Vol. 71, No. 5/May 1981/J. Opt. Soc. Am. 593

Plane-wave expansions used to describe the field diffractedby a grating

J. P. Hugonin, R. Petit, and M. Cadilhac

Facult6 des Sciences et Techniques, Laboratoire d'Optique Electromagn6tique, Equipe de Recherche Associ6edu Centre National de la Recherche Scientifique no. 597, Centre de St.-J6rOme, 13397 Marseille Cedex 13,

France.

Received July 18, 1980; revised manuscript received October 28, 1980

The properties of plane waves are well known, and it is probably because they are that we often try to represent anyunknown field as a combination of such waves. Is such a representation always efficient or even correct? We tryto answer this question for a simple problem arising in the electromagnetic study of gratings. We first summarizein a didactic way some important theoretical results that are probably not well known to those working in optics.Thereafter we report on recently performed numerical experiments. Plane-wave field representations indeed per-mit us to obtain quickly reliable results for some particular profiles. Nevertheless, the methods leading to the solu-tion of an integral equation are the most useful because they are applicable to a larger class of groove profiles.

1. PERFECTLY CONDUCTING-GRATINGPROBLEM

The function f(x) being periodic, with period d = 27r/K, letus consider the surface y = f(x) (the grating surface) illumi-nated by a monochromatic plane wave (wavelength X = 27r/h)under the incidence angle 0 (Fig. 1). We suppose that theelectric field is parallel to the grooves, i.e., to the z axis, andthat the incident wave vector lies in the xy plane. Then,assuming a time dependence exp(-iwt), the determinationof the diffracted field Ed leads to the following problem: finda function Ed (x,y) satisfying an outgoing wave condition fory - + - and such that

AEd + k2Ed =, y > f(X), (1)

Ed[x,f(x)] = -Ei[xf(x)] =-expfih [x sin 0 - f(x)cos 0]}.(2)

The latter equality states that the sum of the incident fieldEi and the diffracted field must be zero on the grating surface.It is easy to prove1 that outside the groove region, i.e., for y >max f(x), Ed may be written as a series

Ed(x,y) = R exp(icnx + i/3ny) = Z Rnl4n(xy),n=-- n=--

(3)

in which an and A, are known coefficients related to thegrating period d and the incidence 0,

a, = k sin b + nK, On = (k2 - an 2 )1 /2, On or /i > 0.(4)

The physical meaning of each term in series (3) is obvious: afunction 4n (xy) represents a propagating reflected wave if

-2-an 2 > or an evanescent wave if k 2 -an 2 < 0. The ef-ficiency en associated with the nth propagating wave isR, 1 7jtp / 0 o.

2. RAYLEIGH HYPOTHESIS

This hypothesis2 consists of assuming that the plane expan-sion [Eq. (3)] is valid not only for y > max f(x) but everywhereabove the grating surface, i.e., in the domain y > f(x), in-cluding in the valley of the groove. The validity of this ex-pansion has been questioned often during the last 20 years,and the topic has provoked heated discussions in meetingsdevoted to electromagnetic waves. The question seems quiteclear today, and the reader interested in theoretical consid-erations can refer to the work of Petit and Cadilhac, 3 Millar,4' 5

Neviere and Cadilhac, 6 and Van den Berg and Fokkema.7 Forour present purpose, it is sufficient to retain only two theo-retical results: for sinusoidal grating [(x) = hi2 cos (Kx)] theRayleigh hypothesis is tenable only for shallow grooves (hid< 0.142); it is generally not valid when f(x) is a piecewise linearfunction. Although interesting from a theoretical point ofview, these conclusions do not settle the dispute; we have noinformation on the error we commit when abusively using theRayleigh expansion [Eq. (3)], and optimists might hope thatthis error is generally small enough to be disregarded in mostpractical problems.

My

VA CUUMAk)'U////

Fig. 1. Notations. The hatched region is filled with a perfectlyconducting'metal. D is a closed subset of the domain y > f(x).

0030-3941/81/050593-06$00.50 ©3 1981 Optical Society of America

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594 J. Opt. Soc. Am./Vol. 71, No. 5/May 1981

3. YASUURA METHOD

In 1973 Ikuno and Yasuura published a paper 8 that we con-sider an important step in the development of electromagneticgrating theories. They indicated that, even when the Ray-leigh hypothesis is not valid (we know that this is generallythe case), it is possible to determine a sequence of linearcombinations of the elementary plane waves 4'n(x,y) ap-pearing in [Eq. (3)] in such a way that this sequence con-verges uniformly to Ed(x,y) in any closed subset of the do-main y > f(x). The reader not yet accustomed to this ques-tion must pay attention to the word sequence, which replacesthe word series used in the previous sections. In order clearlyto understand what follows, he must bear in mind a prelimi-nary comment.

In mathematical physics we often try to approximate afunction u (r) by a linear combination UN of N known func-tions 4', (r). We write

NUN(r) = F cn (N)', 1 (r) (5)

n=l

and hope that the larger N, the better the approximation. Itmust be pointed out that a priori the cn coefficients are al-lowed to depend on N, except if we know from mathematicalconsiderations that u(r) can be expanded in a series of thefunctions I6n (r). In this case and only in this case, we canwrite

NUN= Z an4'n(r), (5')

n=1

in which the first N - 1 coefficients an are the ones associatedwith uN-1.

Let us now summarize the mathematical foundations of theYasuura method. Assuming the amplitude of the incidentfield Ei to be unity, we put

s(x) = -Ei[x,f(x)] = -exp[-iaox - iflof(x)], (6)

and we call Rayleigh functions On those defined by

On (x) = 4n [x,f(x)]- (7)

Now, N being a given positive integer and 1cn I a set of 2N +1 complex numbers, let us consider the integral

+ N 2IN = S - E CnSn(x) dl,

Pn=-N I

in which dl is the arc element on the grating profile (wheneverthe integration interval is not specified the integration mustbe performed over one period). The value of IN is minimalfor a particular set icn} that we call {Y,(N)I. Of course, thisset depends on N; we call its elements Y, (N) the Yasuuracoefficients. In other terms,

+NSN W) = _ Y, (N) Onn

n=-N(8)

is the best least-squares approximation of order N to s (x).It can be shown9"10 by using rather difficult functional analysisthat

lim fSjs(x)-sN(x)1 2 dl = O (9)N-a

and that the desired quantities Rn can be deduced from theYn (N) by a limiting process,

(10)lim Yn(N) = R,.N-

Provided that dl/dx is bounded, these important results re-main true if dl is replaced by dx. Moreover, in any closedsubset of the domain y > f(x), and whichever dl or dx we use,the sequence

+NENd(x,y) = E Yn(N)4"n(x,y)

n=-N(11)

converges uniformly to Ed (x ,y). In other words, this meansthat, for e positive and arbitrarily small, it is possible to findN such that

p E +Nsup Ed(r)- F Y.(N)Oxln(r) < (,r E Di n=-NI

(12)

for any closed subset D of the domain y > f(x). This holdseven if a part of D (or D itself) belongs to the valley of thegroove. Of course the smaller c, the greater N. It must bepointed out that this beautiful result does not imply theconvergence of the Rayleigh series 2nRn'n (r) =7nYn(-)+n (r) at any point of the domain y > f(x). Thisremark must be well understood, which is perhaps difficultfor those who have forgotten the subtleties of functionalanalysis. For them, we end this section with some tutorialremarks that we hope will be sufficiently enlightening.

Let C(n,N) be a family of numbers depending on two in-tegers n and N. We consider the sequence SN = 2'= I C(n,N)and the series with general term un = C(n,c). Clearly theconvergence of the series and that of the sequence may presenttwo different problems:

(1) The sequence may be convergent while the series isdivergent. This occurs if C(nN) = exp[i(27rn/N)], whichimplies that C(n,) = 1. The sequence converges since SN,which is nothing but the sum of the Nth roots of unity, is equalto zero whatever N equals.

(2) The series may be convergent while the sequence isdivergent, as shown by C(n,N) = 1/n2 + 1//INN. Then

C(nco) = 2

SN = N 1 + N ,n=I n2

1 7r2

n1 fl = 6n=i n2 6

lim SN = -N--

(3) The series and the sequence may be both convergent,but they converge toward two different values:

n 1 1n2 N

C(n,) = 2n2

N iSN = Z -+ 1,

n=1 n 2

1 w2

n1 11 = 6n-l n2 62r-

lim SN =-+ 1.N--- 6

4. USE OF PLANE-WAVE EXPANSIONS INGRATING THEORIES

The electromagnetic theory of gratings is now well estab-lished,"', 1,12 and computer codes are available that calculatethe efficiencies not only for perfectly conducting gratings butalso for metal gratings, even when coated with a dielectric.13"14

Hugonin et al.

Vol. 71, No. 5/May 1981/J. Opt. Soc. Am. 595

Usually the problem is reduced to the numerical solution ofan integral equation or a differential system. The situationwas completely different in the early 1960's when the moderntheories of gratings were coming into being: the pioneeringworks were devoted only to perfectly conducting gratings, andthe Rayleigh hypothesis was always clearly or implicitly as-sumed. 15 -19 Starting with the Rayleigh expansion [Eq. (3)],all the authors tried to determine the R, coefficients in orderto satisfy the Dirichlet boundary condition [Eq. (2)]. Ofcourse, because computers do not know series but only finitesums, the authors first had to replace the series appearing onthe left-hand side of [Eq. (2)] by a truncated series with N2P + 1 terms:

+PERn sb (x) E_ R,,i 0x), (13)

n=-- n=-P

W0(x) being defined as in [Eq. (7)].Then they determined the Rn in order to obtain the best fitof the truncated series with the function s(x) = -Ei[x,f(x)].However, the sense of the word best changed from one authorto another. Three main fitting methods have been used.They are briefly described in the next section, and we refer tothem hereafter as the point matching method (PMM), theFourier series method (FSM), and least-squares approxima-tion method (LSAM).

The PMM was abandoned quickly because the other twomethods appeared to be much more efficient. The FSM,which permitted one of us to obtain reliable numerical resultsfor sufficiently shallow groove gratings, 18 later proved to beunable to predict the properties of all commercial gratings,and this failure gave birth to integral and differential meth-ods.20 The LSAM was developed primarily by Meecham, whoused it successfully for some shallow gratings with triangularprofiles.'5 Unfortunately, to the authors' knowledge, nosystematic attempts have been made by Meecham to specifythe limits of the method's domain of applicability, and, as withthe other two methods, the LSAM has been gradually for-gotten because of the amazing successes of integral and dif-ferential methods. It was suddenly brought back into fashionin 1973 when two papers 8' 9 presented it as an exact methodfounded on the nontrivial theorems stated in Section 3. Theuse of plane wave expansion, which, no doubt, is much morefamiliar to opticists than integral equations, seemed to bereinstated. Since the simple algorithm used by Meechammore than 20 years ago had now received the support of the-oreticians, was it still necessary to go on using the much moresophisticated methods proposed by the specialists of gratingtheory since 1965? In our opinion and according to currentpublished results, numerical experiments are needed to an-swer this question. Obviously, among the several methodswhose convergence has been demonstrated, the best is the onethat leads fastest to reliable numerical results. From apractical point of view, the study of a series does not end withproof of its convergence; the speed of this convergence is alsoan important question. Moreover, we must not forget thata method whose theoretical foundations are questionable canbe efficient for solving many practical problems (in optics theHuygens-Fresnel principle is a good example).

We therefore report in Section 6 some computations wehave performed recently with an IBM 370 168 computer inorder to compare, from the practical point of view, two rig-orous methods (the integral method and the LSAM) and twoapproximate methods (the PMM and the FSM). In order to

try to eliminate the influence of round-off errors, we have useddouble precision. Of course, such a comparison is not a simplematter, and evidently the results depend greatly on the criteriaone chooses. For instance, it would be interesting to take intoaccount for each method the possibility of an extension to theother case of polarization (H parallel to the grooves) and toa finite conductivity. This has not been done because, fromthe theoretical point of view and to our knowledge, such ex-tensions are not straightforward generalizations.

5. DESCRIPTION OF PLANE-WAVEEXPANSION METHODS

In the particular case of normal incidence that we use here forthe sake of simplicity, the three methods can briefly be de-scribed as follows:

(1) Point Matching Method. The difference between thetruncated series [Eq. (13)] and s(x) = -Ei[x,f(x)] is writtenas zero at N = 2P + 1 equidistant points of the period interval.One gets the linear algebraic system

+P

E Rn(P).(xi) =S(Xi);n=-P

Xi = X1 ,X2 , ... XN. (14)

(2) Fourier Series Method. Noticing that both thetruncated series and s (x) are periodic functions with periodd, we write the equality of their first N Fourier coefficients.So we again obtain a linear system of N = 2P + 1 equa-tions,

i=-PI... ,..P, (15)+PX, Rn (P) /)n (i = £ (i);

n =-p

in which £(i) and n (i) are complex numbers defined in Table1.

(3) Least-Squares Approximation Method. The least-squares approximation principle has been given in Section 3.As is well known, the best least-squares approximation to s (x)by a linear combination of the N = 2P + 1 functions sb,, (x) isthe projection of s (x) on the subspace spanned byO-p, ... ,00, ... lop (the projection being considered in theusual L2 Hilbert space). The Rn (N), i.e., the Yasuura coef-ficients, are therefore such that

+PE Rn(P) JS4n(x)(i(x)dx = fs(x)0i(x)dx. (16)

no-P(4) Conclusion. Whatever the method, we are led to the

solution of a linear system of equations

OR = S, (17)

where R and S are column matrices and 0 is a N X N square

Table 1. Determination of the Matrices S and 0 a

PMM FSM LSAM

Sn s(xn) 9(n) =ds(x)exp(-inKx)dx dSs(x)_ 0n(x)dx

On, Om (tmXn) L (n) =- f0 - Wh x (In (x) dxd d

X Sf4)(x)exp(-inKx)dx

alf, when using the LSAM, one replaces dx by dl, one obtains different valuesfor the RnI(N), but, in both cases, Rn(N) tends to the same value when N tendsto infinity.

Hugonin et al.

Hugonin et al.596 J. Opt. Soc. Am./Vol. 71, No. 5/May 1981

Table 2. Elements of MatricesS and o as Defined inTerms of Bn (u)

FSM LSAM

S., -B,,( h2) -S. |9(1 + r )

on1 Bn 0-i (9 13n) Bn-ni [9(lnW-IB3)

matrix. The elements of R are the unknown coefficientsR., (N), and Table 1 gives the elements SI, and 1m of S and'p.

We find it convenient to adopt X/27r as the unit of length(k = 1), and we write the function f(x), which describes thegrating profile in the normalized form, as

f(x)=- g(Kx), (18)

where g is a function with period 27r and such that max g -

min g = 2. In this way, h always represents the groove depth.We have

0,x = exp inKx + if3,, hg(Kx)], (19)

s(x) = -exp-i h g(Kx)], (20)

and, by setting

1 p2ffB,,(u) =-T J exp[iug(t) - int]dt, (21)

27r o

we get, after some algebraic manipulations, the results sum-marized in Table 2.

For a given type of profile, i.e., a given function g,it istherefore recommended that we determine the functionsB,,(u). For instance, sinusoidal gratings are linked withBessel functions; indeed, if g(t) = cos t, we obviously get

B. (u) = J. (u), (22)

at least if we use dx (not dl) to perform the integrals appearingin the last column of Table 1.

The determination of B,, (u) is also easy, although perhapsa little more tedious, for triangular profiles. For the sym-metric profile represented in Fig. 2 tg(t) = 1 - (2/7r)lt -

(7r/2)1], the reader verifies that

B, (u) = 4u s- i 2 2 s - 2i) (23)

6. SOME NUMERICAL EXPERIMENTS

For several types of profiles, we have solved the gratingproblem for normal incidence by using the PMM, the FSM,the LSAM, and the integral method (IM). For the IM weemploy a computer program written some years ago by ourcolleague D. Maystre. This program also leads to the solutionof a system of N linear equations; our experience inclines usto trust it insofar as it gives efficiencies e,, that are in goodagreement with the famous energy-balance relation,namely

I-1

I IY

. - -An ir

N1-

Fig. 2. The graph of g(t) for symmetric triangular profiles.

1 E e, = 0,11 C lI

(24)

the sum being performed only for propagating diffractedwaves, i.e., for those values of n that belong to the set U de-fined by k2 - a , 2 > 0.

For each particular profile we start by computing the R,, (N)coefficients using Maystre's program, It appears (Table 3)that, provided that N is sufficiently large, we are always ableto verify relation (24) within an accuracy better than 10-7.We consider the corresponding coefficients R, (N), which wecall R, 0, a good approximation of the exact solution. (In anycase they are, in our opinion, the best approximation we haveobtained.) Then, when we calculate the approximate coef-ficients R0 (N) by one of the methods we want to check, wecompute

(25)e = E IRI(N)-R 00 12 ,

n C Li

and we adopt y as a measure of the error.Our conclusions depend strongly on the profile shape.

They are summarized below for a fixed value of the wave-length-to-period ratio (X/d = K/k = 0.437), which correspondsto five propagating diffracted waves (n = -2,-1,0,+1,+2).The depth-to-period ratio hid has been varied from 0 to0.5.

Sinusoidal Profile: y = h/2 cos(Kx).Let us first recall that, theoretically speaking, the PMM andthe FSM are expected to be efficient only if hid < 0.142 (seeSection 1). In fact, numerical experiments show that thePMM is usable provided that h/d < 0.2 and that the FSM stillconverges even if hid reaches 0.5. This unexpected perfor-mance was observed in 1966, ' but, at that time, probably be-cause we used single precision and because the Bessel func-tions were not so well computed as they are now, we reporteda value of 0.3 (instead of 0.5). The LSAM converges whateverhid equals, as expected from theoretical considerations (seeSection 3). It is worth noting that, from the computation-time point of view and for all the profiles we have studied (hid< 0.5), the FSM is better than the LSAM and even better thanthe integral method (Table 4). We confess that this excep-tional efficiency of FSM remains a mystery to us, whichsuggests a need for new mathematical reflections.

Symmetrical Triangular ProfileIn accordance with theoretical predictions, the LSAM alwaysconverges. Although the PMM and the FSM are generallythought to be poor methods (because the Rayleigh hypothesisis not valid here), they provide reliable results (similar to thoseobtained by the LSAM or the IM) for very shallow gratings

Vol. 71, No. 5/May 1981/J. Opt. Soc. Am. 597

(let us say hid < 0.14 for the PMM and hid < 0.20 for FSM).If hid < 0.20 the performances of FSM and LSAM are roughlythe same, and they are better than the performances of the IM.If 0.20 < hid < 0.30 the IM and the LSAM are practicallyequivalent (Table 5). Finally, for relatively deep gratings hid> 0.30, the IM is again more powerful (Table 6).

Symmetrical Trapezoidal ProfileThis type of profile has been used to check the effect of re-placing dl by dx in the integral IN, which we have to minimizewhen using the LSAM. This effect, which evidently is nullif we deal with symmetrical triangular profiles, is difficult tostudy in the case of a sinusoidal profile because, when we usedl, the integrals in Table 1 do not reduce to Bessel functions.The data that we have obtained for the symmetrical trape-zoidal profile (and that we do not report here for sake of

brevity) show that the capabilities of the LSAM seem prac-tically independent of whether we use dl or dx.

7. CONCLUSION

At least for perfectly conducting gratings and for one of thetwo fundamental cases of polarization (E parallel to thegrooves), there exists a method in which the diffracted fieldis written as a sum of plane waves that, theoretically speaking,permits us to solve the grating problem whatever the grooveprofile is [it has only to be the graph of a periodic functionf(x)]21. This method (LSAM), recently developed by Yasu-ura and his co-workers at Kyushu University, rests on somenontrivial theorems of functional analysis, but it leads to asimple numerical algorithm that a priori looks agreeable for

Table 3. Numerical Experiment Using the Integral Method for a Symmetrical Triangular Profile (h/d =1/2) a

N 1- j en e t eo ei e2n G U

10 8.29 10- 4.58 10-2 0.08 3.69 10-1 1.27 10-1 1.84 10-'20 8.43 10-5 3.35 10-4 0.48 4.86 10-1 9.74 10-2 1.60 10-'30 1.39 10-5 6.09 10-5 1.48 4.72 10-1 1.00 10-' 1.63 10-'40 2.06 10-6 3.57 10-6 3.22 4.75 10-1 9.93 10-2 1.63 10-'60 2.52 10-7 3.02 10-7 10.22 4.76 10-1 9.92 10-2 1.63 10-'80 6.07 10-8 2.42 i0-8 23.98 4.76 10-1 9.92 10-2 1.63 10-'

100 2.10 10-8 0 46.77 4.76 10-1 9.92 10-2 1.63 10-1

a The computation time is given in seconds. N, e, and en are defined in the text. Only three figures have been retained to give the values of the efficienciesen .

Table 4. Comparison of Three Methods for a Sinusoidal Profile (hid = 0.40) a

FSM LSAM IMN 1- Z en e t 1-Z en e t 1-Z en f t

nE U nE U n U

5 1.41 10-2 1.73 10-4 0.02 2.84 10-1 5.75 10-2 0.03 3.89 10-1 8.24 10-1 0.0110 1.10 10-5 2.50 10-8 0.07 6.80 10-2 4.26 10-3 0.07 4.38 10-3 7.12 10-3 0.0815 5.19 10-1t 2.68 10-8 0.17 1.72 10-2 3.58 10-4 0.17 2.49 10-4 4.28 10-4 0.2120 2.83 10-a 2.67 10-8 0.31 8.63 10-3 9.89 10-5 0.31 1.44 10-5 2.33 10-5 0.84

a Again t is the computation time in seconds.

Table 5. Comparison of Three Methods for a Symmetrical Triangular Profile (hid = 0.29)

FSM LSAM LMN 1- en f t 1- Z en f t 1- E en f t

nUt n GU n eU

5 5.59 10-'1 6.70 10-4 0.02 1.22 10-2 7.86 10-4 0.02 1.91 10-2 5.09 10-2 0.0210 9.91 10-2 2.19 10-2 0.07 3.79 10-3 1 43 10-4 0.07 1.99 10-4 1.53 10-3 0.0815 1.90 2.21 0.17 1.99 10- 5.24 10-5 0.17 2.30 10-5 7.33 10-5 0.2320 30.4 26.5 0.31 1.20 10-3 2.47 10-5 0.21 2.80 10-5 7.50 10-5 0.4930 28.0 25.2 0.71 1.07 10-3 2.07 10-5 0.71 1.07 10-6 3.49 10-6 1.41

Table 6. Same Profile as That in Table 5 (hid = 0.4)

FSM LSAM LMN 1- Z en f t 1-Z en e t 1- en f t

n C U nCU nCU

5 6.79 10-2 7.61 10-' 0.02 1.42 10-1 3.10 10-2 0.02 1.97 10-1 6.47 10-1 0.0210 15.8 18.8 0.07 5.22 10-2 7.01 10-3 0.07 2.67 10-3 1.39 10-2 0.0815 33.9 33.6 0.17 2.91 10-2 3.02 10-3 0.17 1.90 10-4 5.60 10-4 0.2320 35.4 34.9 0.31 2.52 10-2 2.47 10-3 0.31 6.23 10-5 1.85 10-4 0.4830 34.0 33.9 0.71 2.35 10-2 2.23 10-3 0.71 7.40 10-6 2.17 10-5 1.49

Hugonin et al.

598 J. Opt. Soc. Am./Vol. 71, No. 5/May 1981

physicists. Unfortunately, the most simple algorithm is notalways the most efficient means for solving a given problem.From our experience it turns out that the LSAM, which isefficient for shallow groove gratings, is not so powerful as theintegral method for numerous profiles we encounter fre-quently in practice. In view of our recent numerical attempts,we cannot recommend using this method to write a computerprogram that attempts to be general. Nevertheless, thetheoretical considerations given in Section 3 must not beforgotten; they are valuable and can be used to clarify manysimilar problems in other fields of physics.

We thank the referee for having drawn our attention onthree papers22 -24 recently published in a French-languagejournal, in which the author also studies the numerical powerof methods based on plane-wave expansions. They containsome conclusions similar to ours.

This work was supported by the Direction des RecherchesEtudes et Techniques under contract no. 78/1148.

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20. R. Petit, "Contribution A l'Atude de la diffraction par un reseaumetallique," Rev. Opt. 45, 249-76 (1966).

21. It is worth noting that a numerical study has been performed byKahlor for the corrugated grating: IEEE Trans. Antennas Pro-pag. AP-2, 657 (1973).

22. A. Wirgin, "Sur la th6orie de Rayleigh de la diffraction d'une ondepar une surface sinusoidale," C. R. Acad. Sci. B 288, 179-182(1979).

23. A. Wirgin, "Aspects numeriques du probleme de la diffractiond'une onde par une surface sinusoidale," C. R. Acad. Sci. B 289,273-276 (1976).

24. A. Wirgin, "Sur trois variantes de la theorie de Rayleigh de ladiffraction d'une onde par une surface sinusoidale," C. R. Acad.Sci. A 289, 259-262 (1979).

Hugonin et al.