2844 J. Opt. Soc. Am. A/Vol. 11, No. 11/November 1994

Scattering from metallic surfaces havinga finite number of rectangular grooves

Ricardo A. Depine and Diana C. Skigin

Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Pabell6n I, 1428 Buenos Aires, Argentina

Received December 22, 1993; revised manuscript received April 25, 1994; accepted June 9, 1994

A modal theory is presented for solving the problem of electromagnetic scattering from a surface consistingof a finite number of one-dimensional rectangular grooves in a metallic plane. The incident plane wave canbe polarized with either its electric or its magnetic field along the grooves. The formalism is applicable toperfectly conducting materials and to real metals with high (but finite) conductivity. Particular attentionis paid to the changes appearing in the scattering pattern when the conductivity of the structure is changedfrom an infinite value (perfect conductor) to a finite value (highly conducting metal). The excitation of surfacewaves when the incident wave is p polarized is illustrated in some numerical examples that demonstrate thedifferences between the spectral amplitudes corresponding to s and p polarizations.

1. INTRODUCTION

Electromagnetic scattering from surfaces having rect-angular profiles has been studied by many authors.1'10

These surfaces are of interest because they can be manu-factured quite easily and they permit accurate controlof their parameters, making it possible to compare theo-retical results with experimental data.2 Most of the in-vestigations have been devoted to ideal gratings, i.e.,those with strictly periodic, unlimited boundaries sepa-rating two media. In particular, the case of perfectlyconducting materials has been studied by Andrewarthaet al. 3 and by Wirgin and Maradudin4 and the case offinitely conducting materials by Botten et al.5'6 Loewenet al.7 present a comparison of theoretical results andexperiments concerning this kind of profile.

An increasing interest in the use of light scattering as ameans of measuring surface microgeometry (inverse scat-tering) has led to many efforts to study the direct prob-lem in depth. To compare experimental with theoreticalresults, Maystre"l developed a rigorous theory to studythe electromagnetic scattering from perfectly conductingfinite gratings, that is, gratings having a finite numberof grooves. The investigations on diffraction from corru-gated surfaces with finite length have not been restrictedto perfect conductors; highly conducting and dielectricmaterials have also been considered. 2 1 3 There has alsobeen great interest in studying the electromagnetic scat-tering from a single groove or protuberance placed on aninfinite plane surface. Several methods for solving thescattering problem by grooves in a plane are mentionedin Refs. 14 and 15. A method employing boundary inte-gral equations to find the current flowing in a groove isgiven in Ref. 16.

Recently Park et al.10 studied the problem of scatteringfrom rectangular grooves in a perfectly conducting planeby means of a method that uses a modal representationfor the field inside the grooves. The analysis in Ref. 10is valid for equally spaced grooves (a finite lamellar grat-ing) and for s polarization (electric field parallel to the

grooves). In this paper we present a rigorous modal the-ory to solve the problem of light scattering from roughsurfaces having a finite number of randomly distributedrectangular corrugations. Our paper differs from Ref. 10on the following points: (1) the width of each rectangulargroove, as well as the distance between each pair of ad-jacent grooves, can be chosen arbitrarily; this fact makesthe formalism suited for dealing with specially preparedpseudorandom surfaces such as those studied by Hollinsand Jordan2 ; (2) we deal not only with perfectly conduct-ing materials but also with metals with losses; (3) in addi-tion to giving the treatment for s polarization, we includethe treatment for p polarization (magnetic field parallel tothe grooves). In particular, points (2) and (3) are to be in-cluded in the modal theory of scattering from rectangulargrooves if we are interested in studying effects associatedwith the excitation of surface plasmons. Furthermore,as shown by Soto-Crespo and Nieto-Vesperinas,1718 thephenomenon of backscattering enhancement from randomsurfaces is intimately connected with the enhancementof antispecular orders from periodic gratings. The struc-tures considered in the present paper seem particularlysuited to the study of how this phenomenon evolves fromthe infinite to the finite grating case.

The organization of the paper is as follows. In Sec-tion 2 the modal theory and the scattering equationsare presented for a perfect conductor and for a highlyconducting metal. The examples are given in Section 3,where we validate our results against those obtained byPark et al.10 and illustrate the behavior of the scatteredfields in the radiative and nonradiative spectral regionswhen surface plasmons are excited in this kind of surface.Concluding remarks are given in Section 4.

2. THEORY

Consider a metallic plane with L grooves of the sameheight h and widths cl (1 = 1, 2, ... , L). The grooves arealong the i direction, and the y direction is normal tothe surface. The origin of the coordinates is set on the

0740-3232/94/112844-07$06.00 © 1994 Optical Society of America

R. A. Depine and D. C. Skigin

Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. A 2845

The fields inside the Ith groove are expressed in terms ofthe corresponding modal functions for each polarization:

f (x, y) = a sin[,um,l(y + h)]sin[ (x - x)

(9)

ff (x, = bn,1 cos[,ut,(y + h)]cos- (x-xi)

(10)

where

Fig. 1. Configuration of the problem.

[k2 - (7/C)2]1I2Imd= L i[(mlT/Cj)2 - 2]1

if k2 > (7r/C) 2

if k2 < (mr/c) 2

top of the grooves, as shown in Fig. 1. The Ith grooveis separated a distance Ax, from groove ( + 1). A planewave of wavelength A is incident upon the surface fromthe region y > 0, forming an angle Oo with the y axis,the plane of incidence being the x-y plane. Assuming aharmonic time dependence of the form exp(-icot), whereXo is the frequency of the incident light, Maxwell's equa-tions are combined to yield the Helmholtz equation thatmust satisfy the fields E and H everywhere. The vec-torial problem can be separated into two independentscalar problems: s polarization (electric field parallel tothe grooves) and p polarization (magnetic field parallelto the grooves). From now on, we assign f/ ( = , p)to the component along z of either the electric field inthe case of s polarization ( = s) or the magnetic fieldin the case of p polarization ( = p). We separate thespace into two regions (see Fig. 1): the region y Ž 0(+),in which the scattered fields are represented by a con-tinuous superposition of plane waves (Rayleigh expan-sion), and the region inside the grooves, -h • y • 0(-),where the fields are represented with modal expansions.In Subsections 2.A and 2.B we solve the problem for theperfect conductor and for the highly conducting metal,respectively.

A. Perfect ConductorIn the upper region we express the total field as a sum ofthe following three terms:

y 0: f4+(X, y) =finc(x, Y) + f pec(X, y) + fatt(X, Y), (1)

where

fi..(x, y) = exp[i(aox - jfoy)],

fsec(X y) = (-1)iexp[i(aox - oy)],1 for s polarization

= 0 for p polarization

fS/att(x, y) = f_ R/L(a)exp[i(ax + 83y)]da

(2)

and am, and bm,1 are the modal amplitudes correspondingto s and p polarization, respectively.

Matching the fields at the plane y = 0, we obtain thefollowing equations for s polarization:

JR'(a)exp(iaex)daz

L aVnx [ m rr xia-,,xsinXsin(ILmlh)W(x1=1 m=1l m~l [( cl cl

(12)

-2i,o exp(iaox) + if RS(a),3 exp(iax)daL M ir(x - xi) l o~ anz )

= E E a,l,. sinI ]cos(/z,1h),1=1 m=_ cl

where W(s) is defined as

W(s)=j1 if 0 < s < 1otherwise

(13)

(14)

In order to drop the x dependence of Eqs. (12)-(14), weproject Eq. (12) in the set of functions {exp(ia'x)} that areorthogonal in [-o, ] and Eq. (13) in the set of modalfunctions {sin[kvr(x - x)/cj]} that are orthogonal in[xj, x + c], thus getting

L 27rRS(a')= am,1 sin(/-mlh)exp(-ia'xl)Jml(-a'),

1=1 m=1(15)

-2i,3 0 exp(iaoxj)Jkj(aO)

+if RS(cr)/ exp(iaxj)Jk,j(a)daCc

= - uAkjakJ COS(Akjh). (16)

(3) Then, substituting Eq. (16) into Eq. (15), we get an inte-gral equation for the amplitudes Rs(a) (s polarization):

(4)

are the incident, specularly reflected, and scattered fields,respectively, RA(a) are the unknown Rayleigh ampli-tudes, and

ao = k sin o,

,8 = k cos 00,

k = o/ic = 2/A,

, = k2 - a2 .

(5)

(6)

(7)

(8)

L2irRS(a') = i R'(a)/8 E exp[i(a - a')xl]

l1=1

2Jm,1(a)Jm,1(-a')tan(tmlh)x CIY ~ dam=l cl jma

L

- 2ifo I exp[i(ao - a')xl]1=1

2Jmi(aco)Jmi(-a')tan(p.lm,1h) (17

m=1 cl w

x

(11)

R. A. Depine and D. C. Skigin

2846 J. Opt. Soc. Am. A/Vol. 11, No. 11/November 1994

We follow the same procedure to find an integral equationfor RP(a) (p polarization):

2 7rif'RP(a') =L

- RP(a)Y exp[i(a - ')xl]L-x 1=1

X Y. /Lm'Im,(a)l,,(-a')tan(utm,ih)dar

L

- 2 E exp[i(ao - a')xl]1=1

m=O (m,8(18)

Smi(x, y) = { sin[um,l(x - xi)] + cos[ul(x

x [-Kl cos(vp,ly) + sin(vPly)]

= mP'lXmPl(Y) >

where

(vA,1)2 = - (Uyi)2,

s iz

7 kZ

where

J.,j (a) = f exp(iax) sin ( dx,

ImI(a) = f| exp(i ax)cos( cm dx,

cI= /2if m = 0if m O 0

(19)

(20)

(21)

B. Highly Conducting MetalTo solve the diffraction problem for the highly (butfinitely) conducting case, we use a surface-impedanceboundary condition'9 :

Ell = Zn X H11, (22)

where Ell and Hl are the tangential components of theelectric and the magnetic field, respectively, and Z isthe surface impedance. For highly conducting materials,Z - 1/v, with v the complex refractive index. It wasrecently demonstrated that the use of this approximationin the infrared region of the spectrum gives good resultsfor gold wire gratings with geometry (rectangular2 0 andtrapezoidal2 l'2 2) similar to that considered here.

In the upper region the incident and the scattered fieldsare expressed in the same way as for the perfect conduc-tor, and the specularly reflected field is written as follows:

,u = s, p, (23)

where Ao (,u = s or p) are the amplitudes of the fieldsspecularly reflected at a plane located at y = 0. Onefinds these amplitudes by imposing the boundary condi-tions on the conducting plane without grooves.

The field inside the Ith groove is represented by meansof the modal functions that satisfy the impedance bound-ary condition at the walls2 0:

f (x, y) = Dm,ik,I(x, y),m=l

(24)

where DMj are the modal amplitudes. When we imposeon the modal functions 0/,i(x, y) and 0 j(x, y) the bound-ary condition at the bottom of the grooves, we obtain

qO4,1(x, y) = {sin[u',(x - Xi)] + 71'u', cos[u', 1(x - xi)}

x [K', 1 cos(v',Iy) + sin(v,y)]

and the quantities u', 1 and uP,, arefollowing eigenvalue equations:

s SU' 12 7 - 1

tan~up, 1ci) = (ufn,1277Pu-Pm )ta~M~l (UP 1)2 (p)2

determined by the

for s polarization,

for p polarization. (30)

We solve these equations by following the same procedureused in Ref. 20, taking into account that we must solve asmany equations as there are different widths.

Matching the fields at the interface y = 0, we obtain twosystems of equations for the unknown amplitudes Rs(a)and RP(a) for s and p polarization, respectively, whichare very similar in form to the equations obtained for aperfect conductor:

2iTRs(a)(1 - ij7 sf3I)L

i | R3(a ),e exp[i(a - a')xl]

_x ~1=1

x sL ~~X 2 Kml' (QS, )+(a!)(Qs j)-(a/) da

- j 8Rs(a )Gj(a - ')da-xL

: ~~- i~lo(l - Aso) exp[i(ao0 - al')xil

X ` Q~ sQmj) +(ao) (Qm,) -(a'); ~~~~~~m= 1 VM'l Umj l

+ io 7s(I - As))GI(ao - a'),

27rRP(a')(if3' - P)L

= -J RP(a) exp[i(a - a')xl]_x1=1

Px M P

X E KVmIP (flP\)+(a)(Qmi) (a')da

- 7P RP(a)Gz(a - a')da

L- (1 + AP) exp[i(ao - a')x1]

1=1

O P

x E pVmll p (Qmj)'(ao)(Qm,I)Y (ai')ml Km~1UmQ

77 P(l + Apo)GI(ao - a'),

(31)

(26)

(27)

(28)

(29)

R. A. Depine and. D. C. Skigin

flp,,, (x, y = A' exp[i (aox - oy)],

(32)= �M"A41,1(y), (25)

Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. A 2847

(Q,)(a) = ,(x)exp(± iax)dx,0

_ 77svsl cos(vs,lh) + sin(vs,lh)

msl - cos(v',ih) - 7svsl sin(vm,lh)

-7 P sin(vp,lh) + vp,l cos(vplh)VPi sin(vp,1h) - 77P cos(vp,lh)L fxl +cl

1=1 Xl

exp[i(a - a')x]dx,

(33)

(34) 0S

0o

(35)0.1

(36)a<

UA, = f (/,i)2(x)dx.M

-90 -60

(37)

I0 0 . . . . . . . .-30 0 30

Observation angle (deg)

(a)

3. NUMERICAL EXAMPLES

To find a numerical solution for a particular example, wediscretize integral equations (31) and (32) and truncatethe series. The integral is performed in a finite inter-val [ao - T, a + T]. It was found that the length ofthe interval in which the integrals are performed (2T),the number of subintervals into which the integrationinterval is divided (NR), and the number of modal func-tions (NM) necessary for the fields to be represented accu-rately are strongly dependent on the number of grooves,the wavelength, and fundamentally on the polarizationof the incident wave. We have not developed any crite-ria for selecting a priori the numerical parameters. In-stead, we study the convergence of the solution whenNR and NM are varied. An example for 21 grooves ina perfectly conducting plane for p polarization is givenin Table 1. It can be observed that the convergenceis well achieved with the sets of numerical parametersshown. It can be noticed that the increase in the num-ber of modal functions considered (NM) does not changethe solution significantly. Therefore we see that, forthe case illustrated here, NR = 251 and NM = 50 areenough to ensure that the error involved in the calcula-tion is less than 0.4%. If the integration interval is in-creased, an increase in NR is also needed to yield thesame degree of confidence of the results. It was veri-fied that the results are not substantially affected whenT is increased; thus it is not worth expanding the inte-gration interval if we take into account the computationtime that these calculations demand.

In order to validate the method presented here, we per-formed calculations for some of the perfectly conductinggratings studied in Ref. 10. In Fig. 2 we plot log(JRs(0)j)as a function of the observation angle for an s polarizedplane wave of wavelength Ad = 0.666 incident upon asurface having rectangular grooves of width cd = 0.386and height hd = 0.3973. The distance between adja-cent grooves is fixed, Ax/d = 0.614, and the incidenceangle is 19.5°. d = c + Ax is the period of the finite grat-ing in the case in which the surface is finitely periodic.

Table 1. RP(aco)j for T = 2

NR = 151 NR = 251 NR =451

NM = 50 3.217959 3.140012 3.135363NM = 100 3.217962 3.140020 3.135370NM = 150 3.217963 3.140022 3.135371

0.1

0.01-

-90 -60 -30 0 30Observation angle (deg)

60 90

(b)

1*

i 0.1

0.01

-90 -60 -30 0 30Observation angle (deg)

60 90

(c)Fig. 2. og(jRs(0)j) versus observation angle for an s-polarizedincident wave of A/d = 0.666, o = 19.50 on a surface withgrooves of cd = 0.386, hd = 0.3973, and Axid = 0.614:(a) 1 groove, (b) 3 grooves, (c) 21 grooves.

This set of parameters is the same as that correspond-ing to the curves shown in Figs. 4a, 4b, and 4c of Ref. 10for surfaces having 1, 3, and 21 grooves, respectively. Itshould be noticed that the two plots are not comparablein the specular direction, because we have subtractedthe specularly reflected field in the expression of f+. An-other feature to take into account is the fact that we aredealing with plane waves, whereas the curves in Ref. 10were performed with consideration of incident Gaussianbeams. That is the reason that, in our plots, we can ob-serve maxima located at angles of ±90°, as is predicted

where

Gi(a - a') =

60 90

. . . . . . . . . . . . . . . . . .

T

R. A. Depine and D. C Skigin

,\n I

2848 J. Opt. Soc. Am. A/Vol. 11, No. 11/November 1994

- Perfec--- v= 1.7. v= 0.3

..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-60 -30 0 30Observation angle (deg)

(a)

t Conductor plasmon) can be excited along the structure, and the pres-

5 + i8.52 ence of losses can introduce new features in the scatter-ing pattern. To complete the analysis presented above,we show in Figs. 4(a) and 4(b) the curves for the same setof parameters used for Figs. 3(a) and 3(b), respectively,but change the incident polarization from s to p. It canbe noted that the presence of losses not only produces adecrease in the values of IRP(O)j but also changes the po-sition of the minimum in the curves for a single groovefrom a positive to a negative value of 0.

It is well known that p- (but not s-) polarized surfacewaves (surface plasmons) can propagate at the surfacebetween a vacuum and a metal. For a perfectly flat in-terface the spatial variation along the direction of propa-

60 90 gation is in the form exp(iax), the propagation constanta given by2 3

- Perfect Conductor--- v= 1.75 + i8.5....... v=0.32+i2.32

X /

a 2k 1+2 (38)

Taking into account that for metals the real part of aikis greater than unity, light cannot be coupled with asurface plasmon along a single flat interface. However,an important property of these surface waves is their

0.01-

0.001 I---90

Fig. 3. Same(b) 3 grooves.

0.1

-60 -30 6 30Observation angle (deg)

60 90

(b)as Fig. 2 for three different metals: (a) 1 groove,

by the grating equation, whereas in the other curves thefinite width of the incident beam makes the peak movetoward the central maximum. We observe that, in spiteof these differences, the agreement between the curves inFig. 2 and the corresponding curves in Fig. 4 of Ref. 10 isquite satisfactory.

Figure 3 shows the changes in the scattering patternwhen we consider a highly conducting metal instead ofthe ideal case of a perfectly conducting material. InFigs. 3(a) and 3(b) we plot the curves in Figs. 2(a) (asingle groove) and 2(b) (three grooves), respectively, to-gether with those corresponding to i, = 0.32 + i2.32(gold at A = 0.55pu) and V2 = 1.75 + i8.5 (aluminum atA = 0.95,u). As happens in this polarization for perfectlyperiodic, infinite gratings, the presence of losses giveslower reflectivity but does not change the scattering pat-tern qualitatively. We observe that, as the losses in-crease, the asymmetry of the curves with respect to theobservation angle is more pronounced.

The most interesting changes in the scattering patternwhen the conductivity of the metal is decreased take placefor p polarization. In this case a surface wave (a surface

b4o

0.01

-9(

1.:

_ 0.1 -o120.01-

0.001

- Perfect conductor-- v=0.32+i2.32v=1.75+i8.5

.....-. 3

-60' -30 0 -I- 30 Observation angle (deg)

(a)

60 90

Perfect conductor- - v= 0.32 + i2.32* v=1.75+i8.5

-90 -60 -30 6 30Observation angle (deg)

60 90

(b)

Fig. 4. log(IRP(0)1) versus observation angle for a p polarizedincident wave of Aid = 0.666, 0o = 19.50 on a surface withgrooves of cid = 0.386, hid = 0.3973, and Ax/d = 0.614:(a) 1 groove, (b) 3 grooves.

_0.1

04

0.05

0.03 _-90

0.1

0

0

. . . . . .~~~~~~~~~~~~~~~~~~~~~~~

1

. . . . . . . . . .

R. A. Depine and D. C. Skigin

o

- 1; 11 : :.. 1. II .... if::

1. I . .

. : . 1:t

Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. A 2849

- Perfect Conductor...... v=0.32+i2.32

0.20

0.15

0.10

0.05

0.00

0.15

R 0.10

0.05

0.00 -0

-2 -1 0 1a/k

(a)

Perfect Conductor..... v= 0.32 +i2.32

/ ... .....I................

for p polarization depend on the refractive index of themetal considered, whereas for s polarization these quan-tities do not change with the refractive index. The factthat the strong peaks observed in the nonradiative regionfor p polarization correspond to surface waves propagat-ing along the scattering structure is clearly seen if wenote that, for v = 0.32 + i2.32, Eq. (38) gives a/k = 1.10,a value in agreement with the positions of the peak inFig. 6(b).

4. CONCLUSION

A modal theory for solving the problem of electromagneticscattering from plane surfaces having a finite number ofrandomly spread rectangular corrugations was presented.

2 Highly and perfectly conducting metals, as well as p-and s-polarized plane incident waves, were considered.We presented some numerical examples in order to con-trast our results with those obtained by Park et al.10 forperfectly conducting metals in s polarization. Particularattention was paid to the differences appearing in thescattering pattern when the conductivity of the metal isno longer infinite. These differences were shown to bemore dramatic in p than in s polarization. In particu-lar, the excitation of p-polarized surface waves in metal-lic surfaces was illustrated.

As was mentioned above, the phenomenon of backscat-

2.5

.8 0.9 1.0 1.1 1.2 1.3a/k

(b)

Fig. 5. (a) IRS(a) 12 versus ak for s polarization and A/d =0.666, o = 19.5°. The surface has three grooves, of c/d = 0.386,hid = 0.3973, and Axid = 0.614; (b) amplification of a zone ofFig. 5(a).

coupling with light via corrugated surfaces.2 3 We illus-trate this phenomenon in Figs. 5 (s polarization) and 6(p polarization), where we plot curves of IRs(a)12 andIRP(a)12 versus a/k for a finite grating with three grooves,the other parameters being the same as those consid-ered in the previous examples. We observe that thediffraction pattern shown in Fig. 5(a) for s polarizationreproduces the features predicted by the scalar theoryof physical optics and that only the intensity, and notthe spectral position of the maxima and minima, is al-tered when the conductivity of the metal is changed.This happens both in the radiative (a I/k < 1) and inthe nonradiative (I a l/k > 1) regions. The behavior of thecurves of IRP(a)12 versus a/k for p polarization is quitedifferent, as shown in Fig. 6, where we plot this quan-tity for the same parameters considered in Fig. 5. Themost obvious differences between the curves in Figs. 5(a)and 6(a) appear in the nonradiative region: whereas forp polarization we observe strong peaks that almost pre-vent us from observing the variations of IRP(a)l2 in theradiative region, for s polarization the most significantcontributions to jRs(a)12 occur in the radiative region.Furthermore, from comparing the curves in Figs. 5(b) and6(b) we note that the position and width of the peaks

2.0

7 1.5

X 1.0

0.5

0.0

2.5

2.0

1.5

S 1.0

0.5

0.0

- Perfect Conductor.v=0.32+i2.32

-2 -1 0 1 2a/k

(a)

Perfect Conductor.v= 0.32 + i2.32

. . .. . . . . .. . . . . .0. 0.9 1.0 1.1 1.2 1.3a/k(b)

Fig. 6. (a) IRP(a)12 versus a/k for p polarization and Aid =0.666, Ho = 19.50. The surface has three grooves, of cid = 0.386,hd = 0.3973, and Axid = 0.614; (b) amplification of a zone ofFig. 6(a).

Fas,;.'....... , ,,,,,,

- ......... ,,,,,,P-o,,:,

R. A. Depine and D. C. Skigin

2850 J. Opt. Soc. Am. A/Vol. 11, No. 11/November 1994

tering enhancement is closely connected to the excitationof surface waves in this kind of structure. A study of thetransition from the infinite to the finite grating and theintensification of the antispecular peak in such surfaceswill be the subject of future research.

ACKNOWLEDGMENTS

This research was supported by grants from the ConsejoNacional de Investigaciones Cientificas y Tecnicas, Fun-daci6n Antorchas, and the Universidad de Buenos Aires.

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R. A. Depine and D. C. Skigin