PHYSICAL REVIEW B 15 OCTOBER 2000-IIVOLUME 62, NUMBER 16

Local electric and magnetic fields in semicontinuous metal films:Beyond the quasistatic approximation

V. A. Shubin,1 Andrey K. Sarychev,2 J. P. Clerc,3 and Vladimir M. Shalaev11Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003

2Center for Applied Problems of Electrodynamics, Moscow 127412, Russia3Institut Universitaire des Syste`mes Thermiques Industriels, 13453 Marseille, France

~Received 14 October 1999; revised manuscript received 29 March 2000!

A theory of optical, infrared, and microwave response of metal-dielectric inhomogeneous films is developed.The generalized Ohm’s law is formulated for the important case, when the inhomogeneity length scale iscomparable with or larger than the skin~penetration! depth in metal grains. In this approach electric andmagnetic fieldsoutsidea film can be related to the currentsinsidethe film. Our computer simulations, with theuse of the generalized Ohm’s law approximation, reproduce the experimentally observed prominent absorptionband near the percolation threshold. Calculations show that the local electric and magnetic fields experiencegiant spatial fluctuations. The fields are localized in small spatially separated peaks: electric and magnetic hotspots. In these hot spots the local fields~both electric and magnetic! exceed the applied field by several ordersof magnitude. It is also shown that transmittance of a regular array of small holes in a metal film is stronglyenhanced when the incident wave is in resonance with surface polaritons in the film. In addition, there is a skinresonance in transmission, which is of a purely geometrical nature.

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I. INTRODUCTION

Random metal-dielectric films, also known as semicotinuous metal films, are usually produced by thermal evaration or spattering of metal onto an insulating substratethe growing process, first, small clusters of metal grainsformed and eventually, at a percolation threshold, a contous conducting path appears between the ends of the samindicating a metal-insulator transition in the system. At higsurface coverage, the film is mostly metallic with voidsirregular shape, and finally the film becomes a uniform mefilm. Over the past three decades, the electric-transport perties of semicontinuous metal films have been a topicactive experimental and theoretical study. The classicalcolation theory had been employed to describe the anolous behavior of the conductivity and other transport propties near the percolation threshold.1–4 Recently it was shownthat quantum effects, such as tunneling between metal cters and electron localization, become important at the pcolation even at room temperature~see Refs. 5–8 and references there in!. The low-frequency divergence of thdielectric-constant was predicted theoretically7,8 and ob-tained then experimentally.9

In this paper we will consider the optical responsemetal-insulator thin films. Although these films have beintensively studied both experimentally and theoretica~see e.g., Refs. 3,4,10–22!, the important role of giant localfield fluctuations was not considered in these earlier stud

A two-dimensional inhomogeneous film is a thin laywithin which the local physical properties are not uniformThe response of such a layer to an incident wave depecrucially on the inhomogeneity length scale compared towavelength and also on the angle of incidence. Usuawhen the wavelength is smaller than the inhomogenscale, the incident wave is scattered in a various directioThe total field that is scattered in certain direction is the sof the elementary waves scattered in that direction by e

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elementary scatterer on the surface. As each elemenwave is given not only by its amplitude, but also by iphase, this sum will be a vector sum. The scattered wavthen distributed in various directions, though certain prileged directions may receive more energy than others.contrast, when the inhomogeneity length scale is musmaller than the wavelength, the resolution of the wave issmall to ‘‘see’’ the irregularities, therefore the wave is threflected specularly and transmitted in well-defined directias if the film were a homogeneous layer with bulk effectiphysical properties~conductivity, permittivity, and permeability! that are uniform. The wave is coupled to the inhmogeneities in such a way that irregular currents are excon the surface of the layer. Strong distortions of the fiethen appear near the surface; however, they decay expotially so that far enough from the surface the wave resumits plane-wave character.

The problem of scattering from inhomogeneous surfahas attracted attention since the time of Lord Rayleigh.23 Dueto the wide range of potential application in, e.g., radiowaand radar techniques, most efforts have been concentratthe regime where the scale of inhomogeneity is larger tthe wavelength.24 In the last decade, a problem of localiztion of surface polaritons25 and other ‘‘internal modes’’ dueto their interaction with surface roughness, attracted a loattention. This localization is found to manifest stronglythe angular dependence of the intensity of nonspecularlyflected light, leading to the peak in the antispecudirection26 and other ‘‘resonance directions.’’27,28The devel-opment of near-field scanning optical microscopy has opethe way to probe the surface polariton field above the surfand visualize its distribution~for review, see, for examplesee Ref. 29.! In this paper we consider another limiting caswhen the inhomogeneity length scale is much smaller tthe wavelength, but can be of the order or even larger tthe skin depth. In other words the coupling of a metal gr

11 230 ©2000 The American Physical Society

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PRB 62 11 231LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

with an electromagneticfield is supposed to be strong ispite of their subwavelength size. In particular, we focusthe high-frequency response~optical, infrared, and micro-wave! of thin metal-dielectric random films.

The optical properties of metal-dielectric films shoanomalous phenomena that are absent for bulk metaldielectric components. For example, the anomalous abstion in the near-infrared spectral range leads to unusualhavior of transmittance and reflectance. Typically, the tramittance is much higher than that of continuous metal filmwhereas the reflectance is much lower~see, e.g., Refs3,4,10,11,16–18!. Near and well-below the percolatiothreshold, the anomalous absorbance can be as hig50%.12–16,20A number of theories were proposed for calclation of the optical properties of semicontinuous randfilms, including the effective-medium approaches,30,31 theirvarious modifications,3,16,17,32–36 and the renormalizationgroup method~see, e.g. Refs. 4,37,38!. In most of these theories the semicontinuous metal-dielectric film is considereda fully two-dimensional system and a quasistatic approximtion is invoked. However, usage of this approximation iplies that both the electric and magnetic fields in the filmassumed to be two dimensional and curl free. That assution ceases to be valid when the fields are changed conerably within the film and in its close neighborhood, whichusually the case for a semicontinuous metal thin film, escially in the regime of strong skin effect.

In the attempt to expand a theoretical treatment beythe quasistatic approximation, a new approach has recebeen proposed that is based on the full set of Maxweequations.18–20 This approach does not use the quasistaapproximation because the fields are not assumed to befree inside the physical film. Although the theory was prposed with metal-insulator thin films in mind, it is, in facquite general and can be applied to any kind of inhomoneous film under appropriate conditions. For the reasonwill be explained below, that theory is referred to as t‘‘generalized Ohm’s law.’’ We use this new theory to finoptical properties and distribution of the local fields insemicontinuous metal film and in a metal film with reguarray of holes in it.

Below we restrict ourselves to the case where all theternal fields are parallel to the plane of the film. This meathat an incident wave, as well as the reflected and transmwaves, are traveling in the direction perpendicular to the fiplane. We focus our consideration on the electric amagnetic-field magnitudes at certain distancesawayfrom thefilm and relate them to the currentsinside the film. We as-sume that inhomogeneities on a film are much smaller inthan the wavelengthl ~but not necessarily smaller than thskin depthd), so that the fields away from the film are cufree and can be expressed as gradients of potential fieldselectric and magnetic induction currents averaged overfilm thickness obey the usual two-dimensional continuequations. Therefore the equations for the fields~e.g.,¹3E50) and the equations for the currents~e.g., ¹3 j50) are thesameas in the quasistatic case. The only diffeence, though important, is that the fields and the averacurrents are now related by new constitutive equationsthat there are magnetic currents as well as electric curre

To determine these constitutive equations, we find

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electric and magnetic-field distributions inside the condtive and dielectric regions of the film. The boundary contions completely determine solutions of Maxwell’s equatiofor the fields inside a grain when the frequency is fixeTherefore the internal fields, which change very rapidly wposition in the direction perpendicular to the film, depelinearly on the electric and magnetic field away from tfilm. The currents inside the film are linear functions of tlocal internal fields given by the usual local constitutiequations. Therefore the currents flowinginsidethe film alsodepend linearly on the electric and magnetic fieldsoutsidethe film. However, the electric current averaged over the fithickness now depends not only on the external electric fibut also on the external magnetic field. The same is truethe average magnetic induction current. Thus we havelinear equations that connect the two types of the averinternal currents to the external fields. These equationsbe considered as a generalization of the Ohm’s law tononquasistatic case and referred to as generalized Ohm’s~GOL!.19,20 The GOL forms the basis of a new approachcalculating the electromagnetic properties of inhomogenefilms.

It is instructive to consider first the electric and magnefields on both sides of the film.19,20 Namely, the electric andmagnetic fields are considered at the distancel 0 behind thefilm E1(r )5E(r ,2d/22 l 0), H1(r )5H(r ,2d/22 l 0), and atthe distancel 0 in front of the film E2(r )5E(r ,d/21 l 0),H2(r )5H(r ,d/21 l 0) as shown in Fig. 1. All the fields andcurrents considered in this paper are monochromatic fiewith the usual exp(2ivt) time dependance. The vectorr5$x,y% in the above equations is a two-dimensional vecin the film plane perpendicular to the ‘‘z’’ axis whered is thethickness of the film. In the case of laterally inhomogeneofilms electric current

jE5E2d/22 l 0

d/21 l 0j ~r ,z!dz

and current of the magnetic induction

FIG. 1. The scheme used in a theoretical model. Electromnetic wave of wavelengthl is incident on a thin metal-insulatofilm with thicknessd. It is partially reflected and absorbed, and tremainder is transmitted through the film. The amplitudes ofelectric and magnetic fields~averaged over the planez52d/22 l 0behind the film! are equal to each other.

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11 232 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

jH5~ iv/4p!E2d/22 l 0

d/21 l 0B~r ,z!dz

are functions of the vectorr . The metal islands in semicontinuous films typically have an oblate shape so that the gdiameterD can be much larger than the film thicknessd ~seee.g., Refs. 11!. When the thickness of a conducting graind~or skin depthd) and distancel 0 are much smaller than thgrain diameterD, the relation of the fieldsE1 andH1 ~or E2andH2) to the currents becomes fully local. The electricjEand magneticjH currents lie in between the planesz52d/22 l 0 and z5d/21 l 0. These currents satisfy to thtwo-dimensional continuity equations

¹• jE~r !50, ¹• jH~r !50, ~1!

which follow from the three-dimensional continuity equtions when thez components ofE1 ,H1 and E2 ,H2 are ne-glected at the planesz52(d/21 l 0) andz5d/21 l 0, respec-tively. This is possible because these components are sin accordance with the fact that the average fields^E1&,^H1&, ^E2&, and ^H2& are parallel to the film plane. Moreover, it is supposed in the GOL approximation that ‘‘z’’components of the local fields vanish on average, so thaintegrals

E2d/22 l 0

d/21 l 0Ezdz50

and

E2d/22 l 0

d/21 l 0Hzdz50

are equal to zero. Then, the second Maxwell’s equacurlH5(4p/c) jE can be written as

R H"dl5~4p/c!~n1• jE!D, ~2!where the integration is over the rectangular contour, whhas sidesd12l 0 andD so that the sidesd12l 0 are perpen-dicular to the film and the sidesD are in the planesz56(d/21 l 0); the vectorn1 is perpendicular to the contoui.e., parallel to the film. WhenD→0 this equation takes thfollowing form

H22H1524p

c@n3 jE#, ~3!

where the vectorn is perpendicular to the film. The firsMaxwell equation curlE5 ikH can be rewritten as

R E"dl5~4p/c!~n1• jH!D, ~4!where the integration contour is the same as in Eq.~2! andjHis the current of the magnetic induction, defined earlierEq. ~1!. Thus we obtain the equation

E22E1524p

c@n3 jH# ~5!

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relating the electric fields in front of and behind the film. Thelectric currentjE and current of the magnetic inductionjHobey GOL~see Refs. 19,20!, namely,

jE~r !5u~r !E~r !, jH~r !5w~r !H~r !, ~6!

whereE5(E11E2)/2, H5(H11H2)/2 and Ohmic param-etersu and w are expressed in terms of the local refractiindex n5A«(r ) as

u52 ic

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tan~Dk/4!1n tan~dkn/2!

12n tan~Dk/4! tan~dkn/2!, ~7!

w5 ic

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n tan~Dk/4!1tan~dkn/2!

n2tan~Dk/4! tan~dkn/2!, ~8!

where the refractive indexn takes valuesnm5A«m and nd5A«d for metal and dielectric regions of the film; the ditance to the reference plane is set asl 05D/4.

18,19 The es-sence of the GOL can be summarized as follows: The enphysics of a three-dimensional inhomogeneous layer, whis described by the full set of Maxwell’s equations, has bereduced to a set of two-dimensional Eqs.~1! and ~6!.

In the quasistatic limit, when the optical thicknessmetal grains is smalldkunmu!1, while the metal-dielectricconstant is large in magnitude,u«mu@1, the following esti-mates hold for the Ohmic parameters of the metal grains

um.2 iv«m4p

d, wm. iv

4p~d1D/2!, ~d/d!1!.

~9!

In the opposite case of a strong skin effect, when the sdepth~penetration depth! d51/k Im nm is much smaller thanthe grain thicknessd and the electromagnetic field does npenetrate into metal grains, the parametersum and wm takevalues

um5 i2c2

pDv, wm5 i

vD

8p, ~d/d@1!. ~10!

For the dielectric region, when the film is thin enough so thdknd!1 and«d;1, Eqs.~7! and ~8! give

ud52 iv«d8

8pD, wd5 i

v

4p~d1D/2!, ~11!

where the reduced dielectric constant«d85112«dd/D is in-troduced. Note that in the limit of the strong skin effect tOhmic parametersum andwm are purely imaginary and theparameterum is of the inductive character, i.e., it has the siopposite to the dielectric parameterud . In contrast, theOhmic parameterw remains essentially the samew; iDv/8p for the dielectric and metal regions, regardlessthe magnitude of the skin effect.

The rest of this paper is organized as follows: In Secwe briefly outline the basic optical properties: reflectantransmittance, and absorbance, found in the GOL approxition; we also present here the self-consistent approachnoted as the dynamic effective-medium theory. Our numcal method and results of calculations for the local elecand magnetic field are described in Sec. III. In Sec. IVtheory is developed for the giant local field fluctuations a

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PRB 62 11 233LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

the spatial high-order moments of the local fields. In Secwe consider the optical properties of a metal film whichperforated with an array of small subwavelength holes;show that the transmittance through such a film canstrongly enhanced in accord with recent experimental obvations. Section VI summarizes and concludes the pape

II. OPTICAL PROPERTIES OF SEMICONTINUOUSFILMS

Since we consider semicontinuous films with an inhomgeneity scale much smaller than the wavelengthl, the fieldsE1(r ) andH1(r ) behind the film are simply the gradientspotential fields, when considered as functions ofx and y inthe fixed reference planez52d/22 l 0. Similarly, the poten-tials for the fieldsE2(r ) andH2(r ) in front of the film can beintroduced. Therefore the fieldsE(r ) and H(r ) in Eq. ~6!can, in turn, be represented as gradients of some potent

E52¹8, H52¹c8. ~12!

By substituting these expressions in Eqs.~6! and then in thecontinuity Eq.~1!, we obtain the following equations

¹•@u~r !¹w8~r !#50, ~13!

¹•@w~r !¹c8~r !#50, ~14!

that can be solved independently for the potentialsw8 andc8. These equations are solved under the following contions

^¹w18&5^E1&[E0 , ^¹c18&5^H1&[H0 , ~15!

where the constant fieldsE0 andH0 are the external~given!fields that are determined by the incident wave. Whenfields E, H and currentsjE , jH are found by solving Eqs~13!, ~14!, and~15!, the local electric and magnetic fieldsthe planez52 l 02d/2 are given by

E15E12p

c@n3 jH#, H15H1

2p

c@n3 jE#, ~16!

as follows from Eqs.~3! and ~5! and the definitions of thefieldsE andH. Note that the fieldE1(r ) can be measured innear-field experiments~see, e.g., Ref. 29!. The effectiveOhmic parametersue andwe are defined in a usual way

^ jE&5ueE0[ue~^E1&1^E2&!/2, ~17!

^ jH&5weH0[we~^H1&1^H2&!/2. ~18!

When these expressions are substituted in Eqs.~3! and ~5!,which are averaged over the film plane~coordinates$x,y%),we obtain the equations

@n3~^H2&2^H1&!#52p

cue~^E1&1^E2&!, ~19!

@n3~^E2&2^E1&!#52p

cwe~^H1&1^H2&!, ~20!

that connect the average fields in front of the film and behit, i.e., we obtain the equations that determine the optresponse of an inhomogeneous film.

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Let us suppose that the wave enters the film from the rihalf-space~see Fig. 1!, so that its amplitude is proportionato e2 ikz. The incident wave is partially reflected and partiatransmitted through the film. The electric-field amplitudethe right half-space, away from the film, can be writtenẼ2(z)5e@e

2 ikz1reikz#, wherer is the reflection coefficientand e is the polarization vector. Well behind the film, thelectric component of the electromagnetic wave acquiresform Ẽ1(z)5ete

2 ikz, wheret is the transmission coefficientIt is supposed for simplicity that the film has no optical ativity, so that the wave polarizatione remains the same aftepassing through the film. At the planesz5d/21 l 0 and z52d/22 l 0 the average electric field equals^E2& and^E1&,respectively. The electric field in the wave is matched wthe average fields in the planesz5d/21 l 0 and z52d/22 l 0, i.e., ^E2&5Ẽ2(d/21 l 0)5e@e

2 ik(d/21 l 0)1reik(d/21 l 0)#and^E1&5Ẽ1(2d/22 l 0)5ete

ik(d/21 l 0). The same matchingwith the magnetic fields giveŝH2&5@n3e#@2e

2 ik(d/21 l 0)

1reik(d/21 l 0)# and ^H1&52@n3e#teik(d/21 l 0) in the planes

z5d/21 l 0 and z52d/22 l 0 respectively. Substitution othese expressions for the fields^E1&, ^E2&, ^H1&, and^H2&in Eqs.~19! and ~20! gives the two linear~scalar! equationsfor the reflectionr and transmissiont coefficients. By solvingthese equations we obtain the reflectance,

R[ur u25U ~2p/c!~ue1we!~11~2p/c!ue!~12~2p/c!we!

U2, ~21!transmittance

T[utu25U 11~~2p/c!!2uewe~11~2p/c!ue!~12~2p/c!we!

U2, ~22!and absorbance

A512T2R ~23!

of the film. Thus, the effective Ohmic parametersue andwecompletely determine the optical properties of an inhomoneous films.

We see that the problem of the field distribution and otical response of the metal-dielectric films reduces to solvthe decoupled quasistatic conductivity problems@Eqs. ~13!and ~14!#, for which a number of theoretical approaches aavailable. Thus efficient analytical and numerical metho~see Sec. III! developed in the frame of the percolatiotheory can be used to find the effective parametersue andweof the film.

We consider now the case of the strong skin effectmetal grains and trace the evolution of the optical properof a semicontinuous metal film with the increase of the mesurface densityp. When p50 the film is purely dielectricand the effective parametersue and we coincide with thedielectric Ohmic parameters given by Eq.~11!. By substitut-ing ue5ud and we5wd in Eqs. ~21!, ~22!, and ~23!, andassuming that the dielectric film has no losses and is ocally thin (dk«d!1), we obtain that the reflectanceR5d2(«d21)

2k2/4, transmittance T512d2(«d21)2k2/4,

and the absorbanceA50 in accord with the well-known re-sults for a thin dielectric film.39,40

It is not surprising that the film without losses has zeabsorbance. When the ratio of the penetration length~skin

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11 234 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

depth! d51/kImnm is negligible in comparison with the filmthicknessd andunmu@1 the losses are also absent in the limof full coverage, when the metal concentrationp51. In thiscase the film is a perfect metal mirror. Indeed substitutingOhmic parametersue5um and we5wm from Eq. ~10! inEqs. ~21!, ~22!, and ~23!, we obtain for the reflectanceR51, while the transmittanceT and absorbanceA are bothequal to zero. Note that the optical properties of the filmnot depend on the particle sizeD for the metal concentrationp50 andp51 since properties of the dielectric and continous metal films do not depend on the shape of the mgrains.

We consider now the film at the percolation thresholdp5pc with pc51/2 for a self-dual system.

3,4 A semicontinu-ous metal film may be thought of as a mirror, which is brken into small pieces with typical sizeD much smaller thanthe wavelengthl. At the percolation threshold the exaDykhne formulasue5Audum andwe5Awdwm hold.41 Thusfollowing equations for the effective Ohmic parametersobtained from Eqs.~11! and ~10!

2p

cue~pc!5A«d8,

2p

cwe~pc!5 i

Dk

4A11 2d

D. ~24!

From this equation it follows thatuwe /ueu;Dk!1 so thatthe effective Ohmic parameterwe can be neglected in comparison withue . By substituting the effective Ohmic parameterue(pc) given by Eq.~24! in Eqs.~21!, ~22!, and~23!, theoptical properties at the percolation can be obtained as

R~pc!5«d8

~11A«d8!2, ~25!

T~pc!51

~11A«d8!2, ~26!

A~pc!52A«d8

~11A«d8!2, ~27!

where «d85112«dd/D. When metal grains are oblatenough so that«dd/D!1 and«d8→1 the above expressionsimplify to the universal result

R5T51/4, A51/2 ~28!

first obtained in Ref. 18. Thus, there is the effective absotion in semicontinuous metal films, even in the case whneither dielectric nor metal grains absorb the light eneri.e., the mirror broken into small pieces effectively absoenergy from the electromagnetic field. The effective absotion in a loss-free film means that the electromagnetic eneis stored in the system and that the amplitudes of the loelectromagnetic field can increase, in principle, up to infinIn any real semicontinuous metal film the local field is finiof course, because of nonzero losses; still, the field fluctions over the film can be very large provided that lossessmall, as discussed below.

To find the optical properties of semicontinuous films farbitrary metal concentrationp, the effective medium theorycan be implemented that was originally developed to prov

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a semiquantitative description of the transport propertiespercolation composites.3 The effective-medium theory beinapplied to Eqs.~13! and~17! and~14! and~18! results in thefollowing equations for the effective parameters

ue22Dpue~um2ud!2udum50, ~29!

we22Dpwe~wm2wd!2wdwm50, ~30!

where the ‘‘reduced’’ concentrationDp5(p2pc)/pc , (pc51/2) is introduced. It follows from Eq.~30! that for thecase of strong skin effect, when the Ohmic parameterswm!c and wd!c @see Eqs.~10!–~11!#, the effective Ohmicparameteruweu!c, for all metal concentrationsp. Thereforethe parameterwe is negligible in Eqs.~21! and ~22!. Forfurther simplification the Ohmic parameterud can be ne-glected in comparison withum in the second term of Eq.~29!@cf. Eqs.~10! and~11!#. Then, by introducing the dimensionless Ohmic parameterue85(2p/c)ue, we rewrite Eq.~29! as

ue8222i

lDp

pDue82«d850. ~31!

Right at the percolation thresholdp5pc51/2, when the re-duced concentrationDp50, Eq. ~31! gives the effectiveOhmic parameterue8(pc)5A«d8 that coincides with the exacEq. ~24! and results in reflectance, transmittance, and abbance given by Eqs.~25!, ~26!, and ~27!, respectively. Forconcentrations different frompc , Eq. ~31! gives

ue85 ilDp

pD1A2~lDp/pD !21«d8, ~32!

that becomes purely imaginary foruDpu.pDA«d8/l. For theimaginary effective Ohmic parameterue8 , Eqs. ~25!, ~26!,and ~27! result in the zero absorbance:A512R2T512uue8u

2/(11uue8u2)21/(11uue8u

2)50 ~recall that the effec-tive Ohmic parameterwe is neglected!. In the vicinity of apercolation threshold, namely, for

uDpu,pD

lA«d8 ~33!

the effective Ohmic parameterue8 has a nonvanishing reapart and, therefore, the absorbance

A52A2~lDp/pD !21«d8

11«d812A2~lDp/pD !21«d8~34!

is nonzero and has a well-defined maximum at the perction threshold; the width of the maximum is inversely prportional to the wavelength. The effective absorption inmost loss-free semicontinuous metal film means thatlocal electromagnetic fields strongly fluctuate in the systas was speculated above. The concentration width forstrong fluctuations should be the same as the width ofabsorption maximum, i.e., it is given by Eq.~33!.

Note that the effective parametersue and we can be de-termined experimentally by measuring the amplitude aphase of the transmitted and reflected waves or by measuthe film reflectance as a function of the fieldsE1 andH1. In

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PRB 62 11 235LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

the latter case, a metal screen placed behind the film caused to control the values of these fields.42,43

III. COMPUTER SIMULATIONS OF LOCAL ELECTRICAND MAGNETIC FIELDS

To find the local electricE(r ) and magneticH(r ) fields,Eqs.~13! and~14! should be solved. Consider the first eqution ~13!, which is convenient to rewrite in terms of threnormalized dielectric constant

«̃5 i4pu~r !

vd~35!

as follows

¹•@ «̃~r !¹f~r !#5E, ~36!wheref(r ) is the fluctuating part of the potentialf8(r ) sothat¹f8(r )5¹f(r )2E0 , ^f(r )&50, andE5¹•@ «̃(r )E0#.We recall that the ‘‘external’’ fieldE0 is defined by Eq.~15!.For the metal-dielectric films considered here local dielecconstant«̃(r ) equals to«̃m54p ium /vd and «̃d5«d8D/2d,for the metal and dielectric regions, respectively. The exnal field E0 in Eq. ~36! can be chosen real, while the locpotentialf(r ) takes complex values since the dielectric costant«̃m is complex«̃m5 «̃m8 1 i «̃m9 .

In the quasistatic limit, when the skin depthd is muchlarger than the film thicknessd, the dielectric constant«̃mcoincides with the metal-dielectric constant«m as followsfrom Eq. ~9!. In the optical and infrared spectral rangessimple model of the Drude metal can be used to reprodsemiquantitatively the basic optical properties of a metalthis approach, the dielectric constant of metal grains canapproximated by the Drude formula

«m~v!5«b2~vp /v!2/@11 ivt /v#, ~37!

where«b is contribution to«m due to the interband transtions, vp is the plasma frequency, andvt51/t!vp is therelaxation rate. In the high-frequency range, losses in mgrains are relatively small,vt!v. Therefore, the real par«m8 of the metal-dielectric function«m is much larger~inmodulus! than the imaginary part«m9 (u«m8 u/«m9 >v/vt@1),and«m8 is negative for the frequenciesv less than the renormalized plasma frequency,

ṽp5vp /A«b. ~38!Thus, the metal conductivity sm52 iv«m/4p>(«bṽp

2/4pv)@ i (12v2/ṽp2)1vt /v# is characterized by

positive imaginary part forṽp.v@vt , i.e., it is of the in-ductive character. This allows us to model the metal graas inductancesL for the frequenciesṽp.v@vt while thedielectric gaps can be represented by capacitancesC. In theopposite case of the strong skin effect, the Ohmic paramum is inductive according to Eq.~10!, for all spectral rangesregardless of the metal properties. Then, the percolametal-dielectric film represents a set of randomly distribuL andC elements forall spectral ranges.

Note that the Ohmic parameterw takes the same sign anis close in magnitude for both metal and dielectric grai

be

-

c

r-

-

ene

al

s

er

nd

,

according to Eqs.~9!, ~10!, and~11!. A film can be thoughtof as a collection ofC elements in ‘‘w’’ space. Therefore,there are no resonances in the solution to Eq.~14!. The fluc-tuations of the potentialc8 can be neglected in comparisowith the w8 fluctuations. For this reason we concentrate oattention on the properties of the ‘‘electric’’ fieldE(r )52¹f8(r )52¹f(r )1E0 when considering the fluctuations of the local fields. The fieldE(r ) can be found bysolving Eq.~36!.

Equation~36! has the same form as usual quasistatic ctinuity equation,¹(«¹f)50. Therefore Eq.~36! being dis-cretized on, say, square lattice acquires the form of Kirhoff’s equations. We employ here the efficient real-sparenormalization method suggested in Ref. 44 to solvediscretized Eq.~36!. In our previous works21,44–55the samemethod was used to find local electric fields for the quastatic case. Solution of Eq.~36! gives the potentialf in thefilm, the local field E(r ) and the electric currentjE(r ) interms of the average fieldE0. The effective Ohmic parameteue is determined by Eq.~17! that can be written aŝjE&5ueE0. The effective dielectric constant«̃e equals4p iue /vd. In the same manner the fieldH(r ), the magneticcurrent jH(r ), and the effective parameterwe can be foundfrom Eq. ~14! and its lattice discretization. Note that thsamelattice should be used to determined the fieldsE(r ) andH(r ).55 The directions of the external fieldsE0 andH0 maybe chosen arbitrary when the effective parametersue andweare calculated since the effective parameters do not depon the direction of the field, for an isotropic, on averagfilm.

Although the effective parameters do not depend onexternal field, the local electricE1(r ) and magnetic andH1(r ) fields do depend on the incident wave. The local fieE1(r ) and H1(r ) are defined in the reference planez52d/22 l 0 ~see Fig. 1!. For calculations below, the electriand magnetic fields of the electromagnetic wave are choin the form ^E1&5$1,0,0% and ^H1&5$0,21,0% in the planez52 l 02d/2. This choice corresponds to the wave vectorthe incident wave ask5(0,0,2k), i.e., there is only a transmitted wave behind the film~see Fig. 1!. It follows from theaveraging of Eq.~16! ~which can be written aŝE1&5E01(2p/c)we@n3H0# and ^H1&5H01(2p/c)ue@n3E0#)that the fieldsE0 andH0 are given by

E05^E1&2~2p/c!we@n3^H1

11~2p/c!2uewe,

H05^H1&2~2p/c!ue@n3^E1

11~2p/c!2uewe. ~39!

These fieldsE0 andH0 are used to calculate the local fieldE(r ) andH(r ). The local electricE1(r ) and magneticH1(r )fields are restored then from the fieldsE(r ) and H(r ) byusing Eqs.~16!.

The local electric and magnetic fields are calculatedsilver-on-glass semicontinuous films, as functions of the sface concentrationp of silver grains. The dielectric constanfor glass is given by«d52.2. The dielectric function forsilver is chosen in the Drude form~37!; the following param-eters are used in Eq.~37!: the interband-transition contribu

ic

T

u

leb

oic

ldigtis

en

eticch

i

ldn,ct,stin

ticheofbecal

iesns-tion

mumds

cal

asion

n-n-

11 236 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

tion «b55, the plasma frequencyvp59.1 eV, and the relax-ation frequencyvt50.021 eV.

56 The metal grains aresupposed to be oblate in shape. The ratio of the grain thnessd ~film thickness! to the grain diameterD has beenchosen asD/d53, the same as the one used in Ref. 18.consider the skin effect of different strength~i.e., differentinteraction between the electric and magnetic fields, occring through the skin effect!, we vary the sized of silverparticles in a wide range,d514100 nm. The size of metagrains in semicontinuous metal films is usually of the ordof few nanometers but it can be increased significantlyusing a proper method of preparation.57 In the microwaveexperiments of Ref. 20, for example, the films were lithgraphically prepared, so that the size of a metal partcould vary in a large range.

The space distribution of the electric and magnetic fieis calculated for two sets of parameters as illustrated in F2 and 3. In Figs. 2 and 3 we show the electric and magnefield distributions forl51 mm and two different thicknessed of the film, d55 nm andd550 nm. The first thickness~Fig. 2! corresponds to a weak skin effect since the dimsionless thickness is small,D[d/d50.2 @where d51/(k Im nm) is the skin depth#. In this case we observe thgiant field fluctuations of the local electric field; the magnefield, although it strongly fluctuates over the film, is musmaller in magnitude compared with the electric field. Thisbecause the film itself is not magnetic,md5mm51, and the

FIG. 2. Distribution of local em field intensities in a semicotinuous silver film at small skin effectd/d50.2, whered is the skindepth andd is the thickness of the film~l51 mm; p5pc).

k-

o

r-

ry

-le

ss.c-

-

s

interaction of the magnetic field with the electric fiethrough the skin effect is relatively small. For comparisoFig. 3 shows the fields in the case of a significant skin effewhen the film thicknessd550 nm and the dimensionlesthickness exceeds one,D52.2. It is interesting to note thathe amplitude of the electric field is roughly the same asFig. 2~a!, despite the fact that the parameterD is increasedby one order of magnitude. In contrast, the local magnefield in Fig. 3~b! is strongly increased in this case so that tamplitude of magnetic field in peaks is of the same ordermagnitude as the electric field maxima. This behavior canunderstood by considering the spatial moments of the lomagnetic field as shown in the next section.

Being given the local fields, the effective parametersueandwe can be found and thus the effective optical propertof the film. In Figs. 4 and 5 we show the reflectance, tramittance, and absorbance as functions of silver concentrap, for wavelengthsl51 mm andl510 mm, respectively.The absorbance in these figures has an anomalous maxiin the vicinity of the percolation threshold that corresponto the behavior predicted by Eq.~34!. This maximum wasdetected first in the experiments.13–16 The maximum in theabsorption corresponds to strong fluctuations of the lofields. We estimated in Eq.~33! the concentration rangeDparoundpc , where the giant local-field fluctuations occur,Dp}1/l. Indeed, the absorbance shrinks at the transitfrom Fig. 4 to Fig. 5, when wavelengthl increases by afactor of 10.

FIG. 3. Distribution of the local-field intensities in a semicotinuous silver film at strong skin effectd/d52.2 ~l51 mm; p5pc).

lu

zeu-eof

p

PRB 62 11 237LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

In Fig. 6~a! and 6~b! we compare results of numericasimulations for the optical properties of silver semicontinous films with the calculations based on our generalieffective-medium approach@Eqs. ~29! and ~30!#, with thenew Ohm’s parametersu and v. Results of such a ‘‘dy-

FIG. 4. Calculated absorptanceA, reflectanceR, and transmit-tanceT for a silver-glass film as functions of metal concentrationpand film thicknessd at l51 mm.

-dnamic’’ effective-medium theory are in accord with our nmerical simulations, for arbitrary-strength skin effect. Wcan see that the theory reproduces well the maximumabsorptance in the vicinity of the percolation threshold.

FIG. 5. Calculated absorptanceA, reflectanceR, and transmit-tanceT for a silver-glass film as functions of metal concentrationand film thicknessd at l510 mm.

l

s

11 238 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

FIG. 6. Results of numericasimulations~dashed line! and thedynamic effective-medium theory~solid line! for absorptanceA, re-flectanceR, and transmittanceT ofsilver-on-glass semicontinuoufilm as function of the metal con-centration p at the followingwavelengthl and film thicknessd: ~1! l51.0 mm and d550 nm;~b! l510.0mm andd55 nm.

he

qtnt

uetio

tric

ed

-r

IV. SPATIAL MOMENTS OF THE LOCAL ELECTRICAND MAGNETIC FIELDS

Below we find the spatial high-order moments for tlocal electric E1 distribution in the reference planez52d/22 l 0 ~see Fig. 1! back of the film. The electric fieldE1is expressed in terms of fieldsE andH by means of Eq.~16!.The fluctuations of the local magnetic currentjH(r )5w(r )H(r ) can be neglected in the first equation in E~16!, as discussed below Eq.~37!. Therefore, the momenMn

E5^uE1(r )un&/u^E1&un equals approximately the momeMn5^uE(r )un&/u^E&un for the fieldE(r ).

We consider now the momentsMnE in the optical and

infrared spectral ranges for arbitrarily strong skin effect. Wassume«m is almost negative and large in the absolute valSince Eq.~36! has the same form as the quasistatic equa¹(«¹f)50 ~investigated in detail in our previous works!

.

e.n

we use the estimate for the moments of the local-elecfield obtained in Refs. 52 and 53

MnE;S u«̃mu3/2A«d«̃m9 D

n21

, ~40!

where the metal permittivity is replaced by the renormalizdielectric constant«̃m given by Eq.~35!. The Drude formulaEq. ~37! is substituted in Eq.~7! to obtain the Ohmic parameterum in the limit vp@v@vt . Then the Ohmic parameteum is substituted in Eq.~35! to obtain«̃m . Finally, the mo-mentMn

E is obtained from Eq.~40! as

MnE;rFvpvtAf 0~x!G

n21

, ~41!

e

ra

ed

ra

en

g

is

u

tic

nrn

inr

eretth

t

ly

-

t-ingan

to

ticnts

tric

asind 3.

re-

g-geralcro-skin

PRB 62 11 239LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

f 0~x!54 tanh3~x!@11D/~4d! tanh~x!#

x$tanh~x!1x@12tanh2~x!#%2, ~42!

wherex5d/2d.dvp/2c is the ratio of the film thicknessdand the skin depthd'c/vp . It follows from these equationsthat the moments of the local-electric field are independof the frequency in the wide frequency bandvp@v.vt ,which typically includes the optical and infrared spectranges. When the skin effect increases, the functionf 0 in Eq.~41! also increases monotonically fromf 0(0)51 to f 0(`)5D/(2d). Provided that the shape of metal grains is fixand they are very oblate, i.e.,D/d@1, the momentsMn

E

increase significantly with increasing the parameterx.Let us consider now the far-infrared, microwave, and

dio frequency ranges, where the metal conductivitysm ac-quires its static value, i.e., it is positive and does not depon frequency. Then it follows from Eqs.~10!, ~11!, and~35!that Eq. ~40! for the field moments acquires the followinform

MnE;S 2psmv D

(n21)/2

, ~43!

in the limit of strong skin effect. Since metal conductivitytypically much larger than frequencyv in the microwaveand radio bands, the moments remain large at these freqcies.

We proceed now with fluctuations for the local magnefield H1(r ) in the reference planez52d/22 l 0. The fluctua-tions of the fieldH(r )5@H1(r )1H2(r )#/2 defined in Eq.~6!can be neglected. Then as follows from the second Eq.~16!,the momentsMn

H5^uH1(r )un&/u^H1(r )&un of the local mag-netic field can be estimated as

MnH.~2p/c!n^u jE~r !un&/u^E1&un, ~44!

where we used the conditionsu^E1&u5u^H1&u that correspondto the wave incident onto the film from the right~see Fig. 1!.Thus the external electric field induces the electric currein a semicontinuous metal film and these currents, in tugenerate the strongly fluctuating local magnetic field.

To estimate the moments^u jE(r )un& of the electric currentdensity in semicontinuous metal films we generalize, toclude the nonlinear case, the approach suggested earlieDykhne41 ~see discussion in Ref. 58!. Since in the consideredcase the electric currentjE is related to the local fieldE viathe first equation in Eqs.~6!, the following equation̂ u jEun&5a(um ,ud)^uE(r )un& is valid @where the coefficienta(um ,ud) is a function of variablesum andud].

We consider now the percolation thresholdp5pc and setpc aspc51/2. It is also supposed that the statistical propties of the system do not change when interreplacing mand dielectric. If all the conductivities are increased byfactor k then the average nonlinear current^u jEun& also in-creases, by the factorukun; therefore, the coefficiena(um ,ud) also increases, by theukun. Then the coefficienta(um ,ud) has an important scaling property, namea(kum ,kud)5ukuna(um ,ud). By takingk51/um the follow-ing equation is obtained

a~um ,ud!5uumuna1~um /ud!. ~45!

nt

l

-

d

en-

ts,

-by

-ale

,

Now we perform the Dykhne transformation

j* 5@n3E#, E* 5@n3 jE#. ~46!

It is easy to verify that thus introduced fieldE* is still po-tential, i.e.,“3E* 50, and the currentj* is conserved, i.e.,“"j* 50. The currentj* is coupled to the fieldE* by theOhm’s formula j* 5u* E* , where the ‘‘conductivity’’ u*takes values 1/um and 1/ud . Therefore, the following equation ^u j* un&5a(1/um,1/ud)^uE* un& holds, from which it fol-lows thata(1/um,1/ud)a(um ,ud)51. Since we suppose thaat the percolation thresholdpc51/2 and the statistical properties of the system do not change when inter-replacmetal and dielectric, the arguments in the first function cbe changed to obtaina(1/ud,1/um)a(um ,ud)51. This equa-tion, in turn, can be rewritten using Eq.~45! asuum /uduna1

2(um /ud)51, where the functiona1 is defined inEq. ~45!. Thus we find thata1(um /ud)5uud /umun/2, and thefinal result is given bya(um ,ud)5uumudun/2, i.e., the follow-ing generalization of the Dykhne’s formula is valid

^u jEun&5uudumun/2^uEun&. ~47!

Now this expression for̂u jEun& can be substituted in Eq.~44!to obtain that

MnH5F S 2pc D

2

uudumuGn/2MnE . ~48!In the optical and infrared spectral ranges it is possiblesimplify this equation as done for Eq.~44! above. Usingagain the Drude formula~37! and assuming thatvt!v!vp , the following estimate is obtained

MnH5F«d8 x tanhx~2d/D !1x tanhxG

n/2

MnE , ~49!

where the momentMnE is given by Eq.~44! and x5d/2d

.dvp/2c has the same meaning as in Eq.~44!. As followsfrom Eq. ~49!, the spatial moments of the local magnefield Mn

H are of the same order of magnitude as the momeof the local electric fieldMn

E in the limit of strong skin effect,i.e., whenx@1.

We can estimate now the moments of the local elecand magnetic fields from Eqs.~41! and ~49! for silver-on-glass semicontinuous films, withvp59.1 eV and vt50.021 eV. The moments of the local electric field areMn

E;(43102)n21, so that the field fluctuations are huge,agreement with the numerical results shown in Figs. 2 anFor sufficiently strong skin effect (x.1), the moments oflocal magnetic fieldMn

H;MnE , which is also in accord with

our simulations.As mentioned, at frequencies much smaller than the

laxation ratevt.3.231013sec21, the silver conductivity ac-quires its static valuevp

2/4pvt.1018sec21. In this case, themoments are given by Eq.~43!. Thus for wavelengthl53 cm (v/2p5n510 GHz! the moments are asMn

H;MnE

;(104)n21. We can conclude that the local electric and manetic field strongly fluctuate in a very large frequency ranfrom the optical down to the microwave and radio spectranges. The fluctuations become even stronger for the miwave and radio bands. This is because for the strong

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11 240 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

effect ~when the penetration depth is much smaller thansize of a metal grain!, losses are small in comparison withe electromagnetic field energy accumulated aroundfilm. This opens a fascinating possibility to observe tAnderson localization of surface plasmons, predictedRefs. 52 and 53, in microwave experiments with localizatlength in the centimeter scale.

V. ENHANCED LIGHT TRANSMISSION THROUGHMETAL FILMS WITH ARRAYS OF SMALL HOLES

Light transmission through a tiny~less than the wavelength! hole in an optically thick metal film is known to bsmall T;(d/l)4.59 However, provided that there is an arraof such subwavelength holes in the metal film and speresonance conditions are fulfilled, the optical transmisscan be enhanced by several orders of magnitude. Forstance, in a silver film of thicknessd5200 nm with an arrayof holes of diameterD5150 nm and lattice constanta05600 nm, large transmission peaks have been observewavelengthsl;300, 400, 500, 700, and 950 nm. The mamum transmission can exceed one, when it is normalizethe projected area of the holes. This corresponds to thehancement of nearly three orders of magnitude, when cpared to what one could expect for the same numbesingle holes.60–62 Since the skin depth in silverd;20 nm ismuch less than the film thickness, these experiments cabe explained using the quasistatic approximation for theface conductivity. Note also that the metal films with peodic arrays of nanoholes represent an example of nanoeneered structures known as electromagnetic crystals.63–66

A. Skin resonance

We use the GOL approximation to calculate the transmtance of an array of subwavelength holes for the casestrong skin effect. It is assumed, for simplicity, that the sface hole concentrationph512p5p(D/2a0)

2 is small,ph!1, which is typical for the experiments mentioned aboTo evaluate the effective parametersue and we in a sparsearray of holes, the Maxwell-Garnet~dipole! approximationcan be used.6,30,67In this approximation the effective Ohmiparameter, say,ue is related to the ‘‘metal’’ (um) and ‘‘di-electric’’ (ud) Ohmic parameters by the following equatio

ue5um~11ph!ud1~12ph!um~12ph!ud1~11ph!um

; ~50!

the same equation connectswe to wm andwd . The substitu-tion of expressions~10! and ~11! in the equations abovegives

ue5 i8p

c

16~12ph!2D~2d1D !k2~11ph!

Dk@16~11ph!2D~2d1D !k2~12ph!#

~51!

and

we5 i8p

c

Dk~d1D1dph!

d1D2dph, ~52!

e

e

nn

alnn-

at

ton--

of

otr-

gi-

t-of-

.

wherek5v/c is the wave vector,d is the thickness of thefilm, and D is the hole diameter. Then, in the considerlimit ph!1, we find the transmittance of the hole array froEq. ~22! in the following form

T51024ph

2

@D~2d1D !k2216#211024ph2

. ~53!

That is the transmittance has a resonance at the wavelel r5AD(2d1D), where T(l r)51 and the film becomestransparent, regardless of the surface concentration of tholes. When the hole surface concentrationph decreases, thewidth of the resonance shrinks, but exactly at the resonathe film remains transparent, even forph→0.

The nature of the described resonance, which can benoted as‘‘skin resonance,’’ is purely geometrical, i.e., theskin resonance does not depend on the metal-dielectric ftion «m . Therefore, this resonance is different from the weknown plasmon resonance of a hole in the metal film.particular, the skin resonance should be also observablthe microwave range. It is also important to emphasize tthe skin resonance does not actually depend on the holeriodicity.

It would be interesting to check out experimentally tposition of the resonance as a function of the holes’ diamD, when the hole concentrationph is fixed. It is possible thatthe skin resonance described above was observed in theexperiments60,61 at wavelengthl.300 nm. Unfortunately,the parameters of the film investigated in Ref. 60 andwere chosen in such a way that the frequency of the sresonance is close to the renormalized plasma frequencyṽpwhere the real part of the metal-dielectric constant«m8 van-ishes. Therefore, the strong skin effect condition mayhold in the vicinity of the resonance and losses becomeportant. It could be the reason why the transmittance,though increased up to 20%, does not achieve 100%.vided that the diameter of the holes is increased, the sresonance can be shifted to larger wavelengths so thaamplitude can be increased, even at much smaller holecentrationph .

Equation ~53! is valid in the vicinity of the skin reso-nance. For wavelengths much larger than the resonancel r ,the equation for the transmittanceT can be simplified sincethe terms}Dk can be neglected. This results in the following expression

T5ph

2k2D4

4~d1D !2. ~54!

The obtained transmittanceT is independent of metal properties ~e.g., it does not depend on the metal dielectric costant«m). It is an anticipated result for the strong skin effewhen the penetration of the electromagnetic field in mecan be neglected. The transmittance is proportional toratio of the hole sizeD and the wavelengthl squared. There-fore, the transmittanceT is much larger thanT;(D/l)4 re-sulting from the Fraunhofer diffraction on single holes in tlimit of D/l!1. It is the ‘‘background’’ transmittanceabove which the considered hereafter resonances areposed.

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PRB 62 11 241LOCAL ELECTRIC AND MAGNETIC FIELDS IN . . .

B. Surface polariton resonance

Consider now surface waves in a metal film. Whenreal part of the metal permittivity is negative«m8 ,0 andlosses are small, it is well known that surface polaritonspropagate along the surface. On metal surfaces, these ware referred to as surface plasmon polaritons.

For the sake of simplicity we neglect losses in the furthconsideration, i.e., we assume that the metal-dielectric cstant is negative«m52n

2; we also suppose thatn.1. Thewave vectorkp of the plasmon polariton in the metal-vacuuinterface is given bykp5kn/A(n221).k ~see e.g., Refs. 28and 39!. There are two kinds of polaritons in the film withfinite thickness, which correspond to symmetric and asymmetric~with respect to reflection in the planez50) so-lutions to the Maxwell equations. The wave vectorsk1 andk2 of these polaritons are determined by the following eqtions

k125~kn!2

n21tanh2@~d/2!A~kn!21k12#n42tanh2@~d/2!A~kn!21k12#

, ~55!

k225~kn!2

11n2 tanh2@~d/2!A~kn!21k22#n4 tanh2@~d/2!A~kn!21k22#21

. ~56!

In the case of the strong skin effect, when exp(dkn)@1, thewave vectors for the symmetric and antisymmetric polaritoare equal to

k125~kn!2

n42124n2exp~dkn2/An221!~n221!~n421!

~57!

and

k225~kn!2

n42114n2 exp~dkn2/An221!~n221!~n421!

, ~58!

respectively. It is important for the further consideration thboth symmetric and antisymmetric polaritons propagateboth interfaces of the film. Moreover, the absolute valuethe electric and magnetic fields are the same on theinterfaces. This consideration holds for arbitrarily thick filmalthough the difference between the two types of polaritbecomes exponentially small for the optically thick films.

Since the polariton wave vectors are larger than the wvectork of an electromagnetic wave~normal to the film! thewave cannot excite the polaritons in a continuous metal fi~see, e.g., Ref. 28!. The situation changes dramatically whethe film is periodically corrugated, with the same spatial priod a0 on both interfaces. The example of such corrugatis a regular array of holes. When wave vectork of the inci-dent wave is such that one of the polariton wavelengl1(k)52p/k1(k) or l2(k)52p/k2(k) coincides witha0,the corresponding polariton is excited on the film. This plariton spreads out onto both sides of the film and interawith the corrugation. As a result of this interaction, the ecited polariton can be transformed back to the plane waThis can also occur on the backside of the film. Thereforetransmittance has a maximum at the resonance cond

e

nves

rn-

i-

-

s

tnfo

s

e

-n

s

-ts-e.e

on

2pm/a05ki $ i 51,2;m51,2, . . .%. Since the polariton am-plitude is the same on both sides of the film we speculateat the resonance the film becomes almost transparent, regless of its thickness, though the width of the resonanshrinks when the film thickness increases. The set of maxin transmittance was indeed observed in experiments of R60 and 61.

For a qualitative analysis of the resonance transmittawe consider a simple model of the periodically inhomogneous film, with the local dielectric constant varying as«52n2@12g cos(qx)# in the film plane$x,y%. Then, the wholespectrum of the spatial harmonics; cos(qmx) is generated inthe film, with the wave vectorsqm5mq. We consider, forsimplicity, small modulations assuming that parameterg!1. Then, we can restrict our consideration to the first tharmonics, i.e., we suggest that the local electric and mnetic fields can be written as

Ẽ~x,z!5E~z!1Eq~z!cos~qx!, ~59!

H̃~x,z!5H~z!1Hq~z!cos~qx!, ~60!

where, as above, the ‘‘z’’ axis is perpendicular to the filmplane. The wavelengthl of the incident wave is supposed tbe larger than the period of the film modulationl.a052p/q. The Maxwell equations in the film can be written a

curl@„11g cos~qx!…curl H̃#52~kn!2H̃. ~61!

We consider the incident wave with the magnetic and eltric fields polarized along the ‘‘y’’ and ‘‘ x’’ axes. Then themagnetic fieldH̃ in the film has the ‘‘y’’ component onlyand Eq.~61! takes the following form

H9~z!1g

2Hq9~z!2k

2n2H~z!50, ~62!

Hq9~z!1gH9~z!2~k2n21q2!Hq~z!50, ~63!

where we neglected the higher spatial harmonics and eqthe terms that have the same dependence on the coordx.23 Solutions to Eqs.~62! and ~63! give the magnetic fieldinside the film. The electric fieldsE(z) and Eq(z) are ob-tained from the second Maxwell equationẼ5 i /@kn2„12g cos(qx)…#curl H̃. The internal fieldsE(z) and H(z) arematched with the field of the incident wave at the front iterface of the film (z5d/2) and these fields, in turn, armatched with the field of the transmitted wave at the bainterface (z52d/2). Note that the fieldsEq(z) and Hq(z)decay exponentially; exp(2uzuAq22k2) outside the filmsince the spatial period of the corrugation is smaller thanwavelength of the incident wavel.a0 (k,q). Thus Eqs.~62! and ~63! connect incident and transmitted wave andsolution gives the transmittanceT of the film. The transmit-tance T, as discussed above, has sharp maxima atplasmon-polariton resonances. At the resonance,T acquiresthe following form

11 242 PRB 62SHUBIN, SARYCHEV, CLERC, AND SHALAEV

T~k!5~gkn3!4

~gkn3!41~n211!~n421!3$114n@2n~n221!1An221~2n221!#%D2~k!, ~64!

gea

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ectheglyge.theldsola-cethe-the

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with the detuning from the resonanceD given by

D~k!5ki2~k!1

2An221~n611!2n~32n412n6!2n~n421!2~n211!

3~gkn3!22q2, ~65!

whereki(k)5k1 or k2 defined in Eqs.~55!–~58!. Thus weobtain the important result that there is a subwavelenresonance in transmittance of a corrugated metal film dusurface polaritons. The resonance has a form of two peclose to each other sincek22k1; exp(2dkn)!1. When theskin effect is so strong that exp(2dkn)!g2 these two peaksmerge together. Exactly at the resonanceT51, that is, thefilm becomes transparent at arbitrary thickness. In a realthe transmittance is limited by losses. Yet, the lossesrather small for a typical metal in the optical and infrarspectral ranges. Note also that the resonance position isfined by the conditionD50 @see Eq.~64!#; it is somewhatshifted with respect to the conditionki(k)5q since the cor-rugation of a film affects the propagation conditions for sface polaritons.

Similar resonances take place whenki(k).mq. We be-lieve that the skin resonance described in Sec. V A is respsible for the sharp short-wavelength peak~near 300 nm! ob-tained in Ref. 60, whereas long-wavelength resonanceRef. 60 occur due to surface polaritons considered herequantitative comparison with the experiments will be psented elsewhere. Now we only note that the quartz subsused in the experiment makes the film asymmetric. Thefore, surface plasmon polaritons propagate in a different won the two sides of the film. This is a possible reason wthe observed resonance transmittance, though stronglyhanced, does not exceed 10%.

VI. SUMMARY AND CONCLUSIONS

In this paper a detailed theoretical consideration ofhigh-frequency response~optical, infrared, and microwave!of thin metal-dielectric films is presented. The generalizGOL approximation was employed. The developed GOLproximation is based on direct solution of Maxwell’s equtions, without having to invoke the quasistatic approximtion. In this approximation the electromagnetic propertiessemicontinuous metal films are described in terms oftwoparametersue and we , in contrast to the usual descriptiowith a single complex conductivity. This approach allowsto calculate field distributions and optical properties of secontinuous metal films in a wide frequency range: fromoptical to the microwave and radio frequencies.

It is shown that metal-dielectric films can exhibit esp

thtoks

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-

cially interesting properties when there is a strong skin effin metal grains. In particular, the theory predicts that tlocal magnetic fields, as well as the electric fields, stronfluctuate within the large optics-to-microwave spectral ranThe obtained equations for the field distributions andhigh-order field moments for the electric and magnetic fieallow one to describe various optical phenomena in perction films, both linear and nonlinear. For example, surfaenhancement for Raman scattering is proportional tofourth field moment46,48 and, therefore, it is strongly enhanced in a wide spectral range. The same is valid forKerr nonlinearity, which is also proportional to the fourmoment.47,52 The giant electric-field fluctuations near thpercolation threshold in metal-dielectric films have beenready observed in the microwave20 and optical21 ranges.

The large electric- and magnetic-field fluctuations eplain, in particular, a number of previously obtained phnomena that remained so far unclear. For example, thesorbance in a percolation film calculated using the quasistapproximation predicts for the maximumA.0.2,35 which istwice less than the measured value13–16,20 and the one fol-lowing from the GOL equations.18 This can occur because oneglecting the magnetic field in the quasistatic calculatioThe energy of the electric field can be converted intoenergy of the magnetic field, so that the magnetic componof the em field is also responsible for the absorption. As sin Figs. 2 and 3, the magnetic and electric fields cancomparable in magnitudes, even at a relatively moderateeffect. This indicates that the magnetic field can carryroughly the same amount of energy as the electric fieldaccordance with this, Figs. 4–6 show that the absorbancthe GOL beyond-the-quasistatic-approximation reachesvalue A.0.45 ~in agreement with experiments!, which isroughly twice larger than that found in the quasistatic aproximation.

We also considered the optical properties of a metal fiwith an array of subwavelength holes and showed that tramittance of such a film is much larger than the Fraunhodiffraction predicts. This result is an agreement with recexperiments.60–62 For such films, a new effect, skin resonance, is predicted; at this resonance the transmittanceincrease up to 100%, regardless of the surface concentraof the holes. We also predict series of the surface plasmpolaritons resonances in the transmittance.

ACKNOWLEDGMENTS

We would like to acknowledge useful discussions withYablonovitch, A. M. Dykhne, T. Thio, and V. PodolskiyThis work was supported in part by NSF~DMR-9810183!,ARO ~DAAG55-98-1-0425!, PRF AVH Foundation, andRFFI ~98-02-17628!.

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