The shift of Brewster's scattering angle

The shift of Brewster's scattering angle

Tetsuya Kawanishi *

Communications Research Laboratory, Ministry of Posts and Telecommunications, 4-2-1 Nukui-Kitamachi, Koganei, Tokyo 184-8795,

Japan

Received 9 August 2000; received in revised form 2 October 2000; accepted 16 October 2000

Abstract

The multiple scattering e�ect at Brewster's scattering angle, which is the direction of a dip in a p-polarized scattering

pro®le from a slightly random dielectric surface, is studied by using the stochastic functional approach. It is shown that

multiple scattering processes cause a shift in Brewster's scattering angle, and that the ratio of co- to cross-polarized

scattering does not depend on the roughness of the surface at Brewster's scattering angle of the ®rst order scattering. We

also investigate Brewster's scattering angle in lossy media, and show that the shift due to the multiple scattering e�ect

depends on the loss of the scatterer. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Recently, this author formulated electromag-netic (EM) scattering from a two-dimensionalslightly random surface by using the stochasticfunctional approach, and also introduced vectorformalism that allows the convenient manipula-tion of vector equations in the same way as inscalar formalism [1]. By using vector formalism,several speci®c phenomena in incoherent scatter-ing from a slightly random dielectric surface havebeen clari®ed, such as the quasi-anomalous scat-tering and Brewster's scattering angle [2±4].

Brewster's scattering angle, which is the direc-tion of a dip in a p-polarized scattering pro®lefrom a slightly random dielectric surface, dependsboth on the incident angle and on the refractiveindex of the scatterer. It is related to the ordinary

Brewster's angle of re¯ection for a ¯at plane, infollowing way. When the incident angle is equal tothe ordinary Brewster's angle, Brewster's scatter-ing angle also is equal to the ordinary Brewster'sangle, as well. In the case of random scatteringfrom a slightly rough metal surface, there is no realsolution to the equation that gives Brewster'sscattering angle. However, by introducing an op-tically denser medium, which can convert anevanescent wave at the random surface into a ra-diative one, the dips due to Brewster's scatteringangle can be seen in the scattering pro®les in theoptically denser medium [5].

For coherent scattering, the shift of Brewster'sangle due to multiple scattering has been reported[6±9]. The coherent scattering factor, the scatteringe�ect on the average scattering wave ®eld, is notzero at Brewster's angle for a ¯at surface, whileFresnel's coe�cient of p-polarized re¯ection is.Similarly, we can consider the shift of Brewster'sscattering angle due to multiple scattering e�ect inincoherent scattering. Brewster's scattering angle

15 December 2000

Optics Communications 186 (2000) 251±258

www.elsevier.com/locate/optcom

* Fax: +81-42-327-6106.

E-mail address: [email protected] (T. Kawanishi).

0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S00 3 0-4 0 18 (0 0 )0 10 7 5- 0

can be derived from the ®rst order scattering anddoes not depend on the roughness or on the cor-relation length of the rough surface [2]. While the®rst order scattering intensity is zero at Brewster'sscattering angle, the higher order scattering in-tensity does not equal zero at this angle. Thus,Brewster's scattering angle may shift when thehigher order scattering, which is corresponding tomultiple scattering, is considered. Furthermore, ifthe scatterer has some degree of loss, the mini-mum value at Brewster's scattering angle is notequal to zero, and a shift of Brewster's scatteringangle may occur, since the equation representingit does not give a real solution for complex re-fractive indices.

In this paper, to investigate the multiple scat-tering e�ects at Brewster's scattering angle, EMscattering from a slightly rough dielectric surface isstudied by using the stochastic functional ap-proach. We demonstrate that Brewster's scatter-ing angle shifts as considering the second orderscattering. We also show that the ratio between co-and cross-polarized scattering intensity at Brew-ster's scattering angle of the ®rst order scatteringangle does not depend on the roughness of thesurface, but does depend on the correlation lengthof the surface. We also discuss the e�ect due to anyloss or gain of the scatterer, and deal with a three-layered structure that has an optically denser me-dium which can convert an evanescent wave at therandom surface into a radiative one. We demon-strate that the relation between the shift due to themultiple scattering e�ect and surface shape pa-rameters strongly depends on the degree of loss ofthe scatterer.

2. Brewster's scattering angle

Consider the scattering of an EM wave, asshown in Fig. 1. The wave is incident from Med. 1to the random interface between Med. 1 and Med.2, which denote the upper and lower media withrespect to the random surface. The re¯ective indexof Med. 2 with respect to Med. 1 is represented byn. The random surface is assumed to be a homo-geneous Gaussian random ®eld, de®ned in our

previous work [2], where l is the correlation length,and r, the roughness of the random surface. Theincident plane wave comes from the direction�h0;/0�, where h0 is the zenith angle, and /0, theazimuth angle. Similarly, �hs;/s� represents thescattering direction. We assume that the incidentwave is monochromatic and the wave number isrepresented by k. We use the coordinates at whichthe incident plane corresponds to the xz plane, i.e./0 � 0°.

In the stochastic functional approach, a scat-tered wave is expressed in terms of Wiener kernels[1,2]. The Nth order (N > 0) Wiener kernel isroughly proportional to rN , so that the ®rst orderscattering, corresponding to the ®rst order Wienerkernel, is dominant and the higher order Wienerkernels are negligible, when the roughness is smalland the multiple scattering e�ect is not signi®cant.If we consider scattering pro®les of the ®rst orderWiener kernels, zero points exist in p-polarizedscattering produced by p-polarized incidence. Thedirection of the zero in Med. 1 is expressed by [2,3]

hs � HB1�h0; n�

� sinÿ1 n sin tanÿ1 1

n2 tan�sinÿ1�nÿ1 sin h0��

!( )" #;

/s � 0°;�1�

and the direction in Med. 2 is

Fig. 1. Coordinate system for random scattering from a rough

surface.

252 T. Kawanishi / Optics Communications 186 (2000) 251±258

hs � H0B2�h0; n�

� sinÿ1 nÿ1 sin tanÿ1 1

tan�sinÿ1�nÿ1 sin h0��

!( )" #;

/s � 180°:�2�

Eqs. (1) and (2) can be rewritten as follows [2]:

tan h00 tan H0B1 � nÿ2; �3�

tan h00 tan H0B2 � 1; �4�where h and h0 denote the angles in Med. 1 andMed. 2, respectively, and they are connected bySnell's law

n sin h0 � sin h �h � h0;HB1;HB2�: �5�If we make a particular choice h0 � HB1, from Eqs.(3) and (5) we obtain

tan h0 � tan HB1 � n; �6�which means that, in the case of incidence from theordinary Brewster's angle (h0 � tanÿ1 n), Brew-ster's scattering angle also equals the ordinaryBrewster's angle. This equation also means thatthe direction of dipoles induced on a mean ¯atsurface between Med. 1 and 2 equals HB1. Theorigin of the ordinary Brewster's angle is explainedas a zero point of the radiation pattern of the in-duced dipoles [10]. However, Brewster's scatteringangles is expressed by a product of two tangentialfunctions, while the expression for the ordinaryBrewster's angle contains only one tangentialfunction, so that it cannot be explained in the sameway as in the ordinary Brewster's angle. In addi-tion, Brewster's scattering angle for vertical inci-dence h0 � 0° has a connection with the criticalangle of the total re¯ection. In the case of n < 1,from Eq. (3), we obtain HB1�0; n� � sinÿ1 n, whichmeans that Brewster's scattering angle in Med. 1equals the critical angle hc � sinÿ1 n. Similarly, inthe case of n > 1, H0B2�0; n� � sinÿ1 nÿ1 is obtainedfrom Eq. (4) which means that Brewster's scat-tering angle in Med. 2 equals the critical anglehc � sinÿ1 nÿ1.

HB1 and H0B2 are decreasing functions of h0, andsatisfy

HB1 HB1�h0; n�; n� � � h0; �7�

H0B2 H0B2�h0; n�; 1=n� �

� h0: �8�

Eqs. (7) and (8) mean the reciprocal theorem inrandom scattering. As a result, Eq. (1) gives a realvalue only when HB1�90°; n� < h0 < 90°. In thefollowing section, Brewster's scattering angles, HB1

and H0B2, given by Eqs. (1) and (2) are called theunperturbed Brewster's scattering angles. We willdiscuss shifts of Brewster's scattering angles due tothe multiple scattering e�ect and to the loss or gainof Med. 2.

3. Multiple scattering e�ect

To investigate the multiple scattering e�ect atBrewster's scattering angle, incoherent scattering iscalculated by using the Wiener kernels up to thesecond order. For numerical calculations, we as-sumed that n � 1:51 and h0 � 60°, so that theunperturbed Brewster's scattering angle HB1 is53:284°. Scattering intensity pro®les are shown inFigs. 2 and 3. There are dips near the unperturbedBrewster's scattering angle, and the minimumvalues do not reach zero, while the scattering in-tensity derived from the ®rst order approximationis zero at Brewster's scattering angle [2]. We thus,rede®ne Brewster's scattering angle as the directionof the dip near the unperturbed Brewster's scat-tering angle. In Fig. 4, Brewster's scattering angleis shown as a function of the roughness, r, and in

Fig. 2. Scattering intensity pro®les for kl � 1:0.

T. Kawanishi / Optics Communications 186 (2000) 251±258 253

Fig. 5, of the correlation length, l. The shift ofBrewster's scattering angle is usually negative.When r increases Brewster's scattering angle de-creases, and when l increases it approaches theunperturbed Brewster's scattering angle. Thesebehaviours are similar to those in the shift ofBrewster's angle [6±9]. Fig. 6 shows the shift ofBrewster's scattering angle as a function of theincident angle, h0. The shift is positive and variesremarkably when h0 ' HB1�90°; 1:51� � 42:159°,while it is negative for h0 > 50°. We note that thenumerical results shown here contain the e�ect ofthe mass operator, which is represented by C inour previous work [2], but this operator does nota�ect the shift of Brewster's scattering angle. Figs.7 and 8 show the ratio of co- to cross-polarized

Fig. 5. Brewster's scattering angle as a function of correlation

length of surface for kr � 0:1.

Fig. 6. The shift of Brewster's scattering angle as a function of

incident angle for kl � 1:0.

Fig. 7. Ratio of co- to cross-polarized scattering intensities for

kl � 1:0.

Fig. 3. Scattering intensity pro®les for kr � 0:1.

Fig. 4. Brewster's scattering angle as a function of roughness of

surface for kl � 1:0.


scattering intensity for h0 � 60° as a function ofthe roughness, r, and the correlation length, l. Atthe unperturbed Brewster's scattering angle, thisratio does not depend on r (see Fig. 7), but doesdepend on l (see Fig. 8). The ®rst order Wienerkernel vanishes at this angle so that the ratioequals that of the second order scattering. Theroughness, r, does not a�ect the ratio between co-and cross-polarized components of the Wienerkernels [1,2]. Thus, this ratio does not depend onthe roughness. This is why all curves in Fig. 7 crossat the same point. However, the ratio may dependon r when the roughness is so large that we cannot disregard the higher order Wiener kernels.

4. Scattering from lossy media

Consider scattering from a medium with somedegree of loss or gain. The refractive index of Med.2 is expressed by a complex number n � nr � ini.Unless ni is zero, Eqs. (1) and (2) do not give realvalues. Fig. 9 shows a p-polarized scattering pro-®le calculated by the ®rst order Wiener kernels fora random surface of n � 1:51� 0:151i, kr � 0:1and kl � 1:0. There is not a zero point due toBrewster's scattering angle from a medium with aloss or gain, even if we disregard the multiplescattering e�ect. We can also rede®ne Brewster'sscattering angle as the direction of a dip nearBrewster's scattering angle for a lossless medium.

The shift of Brewster's scattering angle and scat-tering intensity at this angle are shown in Fig. 10,as functions of ni, where ni < 0 means Med. 2 hasgain. Brewster's scattering angle depends on theloss or gain of the dielectric scatterer, and is aneven function of ni. The shift of Brewster's scat-tering angle is positive, while the shift due tomultiple scattering is negative. In addition, the p-polarized scattering intensity at Brewster's scat-tering angle is also an even function. By contrast,when scattering is from a metal surface, the scat-tering intensity at Brewster's scattering angle is nota symmetric function, but Brewster's scatteringangle is still an even function [6]. Brewster's scat-tering angle in scattering from metal surfaces canbe observed in a three-layered structure shown inFig. 11. Med. 3 is an optically denser medium,which can convert an evanescent wave at therandom surface into a radiative wave. The un-perturbed Brewster's scattering angle on a losslessmetal surface can be expressed by [5]

HBm�h0m; n� � sinÿ1 ang

��n2

g sin2 h0m � a2

n2g sin2 h0m�a4 ÿ 1� ÿ a2

vuut24 35;�9�

where the refractive index of Med. 2 is ia. A part ofor all of the scattering process which showsBrewster's scattering angle of a metal surface isalways evanescent at the random surface, whilethat of a two-layered dielectric scattering structureas shown in Fig. 1 is radiative.

Fig. 9. p-polarized scattering pro®le in Med. 1.

Fig. 8. Ratio of co- to cross-polarized scattering intensities for

kr � 0:1.


To illustrate the e�ect of evanescent waves on arandom surface, we consider scattering from adielectric rough surface with the three-layeredstructure shown in Fig. 11. The rough surface canbe illuminated by an evanescent wave, and thescattered wave into an evanescent wave can beconverted into an radiative wave in Med. 3. Thezero due to Brewster's scattering angle can be seenin the scattering pro®les in Med. 3. By using of Eq.(1) and Snell's law

ng sin h0m � sin h0; ng sin HBm � sin HB1; �10�

where h0m denotes the incident angle in Med. 3, weget the expression for Brewster's scattering anglein Med. 3 as follows:

HBm�h0m; n�

� sinÿ1 nnÿ1g sin tanÿ1 1

n2 tan�sinÿ1�nÿ1ng sin h0m��

!( )" #:

�11�The refractive index of Med. 3 with respect toMed. 1, denoted by ng, should be greater thanunity. As shown in Fig. 12, Brewster's scatter-ing angle in Med. 3 exists even if h0m or HBm >

Fig. 12. Brewster's scattering angles HB1 and HBm as functions

of incident angle, for refractive index, n � 1:51.

Fig. 10. The shift of Brewster's scattering angle in Med. 1, for h0 � 60°, kr � 0:1 and kl � 1:0. Solid line and dashed line denote

Brewster's scattering angle and the scattering intensity at Brewster's scattering angle, respectively.

Fig. 11. Scattering structure with an optically denser medium.


sinÿ1�1=ng�. In these cases, the incident or thescattered wave in Med. 1 corresponds to an eva-nescent wave. Fig. 13 shows Brewster's scatteringangle and the scattering intensity at this angle asfunctions of ni calculated by using the ®rst orderWiener kernels. The refractive index of Med. 3, ng,is assumed to be 1:255, so that the critical angle ofthe total re¯ection on the interface between Med. 1and 3 is 52:827°. We consider two cases: h0m � 50°and h0m � 60°. In the case of h0m � 50°, HBm �34:36°, so that both incident and scattered wavesin Med. 1 are radiative. As shown in Fig. 13, bothBrewster's scattering angle and the intensity areeven functions of ni. On the other hand, in thecase of h0m � 60°, which corresponds to evanescentwave incidence at the rough surface, HBm � 27:95°,and the intensity at Brewster's scattering angle isnot symmetric. This is similar to scattering from ametal surface [5]. Thus, we may say that this dif-ference is caused by evanescent waves at the ran-dom surface.

We also investigated the multiple scattering ef-fect in scattering from lossy media by using the®rst and the second order Wiener kernels. In Figs.14 and 15, Brewster's scattering angle for h0 � 60°and nr � 1:5 is shown as a function of r and l.Brewster's scattering angle of a lossless medium isa monotonically decreasing function of r and amonotonically increasing function of l (see Figs. 4and 5). However, if the loss is not small, Brewster's

scattering angle is an increasing function of r andis not a monotonic function of l. The relation be-tween the shift of Brewster's scattering angle andthe parameters of the surface shape �l; r�, stronglydepends on the imaginary part of the refractiveindex.

5. Conclusion

By using the stochastic functional approach,we studied incoherent scattering from a slightly

Fig. 13. Brewster's scattering angle and scattering intensity at

Brewster's scattering angle for a three-layered structure as

functions of imaginary part of refractive index of Med. 1.

kd � 1:0, where d is the thickness of Med. 1.

Fig. 14. Brewster's scattering angle on lossy surfaces as a

function of roughness of surface for kl � 1:0.

Fig. 15. Brewster's scattering angle on lossy surfaces as a

function of correlation length of surface for kr � 0:1.


random dielectric surface. The shift of Brewster'sscattering angle due to multiple scattering e�ectwas shown by using numerical calculations. Weconcluded that the ratio of co- to cross-polarizedscattering does not depend on the roughness of therandom surface at the unperturbed Brewster'sscattering angle. We also illustrated the shift dueto a loss or gain of the scatterer, and the e�ect ofevanescent wave at the rough surface. In addition,the shift due to multiple scattering has been shownto be highly dependant on the loss of the scatterer.

Acknowledgements

We would like to express our appreciation toProf. H. Ogura and Dr. M. Izutsu for their en-couragement.

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The shift of Brewster's scattering angle