that Eqs. (24) constitute a completeclosed-form solution for the nonevanescent plane-wave coef-ficients (those for which n e Y1/, n e Y3) in regions i1 andQ3. To do this, we first note that
Eqs. (10)-(12) tell us that only the zeroth (specular) reflect-ed and transmitted spectral orders are involved in the ener-gy balance. Thus Eqs. (27) are closed-form solutions for allthe nonevanescent spectral orders if condition (29) is satis-fied. When, on the other hand, this is not the case, thenother (nonevanescent) spectral orders can theoreticallycompete for the incident energy; but since we already estab-lished that Ro = ro and To = to satisfy Eq. (15), than we mustconclude, from the energy balance relation [Eq. (10)], thatno energy whatsoever is injected into the other-than-specu-lar nonevanescent orders. This means, in all cases, that
Rno = for n W1,
Tno = for n E 3- (30)
This is the second fundamental result of this paper.The technique used above does not enable us to say any-
thing about the coefficients of the evanescent spectral or-ders. In general, these should not vanish, because of diffrac-tion. Thus we cannot conclude that the fields in the neigh-borhood of the strip-grating-loaded slab are identical tothose of the bare slab [regardless of whether condition (29) isverified]. However, we have shown, as concerns the reflect-
Armand Wirgin
218 J. Opt. Soc. Am. A/Vol. 5, No. 2/February 1988
ed and transmitted amplitudes and intensities in the far-field region, that the response of the strip-grating-loadedslab is identical to that of the bare slab when y = 1 and Oi =00. This is consistent with our previous finding, which wasrestricted to high frequencies of the incident radiation.
Since r and to are functions of the slab thickness, theanswer to the question that is the title of this paper is nega-tive [note from Eqs. (14) that r and to do not depend on hsimply by phase terms, so that the reflectivity and transmis-sivity necessarily depend on h]. However, just as for thebare slab, there are specific values of h > 0 satisfying
sin(k2h) = -* k2h = sr, s = 1, 2., (31)
which lead to po(h) = p0(O) and ro(h) = ro(O), with po(O) andro(0) the reflectivity and transmissivity of a plane interfacebetween media of refractive indices nl and n3. This state-ment is not sufficient to corroborate the finding of Lakhta-kia et al., 3 since the latter applies for all h.
DISCUSSION
It can be seen from Tables 1-4 of Ref. 3 that the reflectedand transmitted powers of the strip-grating-loaded slab arealways equal to those of the (bare) slab of zero thickness (i.e.,of the plane interface between the two media flanking thestrip-grating-loaded slab). This is why we can summarizethe contents of Ref. 3 in a single statement: a strip-grating-loaded (nonlossy) slab (of arbitrary thickness h > 0) re-sponds (in terms of reflected and transmitted power) tobroadside plane-wave radiation of widely ranging frequencyin a manner identical to that of a plane interface between thesame two media as those flanking the strip-grating-loadedslab. Assuming its truth, this statement would mean thatthe grating produces the effect of wiping out the slab (andleaving only the interface) from the incident beam of light(as concerns reflected and transmitted power).
The result obtained in this paper, which is more plausible,can also be summarized in a single statement: the far-fieldresponse of a strip-grating-loaded slab to broadside TMplane-wave radiation is identical to that of a bare slab of thesame thickness h, flanked by the same two media, for all h,frequencies, and grating periods. It should be noted thatthis statement makes no allusion to the near field, whosebehavior is largely conditioned by the evanescent compo-nents in the plane-wave and modal expansions in Eqs. (7)-(9). We have said nothing of the near field because we havenot attempted to evaluate the coefficients of the evanescentwaves and modes (this could be done by employing, forinstance, the numerical methods outlined in Refs. 1,2,4, and5 or, better yet, those that are explained in Ref. 7).
All that we have discussed until now does not prove thatthe conclusions of Lakhtakia et al.3 are false, nor does itprove that our own conclusions are true (for y = 1 and Oi =00). To accomplish the second of these two tasks, we mustdemonstrate that our solution satisfies the conditions thatactually defined the problem (these were given in the secondsection of this paper). Unfortunately, we cannot do thisbecause we have not found the complete solution for thefield. Nevertheless, we can try out a field ansatz that incor-porates our partial solution. We simply guess that (when y= 1 and Oi 0°) there are no evanescent waves (define C1 and
C3 as the sets of evanescent wave indices in Ml and M 3,respectively) in the half-spaces that flank the slab,
Rno = for n 1,
Tnso = for n 63, (32)
and that all the odd-order modes of the field in the regionbetween the strips disappear; i.e.,
A2p+l = B2p+l = , p = 0, 1, 2.... (33)
If Eqs. (13), (14), (23), (27), and (30) are also taken intoaccount, then
Rn = rn,0, Tn = t0kon, n 0, i1, 2,...,Am = om,o, Bm = 0O6m,O, m 0, 1, 2,...
so that, by virtue of Eqs. (1), (7), and (9),
ul(x) = exp(-iklx 2 ) + Ro exp(iklx 2 ),
u2(x) = a0 exp(ik 2x 2) + fo exp(-ik 2x 2 ),
u 3 (x) = To exp(-ik 3x2 )-
(34)
(35)
This ansatz satisfies Eq. (6), the radiation (outgoing wave)condition, and the condition of local integrability. Since = 1, the boundary conditions in Eq. (4) reduce simply to
12 Ox 2 /
which is satisfied by our ansatz [ 2 in Eqs. (35)], since thelatter does not depend on xl. It will be noticed that Eqs.(35) have the same structure as the field in and around a bareslab. Thus the strips do not make themselves felt withinand outside the slab. This is a peculiarity of the fact thatthe polarization is TM, the incidence is broadside, and thestrips are infinitely thin and perfectly conducting. [For TEpolarization, with uj representing the X 3 component of theelectric field, we would have the boundary condition u 2 (+d/2, x2) = 0; this would not be satisfied by our ansatz unless u 2
= 0 throughout Q2, a condition that would not permit thecontinuity conditions on L1 and L2 to be satisfied.] The onlyremaining conditions [Eq. (3), with b = d, since y = 1] arethose of P1, of which we have already shown our ansatz to bethe solution [with a 0, fib, Ro, and To as defined in Eqs. (13)and (14)]. Thus our ansatz satisfies all the conditions of theproblem and is therefore the desired solution. Since thesolution of Lakhtakia et al.
3 reduces to our solution only forspecial values of k2h (again for y = 1 and Oi = 0), it cannotbe, in general, the right solution for TM polarization. It isalso incorrect for TE polarization, since the strips makethemselves felt in the slab region and therefore diffract whenthe field is oriented in this way.
It should be noted that our ansatz is consistent with thepreviously mentioned results of Kent and Lee and Kobaya-shi for O = 0, since both we and they find that the powertransmitted by the structure, when by = 1 and ni = n2 = n 3, isthe same with or without the grating.
Kobayashi has also made plots of the transmitted powerversus kd for O = 300 and 600 and n 2/n = n 2/n 3 = (1.5)1/2.It can be observed from Figs. 2 and 3 of Ref. 7 that, when hi2d = 0.5 and 1.0, there are several stretches of kd (andtherefore of k 1h, since h/2d is held fixed) over which T = To
Armand Wirgin
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. A 219
1. When Oi = 300, these stretches appear to be somewhatwider that when Di = 600, a fact that may suggest (by extrap-olation) that when Oi = 00 the transmissivity will be equal to1 for all k1 h, in conformity with the prediction of Lakhtakiaet al.3 and-in contradiction to the result presented in thispaper. However, the same results also reveal that a numberof sharp dips occur in the curves and that the number andsharpness of these dips are not significantly different be-tween the cases Oi = 300 and Ds = 600. Extrapolation would,once again, induce us to predict that a certain number ofsharp.dips will also appear in the T-versus-kid curves for Di =
00. This prediction is in contradiction to the results ofLakhtakia et al. (and also to the results presented in thispaper). Thus extrapolation, as it applies to Kobayashi'sresults, is clearly not a sufficient means of proving the truth(or falsity) of the result of Lakhtakia et al. Of course, thisextrapolation is not necessary, since we have proved above,by simple theoretical means, that our solution [embodied inEqs. (34) and (35)] is correct and that the solution of Lakhta-kia et al. is generally false.
Classical optics is usually concerned with the so-calledradiation field, i.e., the field at distances from the interac-tion region that are large compared with the wavelength.This far field plays an important role in an inverse-scatter-ing context. The problem is then to obtain information(geometry, size, composition) about the interaction domainfrom measurements of the light it scatters into the far-fieldzone. A common situation is one in which the light source isalso in the far-field zone (i.e., the incident field is that of aplane wave) and the angle of incidence is held fixed while thedetector is rotated around the interaction domain and/or thefrequency of the incident light is varied. If this method is
used to determine whether a slab contains a strip grating,and if one makes the (unfortuante) choice of a TM wave atbroadside incidence, then the present analysis shows that itis not possible to make this determination unambiguously,since both the loaded slab and the unloaded slab give thesame far-field response. Although this is a particular in-verse-scattering problem, it suggests that, in a more generalcontext, it might be useful (in addition to varying the fre-quency) to modify the incident angle and/or the polarizationto reduce and possibly to eliminate the ambiguity in thedetermination of the physical and geometrical characteris-tics of the scatterer.
REFERENCES
1. A. Wirgin, "On the possibility of non-redundant spectral tunnel-ing through a periodic transmission structure at total reflec-tion-applications to the study of the Smith-Purcell effect,"Opt. Commun. 26, 148-152 (1978).
2. A. Wirgin, "High-efficiency frequency-modulation nonredun-dant scanning of a light beam in a half space," Opt. Commun. 26,153-157 (1978).
3. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Nontrivialgrating that possesses only specular characteristics: normal in-cidence," J. Opt. Soc. Am. A 3, 1788-1793 (1986).
4. R. J. Ikola, A. Hessel, and T. Tamir, "Asymmetry and blazingeffects in corrugated structures," J. Opt. Soc. Am. 63, 408-415(1973).
5. J. W. Heath and E. V. Hull, "Perfectly blazed reflection gratingswith rectangular grooves," J. Opt. Soc. Am. 68, 1211-1217 (1978).
6. W. H. Kent and S. W. Lee, "Diffraction by an infinite array ofparallel strips," J. Math. Phys. 13, 1926-1930 (1972).
7. K. Kobayashi, "Diffraction of a plane electromagnetic wave by aparallel plate grating with dielectric loading: the case of trans-verse magnetic incidence," Can. J. Phys. 63, 453-465 (1985).
