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1288 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
Scattering of light from the random interfacebetween two dielectric media with low contrast
Tamara A. Leskova
Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142092, Russia
Alexei A. Maradudin and Igor V. Novikov
Department of Physics and Astronomy and Institute for Surface and Interface Science,University of California, Irvine, Irvine, California 92697
Received August 31, 1999; revised manuscript received March 1, 2000; accepted March 28, 2000
We calculate the coherent and incoherent scattering of p- and s-polarized light incident from a dielectric me-dium characterized by a real, positive, dielectric constant e0 onto its one-dimensional, randomly rough inter-face with a dielectric medium characterized by a real, positive, dielectric constant e. We use a perturbationtheory with a new small parameter, namely, the dielectric contrast h 5 e0 2 e between the medium of inci-dence and the scattering medium. The proper self-energy entering the expression for the reflectivity is ob-tained as an expansion in powers of h through the second order in h, and the reducible vertex function in termsof which the scattered intensity is expressed is obtained as an expansion in powers of h through the fourth.The roughness-induced shifts of the Brewster angle (in p polarization) and of the critical angle for total inter-nal reflection (e0 . e) are obtained. The angular dependence of the intensity of the incoherent component ofthe scattered light displays an enhanced backscattering peak, which is due to the coherent interference of mul-tiply scattered lateral waves supported by the interface and their reciprocal partners. Analogs of the Yonedapeaks observed in the scattering of x rays from solid surfaces are also present. The results obtained by oursmall-contrast perturbation theory are in good agreement with those obtained in computer simulation studies.© 2000 Optical Society of America [S0740-3232(00)01207-2]
OCIS codes: 240.5770, 290.5880, 290.1350, 290.4210, 260.6970.
1. INTRODUCTIONEach perturbation approach to rough-surface scatteringhas its own small parameter. Thus in small-amplitudeperturbation theory,1 it is d/l, where d is the rms height ofthe surface and l is the wavelength of the incident field;in the small-slope approximation,2 the small parameter isthe rms slope of the surface. In this paper we present anew perturbation theory for rough-surface scattering witha new small parameter, namely, the dielectric contrastbetween the medium of incidence and the scattering me-dium.
To illustrate this new perturbation theory, we apply itto calculations of the coherent (specular) and incoherent(diffuse) scattering of light incident from a dielectric me-dium characterized by an isotropic, real, positive,frequency-independent dielectric constant e0 onto its one-dimensional, randomly rough interface with a second di-electric medium characterized by an isotropic, real, posi-tive, frequency-independent dielectric constant e. Inthese calculations we assume that the dielectric contrasth 5 e0 2 e is small—of the order of a few tenths. Theincident light is either p polarized or s polarized, with theplane of incidence normal to the generators of the randominterface. In this geometry there is no cross-polarizedscattering, and the plane of scattering coincides with theplane of incidence.
In calculating the reflectivities of this interface for p-and s-polarized light we have chosen to obtain themwithin the framework of self-energy perturbation theory.3
0740-3232/2000/071288-13$15.00 ©
However, in contrast with our earlier use of self-energyperturbation theory in calculating the reflectivities of ran-dom dielectric surfaces,4,5 in which the self-energy thatenters the expression for the reflectivities of p- ands-polarized light was calculated perturbatively as an ex-pansion in powers of d/l, in this paper we calculate it asan expansion in powers of h.
The choice of self-energy perturbation for calculatingreflectivities was made for two reasons. It was shown re-cently that self-energy perturbation theory is well suitedto the determination of the roughness-induced shift of theBrewster angle (in p polarization) from its value for a pla-nar interface.4 In addition, we will show below that self-energy perturbation theory is also well suited to the studyof the roughness-induced shift of the critical angle for to-tal internal reflection (when e0 . e) from its value for aplanar interface in both p and s polarization. The studyof these two roughness-induced angular shifts is the firstmotivation for the present work.
In studying the incoherent scattering of p- ands-polarized light from a low-contrast random dielectric in-terface, we calculate the reducible vertex function, whichdetermines the intensity of the light that has been scat-tered incoherently, as an expansion in powers of the di-electric contrast, through terms of the fourth order in hrather than in powers of d/l. We show that, as was sug-gested in Ref. 6, in the scattering of light from a compara-tively weakly rough interface between two dielectric me-dia the lateral waves supported by the interface are
2000 Optical Society of America
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1289
excited (see, e.g., Ref. 7), and the coherent interference ofeach multiply scattered lateral wave path with its recip-rocal partner leads to the appearance of an enhancedbackscattering peak. Thus in this case it is the weak lo-calization of lateral waves that leads to the backscatter-ing enhancement in our results. The demonstration thatlateral waves can give rise to enhanced backscattering oflight from weakly rough random surfaces that supportneither surface nor guided waves is the second main mo-tivation for the present work. We also show that the an-gular dependence of the intensity of the light scattered in-coherently displays asymmetric peaks when thescattering angle equals the critical angle for total internalreflection (for e0 . e), which are analogous to the Yonedapeaks observed in the scattering of x rays from a solidsurface.8
We note that perturbative expansions in powers of thedielectric contrast have been used recently in a study ofthe coherent and incoherent scattering of x rays from ran-domly rough metal surfaces,9 in which case the value ofthe contrast is of the order of h 5 1026 2 1023. How-ever, in that study the intensity of the light scatteredcoherently—the reflectivity—was calculated directly asan expansion in powers of the dielectric contrast, withterms of the zeroth and second orders in h retained, in-stead of by the use of self-energy perturbation theory.This result, therefore, lacked the contribution from thehigher-order scattering processes that are alreadypresent in the latter approach when a low-order expan-sion of the proper self-energy in powers of h is made. Inaddition, the intensity of the x rays scattered incoherentlywas also expanded in powers of the dielectric contrast,with terms of the zeroth and second orders in h obtained.In this approximation no backscattering enhancementcan be obtained. The presentation of the present im-proved version of small-contrast perturbation theory, to-gether with results obtained by its use, is the third mainmotivation for the present work.
2. CHARACTERIZATION OF A ROUGHINTERFACEThe physical system that we consider in this paper con-sists of a dielectric medium, characterized by a real, posi-tive, dielectric constant e0 in the region x3 . z(x1) and asecond dielectric medium, characterized by a real, posi-tive, dielectric constant e in the region x3 , z(x1) (Fig. 1).The profile function z(x1) is assumed to be a single-valuedfunction of x1 that is differentiable as many times as isnecessary and that constitutes a stationary, zero-mean,Gaussian random process, defined by the properties
^z~x1!& 5 0, (2.1)
^z~x1!z~x18 !& 5 d 2W~ ux1 2 x18u!, (2.2)
d 2 5 ^z2~x1!&. (2.3)
The angle brackets in Eqs. (2.1)–(2.3) denote an averageover the ensemble of realizations of the profile functionz(x1), and d is the rms height of the surface.
It is also necessary to introduce the Fourier integralrepresentation of z(x1):
z~x1! 5 E2`
` dQ
2pz~Q !exp~iQx1!. (2.4)
The Fourier coefficient z(Q) is also a zero-mean, Gauss-ian random process, defined by
^z~Q !& 5 0, (2.5)
^z~Q !z~Q8!& 5 2pd ~Q 1 Q8!d 2g~ uQu!, (2.6)
where g(uQu), the power spectrum of the surface rough-ness, is related to the surface height autocorrelation func-tion W(ux1u) by
g~ uQu! 5 E2`
`
dx1W~ ux1u!exp~2iQx1!. (2.7)
In this paper we will present results calculated for arandom interface characterized by a power spectrum thathas the Gaussian form
g~ uQu! 5 Apa exp~2Q2a2/4!, (2.8)
where a is the transverse correlation length of the inter-face roughness, as well as for an interface characterizedby a power spectrum given by
g~ uQu! 5 ~p/Dk !(rect$@Q 2 ~Aev/c !#/Dk%
1 rect$@Q 1 ~Aev/c !#/Dk%), (2.9)
where Dk 5 2Ae(v/c)sin umax , and rect denotes the rect-angle function. The latter type of surface has been usedin recent experimental studies,10 where its use increasedthe strength of the enhanced backscattering caused bythe surface plasmon polariton mechanism11 by an order ofmagnitude compared with that for a surface with thesame rms height and rms slope characterized by a Gauss-ian power spectrum.
3. REDUCED RAYLEIGH EQUATIONSWe will study the scattering of both p- and s-polarizedlight, incident from the medium whose dielectric constantis e0 on its interface x3 5 z(x1) with the medium whosedielectric constant is e. We can treat both polarizations
Fig. 1. Physical system studied in this paper.
1290 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
together by introducing the function F(x1 , x3uv), whichis H2(x1 , x3uv) when the incident light is p polarized, andE2(x1 , x3uv) when the incident light is s polarized. Thenin the region x3 . z(x1)max the function F(x1 , x3uv) is thesum of an incoming incident wave and outgoing scatteredwaves,
F.~x1 , x3uv! 5 exp@ikx1 2 ia0~k !x3# 1 E2`
` dq
2pR~quk !
3 exp@iqx1 1 ia0~q !x3#, (3.1)
while in the region x3 , z(x1)min it consists of outgoingtransmitted waves,
F,~x1 , x3uv! 5 E2`
` dq
2pT ~quk !exp@iqx1 2 ia~q !x3#,
(3.2)
where
a0~q ! 5 @e0~v2/c2! 2 q2#1/2, Re a0~q ! . 0,
Im a0~q ! . 0,
a~q ! 5 @e~v2/c2! 2 q2#1/2, Re a~q ! . 0,
Im a~q ! . 0. (3.3)
In the scattering of electromagnetic waves from a one-dimensional randomly rough surface or interface, what ismeasured is a mean differential reflection coefficient.This is defined as the fraction of the total energy incidenton the surface per unit time that is scattered into an an-gular interval dus about the scattering direction definedby the scattering angle us (Fig. 1), averaged over the en-semble of realizations of the profile function z(x1).
In the present case the contribution to the mean differ-ential reflection coefficient from the coherent componentof the scattered light is given by
K ]R
]usL
coh
51
L1
Ae0
v
2pc
cos2 us
cos u0u^R~quk !&u2, (3.4a)
where L1 is the length of the x1 axis covered by the ran-dom interface, while the contribution from the incoherentcomponent of the scattered light is
K ]R
]usL
incoh
51
L1
Ae0
v
2pc
cos2 us
cos u0@^uR~quk !u2&
2 u^R~quk !&u2#. (3.4b)
In Eqs. (3.4), q and k are given by q 5 Ae0 (v/c)sin us andk 5 Ae0 (v/c)sin u 0 . The scattering amplitude R(quk)satisfies the reduced Rayleigh equation9
E2`
` dq
2pM~ puq !R~quk ! 5 2N~ puk !, (3.5)
where
Mp~ puq ! 5pq 1 a~ p !a0~q !
a~ p ! 2 a0~q !I@a~ p ! 2 a0~q !u p 2 q#,
Np~ puk ! 5pk 2 a~ p !a0~k !
a~ p ! 1 a0~k !I@a~ p ! 1 a0~k !u p 2 k#,
(3.6a)
and
Ms~ puq ! 5I@a~ p ! 2 a0~q !u p 2 q#
a~ p ! 2 a0~q !,
Ns~ puk ! 5I@a~ p ! 1 a0~k !u p 2 k#
a~ p ! 1 a0~k !, (3.6b)
while I(guQ) is defined by
I~guQ ! 5 E2`
`
dx1 exp$2i@Qx1 1 gz~x1!#%. (3.7)
We will seek the solution of Eq. (3.5) in the form
R~quk ! 5 2pd ~q 2 k !R ~0 !~k !
2 2iG ~0 !~q !T~quk !G ~0 !~k !a0~k !, (3.8)
where
Rp~0 !~k ! 5
ea0~k ! 2 e0a~k !
ea0~k ! 1 e0a~k 
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1291
From Eqs. (3.8)–(3.10) and (3.13b) we obtain the usefulrelation
R~quk ! 5 22pd ~q 2 k ! 2 2iG~quk !a0~k !. (3.14)
In what follows we will need to be able to expandV(quk) in powers of the dielectric contrast h 5 e0 2 e in-stead of in powers of z(x1). To do this, we must trans-form Eq. (3.12) into a form in which the dependence on his explicit. This can be achieved by making the replace-ment
I~guQ ! 5 2pd ~Q ! 1 J~guQ ! (3.15)
in Eqs. (3.6), where
J~guQ ! 5 E2`
`
dx1 exp~2iQx1!$exp@2igz~x1!# 2 1%.
(3.16)
If we introduce the definition
V~quk ! 5 2iv~quk !/ e, (3.17)
where e 5 e is in p polarization and e 5 1 is in s polar-ization, the reduced potential v(quk) is the solution of theequation
v~quk ! 5 hA~quk ! 1 hE2`
` dp
2pF~qu p !v~ puk !. (3.18)
The matrices A(quk) and F(quk) entering this equationare given by
A~quk ! 5 @n~quk !d~k ! 1 m~quk !D~k !#1
2 ea0~k !, (3.19)
F~quk ! 5 @m~quk ! 2 n~quk !#1
2 ea0~k !, (3.20)
where
mp~quk ! 5qk 1 a~q !a0~k !
a~q ! 2 a0~k !J@a~q ! 2 a0~k !uq 2 k#,
np~quk ! 5qk 2 a~q !a0~k !
a~q ! 1 a0~k !J@a~q ! 1 a0~k !uq 2 k#,
(3.21a)
and
ms~quk ! 5v2/c2
a~q ! 2 a0~k !J@a~q ! 1 a0~k !uq 2 k#,
ns~quk ! 5~v2/c2!
a~q ! 1 a0~k !J@a~q ! 1 a0~k !uq 2 k#.
(3.21b)
The iterative solution of Eq. (3.18) yields the reducedpotential v(quk) formally as an expansion in powers of h:
v~quk ! 5 hA~quk ! 1 h2E2`
` dp1
2pF~qu p1!A~ p1uk !
1 h3E2`
` dp1
2pE
2`
` dp2
2pF~qu p1!F~ p1u p2!
3 A~ p2uk ! 1 ¯ . (3.22)
4. COHERENT SCATTERINGWe see from Eq. (3.4a) that for the calculation of the con-tribution to the mean differential reflection coefficientfrom the coherent component of the scattered light weneed the ensemble average of R(quk). In view of Eq.(3.14) this is given by
^R~quk !& 5 22pd ~q 2 k ! 2 2i^G~quk !&a0~k !. (4.1)
Owing to the stationarily of z(x1), the averaged Green’sfunction ^G(quk)& appearing in Eq. (4.1) has the form12
^G~quk !& 5 2pd ~q 2 k !G~k !
5 2pd ~q 2 k !1
G ~0 !~k !21 2 M~k !, (4.2)
where G(k) is the averaged Green’s function and M(k) isthe averaged proper self-energy. The latter is given by
^M~quk !& 5 2pd ~q 2 k !M~k !, (4.3)
where the (unaveraged) proper self-energy M(quk) is thesolution of 12
M~quk ! 5 V~quk ! 1 E2`
` dp
2pE
2`
` dr
2pM~qu p !
3 ^G~ pur !&w~ruk !, (4.4)
where we have introduced the notation
w~quk ! 5 V~quk ! 2 ^M~quk !&. (4.5)
When the explicit expressions for G (0)(k) given by Eqs.(3.10) are used in Eqs. (4.1) and (4.2), we find that
^R~quk !& 5 2pd ~q 2 k !r~u0!, (4.6)
where
rp~u0! 5eAe0 cos u0 2 e0~e0 cos2 u0 2 h!1/2 1 ie~c/v!Mp@Ae0~v/c !sin u0#
eAe0 cos u0 1 e0~e0 cos2 u0 2 h!1/2 2 ie~c/v!Mp@Ae0~v/c !sin u0#, (4.7a)
rs~u0! 5Ae0 cos u0 2 ~e0 cos2 u0 2 h!1/2 1 i~c/v!Ms@Ae0~v/c !sin u0#
Ae0 cos u0 1 ~e0 cos2 u0 2 h!1/2 2 i~c/v!Ms@Ae0~v/c !sin u0#. (4.7b)
1292 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
It follows from Eq. (3.4a) that the reflectivities for p- ands-polarized light are given by
Rp,s~u0! 5 urp,s~u0!u2. (4.8)
We will calculate the self-energies Mp,s(k) as expan-sions in powers of h, through terms of second order. Thisis done in two steps. Equation (4.4) is solved iterativelyas an expansion in powers of V(quk), and the result is av-eraged term by term. The result can be written in theform
^M~quk !& 5 (n51
`
^Mn~quk !&, (4.9)
where the subscript denotes the order of the correspond-ing term in V(quk). The terms in this expansion that weneed are
^M1~quk !& 5 $V~quk !%, (4.10a)
^M2~quk !& 5 E2`
` dp
2p
$V~qu p !V~ puk !%
G ~0 !~ p !21 2 M~ p !, (4.10b)
where the bracket symbol $...% denotes the cumulantaverage.9,13 The expansion of M(quk) in powers ofV(quk) does not, of course, imply an expansion in powersof h. To obtain ^M(quk)& as an expansion in powers of h,we must use the expansion of V(quk) in powers of h givenby Eq. (3.22) and evaluate the cumulant averages appear-ing in ^Mn(quk)& as expansions in powers of h. The con-tribution to ^M1(quk)& of lowest order in h, i.e., the con-tribution to ^V(quk)& of lowest order, is of O(1), while thecontribution to ^Mn(quk)&, n 5 2,3,..., of lowest order in his of O(hn). This will be clearly seen from the following.
From Eqs. (4.10a), (4.10b), and (3.22), we see that tocalculate the ensemble average of V(quk) and products ofany number of V(quk)’s we need the average of J(guQ)and of products of an arbitrary number of J(guQ)’s.With the assumption that z(x1) is a Gaussian randomprocess, we find that
^J1J2 ...Jn& 5 )i51
n
$Ji% 1 u~n 2 2 ! (i, j51~i,j !
n
$JiJj% )k51
~k Þ i, j !
n
$Jk%
1 u~n 2 3 ! (i, j,k51~i,j,k !
n
$JiJjJk% )l51
~lÞi, j, k !
n
$Jl%
1 u~n 2 4 ! (i, j,k,l51~i,j,k,l !
n
$JiJjJkJl%
3 )m51
~mÞi, j, k, l !
n
$Jm% 1 u~n 2 4 !
3 (i, j51~i,j !
n
(k,l51~k,l !
n
$JiJj%$JkJl% )m51
~mÞi, j, k, l !
n
$Jm%
~i Þ k, l; j Þ k, l !
1terms with fewer than n 2 4 factors of $Jp%,(4.11)
where u(n) 5 1 for n > 0 and u(n) 5 0 for n , 0. Thereason for organizing the contributions to ^J1J2 ...Jn& ac-cording to the number of factors of $Jp% they contain, indecreasing order, is that when we come to average, for ex-ample, the series (3.22) term by term, the results given byEq. (4.11) tell us that the actual order in h of each of thecontributions to the average of a term formally propor-tional to hn in fact depends on the number of factors of^A(quk)& and ^F(quk)& contained in that average when itis evaluated with the use of Eq. (4.11). This is becausethese averages are diagonal in q and k and, more impor-tant, are proportional to 1/h,
^A~quk !& 5 2pd ~q 2 k !a~k !
h, (4.12a)
^F~quk !& 5 2pd~q 2 k !F2f~k !
hG , (4.12b)
where
a~k ! 5d~k !D~k !
2 ea0~k !@Y~k ! 2 X~k !#, (4.13a)
f~k ! 5d~k !X~k ! 1 D~k !Y~k !
2 ea0~k !2 1, (4.13b)
with
X~k ! 5 expH 2d 2
2@a~k ! 2 a0~k !#2J , (4.14a)
Y~k ! 5 expH 2d 2
2@a~k ! 1 a0~k !#2J . (4.14b)
The cumulant averages of products of two or more F(quk)or A(quk), or both, do not possess an explicit dependenceon h.
Thus the dominant contribution to the expansion of theaverage of a product of n factors of F(quk) and A(quk),e.g., ^F(qu p1)F( p1u p2)... A( pn21uk)&, in powers of h isobtained by replacing each of the factors by its average.Each of these averages is proportional to 1/h, so that thiscontribution is of O(h2n). The next most important termin this expansion is given by the sum of the contributionsobtained by pairing two of the factors in all possible waysirrespective of their order [there are n(n 2 1)/2 suchpairs], evaluating the cumulant average of their product,and then multiplying the result by the product of the av-erage of each of the remaining unpaired factors. Sincethere are n 2 2 such factors in an nth-order product, theresulting contribution to ^F(qu p1)F( p1u p2)...A( pn21uk)&is of O(h22n). Terms of O(h32n) and higher are obtainedin an analogous way.
The consequences of the preceding results for the cal-culations of the cumulant averages of products of the scat-tering potential V(quk) are two in number. The first isthat the dominant contribution to the nth-order cumulantaverage $V(q1u p1)...V(qnu pn)% for n > 2 is of O(hn). Itis also the case that each cumulant average, e.g.,$V(q1u p1)V(q2u p2)%, is proportional to a delta functionwhose argument is the sum of the wave-number transfersin each scattering process occurring within it, viz., to
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1293
2pd (q1 2 p1 1 q2 2 p2) for the cumulant average indi-cated. The latter result leads to ^M(quk)& having theform given by Eq. (4.3). These results have the conse-quence that the leading two terms in the expansion of theright-hand side of Eq. (4.10a) in powers of h are of O(h0)and of O(h2); the leading term on the right-hand side ofEq. (4.10b) is of O(h2). Thus if we wish to calculate theself-energy M(k) to second order in h, we need to evaluatethe contributions to $V(quk)% of O(h0) and of O(h2) andadd the result to that obtained from Eq. (4.10b) by evalu-ating $V(qu p)V( puk)% in the numerator of the integrandto O(h2) and replacing M( p) in the denominator of theintegrand by the contribution of O(h0) obtained from Eq.(4.10a). After the summation of all the contributions isdone (see the detailed description in Appendix A), we ob-tain the expression for the self-energy M(k) in the form
M~k ! 51
i e
a~k !
1 1 f~k !1 h2S 1
i e D2 1
@1 1 f~k !#2
3 E2`
` dp
2p$m~ku p !G ~0 !~ p !X21~ p !A1~ puk !%,
(4.15)
where
A1~quk ! 5 @n~quk !d~k !X~k !
1 m~quk !D~k !Y~k !#1
2 ea0~k !. (4.16)
The evaluation of the cumulant average in the inte-grand of the second term in brackets on the right-handside of Eq. (4.15) utilizes the result that
$J~g1uQ1!J~g2uQ2!%
5 2pd ~Q1 1 Q2!expF21
2~g1
2 1 g22!d 2G
3 E2`
`
du exp~2iQ1u !$exp@2g1g2d 2W~ uuu!#21%.
(4.17)
The resulting integrals in M(k) have to be evaluated nu-merically, recalling that k 5 Ae0(v/c)sinu0 .
5. INCOHERENT SCATTERINGFrom Eq. (3.14), we find that
@^uR~quk !u2& 2 u^R~quk !&u2#
5 4a02~k !@^uG~quk !u2& 2 u^G~quk !&u2#. (5.1)
The equation for the averaged two-particle Green’s func-tion ^uG(quk)u2& can be obtained starting from the equa-tion for the Green’s function written in a form12
G~quk ! 5 G~q !2pd ~q 2 k ! 1 G~q !t~quk !G~k !, (5.2)
which incorporates into its calculation the averagedGreen’s function G( p) instead of the unperturbed Green’sfunction G (0)( p). In Eq. (5.2) the operator t( pur) was in-troduced to satisfy
E2`
` dp
2pw~qu p !G~ puk ! 5 t~quk !G~k !, (5.3)
where w has been defined by Eq. (4.5) and is the solutionof the equation
t~quk ! 5 w~quk ! 1 E2`
` dp
2pw~qu p !G~ p !t~ puk !. (5.4)
It follows from Eq. (5.2) that ^t(quk)& 5 0. When wemultiply Eq. (5.2) by its complex conjugate and use theprocedure outlined in Ref. 12, we obtain
^G~quk !G* ~quk !& 5 ~2p!2d 2~q 2 k !uG~k !u2
1 uG~q !u2$t~q, kuq, k !%uG~k !u2,
(5.5)
where t(q, kuq8, k8) 5 t(quk)t* (q8uk8) is the reduciblevertex function, and we have used the result that^G(quk)& 5 2pd (q 2 k)G(k) and ^t(quk)& 5 0. The useof Eq. (5.5) in Eqs. (5.1) and (3.4b) yields the equation onwhich our calculation of ^]R/]us& incoh is based:
K ]R
]usL
incoh
51
L1
Ae0
2
pS v
c D 3
cos2 us cos u0uG~q !u2
3 ^t~q, kuq, k !&uG~k !u2. (5.6)
In this paper we will calculate ^t(q, kuq, k)& throughterms of fourth order in h, since it is in the fourth-ordercontribution that the phenomenon of enhanced back-scattering first arises. To evaluate ^t(q, kuq, k)& we needto evaluate the cumulant average $t(quk)t* (quk)% tofourth order in h. To do this we multiply the iterative so-lution of Eq. (5.4),
t~quk ! 5 w~quk ! 1 E2`
` dp
2pw~qu p !G~ p !w~ puk !
1 E2`
` dp
2pE
2`
` dr
2pw~qu p !G~ p !w~ pur !
3 G~r !w~ruk ! 1 ¯ , (5.7)
by its complex conjugate and average the resulting ex-pression term by term.
Following the procedure of averaging described in Sec-tion 4 and Appendix A, we obtain to fourth order in h,
$t~q, kuq, k !% 5h2
e2
1
u1 1 f~q !u2 @ t2~quk ! 1 ht3~quk !
1 h2t4~quk !#1
u1 1 f~k !u2 , (5.8)
where, to make the expressions for tn(quk) clearer, we in-troduced a new notation for the cumulant averages, viz.,
$J~g1uQ1!...J~gnuQn!%
5 2pd ~Q1 1 Q2 1 ¯ 1 Qn!$J~g1uQ1!...J~gnuQn!%0 ,
so that after making use of the delta functions in carryingout the integrals, we obtained
t2~quk ! 5 L1$A1~quk !A1* ~quk !%0 , (5.9a)
1294 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
t3~quk ! 5 2L1 ImH m~qu p !G~ p !
1 1 f~ p !
3 A1~ puk !A1* ~quk !J0
, (5.9b)
t4~1 !~quk ! 5 L1FH m~qu p !G~ p !
1 1 f~ p !A1~ puk !m* ~qur !
3G* ~r !
1 1 f* ~r !A1* ~ruk !J
0
1 2 ReH m~qu p !G~ p !
1 1 f~ p !
3 m~ pur !G~r !
1 1 f~r !A1~ruk !A1* ~quk !J
0
G,(5.9c)
t4~2 !~quk ! 5 L1FH m~qu p !G~ p !
1 1 f~ p !$A1~ puk !
3 m* ~quq 1 k 2 p !%0
G* ~q 1 k 2 p !
1 1 f* ~q 1 k 2 p !
3 A1* ~q 1 k 2 puk !J0
1 H m~qu p !G~ p !
1 1 f~ p !m* ~qu p !J
0
3 H A1~ puk !G* ~ p !
1 1 f* ~ p !A1* ~ puk !J
0
1 2 ReH m~qu p !G~ p !
1 1 f~ p !
3 A1~ p 2 q 1 kuk !J0H m~ pu p 2 q 1 k !
3G~ p 2 q 1 k !
1 1 f~ p 2 q 1 k !A1* ~quk !J
0
G, (5.9d)
where A1(quk) is given by Eq. (4.16).Having in hand the explicit expressions for the various
cumulant averages $...%0 in Eq. (5.8) and the averagedGreen’s function, obtained in Section 4, we can calculatenumerically the angular distribution of the intensity ofthe scattered light.
6. DISCUSSIONA. Brewster Angle and Critical Angle for Total InternalReflectionIn Figs. 2(a) and 2(b) are plotted the logarithms of the re-flectivities Rp(u0) and Rs(u0), respectively, calculated onthe basis of Eqs. (4.7a), (4.7b), and (4.8), when the me-dium of incidence (e0 5 2.4336) is optically less densethan the scattering medium (e 5 2.6569); in Figs. 2(c)
and 2(d) the same quantities are plotted for the casewhere e0 5 2.6569 and e 5 2.4336. The dielectric con-trast is therefore h 5 20.2233(10.2233) in the former(latter) case. The random roughness of the interface ischaracterized by the Gaussian height autocorrelationfunction, Eq. (2.8), and the roughness parameters in bothcases are d 5 0.2l and a 5 5l, where l is the vacuumwavelength of the incident light.
In the absence of the interface roughness, the reflectivi-ties of p- and s-polarized light are given by Eqs. (4.8) withMp,s 5 0. Rp
(0)(u0) is seen to vanish at an angle of inci-dence given by
u0 5 uB~0 ! 5 tan21S e
e0D 1/2
(6.1)
and called the Brewster angle. A corresponding angledoes not exist in the case of s polarization.
In the presence of interface roughness the reflectivityRp(u0) does not vanish at any real value of u0 because theterm ie(c/v)Mp@Ae0(v/c)sin u0#, appearing in the nu-merator of the expression (6.13a) for rp(u0), is complex.In this case we can define the Brewster angle uB as theangle of incidence at which Rp(u0) possesses aminimum.4,14 Note that this definition of the Brewsterangle reduces to the usual one, Eq. (6.1), when the ran-dom roughness tends to zero, as it should.
The Brewster effect is clearly seen in the plots pre-sented in the insets in Figs. 2(a) and 2(c). For the valuesof e0 and e used in obtaining these plots, we find from Eq.(6.1) that the Brewster angle in the case of a planar in-terface has the values uB
(0) 5 46.257° and uB(0) 5 43.743°,
respectively. From these figures we see that in the pres-ence of the random roughness the Brewster angle isshifted toward smaller values from its value for a planar
Fig. 2. Logarithm of the reflectivity of a one-dimensional ran-domly rough dielectric–dielectric interface (solid curves) and of aplanar interface (dashed curves) as functions of the angle of in-cidence, u0 , when e0 5 2.4336 and e 5 2.6569 for (a) p polariza-tion and (b) s polarization, and when e0 5 2.6569 and e5 2.4336 for (c) p polarization and (d) s polarization.
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1295
interface when e0 , e and toward larger values when e0. e. The former result is in agreement with results ob-tained earlier for the roughness-induced shift of theBrewster angle obtained by several different approachesin the scattering of p-polarized light, incident fromvacuum, on a one-dimensional random dielectricsurface.4,5,14–16 The latter result seems unexpected atfirst sight, since we kept only the contributions to the av-eraged self-energy of zeroth and second orders in h. Itsexplanation lies in the fact that, according to our defini-tion of the Brewster angle in the presence of interfaceroughness and to Eq. (4.7a), its value is given approxi-mately by the solution of the equation
eAe0 cos u0 2 e0~e0 cos2 u0 2 h!1/2
5 2Re$ie~c/v!Mp@Ae0~v/c !sin u0#%, (6.2)
where the right-hand side is positive for either sign of h.In the vicinity of the Brewster angle for a planar inter-face, uB
(0) 5 tan21(e/e0)1/2, the left-hand side of Eq. (6.2)
can be expanded as h@(Ae0/e)(e0 1 e)sin u B(0)#(u0 2 u B
(0)).Thus the roughness-induced shift of the Brewster angle isgiven approximately by
u0 2 uB~0 ! 5
e~c/v!Im Mp@Ae0~v/c !sin uB~0 !#
h@~Ae0/e!~e0 1 e!sin uB~0 !#
. (6.3)
Since the sign of the remaining factors is positive irre-spective of the sign of h, the presence of the factor 1/h onthe right-hand side of Eq. (6.3) shows that the shift ispositive when h is positive and negative when h is nega-tive. From a comparison of the results presented in theinsets to Figs. 2(a) and 2(c), we also see that the magni-tude of the shift of the Brewster angle from its value for aplanar interface is larger when e0 . e than when e0, e. The shift of the Brewster angle depends on theroughness parameters through the imaginary part of theself-energy. In general, this dependence cannot be writ-ten out explicitly. However, in the limit of a small corre-lation length, a → 0, the shift is inversely proportional tothe correlation length, uB 2 uB
(0) } d 2/(al), as was indi-cated by Greffet in Ref. 15. It should be pointed out,however, that the use of small-amplitude perturbationtheory in Ref. 15 leads to an overestimation of the shift ofthe Brewster angle.4,5 In the opposite limit of a large cor-relation length, the shift becomes independent of it.
Returning to Eqs. (4.7a) and (4.7b), we see that in bothpolarizations the reflectivity in the absence of surfaceroughness equals unity at an angle of incidence definedby e 2 e0 sin2 u0 5 0, i.e., for
u0 5 uc~0 ! 5 sin21S e
e0D 1/2
, (6.4)
and remains equal to unity as u0 increases beyond uc(0) up
to p/2, because (e 2 e0 sin2 u0)1/2 is pure imaginary for
uc(0) , u0 < p/2. The angle uc
(0) is called the criticalangle for total internal reflection. From Eq. (6.4) we seethat in order for such an angle to exist the medium of in-cidence must be the optically more dense medium; i.e., wemust have e0 . e.
In the case of a randomly rough interface, we see fromEqs. (4.7a) and (4.7b) that the existence of a true criticalangle for total internal reflection would require the satis-faction of the condition
Cp~u0! [ ~e 2 e0 sin2 u0!1/2 2 i~e/e0!~c/v!
3 Mp@Ae0~v/c !sin u0# 5 0 (6.5)
in the case of p polarization and the satisfaction of thecondition
Cs~u0! [ ~e 2 e0 sin2 u0!1/2 2 i~c/v!
3 Ms@Ae0~v/c !sin u0# 5 0 (6.6)
in the case of s polarization. However, because Mp,s(k)is complex, these conditions can be satisfied only by com-plex angles of incidence. To obtain a real critical angle inthe presence of a randomly rough interface, we define itas the angle uc
p,s at which uCp,s(u0)u2 has a minimum in p,s polarization. Again, this definition reduces to the usualone, Eq. (6.4), in the limit as the random roughness tendsto zero.
The critical angle uc is clearly seen in the results plot-ted in Figs. 2(c) and 2(d). For the values of e0 and e usedin obtaining these plots, we find that in the case of a pla-nar interface uc
(0) 5 73.148°. To examine the behavior ofRp(u0) and Rs(u0) for u0 in the vicinity of uc
(0) , we haveplotted these functions in this range of u0 values, using agreatly expanded horizontal scale, in Figs. 3(a) and 3(b),respectively. In the inset to each of these figures we havealso plotted uCp(u0)u2 and uCs(u0)u2, respectively. The ar-rows give the position of the minimum of each of the lat-ter functions, viz., the value of the critical angle for totalinternal reflection in the presence of the interface rough-ness. We see that, although the reflectivity variesstrongly as a function of u0 in the vicinity of the criticalangle, the shift of the critical angle is small. However, aswas noted earlier in Ref. 17, the shift depends strongly onthe roughness parameters, the power spectrum, etc., andcan have either sign depending on them. This depen-dence will be discussed in detail in a separate study. Inthe present study, the value of the critical angle for theroughness parameters characterizing the weakly roughinterfaces studied here is shifted by the roughness to val-ues larger than its value for a planar interface for eachpolarization. The shift of the critical angle for total in-ternal reflection of p-polarized light is larger than that fors-polarized light.
From an experimental standpoint, the measurement ofthe roughness-induced shift of uc could be accomplishedby means of a determination of the complex scatteringamplitude rp,s(u0). It then follows that
Cp~u0! 5e
Ae0
cos u0
1 2 rp~u0!
1 1 rp~u0!, (6.7)
Cs~u0! 5 Ae0 cos u0
1 2 rs~u0!
1 1 rs~u0!, (6.8)
from which the minima of uCp(u0)u2 and uCs(u0)u2 can bedetermined. The magnitude of the roughness-induced
1296 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
shift of uc( p, s) , however, is smaller than the magnitude of
the corresponding shift of the Brewster angle.
B. Enhanced Backscattering PhenomenonIn Fig. 4 we present the contribution to the mean differ-ential reflection coefficient from the incoherent compo-nent of p-polarized [Figs. 4(a) and 4(c)] and s-polarized[Figs. 4(b) and 4(d)] scattered light plotted as a function ofthe scattering angle us for a low-contrast, one-dimensional, randomly rough dielectric–dielectric inter-face characterized by the Gaussian height autocorrelationfunction Eq. (2.8). The roughness parameters are d5 0.1l and a 5 0.2l, where l is the vacuum wavelengthof the incident light, when the medium of incidence (e05 2.4336) is optically less dense than the scattering me-dium (e 5 2.6569) [Figs. 4(a) and 4(b)] and when the me-dium of incidence (e0 5 2.6569) is optically more densethan the scattering medium (e 5 2.4336) [Figs. 4(c) and4(d)]. The values of h in this case are therefore h5 70.2233, respectively. For comparison the contribu-tion to the mean differential reflection coefficient from theincoherent component of p- and s-polarized scattered lightcalculated by means of the formally exact numericalsimulation approach18 is plotted by dashed curves.
As expected, the scattering of light from a rough inter-face with a low dielectric contrast is extremely weak. As
Fig. 3. Reflectivity of a one-dimensional randomly roughdielectric–dielectric interface characterized by the Gaussianpower spectrum (solid curves) and of a planar interface (dashedcurves) as functions of the angle of incidence, u0 , for u0 in theimmediate vicinity of the critical angle for total internal reflec-tion uc , when e0 5 2.6569 and e 5 2.4336 for (a) p polarizationand (b) s polarization. The insets show plots of uCp(u0)u2 anduCs(u0)u2 as functions of u0 , for u0 in the immediate vicinity ofuc . The arrows indicate the critical angle in the presence of therandom roughness.
seen from Fig. 4 the s-polarized light is scattered morestrongly than the p-polarized light. In the case in whichlight is incident onto the interface from the optically moredense medium, ^]R/]us& incoh displays asymmetric peaksat us 5 uc , analogous to the Yoneda peaks occurring inthe scattering of x rays from solid surfaces.8,9 Thesepeaks come from the factors cos usuG(q)u2 and cos u0uG(k)u2
in Eq. (5.6). These functions increase with q and k andreach their maxima at the branch points of the Green’sfunctions, that is, when the angles of incidence or scatter-ing reach the critical angle uc for total internal reflection;the functions vanish at u0 5 p/2, us 5 p/2, owing to thepresence of the factors cos u0 and cos us . In the case ofthe scattering of p-polarized light incident from the opti-cally more dense medium [Fig. 4(c)], the minimum analo-gous to that obtained in Refs. 6 and 19 at an angle of scat-tering that was called the Brewster angle for diffusescattering cuts the peaks from the side of the smallerangles of scattering. It is easy to see from the analyticalresults of the small-amplitude perturbation approach tothe scattering of p-polarized light20 that the contributionto the mean differential reflection coefficient from theterm of the lowest order in the surface profile function,which describes the single-scattering processes, vanisheswhen eqk 2 e0a(q)a(k) 5 0, where
q 5 ~v/c !Ae0 sin us , k 5 ~v/c !Ae0 sin u0 ,
a~q ! 5 ~v/c !~cos2 us 2 h!1/2,
a~k ! 5 ~v/c !~cos2 u0 2 h!1/2.
Fig. 4. Contribution to the mean differential reflection coeffi-cient from the incoherent component of the scattered light as afunction of the scattering angle us when (a) p-polarized light and(b) s-polarized light are incident from the optically less dense me-dium and when (c) p-polarized light and (d) s-polarized light areincident from the optically more dense medium. The angle of in-cidence u0 5 0°. The interface roughness is characterized bythe Gaussian power spectrum Eq. (2.8) with d 5 0.1l and a5 0.2l. Dashed curves show the results of rigorous numericalsimulations.
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1297
However, when the interface roughness is increased, thescattering processes of higher order smear out the mini-mum.
In Fig. 5 we present the results for the contribution tothe mean differential reflection coefficient from the inco-herent component of the p- and s-polarized scattered lightwhen the roughness of the interface is characterized bythe rectangular power spectrum given by Eq. (2.9) cen-tered at @min(e, e0)#
1/2(v/c), when the medium of inci-dence is optically less dense than the scattering medium[Figs. 5(a) and 5(b), respectively] and when the medium ofincidence is optically more dense than the scattering me-dium [Figs. 5(c) and 5(d), respectively]. The roughnessparameters in this case are d 5 0.1l and umax 5 12°. Asin the case of the scattering from the surface with theGaussian power spectrum, the s-polarized light is scat-tered more strongly than the p-polarized light and pro-duces a stronger enhanced backscattering peak. Thesituation is similar to that encountered in the scatteringof light from a rough dielectric surface with large rmsheights and large rms slopes, when the multiple reflectionof light from the slopes is important for producing the en-hanced backscattering peak.21 Since the p-polarizedlight is reflected considerably more weakly than thes-polarized light, the diffuse scattering of the former isalso weaker. The phenomenon of total internal reflec-tion, which amplifies the diffuse scattering and producesthe enhanced backscattering peak present in the resultsof Ref. 22, increases the scattered intensity in the caseconsidered in this paper but is not able to overcome theeffects of the low dielectric contrast.
Fig. 5. Contribution to the mean differential reflection coeffi-cient from the incoherent component of the scattered light as afunction of the scattering angles us when (a) p-polarized light and(b) s-polarized light are incident from the optically less dense me-dium and when (c) p-polarized light and (d) s-polarized light areincident from the optically more dense medium. The interfaceroughness is characterized by the rectangular power spectrumEq. (2.9) centered at (min$e,e0%)
1/2v/c with d 5 0.1l and umax5 12°.
We recall that the expression for the contribution to themean differential reflection coefficient from the incoher-ent component of the scattered light [Eqs. (5.6) and (5.8)]contains contributions of second, third, and fourth ordersin h. However, each contribution contains all orders inthe interface profile function. Therefore there are no re-strictions on the degree of the interface roughness exceptthose imposed by the Rayleigh hypothesis, and we canstudy the scattering of light from fairly rough interfaces.The form of the expression for ^]R/]us& incoh is similar tothat which is usually obtained in the framework of many-body perturbation theory. The first two contributions tot4(2)(qu k) are the contributions from the ladder and themaximally crossed diagrams, respectively. In the poleapproximation, if we had one, the first term in Eq. (5.9d)would produce an enhanced backscattering peak with ahalf-width equal to the inverse mean free path of the ex-citations associated with poles of the Green’s functionG( p). At the same time, the second term would producea background of the same magnitude as the magnitude ofthe peak in the first term. We consider in this paper theinterface between two dielectric media, characterized bypositive dielectric constants. In this situation there areno surface polaritons, and the Green’s function G( p) doesnot have poles. However, it has branch points at p5 6Ae0v/c and p 5 6Aev/c. In the case of thedielectric–dielectric interface, the branch points lead topeaks in the product of the Green’s functions G( p)3 @G(x 2 p)#* exactly as the poles do in the case of ametal surface.20 They are much wider and weaker, ofcourse, than those associated with the surface polaritons,but, nevertheless, they play exactly the same role. It isknown (see, e.g., Ref. 7) that lateral waves are associatedwith the branch point in the Green’s function at p5 6@min(e, e0)#
1/2v/c. These are waves that propagatealong the interface in a wavelike manner, are radiativewaves in the optically more dense medium, and are inde-pendent of the distance from the interface in the opticallyless dense medium. In the case in which both media incontact are lossless, the amplitude of the lateral wavefield is determined by the difference (Ae0 2 Ae)(v/c) anddecays algebraically as L23/2, where L is the distancealong the interface. As a result, the width of the en-hanced backscattering peak is proportional to the dielec-tric contrast. The interface roughness induces additionallosses, so the enhanced backscattering peak in this case isquite broad. In addition, the peak is non-Lorentzian,since the peaks associated with the branch points arenon-Lorentzian. It should also be mentioned here thatthe power spectrum of the interface roughness should besufficiently large in the vicinity of the branch points, as itshould also be in the vicinity of the poles associated withsurface plasmon polaritons, to allow efficient excitation ofthe lateral waves. For a Gaussian power spectrum of theinterface roughness this means that the correlationlength a should be small, a < hl@min(e, e0)#
1/2, where l isthe vacuum wavelength of the incident light. The use ofthe rectangular power spectrum g(uQu) of the interfaceroughness centered at Q 5 @min(e, e0)#
1/2v/c, Eq. (2.9),greatly amplifies the enhanced backscattering in the casein which light is scattered from the random interface be-tween two dielectric media, as it does in the case in which
1298 J. Opt. Soc. Am. A/Vol. 17, No. 7 /July 2000 Leskova et al.
the enhanced backscattering is caused by the surfaceplasmon polariton mechanism.11,20
7. CONCLUSIONSIn the present paper we have described a perturbation ap-proach to the problem of the scattering of light from aone-dimensional randomly rough interface between twodielectric media with low dielectric contrast. In this ap-proach we have calculated the self-energy and the two-particle Green’s function as expansions not in powers ofthe profile function z(x1) of the interface but as an expan-sion in powers of the dielectric contrast h 5 e0 2 e.Therefore there are no restrictions on the strength of thesurface roughness except those required for the validity ofthe Rayleigh hypothesis. Several conclusions can bedrawn from the results described here.
The first is that at a one-dimensional randomly roughdielectric–dielectric interface the Brewster angle (in p po-larization) is shifted from the value it has when the inter-face is planar. The shift is in the direction of smallerangles when the medium of incidence is optically lessdense than the scattering medium (e0 , e) and is in thedirection of larger angles when the medium of incidence isoptically more dense than the scattering medium (e0. e). The former result is consistent with results ob-tained for the roughness-induced shift of the Brewsterangle obtained by several different approaches in thescattering of p-polarized light, incident from vacuum, on aone-dimensional random dielectric surface.4,14–16 It isalso consistent with the theoretical results obtained byBaylard et al.5 and Kawanishi et al.,19 as well as with theexperimental results of De Roo and Ulaby,23 for the scat-tering of p-polarized electromagnetic waves, incident fromvacuum, on a two-dimensional random dielectric surface.For the dielectric contrasts assumed in the present work,uhu 5 0.223, and the roughness parameters adopted, d5 0.2l and a 5 5l, the shifts of the Brewster angle cal-culated are of the order of 0.1°. This is a small value butone that should be measurable. The magnitude of theshift of the Brewster angle from its value for a planar in-terface is larger in the presence of total internal reflectionthan when it is absent.
The second conclusion we can draw is that in the casein which the medium of incidence is optically more densethan the scattering medium (e0 . e), so that the phe-nomenon of total internal reflection occurs, the value ofthe critical angle for total internal reflection is shifted bythe roughness of the interface from its value when the in-terface is planar. In the case in which the interfaceroughness is characterized by the Gaussian power spec-trum, Eq. (2.8), and for the roughness parameters as-sumed here, the shift is in the direction of larger anglesfor incident light of both p and s polarization.
In the scattering of light from a one-dimensional ran-domly rough dielectric–dielectric interface, the angulardependence of the intensity of the incoherent componentof the scattered light displays an enhanced backscatteringpeak, which is caused by the coherent interference of eachmultiply scattered lateral wave path with its reciprocalpartner. When light is scattered from a rough interfacewhose roughness is characterized by a rectangular power
spectrum centered at @min(e, e0)#1/2v/c, the enhanced
backscattering peak is amplified considerably. As ex-pected, the enhanced backscattering peak as well as thediffuse background is stronger when light is incident ontothe interface from the optically more dense medium. Themost favorable situation for observing enhanced back-scattering from a dielectric–dielectric interface is whens-polarized light is incident from the optically more densemedium onto a random interface whose roughness ischaracterized by a rectangular power spectrum. In thiscase the angular distribution of the intensity of the inco-herent component of the scattered light also displaysstrong asymmetric peaks when the scattering angleequals the critical angle for total internal reflection,which are analogous to the Yoneda bands observed in thescattering of x rays from a solid surface.
Finally, we point out that a variant of small-contrastperturbation theory has been explored recently in connec-tion with the scattering of light from, and its transmissionthrough, a random interface between two different dielec-tric media.24–26 It differs from the approach presentedhere in several respects. In application to a one-dimensional random interface, the expansion parameteris not e0 2 e but rather the difference between the dielec-tric constant of the system, e(x1 , x3) 5 e0u@x3 2 z(x1)#1 eu@z(x1) 2 x3# in our notation, where u(x) is theHeaviside unit step function, and its ensemble averageeeff (x3) 5 ^e (x1 , x3)&. In addition, it is an essentially nu-merical approach. It can yield accurate results for thereflectivity24 and transmissivity25 for values of e0 2 e upto 2.25 and 5.0, respectively, for rms heights of the rough-ness up to l/2 and transverse correlation lengths of up to2l. In its present form, however, it cannot predict en-hanced backscattering and enhanced transmission, norcross-polarized scattering from two-dimensional randomsurfaces, but this appears to be due more to computa-tional reasons than to any fundamental limitation of thetheory.
APPENDIX AFrom Eqs. (3.17) and (3.22) we see that to calculate$V(quk)% we have to calculate the average:
$v~quk !%
5 ^v~quk !&
5 h^A~quk !& 1 h2E2`
` dp1
2p^F~qu p1!A~ p1uk !&
1 h3E2`
` dp1
2pE
2`
` dp2
2p^F~qu p1!F~ p1u p2!A~ p2uk !&
1 h4E2`
` dp1
2pE
2`
` dp2
2pE
2`
` dp3
2p^F~qu p1!
3 F~ p1u p2!F~ p2u p3!A~ p3uk !& 1 ¯ . (A1)
To calculate the contribution to ^v(quk)& of zeroth order inh we have to calculate the contribution of this order from
Leskova et al. Vol. 17, No. 7 /July 2000 /J. Opt. Soc. Am. A 1299
each term on the right-hand side of Eq. (A1) and sum theresults. To obtain the contribution of zeroth order in hfrom any term, it suffices to replace the average of thatterm by the product of the averages of the individual fac-tors in it. In this way we obtain
$v~quk !%~0 ! 5 2pd ~q 2 k !@a~k ! 2 f~k !a~k !
1 f 2~k !a~k ! 2 ¯#
5 2pd ~q 2 k !a~k !
1 1 f~k !. (A2)
The leading contribution to $v(quk)% that is of nonze-roth order in h is of O(h2) and is obtained by summing ineach term on the right-hand side of Eq. (A1) the resultsobtained by pairing two of the factors in all possible ways,irrespective of order, evaluating their cumulant average,and then multiplying the latter by the product of the av-erages of each of the remaining, unpaired, factors. In thenth-order term (n > 2) two types of contributions arise.The first consists of the n 2 1 terms in which one of theF( piu pj) is paired with A( pn21uk), while the remainingn 2 2 factors of F( piu pj) are averaged individually; thesecond consists of the (n 2 1)(n 2 2)/2 terms in whichtwo of the F( piu pj) are paired, while the remainingn 2 2 factors [including A( pn21)] are averaged individu-ally. The contribution to $v(quk)% from the first type ofpairing is
$v~quk !%~21! 5 h2@1 2 f~q ! 1 f 2~q ! 2 ¯#
3 E2`
` dp
2p$F~qu p !@1 2 f~ p !
1 f 2~ p ! 2 ¯#A~ puk !%
5 h21
1 1 f~q !E
2`
` dp
2p
$F~qu p !A~ puk !%
1 1 f~ p !.
(A3)
The contribution from the second type of pairing is
$v~quk !%~22! 5 h2@1 2 f~q ! 1 f 2~q ! 2 ¯#
3 E2`
` dp
2p$F~qu p !@1 2 f~ p !
1 f 2~ p ! 2 ¯#F~ puk !%
3 @1 2 f~k ! 1 f 2~k ! 2 ¯#a~k !
5 h21
1 1 f~q !E
2`
` dp
2p
$F~qu p !F~ puk !%
1 1 f~ p !
3a~k !
1 1 f~k !. (A4)
From Eq. (3.17) we also see that to evaluate the cumu-lant average $V(qu p)V( puk)% appearing in the integrandon the right-hand side of Eq. (4.10b) to second order in h,we have to evaluate the cumulant average,
$v~qu p !v~ puk !% 5 ^v~qu p !v~ puk !& 2 ^v~qu p !&^v~ puk !&,(A5)
to the same order. To do this we multiply two iterativesolutions to Eq. (3.18) and average the result term byterm. The lowest-order contribution to ^v(qu p)v( puk)&as an expansion in powers of h, which is of zeroth order inh, is obtained by replacing each average of the product bythe product of the averages of the individual factors.However, since what we need is not ^v(qu p)v( puk)& itself,but the difference ^v(qu p)v( puk)& 2 ^v(qu p)&^v( puk)&,the zeroth-order contribution to ^v(qu p)v( puk)& can beomitted since it does not contribute to this difference.The contribution to ^v(qu p)v( puk)& of second order in h isobtained from the sum of terms in which two factors arepaired and the cumulant average of their product is mul-tiplied by the product of the averages of the remainingfactors. However, the only terms of this type that con-tribute to the difference ^v(qu p)v( puk)& 2 ^v(qu p)&3 ^v( puk)& are those in which one of the paired factorswhose cumulant average is calculated comes from the ex-pansion of v(qu p) while the other comes from the expan-sion of v( puk).
Four classes of terms contribute to ^v(qu p)v( puk)&2 ^v(qu p)&^v( puk)& to order h2. They are defined bythe two factors that appear in the pair whose cumulantaverage is evaluated. They can be written schematicallyas $AA%, $FA%, $AF%, and $FF%. These four categories ofterms can be summed to yield the result that to O(h2),
$v~qu p !v~ puk !% 5 h2H F A~qu p !
1 1 f~q !1
F~qu p !
1 1 f~q !
a~ p !
1 1 f~ p !G
3 F A~ puk !
1 1 f~ p !1
F~ puk !
1 1 f~ p !
3a~k !
1 1 f~k !G J . (A6)
ACKNOWLEDGMENTThis research was supported in part by the U.S. Army Re-search Office grant DAAD 19-99-1-0321.
The corresponding author, A. Maradudin, can bereached by e-mail at [email protected].
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