Impedance Boundary Conditions for a Metal Film with a Rough Surface

Impedance boundary conditions for a metal filmwith a rough surface

Tamara A. Leskova, Alexei A. Maradudin, and Igor V. Novikov

We have obtained local impedance boundary conditions for a metal film characterized by an isotropic,frequency-dependent, complex dielectric function e~v! that occupies the region z~x1! , x3 , D. Thesurface-profile function z~x1! is assumed to be a single-valued function of x1 that is differentiable as manytimes as is necessary. The electromagnetic field in the system is assumed to be p polarized with theplane of incidence, the x1x3 plane. The results are used to study the scattering of p-polarized light fromand its transmission through the metal film when the surface-profile function z~x1! in these calculationsis assumed to be a stationary, zero-mean, Gaussian random process. These calculations are approxi-mately four times faster than rigorous computer simulations, and their results are qualitatively andquantitatively in good agreement with those of the latter simulations. © 1999 Optical Society ofAmerica

OCIS codes: 240.0310, 240.4350, 240.5770, 290.5880.

4

a

zflz

adv

1. Introduction

Impedance boundary conditions can be both usefuland necessary in computer simulation approaches tothe scattering of electromagnetic waves from roughsurfaces. Useful, because by eliminating the elec-tromagnetic field inside the scattering medium, theyreduce the size of the computational problem by afactor of 2. Necessary, because in the range of fre-quencies of the incident electromagnetic field forwhich the dielectric function of the scattering me-dium is large in magnitude and the standard methodfor the transformation of the integral equations forthe surface values of the fields and their normal de-rivatives into matrix equations requires the use ofprohibitively large matrices, the use of an impedanceboundary condition, which exploits the largeness ofthe dielectric function, overcomes this difficulty.

At the present time impedance boundary conditionshave been derived for the rough interface betweenvacuum and a semi-infinite metal characterized by aone-dimensional single-valued profile function,1–3 by a

T. A. Leskova is with the Institute of Spectroscopy, RussianAcademy of Sciences, Troitsk, Moscow Region 142092, Russia.A. A. Maradudin and I. V. Novikov are with the Department ofPhysics and Astronomy and Institute for Surface and InterfaceScience, University of California, Irvine, California 92697.

Received 25 November 1998.0003-6935y99y071197-16$15.00y0© 1999 Optical Society of America

one-dimensional multivalued profile function, and bytwo-dimensional single-valued profile function.5,6

It has also been shown that impedance boundary con-ditions derived for rough vacuum–metal interfaces canbe applied to rough vacuum–dielectric interfaces aswell, if the dielectric constant of the dielectric mediumis large enough.7 The utility and the necessity of im-pedance boundary conditions have been confirmed inthese investigations.

Up to the present time, however, impedanceboundary conditions do not appear to have been de-rived for systems possessing more than one interface,e.g., metal ~or dielectric! films. In this paper wepresent a derivation of such boundary conditions.The system we consider consists of a metal charac-terized by an isotropic, frequency-dependent, com-plex dielectric function e~v! that occupies the region~x1! , x3 , D ~Fig. 1!. We are interested in therequency range in which Re e~v! is negative andarge in magnitude. The surface-profile function~x1! is assumed to be a single-valued function of x1

that is differentiable as many times as is necessary.The motivation for this study is the desire to study

by a computer simulation approach multiple-scattering phenomena occurring in the second-harmonic generation of light from a nonlinear metalsurface in the Kretschmann attenuated total reflec-tion geometry. In this geometry the region x3 . D is

prism characterized by an isotropic, real, positiveielectric constant ep, while the region x3 , z~x1! isacuum. This structure is illuminated through the

1 March 1999 y Vol. 38, No. 7 y APPLIED OPTICS 1197

qt

cfwcpha

c

W

1

prism by a p-polarized electromagnetic wave of fre-uency v, and the fields scattered from and transmit-ed through the metal film at frequency 2v are

measured. This geometry has been used in severalexperimental studies of second-harmonic generationat random metal surfaces,8–12 but no rigorous theoryof this effect exists at the present time. Such a the-ory is currently being developed.13 It requires thenumerical solution of a system of four coupled inte-gral equations, which is computationally intensive.In addition, the frequency of the incident field v is inthe infrared, which further increases the computa-tional problems encountered in these calculations.It seemed attractive therefore to explore the extent towhich the use of impedance boundary conditions cansimplify the calculations described. The derivationof such boundary conditions must precede their ap-plication, and it is to that derivation that this paperis devoted.

The outline of this paper is as follows. In Section2, by the use of the Green’s second integral identity inthe plane,14 the matrix integral equation for the ele-ments of the 2 3 2 nonlocal impedance matrix thatarises for a film geometry is derived. The integralequation satisfied by the local impedance matrix isthen obtained from it. An approximate analytic so-lution of the latter equation is obtained in Section 3.In Section 4 an integral equation for the Fouriertransform of the local impedance matrix is derived.This result is used in Section 5 to obtain an exactmatrix equation for the elements of the impedancematrix in the case in which the rough surface-profilefunction z~x1! is a periodic function of x1. The re-sults are of interest in themselves since periodicallycorrugated surfaces have been used recently15 inomputer simulation studies of the scattering of lightrom surface and guided wave-bearing structures as aay of overcoming convergence problems in such cal-

ulations that can arise from the long mean freeaths of these excitations. In the present paper,owever, they are used, in Section 6, as a standardgainst which the accuracy of the analytic results

Fig. 1. Metal film for which the impedance boundary conditionson the surfaces x3 5 D and x3 5 z~x1! are sought.

198 APPLIED OPTICS y Vol. 38, No. 7 y 1 March 1999

obtained in Section 3 can be determined. Discussionof the results obtained and the conclusions that canbe drawn from them are presented in Section 7. Anappendix in which results needed in the text are de-rived concludes this paper.

2. Impedance Matrix

For specificity, in deriving the integral equation sat-isfied by the impedance matrix for the metal film inthe structure displayed in Fig. 1, we assume ap-polarized electromagnetic field in that structurewhose plane of incidence is the x1x3 plane. The elec-tric and the magnetic fields in the system then havethe forms

H~x; t! 5 (0, H2~x1, x3uv!, 0)exp~2ivt!, (2.1a)

E~x; t! 5 (E1~x1, x3uv!, 0, E3~x1, x3uv!)exp~2ivt!,

(2.1b)

where

E1~x1, x3uv! 5 2icve

]

]x3H2~x1, x3uv!, (2.2a)

E3~x1, x3uv! 5icve

]

]x1H2~x1, x3uv!, (2.2b)

and e is the dielectric constant of the medium inwhich the fields are being calculated.

In the region z~x1! , x3 , D the magnetic-fieldomponent is the solution of the Helmholtz equation,

F ]2

]x12 1

]2

]x32 1 e~v!

v2

c2GH2~x1, x3uv! 5 0. (2.3)

e also introduce the Green’s function Ge~x1, x3ux19,x39! as the solution of the equation

F ]2

]x12 1

]2

]x32 1 e~v!

v2

c2GGe~x1, x3ux19, x39! 5

24pd~x1 2 x19!d~x3 2 x39!, (2.4)

subject to the condition that it vanish at infinity @be-cause Re e~v! , 0#. This Green’s function has thefollowing representations:

Ge~x1, x3ux19, x39! 5 2K0H 1d~v!

@~x1 2 x19!2

1 ~x3 2 x39!2#1y2J (2.5a)

5 *2`

`

dkexp@ik~x1 2 x19! 2 b~k, v!ux3 2 x39u#

b~k, v!, (2.5b)

where K0~z! is a modified Bessel function,

d~v! 5cv

1@2e~v!#1y2 Re d~v! . 0, Im d~v! . 0,

(2.6)

i

d

b

wz

g

,

is the optical skin depth of the metal at the frequencyv, and

b~k, v! 5 Fk2 2 e~v!v2

c2G1y2

Re b~k, v! . 0,

Imb~k, v! , 0. (2.7)

We next invoke Green’s second integral indentityn the plane, which can be stated as follows14: Let

f ~x1, x3! and g~x1, x3! be continuous with continuouserivatives in an open region D in the x1x3 plane.

Let R be a closed simply connected region in D withoundary C, a closed regular curve. Let n be the

outer unit normal to C. Then

*R

dx1 *R

dx3F f ~x1, x3!S ]2

]x12 1

]2

]x32Dg~x1, x3! 2 g~x1, x3!

3 S ]2

]x12 1

]2

]x32Df ~x1, x3!G

5 *C

dsF f ~x1, x3!]

]ng~x1, x3! 2 g(x1,x3)

]

]nf ~x1, x3!G,

(2.8)

where ds is the element of arc length along the curveC and

]

]n5 n z Sx1

]

]x11 x3

]

]x3D . (2.9)

We assume that R is the region z~x1! , x3 , D.The curve C is then the union of the curve x3 5 D, thecurve x3 5 z~x1!, which we denote by G, and twocircular arcs of infinite radius that close the contour.Then if we set f ~x1, x3! [ H2~x1, x3uv! and g~x1, x3! 5Ge~x1, x3ux19, x39!, and use Eqs. ~2.3! and ~2.4!, weobtain from Eq. ~2.8!

24pH2~x19, x39uv!u~D 2 x39!u@x39 2 z~x19!#

5 *2`

`

dx1FH2~x1, x3uv!]

]x3Ge~x1, x3ux19, x39!

2 Ge~x1, x3ux19, x39!]

]x3H2~x1, x3uv!G

x35D

2 *G

dsHH2~x1, x3uv!1

f~x1!F2z1~x1!

]

]x11

]

]x3G

3 Ge~x1, x3ux19, x39!

2 Ge~x1, x3ux19, x39!1

f~x1!F2z1~x1!

]

]x11

]

]x3G

3 H2~x1, x3uv!J , (2.10)

where u~z! is the Heaviside unit step function and

f~x1! 5 $1 1 @z1~x1!#2%1y2. (2.11)

In writing Eqs. ~2.10! and ~2.11! we have introduceda notation that will be used throughout this paper,namely that

zn~x1! ; dnz~x1!ydx1n. (2.12)

The contribution to the integral over C from the twocircular arcs at infinity vanishes because the Green’sfunction Ge~x1, x3ux19, x39! decays faster than expo-nentially as @~x1 2 x19!2 1 ~x3 2 x39!2#1y23 `. Since

e have assumed that the surface-profile function~x1! is a single-valued function of x1, we can replace

integration along the curve G in Eq. ~2.10! by inte-ration along the x1 axis with the aid of the relation

ds 5 f~x1!dx1. (2.13)

When we do this and interchange the primed and theunprimed variables for convenience, the resultingequation can be written in the form

u~D 2 x3!u@x3 2 z~x1!#H2~x1, x3uv!

5 21

4p *2`

`

dx19HF ]

]x39Ge~x1, x3ux19, x39!G

x395D

3 H1~x19uv! 2 @Ge~x1, x3ux19, x39!#x395DL1~x19uv!J1

14p *

2`

`

dx19HF ]

]N9Ge~x1, x3ux19, x39!G

x395z~ x19!

3 H2~x19uv! 2 @Ge~x1, x3ux19, x39!#x395z~ x19!L2~x19uv!J(2.14)

where

]

]N5 2z1~x1!

]

]x11

]

]x3(2.15)

and the source functions H1~x1uv!, L1~x1uv!, H2~x1uv!,and L2~x1uv! are defined by

H1~x1uv! 5 H2~x1, x3uv!ux35D, (2.16a)

L1~x1uv! 5]

]x3H2~x1, x3uv!ux35D, (2.16b)

H2~x1uv! 5 H2~x1, x3uv!ux35z~ x1!, (2.16c)

L2~x1uv! 5]

]NH2~x1, x3uv!ux35z~ x1!. (2.16d)

The symmetry of the Green’s function Ge~x1, x3ux19,x39! in its arguments, Ge~x1, x3ux19, x39! 5 Ge~x19,x39ux1, x3!, was used in obtaining Eq. ~2.14!.

We obtain one relation among the four source func-tions by setting x3 5 D 1 h and x3 5 D 2 h in Eq.


wob

H

T

M

M

1

~2.14!, where h is a positive infinitesimal, and addingthe resulting equations2:

H1~x1uv! 5 21

2pP *

2`

`

dx19N11~x1ux19!H1~x19uv!

11

2p *2`

`

dx19M11~x1ux19!L1~x19uv!

11

2p *2`

`


21

2p *2`

`

dx19M12~x1ux19!L2~x19uv!, (2.17a)

here P denotes the Cauchy principal value. Webtain a second relation among the source functionsy setting x3 5 z~x1! 1 h and x3 5 z~x1! 2 h in Eq.

~2.14! and adding the resulting equations:

2~x1uv! 5 21

2p *2`

`


11

2p *2`

`

dx19M21~x1ux19!L1~x19uv!

11

2pP *

2`

`


21

2p *2`

`

dx19M22~x1ux19!L2~x19uv!. (2.17b)

he kernels in these equations are given by

11~x1ux19! 5 @Ge~x1, x3ux19, x39!#x35Dx395D

5 2K0S1d

ux1 2 x19uD , (2.18a)

12~x1ux19! 5 @Ge~x1, x3ux19, x39!#x35Dx395z~ x19!

5 2K0(1d

$~x1 2 x19!2 1 @D 2 z~x19!#

2%1y2) ,

(2.18b)

M21~x1ux19! 5 @Ge~x1, x3ux19, x39!#x35z~ x1!x395D

5 2K0(1d

$~x1 2 x19!2 1 @z~x1! 2 D#2%1y2) ,

(2.18c)

M22~x1ux19! 5 @Ge~x1, x3ux19, x39!#x35z~ x1!x395z~ x19!

5 2K0(1d

$x1 2 x19!2 1 @z~x1! 2 z~x19!#

2%1y2) ,

(2.18d)


N11~x1ux19! 5 F ]

]x39Ge~x1, x3ux19, x39!G

x35Dx395D

5 0, (2.19a)

N12(x1ux19! 5 F ]

]N9Ge~x1, x3ux19, x39!G

x35Dx395z~ x19!

52d

K1(1d

$~x1 2 x19!2 1 @D 2 z~x19!#

2%1y2)$~x1 2 x19!

2 1 @D 2 z~x19!#2%1y2

3 $~x19 2 x1!z1~x19! 1 @D 2 z~x19!#%,(2.19b)

N21~x1ux19! 5 F ]

]x39Ge~x1, x3ux19, x39!G

x35z~ x1!x395D

52d

K1(1d

$~x1 2 x19!2 1 @z~x1! 2 D#2%1y2)

$~x1 2 x19!2 1 @z~x1! 2 D#2%1y2

3 @z~x1! 2 D#, (2.19c)

N22~x1ux19! 5 F ]

]N9Ge~x1, x3ux19, x39!G

x35z~ x1!x395z~ x19!

52d

K1(1d

$~x1 2 x19!2 1 @z~x1! 2 z~x19!#

2%1y2)$~x1 2 x19!

2 1 @z~x1! 2 z~x19!#2%1y2

3 $~x19 2 x1!z1~x19! 1 @z~x1! 2 z~x19!#%.

(2.19d)

Equations ~2.17! can be rewritten as

*2`

`

dx19FM11~x1ux19! 2M12~x1ux19!2M21~x1ux19! M22~x1ux19!

GFL1~x19uv!L2~x19uv!G

5 *2`

`

dx19F2pd~x1 2 x19! 2N12~x1ux19!2N21~x1ux19! 22pd~x1 2 x19!

1 PN22~x1ux19!GFH1~x19uv!

H2~x19uv!G , (2.20)

i.e., a linear relation exists between L1,2~x1uv! andH1,2~x1uv!. The most general relation of this typecan be written in the form

FL1~x1uv!L2~x1uv!G 5 *

2`

`

dx19FK11~x1ux19! K12~x1ux19!K21~x1ux19! K22~x1ux19!

G3 FH1~x19uv!

H2~x19uv!G . (2.21)

The matrix K~x1ux19! entering this relation is the non-local impedance matrix. When we substitute Eq.~2.21! into Eq. ~2.20! and use the arbitrariness of

w

wK

It

im

W

n

I

H1,2~x1uv!, we obtain as the equation for the imped-ance matrix

*2`

`


G3 FK11~x10ux19! K12~x10ux19!

K21~x10ux19! K21~x10ux19G

5 F2pd~x1 2 x19! 2N12~x1ux19!2N21~x1ux19! 22pd~x1 2 x19! 1 PN22~x1ux19!

G .

(2.22)

We can write nonlocal relation ~2.21! betweenL1,2~x1uv! and H1,2~x1uv! in an alternative way byexpanding H1,2~x19uv! about x19 5 x1:

Li~x1uv! 5 (j51

2

*2`

`

dx19Kij~x1ux19!Hj~x19uv!

5 (j51

2

*2`

`

dx19Kij~x1ux19!FHj~x1uv!

1 ~x19 2 x1!d

dx1Hj~x1uv! 1

12

~x19 2 x1!2

3d2

dx12 Hj~x1uv! 1 · · · G

5 (j51

2

(n50

` 1n!

Kij~n!~x1uv!

dn

dx1n Hj~x1uv!, (2.23)

here

Kij~n!~x1uv! 5 *

2`

`

dx19Kij~x1ux19!~x19 2 x1!n. (2.24)

The zero-order term on the right-hand side of Eq.~2.23! expresses a local relation between L1,2~x1uv!and H1,2~x1uv!:

FL1~x1uv!L2~x1uv!G5 FK11

~0!~x1uv! K12~0!~x1uv!

K21~0!~x1uv! K22

~0!~x1uv!GFH1~x1uv!H2~x1uv!G ,

(2.25)

here the elements of the local impedance matrixij~0!~x1uv! are given by

Kij~0!~x1uv! 5 *

2`

`

dx19Kij~x1ux19!. (2.26)

The remaining terms on the right-hand side of Eq.~2.23! give nonlocal corrections to local relation ~2.25!.n the remainder of this paper we restrict our atten-ion to local impedance boundary condition ~2.25!.

On combining Eqs. ~2.22! and ~2.26! we obtain thentegral equation satisfied by the local impedanceatrix K~0!~x1uv!,

*2`

`

dx19FM11~x1ux19! 2M12~x1ux19!2M21~x1ux19! M22~x15x19!

G3 FK11

~0!~x19uv! K12~0!~x19uv!

K21~0!~x19uv! K22

~0!~x19uv!G5 F2p 0

0 22pG 1 *2`

`

dx19

3 F 0 2N12~x1ux19!2N21~x1ux19! PN22~x1ux19!

G .

(2.27)

e now turn to the solution of this equation.

3. Impedance Matrix in the Coordinate Representation

To obtain an approximate analytic solution of Eq.~2.27!, we note that the kernels Mij~x1ux19!, regardedas functions of x19, are sharply peaked around x19 5x1. We therefore expand Kij

~0!~x19uv! on the left-handside of Eq. ~2.27! about x19 5 x1 and obtain the equa-tion

(50

` 1n! *

2`

`


G~x19 2 x1!n

3dn

dx1n FK11

~0!~x1uv! K12~0!~x1uv!

K21~0!~x1uv! K22

~0!~x1uv!G 5 F2p 00 22pG

1 *2`

`

dx19F 0 2N12~x1ux19!2N21~x1ux19) PN22~x1ux19!

G .(3.1)

t is convenient to rewrite this equation in the form

FA11~0!~x1uv! 2A12

~0!~x1uv!2A21

~0!~x1uv! A22~0!~x1uv! GFK11

~0!~x1uv! K12~0!~x1uv!

K21~0!~x1uv! K22

~0!~x1uv!G5 FB11~x1uv! B12~x1uv!

B21~x1uv! B22~x1uv!G2 (

n51

` 1n! FA11

~n!~x1uv! 2A12~n!~x1uv!

2A21~n!~x1uv! A22

~n!~x1uv! G dn

dx1n

3 FK11~0!~x1uv! K12

~0!~x1uv!K21

~0!~x1uv! K22~0!~x1uv!G , (3.2)

where

Aij~n!~x1uv! 5 *

2`

`

dx19Mij~x1ux19!~x19 2 x1!n, (3.3)

B11~x1uv! 5 2p, (3.4a)

B12~x1uv! 5 2*2`

`

dx19N12~x1ux19!, (3.4b)


o

snz

oB2

b

A

A

A

A

3 2 2

w

B

K

1

B21~x1uv! 5 2*2`

`

dx19N21~x1ux19!, (3.4c)

B22~x1uv! 5 22p 1 P *2`

`

dx19N22~x1ux19!. (3.4d)

The matrix elements A11~n!~x1uv! and Bij~x1uv! can be

btained as expansions in powers of d~v!z2~x1!, wherez2~x1! is the reciprocal of the local radius of curvatureof the surface. In what follows we assume thatthe product ud~v!iz2~x1!maxu is small, in the case ofa deterministic surface-profile function and thatud~v!u^@z2~x1!#2&1y2 is small in the case of a randomurface-profile function, where the angle brackets de-ote an average over the ensemble of realizations of~x1!. These conditions are easily satisfied even for

large rms height ~relative to l! and large rms slopemetal surfaces in the visible region of the optical spec-trum3 and certainly in the infrared region. The diag-nal elements A11

~n!~x1uv!, A22~n!~x1uv!, B11~x1uv!, and

22~x1uv!, can be obtained from results derived in Ref.; the off-diagonal elements A21

~n!~x1uv! and B21~x1uv!are exactly integrable; and the elements A12

~n!~x1uv! cane evaluated as expansions in powers of d~v!z2~x1! by

the methods used in Appendix A for the evaluation ofA21

~0!~x1uv! and B12~x1uv!. The leading terms in the ex-pansions of Aij

~n!~x1uv! ~n 5 0, 1, 2! and Bij~x1uv! are

11~0!~x1uv! 5 2pd, (3.5a)

A12~0!~x1uv! 5 2p

da

exp~2b!

3 H1 2dz2

2bz1

2

a4 F3 11b Sbz1

a D2G 1 O~d2!J ,

(3.5b)

A21~0!~x1uv! 5 2pd exp~2b!, (3.5c)

A22~0!~x1uv! 5

2pdf

12pd3

f 4 S32

z12z2

2

f 3 238

z22

f 3 212

z1z3

f D1 O~d5!, (3.5d)

A11~1!~x1uv! 5 0, (3.6a)

12~1!~x1uv! 5 2p

d2bz1

a3 exp~2b! 2 2pd3z1z2

2a5 exp~2b!

3 F3~1 1 b! 16b2z1

2

a2

b4z14

ba4 G 1 O~d4!,

(3.6b)

21~1!~x1uv! 5 0, (3.6c)

22~1!~x1uv! 5 22p

32

d3z1z2

f 5 1 O~d5!, (3.6d)

A11~2!~x1uv! 5 2pd3, (3.7a)


A12~2!~x1uv! 5 2p

da3 exp~2b!F~1 1 b! 1

b z1

a2 G1 O~d4!, (3.7b)

A21~2!~x1uv! 5 2pd3 exp~2b!~1 1 b!, (3.7c)

A22~2!~x1uv! 5 2p

d3

f 3 1 O~d5!; (3.7d)

B11~x1uv! 5 2p, (3.8a)

B12~x1uv! 5 22pb

abexp~2b!H1 1

dz2

2a2

b

b F1 2 Sbz1

a D2

3 S2b

23b2D 2 Sbz1

a D4S1b

11b3DG 1 O~d2!J ,

(3.8b)

B21~x1uv! 5 2p exp~2b!, (3.8c)

B22~x1uv! 5 22p 1 pdz2

f 3 1 3pd3

f 5 S154

z12z2

3

f 4 258

z23

f 4

252

z1z2z3

f 2 1z4

4D 1 O~d5!, (3.8d)

here

a~x1! 5 $1 1 z12~x1! 2 @D 2 z~x1!#z2~x1!%

1y2, (3.9a)

b~x1uv! 5D 2 z~x1!

d~v!, (3.9b)

b~x1uv! 5 b~x1uv!F1 2z1

2~x1!

a2~x1!G1y2

5 b~x1uv!$1 2 @D 2 z~x1!#z2~x1!%

1y2

a~x1!. (3.9c)

From the results given by Eqs. ~3.5!–~3.8! we seethat the off-diagonal elements A12

~n!~x1uv!, A21~n!~x1uv!,

12~x1uv!, and B21~x1uv!, in addition to possessing ex-pansions in powers of dz2, contain factors of exp~2b!and exp~2b!. In what follows we assume that thelatter factors are also small parameters of the theory,of the order of dz2. This assumption means that themean thickness of the film is sufficiently larger thanthe magnitude of the optical skin depth d~v! that thevalue of the normal derivative of the field on one sur-face of the film is determined primarily by the value ofthe field itself on the same surface and only secondarilyby the field on the other surface. With the aid of thisassumption we can seek a solution of Eq. ~3.2! for

~0!~x1uv! as an expansion in powers of the small pa-

ˆ

Ht

tr

wtcpp

rameters dz2, exp~2b!, and exp~2b!. The first twoterms in this expansion are

K~0!~x1uv! 51d F1 0

0 2fG1 30 2

1a

exp~2b!

d Sb

b1 fD

2fexp~2b!

dz2

2f 24

1 · · · . (3.10)

igher-order terms can be obtained, but the calcula-ions and the results quickly become cumbersome.

However, a better solution for K~0!~x1uv! is obtainedif we neglect all the derivative terms on the right-hand side of Eq. ~3.2! and approximate the matrixelements Aij

~0!~x1uv! and Bij~x1uv! by the first twoerms in their expansions in powers of the small pa-ameters dz2,exp~2b!, and exp~2b!. The resulting

expression for K~0!~x1uv! is

`
The result expressed by Eq. ~3.11! coincides withthat given by Eq. ~3.10! through terms of the ~21! andthe ~0! orders in the small parameters. However, ithas the advantage over the latter result in that in thelimit z~x1! [ 0 it reduces to
K~0!~x1uv! 51

d~v! 3cothD

d~v!2csch

Dd~v!

cschD

d~v!2coth

Dd~v!

4 , (3.12)

which is the exact result for the local impedance ma-trix for a metal film bounded by the planes x3 5 0 andx3 5 D. Solution ~3.10! does not have this property.This result indicates that all terms past the first on theright-hand side of Eq. ~3.2! vanish when z~x1! vanishesand hence provide small corrections to the result ob-tained through their neglect when z~x1! is not identi-cally zero but is small and a slowly varying function ofx1. The result given by Eq. ~3.11! is the impedancematrix in the coordinate representation that we use inthis paper.

4. Impedance Matrix in the Fourier Representation

In this section we derive the integral equation satis-fied by the Fourier-transformed local impedance ma-trix. The result may be of interest in itself becauseit provides a convenient starting point for obtainingan expansion of this matrix in powers of the surface-profile function z~x1!. Such an expansion can be use-

ful in the case in which the surface x3 5 z~x1! iseakly rough. However, our interest in obtaining

his equation is due to the fact that it provides aonvenient starting point for obtaining the local im-edance matrix in the case in which the surface-rofile function z~x1! is a periodic function of z~x1!.

This derivation is carried out in Section 5.We now return to Eq. ~2.14! and set x3 . D in it,

whereupon the left-hand side vanishes. We then in-troduce Fourier representation ~2.5b! for the Green’sfunction Ge~x1, x3ux19, x39! into the resulting equation,together with the representations

H1,2~x1uv! 5 *2`

` dq2p

H1,2~q!exp~iqx1!, (4.1a)

L1,2~x1uv! 5 *2`

` dq2p

L1,2~q!exp~iqx1!;

(4.1b)

exp@gz~x1!# 5 *2`

` dQ2p

I~guQ!exp~iQx1!, (4.2a)

z9~x1!exp@gz~x1!# 5 *2`

dQ2p

iQg

I~guQ!exp~iQx1!.

(4.2b)The result is the equation

0 5 *2`

` dk2p

exp@ikx1 2 b~k, v!x3#

b~k, v!

3 H2exp@b~k, v!D#@b~k, v!H1~k! 2 L1~k!#

1 *2`

` dq2p

I(b~k, v!uk 2 q) Fkq 2 ε~v!~v2yc2!

b~k, v!

3 H2~q! 2 L2~q!GJ. (4.3)

On equating to zero the kth Fourier coefficient, weobtain the equation

0 5 2exp@b~k, v!D# *2`

` dq2p

2pd~k 2 q!@b~q, v!H1~q!

2 L1~q!# 1 *2`

` dq2p

I(b~k, v!uk 2 q)

3 Fkq 2 e~v!~v2yc2!

b~k, v!H2~q! 2 L2~q!G . (4.4)

K~0!~x1uv! 51d

af

a 2 f exp@2~b 1 b!#31

af$a 1 f exp@2~b 1 b!#% 2

1a S1 1

b

fb2

dz2

2f 3Dexp~2b!

2exp~2b! 21 1dz2

2f 3 2b

abexp@2~b 1 b!#4 . (3.11)


Em

Lh

Ot

1

We next set x3 , z~x1!min in Eq. ~2.13!, whereuponthe left-hand side again vanishes. Use of Eqs.~2.5b!, ~4.1!, and ~4.2! in the resulting equation yields

0 5 *2`

` dk2p

exp@ikx1 1 b~k, v!x3#

b~k, v!$exp@2b~k, v!D#

3 @b~k, v!H1~k! 1 L1~k!# 2 *2`

` dq2p

I(2b~k, v!uk 2 q)

3 Fkq 2 e~v!~v2yc2!

b~k, v!H2~q! 1 L2~q!G . (4.5)

On equating to zero the kth Fourier coefficient, weobtain the equation

0 5 exp@2b~k, v!D# *2`

` dq2p

3 2pd~k 2 q!@b~q, v!H1~q! 1 L1~q!#

2 *2`

` dq2p

I(2b~k, v!uk 2 q)Fkq 2 e~v!~v2yc2!

b~k, v!

3 H2~q! 1 L2~q!G . (4.6)

quations ~4.4! and ~4.6! can be rearranged into theatrix integral equation

ˆi

atK

Equation ~4.7! implies a linear relation betweenˆ

1,2~q! and H1,2~q!. The most general such relationas the form

FL1~q!

L2~q!G5 *

2`

` dp2p [K11~qup! K12~qup!

K21~qup! K22~qup!]FH1~p!

H2~p!G . (4.8)

*2`

` dq2pF22pd~k 2 q!exp@b~k, v!D# I(b~k, v!uk 2 q)

22pd~k 2 q!exp@2b~k, v!D# I( 2 b~k, v!uk 2

5 322pd~k 2 p!b~k, v!exp

2pd~k 2 p!b~k, v!exp@2


If we use the inverses of definitions ~4.1!,

H1,2~q! 5 *2`

`

dx1 exp~2iqx1!H1,2~x1uv!, (4.9a)

L1,2~q! 5 *2`

`

dx1 exp~2iqx1!L1,2~x1uv!, (4.9b)

in Eq. ~4.8!, we are led to the relation

Li~x1uv! 5 (j51

2

*2`

`

dx19

3 F*2`

` dq2p *

2`

` dp2p

exp~iqx1!Kij~qup!

3 exp~2ipx19!GHj~x9uv!, i 5 1, 2. (4.10)

n comparing Eq. ~4.10! with Eq. ~2.21!, we obtainhe relation

Kij~x1ux19! 5 *2`

` dq2p *

2`

` dp2p

exp~iqx1!Kij~qup!

3 exp~2ipx19!. (4.11)

Thus the matrix Kij~qu p! can be called the nonlocalmpedance matrix in the Fourier representation.

When relation ~4.8! is substituted into Eq. ~4.7!nd the arbitrariness of H1,2~q! is used, we obtainhe integral equation satisfied by the matrixˆ

ij~qu p!:

K11~qup! K12~qup!

K21~qup! K22~qup!G

, v!D# I(b~k, v!uk 2 p)kp 2 e~v!~v2yc2!

b~k, v!

, v!D# 2I(2b~k, v!uk 2 p)kp 2 e~v!~v2yc2!

b~k, v!4 . (4.12)

*2`

` dq2p [22pd~k 2 q!exp@b~k, v!D# I(b~k, v!uk 2 q)

22pd~k 2 q!exp@2b~k, v!D# I(2b~k, v!uk 2 q)] FL1~q!

L2~q!G

5 *2`

` dq2p 322pd~k 2 q!b~k, v!exp@b~k, v!D# I(b~k, v!uk 2 q)

kq 2 e~v!~v2yc2!

b~k, v!

2pd~k 2 q!b~k, v!exp@2b~k, v!D# 2I(2b~k, v!uk 2 q)kq 2 e~v!~v2yc2!

b~k, v!4 FH1~q!

H2~q!G . (4.7)

q)GF@b~k

b~k

5.1!

If we wish to obtain the local impedance matrixKij

~0!~x1uv! we see on combining Eqs. ~2.26! and ~4.11!that it is given by

Kij~0!~x1uv! 5 *

2`

`

dx19Kij~x1ux19!

5 *2`

` dq2p

exp~iqx1!Kij~qu0!. (4.13)

We find from Eq. ~4.12! that the equation satisfiedby Kij~qu0! is

abe

n

where we have used the result that b~0, v! 5d21~v!.

5. Periodic Surface

The results of Section 4 are used in this section toobtain the equations for the elements of the localimpedance matrix in the case in which the surface-profile function of the rough surface z~x1! is a peri-odic function of x1 with period p. This case is ofinterest for at least two reasons. The first is that,as we see below, it is possible to obtain formallyexact results for the elements of the local imped-ance matrix when the rough surface of the film isperiodic, and these results can be used to assess thequality of the approximate analytic result obtainedin Section 3. The second is that, as we have al-ready noted in Section 1, periodically corrugatedsurfaces with long periods have been used recent-ly15 in computer simulation studies of the light scat-tered from surface and guided wave-bearingstructures as a way of overcoming convergenceproblems that can arise in such calculations be-cause of the long mean free paths of these excita-tions. The use of an impedance boundarycondition can simplify such calculations for film sys-tems.

We begin with the observation that if z~x1! is aperiodic function of x1, with period p, so is Kij

~0!~x1uv!,

and we expand it in a Fourier series:

Kij~0!~x1uv! 5 (

n52`

`

Kij~n!~v!expSi

2pnp

x1D. (5.1)

It follows from the inverse of Eq. ~4.13! and from Eq. ~that

Kij~qu0! 5 *2`

`

dx1 exp~2iqx1!Kij~0!~x1uv!

5 (n52`

`

2pdSq 22pn

p DKij~n!~v!. (5.2)

At the same time, the function I~guQ! defined by Eq.~4.2a! becomes

I~guQ! 5 *2`

`

dx1 exp@2iQx1 1 gz~x1!#

5 (n52`

`

*~n21y2! p

~n11y2! p

dx1 exp@2iQx1 1 gz~x1!#

5 (n52`

`

exp~2iQnp! *2 py2

py2

dx1 exp@2iQx1

1 gz~x1!#

5 (n52`

`

2pdSQ 22pn

p D(n~g!, (5.3)

where

(n~g! 51p *

2py2

py2

dx1 expF2i2pn

px1 1 gz~x1!G. (5.4)

When we substitute Eqs. ~5.2! and ~5.3! into Eq. ~4.14!nd equate the coefficients of 2pd@k 2 ~2pmyp!# onoth sides of the resulting equation, we obtain thequation satisfied by Kij

~n!~v!:

(52`

` F2dm,n exp@bm~v!D# (m2n@bm~v!#2dm,n exp@2bm~v!D# (m2n@2bm~v!G

3 FK11~n!~v! K12

~n!~v!

K21~n!~v! K22

~n!~v!G

*2`

` dq2p F22pd~k 2 q!exp@b~k, v!D# I(b~k, v!uk 2 q)

22pd~k 2 q!exp@2b~k, v!D# I(2b~k, v!uk 2 q)GFK11~qu0! K12~qu0!

K21~qu0! K22~qu0!G

5 322pd~k!exp@Dyd~v!#

d~v!2I(b~k, v!uk)

e~v!~v2yc2!

b~k, v!

2pd~k!exp@2Dyd~v!#

d~v!I(2b~k, v!uk)

e~v!~v2yc2!

b~k, v!4 , (4.14)


w

b

afi

t

ttap

e

dcg

sf

c

1

5 32dm,0

1d~v!

exp@Dyd~v!#1

d2~v!

(m@bm~v!#

bm~v!

dm,0

1d~v!

exp@2Dyd~v!# 21

d2~v!

(m@2bm~v!#

bm~v!4,

m 5 0, 6 1, 6 2, . . . , (5.5)

here

m~v! 5 bS2pmp

, vD 5 FS2pmp D2

11

d2~v!G1y2

Re bm~v! . 0, Im bm~v! , 0. (5.6)

6. Results

In this section we present the results of several nu-merical calculations intended to demonstrate the ac-curacy and the utility of local impedance matrix~3.11! in calculations of the scattering of light fromnd its transmission through a free-standing metallm with a single rough surface.We begin by assessing the accuracy of approximate

analytic result ~3.11! for the local impedance matrixby comparing the values of its elements for the case ofa periodic surface-profile function,

z~x1! 5 z0 cos2px1

p, (6.1)

with the formally exact results obtained from thesolution of Eq. ~5.5!. In Figs. 2, 3, and 4 we presentplots of @Re Kij

~0!~x1uv!exact 2 Re Kij~0!~x1uv!imp#y

Re Kij~0!~x1uv!exact and @Im Kij

~0!~x1uv!exact 2 ImKij

~0!~x1uv!impyIm Kij~0!~x1uv!exact as functions of p for

x1 5 0, x1 5 0.392p, and x1 5 0.5p, respectively.These calculations have been carried out for a silverfilm in the presence of a p-polarized electromagneticfield of wavelength l 5 394.7 nm, at which its dielec-ric function is e~v! 5 24.28 1 i0.21.16 The ampli-

tude z0 has the value z0 5 14.142 nm, and the meanhickness of the film is D 5 88 nm. We see fromhese results that the relative errors in the imped-nce approximation to the real and the imaginaryarts of the diagonal elements Kii

~0!~x1uv! of the localimpedance matrix are significantly smaller than theyare for the off-diagonal elements. Nevertheless, therelative errors in the impedance approximation to thereal and the imaginary parts of all elements Kij

~0!~x1uv!are smaller than 1022 for p $ 800 nm, with thelargest relative errors occurring in Re K12

~0!~x1uv! andIm K12

~0!~x1uv! for x1 5 0.392 p. The increasing rela-tive error in this case as p decreases below 800 nm isdue to the fact that a~x1 5 0.392p! becomes signifi-cantly smaller than unity as p becomes smaller than800 nm ~it vanishes at p > 180 nm!, which results inxp@2b~x1uv!#’s no longer being a small quantity of

the order of exp@2b~x1uv!#, as was assumed in theerivation of Eq. ~3.11!. More generally, the in-reasing relative error in all matrix elements at aiven value of x1yp~#0.5! as p decreases below 800

nm is related to the departure of a~x1!, which entersall the matrix elements, from unity, either in the


direction of smaller or larger values. Thus it ap-pears as if a further condition for the applicability ofEq. ~3.11! is that a~x1! does not depart significantlyfrom unity.

To illustrate the use of result ~3.11! for the imped-ance matrix in computer simulation calculations ofthe scattering of light from and its transmissionthrough a thin rough metal film, we consider the caseof a beam of p-polarized light incident upon a free-standing silver film whose illuminated surface is pla-nar while its back surface is a one-dimensional

Fig. 2. ~a! @Re Kij~0!~x1uv!exact 2 Re Kij

~0!~x1uv!impyRe Kij~0!~x1uv!exact,

~b! @Im Kij~0!~x1uv!exact 2 Im Kij

~0!~x1uv!imp#yIm Kij~0!~x1uv!exact for a

ilver film as functions of the period p when the surface profileunction is given by z~x1! 5 z0 cos~2px1yp!. e~v! 5 24.28 1 i0.21,

z0 5 14.142 nm, D 5 88 nm, and x1 5 0. The Kij~0!~x1uv!imp are

calculated on the basis of Eq. ~3.11!, while the Kij~0!~x1uv!exact are

alculated on the basis of Eqs. ~5.1! and ~5.5!.

i

Ts

we

random surface defined by x3 5 z~x1!. The plane ofincidence is the x1x3 plane. The calculations arebased on the rigorous scattering theory for a three-layer system presented in Ref. 17. The wavelengthof the incident light is l 5 394.7 nm. The dielectricconstant of silver at this wavelength is e~v! 524.28 1 i0.21.16 The angle of incidence u0 5 10° ands measured counterclockwise from the positive x3

axis. The mean thickness of the film is D 5 88 nm.he length L of the x1 axis covered by the randomurface is L 5 12.6 mm. It is divided into N 5 500

equally spaced sampling points. The half-width ofthe intercept of the incident beam with the x1 axis isg 5 3.15 mm.

The surface-profile function z~x1! of the lower sur-face of the film is assumed to be a single-valued func-tion of x1 that is differentiable as many times as is

necessary and to constitute a stationary, zero-meanGaussian random process defined by

^z~x1!z~x19!& 5 d2W~ux1 2 x19u!, (6.2)

here the angle brackets denote an average over thensemble of realizations of z~x1!. In Eq. ~6.2! d 5

^z2~x1!&1y2 is the rms height of the surface. Thesurface-height autocorrelation function W~ux1u! is de-fined by its power spectrum g~uku!, defined by

g~uku! 5 *2`

`

dx1 exp~2ikx1!W~ux1u!. (6.3)

Fig. 3. Same as Fig. 2, but for x1 5 0.392p.
Fig. 4. Same as Fig. 2, but for x1 5 0.5p.

lcasgA

wtE

tsd

itls

tf

cttHt f

aitt

rwfiaaci

1

The numerical calculations are carried out for a ran-dom surface characterized by the Gaussian powerspectrum,

g~uku! 5 Îpa exp~2a2k2y4!, (6.4)

where the parameter a is the transverse correlationength of the surface roughness. In the present cal-ulations the values of d and a used are d 5 10 nm and5 500 nm. The surface-profile functions z~x1! pos-

essing the statistical properties just enumerated areenerated by the method described, e.g., in Appendixof Ref. 18.In relating the results of these calculations to thosehose results are presented in Figs. 2–4, we note

hat the rms height of the periodic surface defined byq. ~6.1! is d 5 z0y=2, while the mean distance be-

tween consecutive maxima on the random surface,which is the analog of the period p in Eq. ~6.1!, is closeo twice the transverse correlation length a. For aurface defined by power spectrum ~6.4!, the meanistance between consecutive maxima is 2.5674a.19

This yields the equivalence p 5 2.5674a. Thus avalue of d 5 10 nm corresponds to a value of z0 514.142 nm, while a value of a 5 500 nm correspondsto a value of p 5 1283.7 nm.

What has been calculated in these simulation stud-es is the contribution to the mean differential reflec-ion coefficient from the incoherent component of theight scattered from the film as a function of thecattering angle us, ^]Ry]us&incoh, and the contribution

to the mean differential transmission coefficient fromthe incoherent component of the light transmittedthrough the film as a function of the angle of trans-mission ut, ^]Ty]ut&incoh. The differential reflection~transmission! coefficient is defined as the fraction ofhe total time-averaged flux incident upon the sur-ace that is scattered ~transmitted! into an angular

interval dus~dut! about the angle of scattering us~transmission ut!. The angle of scattering us is mea-sured clockwise from the positive x3 axis, while theangle of transmission is measured counterclockwisefrom the negative x3 axis.

Plots of ^]Ry]us&incoh and ^]Ty]ut&incoh are presentedin Figs. 5~a! and 5~b!, respectively. The results ob-tained from Np 5 500 different realizations of therandom surface were used in the calculations whoseresults are plotted in these figures. In each figurethe solid curve is the rigorous result obtained fromthe numerical solution of a system of four coupledinhomogeneous integral equations for the four sourcefunctions L1,2~x1uv! and H1,2~x1uv!. The dashedurve is the result obtained from the numerical solu-ion of the system of two coupled inhomogeneous in-egral equations for the two source functions

1,2~x1uv! when L1,2~x1uv! have been expressed inerms of H1,2~x1uv! by use of Eqs. ~2.25! and ~3.11!.

The agreement between the results of the rigorouscalculations and those based on use of the impedanceboundary condition is entirely satisfactory.

It is seen from Figs. 5~a! and 5~b! that ^]Ty]ut&incohis approximately 60 times larger than ^]Ry]us&incoh.


This is readily understood. The incoherent compo-nent of the transmitted and the scattered light arisesfrom the interaction of the incident light with therandom surface. In the case of transmission, thelight has to pass through the film only once for thisinteraction to occur. In the case of scattering, how-ever, the light passes through the film twice in thecourse of this interaction, and the film is a stronglyattenuating medium at the wavelength of the inci-dent light @d~v! 5 ~30.32 1 i0.7433!nm#.

When the transverse correlation length is reducedrom 500 to 250 nm, with the remaining experimentalnd roughness parameters keeping the values usedn obtaining the results plotted in Fig. 5, we obtainhe results plotted in Fig. 6. We see that in this casehe agreement between the exact result for ^]Ry

]us&incoh and the one calculated with the use of the

Fig. 5. ~a! ^]Ry]us&incoh, ~b! ^]Ty]ut&incoh as functions of us and ut,espectively, in the case in which a p-polarized beam of light ofavelength l 5 394.7 nm is incident upon a free-standing silverlm whose illuminated surface is planar and whose back surface isone-dimensional random surface characterized by d 5 10 nm and5 500 nm. e~v! 5 24.28 1 i0.21, u0 5 10°, D 5 88 nm; solid

urve, exact result; dashed curve, result obtained with the use ofmpedance matrix ~3.11!.

bif

d

feecptl

thtetpocwtlsits

wwctlwsl

impedance matrix is less good than it is for the largertransverse correlation length, while the agreementbetween the exact and the approximate results for^]Ty]ut&incoh is still both qualitatively and quantita-tively good. We surmise that the reason for the dif-ference between the results plotted in Fig. 6~a! isecause the conditions for the validity of approximatempedance matrix ~3.11! are beginning to be violatedor the values of z0 and a assumed. If we note that

a value of a 5 250 nm corresponds to a value of p 5641.85 mm, the latter falls in the range of p values~,800 nm! in which the relative error in the off-

iagonal elements of K~0!~x1uv! can exceed 1022, es-pecially for values of x1yp . 0.392. In view of thesmallness of ^]Ry]us&incoh, which is due both to theact that it is the small difference between two nearlyqual quantities, the mean differential reflection co-fficient and the contribution to it from the coherentomponent of the scattered light, and to the doubleassage of the light through the film, a small error inhe impedance matrix can be magnified in the calcu-ation of ^]Ry]us&incoh. This is less the case for ^]Ty

]ut&incoh, which is a significantly larger quantity.

Fig. 6. Same as Fig. 4, but with a 5 250 nm.

The calculations of the results plotted in Figs. 5and 6 were significantly faster when impedance ma-trix ~3.11! was used than when they were carried outrigorously. For example, the calculation for one re-alization of the surface profile function required 576.2CPU seconds on an Alpha 21064 machine running at236 Mhz, but only 157.2 CPU seconds when imped-ance matrix ~3.11! was used.

The failure of ^]Ry]us&incoh to vanish at us 5 690° inthe approximate results plotted in Fig. 6~a! ~althoughhe magnitude of this function should also be noted!,owever, is not due to a deficiency of impedance ma-rix ~3.11!, since this feature is also present in thexact result plotted in this figure. Instead, we at-ribute it to the scattering of the surface-plasmonolaritons supported by the metal film from the endsf the finite segment of random surface used in theomputer simulations. For the weakly lossy andeakly rough surfaces used in the present simula-

ions, the mean free path of the surface-plasmon po-aritons is longer than the length L of the randomurface assumed, so that after their excitation by thencident light, through the roughness of the surface ofhe film, they can propagate to the ends of the roughegment of the surface at x1 5 6Ly2 and be scattered

by them. The effectiveness with which the surface-plasmon polaritons are excited through the surfaceroughness is proportional to g~uk 7 k sp~v!u! for theforward- and the backward-propagating surfacewaves, where g~uku! is the power spectrum of the sur-face roughness and ksp~v! . vyc is the wave numberof the surface-plasmon polaritons whose frequency vis that of the incident light. When the roughness ischaracterized by Gaussian power spectrum ~6.4!,surface-plasmon polaritons are strongly excited whena , l, as is the case here. It is known20 that thisscattering slows down or even, in some cases, pre-vents obtaining convergent results when averagingresults for the intensity of the scattered or the trans-mitted light over the ensemble of realizations of therandom surface, especially for grazing angles of scat-tering for which the contribution of the surface wavesis significant. This difficulty is much less marked inthe results displayed in Fig. 5~a! because in this casea . l and the surface-plasmon polaritons are more

eakly excited than in the case for which Fig. 6~a!as calculated. We can overcome the difficulties

aused by the excitation of surface-plasmon polari-ons whose mean free paths are comparable with theength of the random surface by increasing the latter,hich is sometimes impractical, or by generating a

egment of random surface of length L and then rep-icating it periodically to cover the entire x1 axis.15

7. Conclusions

Several conclusions can be drawn from the resultsobtained in this paper. The first is that the approx-imate analytic result for the local impedance matrixK~0!~x1uv! given by Eq. ~3.11! is an accurate represen-tation of that matrix when the conditions for its va-lidity are satisfied, namely that d~v!z2~x1! is smalland that exp@2b~x1uv!# and exp@2b~x1uv!# are also


aK

aa

iSetbddj t

T

1

small and of the same order of magnitude asd~v!z2~x1!. This conclusion follows from the goodgreement between the results for the elementsij~0!~x1uv! for a metal film with one periodically cor-

rugated surface and one planar surface obtained fromEq. ~3.11! and the exact results obtained through thesolution of Eq. ~5.5!.

A second conclusion is that, as a consequence of theaccuracy of the result for K~0!~x1uv! given by Eq.~3.11!, it yields accurate results for the differentialreflection and transmission coefficients when it isused in computer simulation calculations of the scat-tering of light from and its transmission through afree-standing metal film with a one-dimensional ran-domly rough surface. This is demonstrated by theresults of computer simulation calculations of thesecoefficients presented in Section 6.

In view of these good results obtained by the use ofthe local impedance matrix, one would like to knowthe criterion for the applicability of the local imped-ance boundary conditions, i.e., when is it permissibleto neglect the terms with the derivatives dnHj~x1uv!ydx1

n~n $ 1! on the right-hand side of Eq. ~2.23!? Be-cause of the matrix nature of the present impedanceboundary conditions this is not an easy criterion toestablish. The results of Ref. 3 for the surface im-pedance of a semi-infinite metal yield the conclusionthat if one requires that the leading nonlocal term, inan expansion in powers of d~v!z2~x1!, be small com-pared with the leading local term, the condition

udz2u ,, ufydu (7.1)

must be satisfied. Here k is a wave number thatcharacterizes the rate of change of H~x1uv! as a func-tion of x1, so that dH~x1uv!ydx1 5 O(kH~x1uv!). For

periodically corrugated surface whose period is p,n upper bound for k will be 2pyp, the width of the

first Brillouin zone of the grating, and the neglect ofthe nonlocal terms will therefore be justified if p ..2puz1z2idu2yf4. For a randomly rough surface wecan replace p with 2^d&, where ^d& is the averagedistance between consecutive maxima and minimaon the surface, and z1 and z2 with their rms values.Then the neglect of the nonlocal terms will be justi-fied if ^d& .. p^z1

2&1y2^z22&1y2udu2yf4. These inequal-

ties are well satisfied for the surfaces considered inection 6. It would seem that these criteria are nec-ssary for the validity of the local approximation tohe impedance matrix obtained in the present work,ut they may not be sufficient because of the off-iagonal elements that couple the fields and theirerivatives on the two surfaces. However, we con-

ecture that when exp~2b! and exp~2b! are of theorder of udz2u or udu^z2

2&1y2, the inequalities presentedabove will be sufficient as well. This conjecture issupported by the good agreement between the rigor-ous and the approximate results presented in Figs. 5and 6.

Finally, when the number of coupled integral equa-tions for the source functions H1,2~x1uv! and L1,2~x1uv!that have to be solved in computer simulation studies


of optical interactions with a rough free-standingmetal film are reduced from four to two, the use of theimpedance boundary conditions obtained in this pa-per reduces the computation time significantly, bynearly a factor of 4 for a single realization of therough surface profile.

Appendix A

In this Appendix we outline the evaluation of theexpressions for the matrix elements A12

~0!~x1uv! andB12~x1uv! given by Eqs. ~3.5b! and ~3.8b!, respectively.

From Eqs. ~2.17b! and ~3.3! we find that the matrixelement A12

~0!~x1uv! is given by

A12~0!~x1uv! 5 2 *

2`

`

dx19K0(1d

$~x1 2 x19!2

1 @D 2 z~x19!#2%1y2) . (A1)

We begin the evaluation of this expression by makingthe change of variable x91 5 x1 1 d~v!u. We thenrewrite the argument of the modified Bessel functionin the integrand by extracting the square root of aquadratic form in u and expanding the remainder inpowers of the skin depth d~v! as

1d

$~x1 2 x19!2 1 @D 2 z~x19!#

2%1y2 5 ~a2u2 2 2bz1u 1 b2!1y2

1d2

z1z2u3

~a2u2 2 2bz1u 1 b2!1y2

1 d2312

F13 z1z3 114

z22Gu4 2

13

bz3u3

~a2u2 2 2bz1u 1 b2!1y2

218

z12z2

2u6

~a2u2 2 2bz1u 1 b2!3y241 O~d3!, (A2)

where

a~x1! 5 $1 1 z12~x1! 2 @D 2 z~x1!#z2~x1!%

1y2, (A3a)

b~x1uv! 5D 2 z~x1!

d~v!, (A3b)

zn~x1! 5dnz~x1!

dx1n . (A3c)

The integral on the right-hand side of Eq. ~A1! canhen also be expanded in powers of the skin depth.o the second nonzero order in d~v! we obtain

A12~0!~x1uv! 5 2dI1~x1uv! 2 d2z1z2 I2~x1uv! 1 O~d3!, (A4)

where

I1~x1uv! 5 *2`

`

duK0@~a2u2 2 2bz1 u 1 b2!1y2#, (A5a)

t

i

I2~x1uv! 5 *2`

`

duu3 K1@~a2u2 2 2bz1 u 1 b2!1y2#

~a2u2 2 2bz1 u 1 b2!1y2 . (A5b)

The further change of variable u 5 ~zya! 1 ~bz1ya2!ransforms Eqs. ~A5! into

I1~x1uv! 52a *

0

`

dzK0@~z2 1 b2!1y2#, (A6a)

I2~x1uv! 52a4 *

0

`

dzF3bz1

az2 1 Sbz1

a D3G K1@~z2 1 b2!1y2#

~z2 1 b2!1y2 ,

(A6b)

where

b 5 bS1 2z1

2

a2D1y2

5 b~1 2 bdz2!

1y2

a. (A7)

By expanding both sides of the Fourier cosine trans-form21

*0

`

dxKn@~x2 1 b2!1y2#

~x2 1 b2!1y2n cos xy

5 Îp

2b1y22n~1 1 y2!1y2n21y4Kn21y2@b~1 1 y2!1y2# (A8)

n powers of y2 and equating terms of the same order,we can evaluate the integrals appearing in Eqs. ~A6!:

I1~x1uv! 5p

aexp~2b!, (A9a)

I2~x1uv! 5 pbz1

a5 exp~2b!F3 11b Sbz1

a D2G . (A9b)

We obtain finally

A12~0!~x1uv! 5 2p

da

exp~2b!H1 2dz2

2bz1

2

a4

3 F3 11b Sbz1

a D2G 1 O~d2!J . (A10)

The evaluation of the matrix element B12~x1uv! pro-ceeds in a similar fashion. From Eqs. ~2.18b! and~3.4b! we find that B12~x1uv! is given by

2B12~x1uv! 52d2 *

2`

`

dx19

3

K1(1d

$~x1 2 x19!2 1 @D 2 z~x19!#

2%1y2)1d

$~x1 2 x19!2 1 @D 2 z~x19!#

2%1y2

3 $~x19 2 x1!z9~x19! 1 @D 2 z~x19!#%.

(A11)

We again use the change of variable x19 5 x1 1 d~v!u,expansion ~A2!, and the expansion

$~x19 2 x1!z1~x19! 1 @D 2 z~x19!#%

5 dFb 112

dz2 u2 113

d2z3 u3 1 O~d3!G (A12)

to obtain an expansion of the right-hand side of Eq.~A11! in powers of the skin depth d~v!:

2B12~x1uv! 5 2bJ1~x1uv! 1 dz2@bz1 J2~x1uv!

1 J3~x1uv!# 1 O~d2!, (A13)

where

J1~x1uv! 5 *2`

`

duK1@~a

2u2 2 2bz1 u 1 b2!1y2#

~a2u2 2 2bz1 u 1 b2!1y2 ,

(A14a)

J2~x1uv! 5 *2`

`

duu3HK19@~a2u2 2 2bz1 u 1 b2!1y2#

a2u2 2 2bz1 u 1 b2

2K1@~a

2u2 2 2bz1 u 1 b2!1y2#

~a2u2 2 2bz1 u 1 b2!3y2 J5 2*

2`

`

duu3 K1@~a2u2 2 2bz1 u 1 b2!1y2#

a2u2 2 2bz1 u 1 b2 ,

(A14b)

J3~x1uv! 5 *2`

`

duu2 K1@~a2u2 2 2bz1 u 1 b2!1y2#

~a2u2 2 2bz1 u 1 b2!1y2 .

(A14c)

We next make the change of variable u 5 ~zya! 1~bz1ya2! and obtain

J1~x1uv! 52a *

0

`

dzK1@~z2 1 b2!1y2#

~z2 1 b2!1y2 , (A15a)

J2~x1uv! 5 22bz1

a5 *0

`

dzFSbz1

a D2

1 3z2G3

K2@~z2 1 b2!1y2#

z2 1 b2 , (A15b)

J3~x1uv! 52a3 *

0

`

dzFSbz1

a D2

1 z2G K1@~z2 1 b2!1y2#

~z2 1 b2!1y2 .

(A15c)

The use of Eq. ~A8! with n 5 1 and y 5 0 yields

J1~x1uv! 5p

abexp~2b!. (A16a)


Spektrosk. 80, 459–470 ~1996! @Opt. Spectrosc. 80, 409–420

1

In a similar fashion we obtain

J2~x1uv! 5 2pbz1

a5bexp~2b!F3S1 2

1bD1 Sbz1

a D2S1b 11b2DG ,

(A16b)

J3~x1uv! 5p

a3 exp~2b!F1 1 Sbz1

a D21bG . (A16c)

On combining the results given by Eqs. ~A13! and~A16!, we obtain finally

B12~x1uv! 5 22pb

abexp~2b! 2 p

dz2

a3 exp~2b!

3 F1 2 Sbz1

a D2S2b

23b2D 2 Sbz1

a D4S 1b2 1

1b3DG

1 O~d2!. (A17)

This work was supported by U.S. Army ResearchOffice grant DAAH 04-96-1-0187.

References1. R. Garcia-Molina, A. A. Maradudin, and T. A. Leskova, “The

impedance boundary condition for a curved surface,” Phys.Rep. 194, 351–359 ~1990!.

2. A. A. Maradudin, “The impedance boundary condition for aone-dimensional, curved metal surface,” Opt. Commun. 103,227–234 ~1993!.

3. A. A. Maradudin and E. R. Mendez, “Theoretical studies of theenhanced backscattering of light from one-dimensional ran-domly rough metal surfaces by the use of a nonlocal impedanceboundary condition,” Physica A 207, 302–314 ~1994!.

4. A. Mendoza-Suarez and E. R. Mendez, “Derivation of an im-pedance boundary condition for one-dimensional, curved, re-entrant surfaces,” Opt. Commun. 134, 241–250 ~1997!.

5. T. T. Ong, V. Celli, and A. A. Maradudin, “The impedance of acurved surface,” Opt. Commun. 95, 1–4 ~1993!.

6. A. A. Maradudin, “The impedance boundary condition at atwo-dimensional rough surface,” Opt. Commun. 116, 452–467~1995!.

7. A. A. Maradudin and E. R. Mendez, “The utility of an imped-ance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt.


~1996!#.8. X. Wang and H. J. Simon, “Directionally scattered optical

second-harmonic generation with surface plasmons,” Opt.Lett. 16, 1475–1477 ~1991!.

9. H. J. Simon, Y. Wang, L. B. Zhou, and Z. Chen, “Coherentbackscattering of optical second-harmonic generation withlong-range surface plasmons,” Opt. Lett. 17, 1268–1270 ~1992!.

10. O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Les-kova, and N. N. Novikova, “Anderson localization effects in thesecond harmonic generation at a weakly rough metal surface,”Phys. Lett. A 170, 231–234 ~1992!.

11. Y. Wang and H. J. Simon, “Coherent backscattering of opticalsecond-harmonic generation in silver films,” Phys. Rev. B 47,13695–13699 ~1993!.

12. L. Kuang and H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plas-mons,” Phys. Lett. A 197, 257–261 ~1995!.

13. Further information will be available in a paper entitled“Multiple-scattering effects in the second-harmonic generationof light in scattering from a random metal surface in theKretschmann ATR geometry,” which is currently in prepara-tion by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R.Mendez, and I. V. Novikov.

14. A. E. Danese, Advanced Calculus ~Allyn and Bacon, Boston,1965!, Vol. I, p. 123.

15. A. Madrazo and A. A. Maradudin, “Numerical solutions of thereduced Rayleigh equation for the scattering of electromag-netic waves from rough dielectric films on perfectly conductingsubstrates,” Opt. Commun. 134, 251–263 ~1997!.

16. P. W. Johnson and R. W. Christy, “Optical constants of thenoble metals,” Phys. Rev. B 6, 4370–4379 ~1972!.

17. V. Freilikher, E. Kanzieper, and A. A. Maradudin, “Coherentscattering enhancement in systems bounded by rough surfac-es,” Phys. Rep. 288, 127–204 ~1997!.

18. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez,“Enhanced backscattering of light from a random grating,”Ann. Phys. 203, 255–307 ~1990!.

19. A. A. Maradudin and T. Michel, “The transverse correlationlength for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 ~1990!.

20. M. Saillard and D. Maystre, “Scattering from metallic anddielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990~1990!.

21. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Tables of Integral Transforms ~McGraw-Hill, New York, 1954!,Vol. I, p. 56, formula 45.

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Impedance Boundary Conditions for a Metal Film with a Rough Surface