Conical propagation of a surface polariton across a grating

Optics Communications 215 (2003) 205–223 www.elsevier.com/locate/optcom

Conical propagation of a surface polariton across a grating M. Kretschmann, T.A. Leskova *, A.A. Maradudin Department of Physics and Astronomy, Institute for Surface and Interface Science, University of California, Irvine, CA 92697, USA Received 19 March 2002; received in revised form 6 November 2002; accepted 27 November 2002

Abstract By the use of the homogeneous form of the reduced Rayleigh equation for the electromagnetic field above and on a two-dimensional rough surface we obtain the dispersion relation for a surface polariton propagating across a classical diffraction grating when the sagittal plane is not perpendicular to the generators of the surface. This dispersion relation is exact within the domain of validity of the Rayleigh hypothesis upon which it is based. It is solved numerically, and dispersion curves are determined for several directions of propagation of the surface polariton for three different choices for the grating profile function. Particular attention is paid to the dependence on the direction of propagation of the position and width of the gap in the surface polariton dispersion curve that occurs at the boundary of the first Brillouin zone defined by the periodicity of the grating. The results obtained are compared with those of a recent experimental determination of this gap. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.Fx; 41.20.Jb Keywords: Surface polaritons; Gratings; Conical propagation

1. Introduction Although there have been several theoretical [1–8] and experimental [9–17] studies of the propagation of surface polaritons across a classical grating in the case that the sagittal plane is perpendicular to the generators of the surface, comparatively few studies have been carried out in the case where the sagittal plane is not perpendicular to the generators of the surface. We refer to the latter case as the conical propagation of surface polaritons across a grating, by analogy with the conical scattering of light from a classical grating [18]. The first theoretical study of the conical propagation of a surface polariton across a one-dimensional periodically corrugated surface was carried out by Mills [19]. In this work the periodic corrugations of the surface were caused by the passage of a Rayleigh surface acoustic wave along the surface, and the calculations were carried out by a coupled mode approach. One of the major results of MillsÕ analysis was an *

Corresponding author. Tel.: +194-9824-5453; fax: +194-9824-2174. E-mail address: [email protected] (T.A. Leskova).

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 2 3 6 - 8

206

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

expression for the width of the gap in the surface plasmon polariton dispersion curve as a function of the angle h between the direction of the surface plasmon polariton propagation and the grooves and ridges of the surface, that showed a cos2 h dependence of this width on h. More recently, Seshadri [20] has also used a coupled-mode approach to study the dependence of the dispersion relation in the vicinity of a band gap of a surface plasmon polariton propagating across a classical metallic grating on the angle the wave vector of the surface plasmon polariton makes with respect to the grooves and ridges of the grating. When differences in notation are taken into account, and a misprint in [19] is corrected, his result agrees with that of Mills [19], and both results agree with that of Laks et al. [3], of Maradudin [4], and of Barnes et al. [8] in the case of surface plasmon polariton propagation normal to the grooves and ridges of the grating. The problem of the conical propagation of a surface polariton across a classical grating is of interest because the extra degree of freedom provided by having a component of its wave vector parallel to the grooves and ridges of the surface provides a means of varying the position and width of the gap in the surface polariton dispersion curve that occurs at the boundary of the first Brillouin zone defined by the periodicity of the grating. This could be useful, for example, in the design of filters for surface wave propagation. In this paper, on the basis of the homogeneous form for the reduced Rayleigh equation [21] for electromagnetic waves above and on a two-dimensional rough surface, we derive a dispersion relation for a surface polariton propagating in an arbitrary direction across a classical grating that is exact within the domain of validity of the Rayleigh hypothesis [22–25] on which it is based. This dispersion relation is solved nonperturbatively by a numerical approach, and complete dispersion curves are determined for several directions of propagation of the surface wave. The outline of this paper is as follows. In Section 2 the dispersion relation for the conical propagation of a surface polariton across a classical grating is derived. This equation is solved numerically in Section 3 for three different forms of the surface profile function of the grating surface. The results of these calculations are discussed in Section 4, where they are also compared with the predictions of small-amplitude perturbation theory, where such a comparison is possible. The conclusions drawn from the present work are presented in Section 5. 2. The dispersion relation The physical system underlying the present work consists of vacuum in the region x3 > fðxk Þ, and a metal or dielectric medium characterized by a real, isotropic, frequency-dependent, dielectric function ðxÞ in the region x3 < fðxk Þ. Here xk ¼ ðx1 ; x2 ; 0Þ is a position vector in the plane x3 ¼ 0. The surface profile function fðxk Þ is assumed to be a single-valued function of xk . The dielectric function ðxÞ is assumed to be negative and to satisfy the condition ðxÞ < 1 in some frequency range. It is within this frequency range that surface polaritons exist. The electric field in the region x3 > fðxk Þ can be written in the form Z 2 n o d qk c ½i qbk b0 ðqk Þ  b Eðx; tÞ ¼ x 3 qk Ap ðqk Þ þ ðb x 3 qbk ÞAs ðqk Þ expðiqk xk  b0 ðqk Þx3  ixtÞ; ð2:1Þ 2 ð2pÞ x where qk ¼ ðq1 ; q2 ; 0Þ, and 2

1

b0 ðqk Þ ¼ ½q2k  ðx=cÞ 2 ; 2

qk > x=c 1 2

¼  i½ðx=cÞ  q2k  ;

qk < x=c:

ð2:2aÞ ð2:2bÞ

The subscripts p and s to the coefficients Ap ðqk Þ and As ðqk Þ in Eq. (2.1) denote the p- and s-polarized components, respectively, of the field with respect to the sagittal plane. A caret over a vector denotes a unit vector.

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

207

The amplitudes Ap ðqk Þ and As ðqk Þ satisfy the homogeneous reduced Rayleigh equation [21] ! ! Z 2 Ap ðqk Þ d qk Iðbðpk Þ  b0 ðqk Þjpk  qk Þ pk qk  bðpk Þ pbk qbk b0 ðqk Þ iðx=cÞbðpk Þð pbk qbk Þ3 ð2pÞ

2

bðpk Þ  b0 ðqk Þ

2 ðx=cÞ pbk qbk

iðx=cÞð pbk qbk Þ3 b0 ðqk Þ

As ðqk Þ

¼ 0: ð2:3Þ

In this equation we have introduced the following functions: 1

bðqk Þ ¼ ½q2k  ðxÞðx=cÞ2 2 ; and IðcjQk Þ ¼

Z

Re bðqk Þ > 0; Imbðqk Þ < 0

ð2:4Þ

d2 xk expðiQk xk Þ expðcfðxk ÞÞ:

ð2:5Þ

The functions b0 ðqk Þ and bðqk Þ in the case that qk > ðx=cÞ are the inverse decay lengths of a surface polariton into the vacuum and into the substrate, respectively. The sign of their imaginary part is chosen so that the radiative part of the fields are outgoing waves into the vacuum and into the substrate. We can now specialize to the case of interest to us, namely the case in which the surface profile function fðxk Þ is independent of the coordinate x2 and is a periodic function of the coordinate x1 with period a fðx1 þ aÞ ¼ fðx1 Þ:

ð2:6Þ

In this case the function Iðbðpk Þ  b0 ðqk Þjpk  qk Þ in Eq. (2.5) becomes Z 1 Z Iðbðpk Þ  b0 ðqk Þjpk  qk Þ ¼ dx1 expðiðp1  q1 Þx1 þ ðbðpk Þ  b0 ðqk ÞÞfðx1 ÞÞ 1

1

dx2 eiðp2 q2 Þx2 1

¼ 2pdðp2  q2 Þ

Z 1 X

ðn12Þa

n¼1

¼ 2pdðp2  q2 Þ

1 X

ðnþ12Þa

e

iðp1 q1 Þna

2

Z

1a 2

dx eiðp1 q1 Þx eðbðpk Þb0 ðqk ÞÞfðx1 Þ

12a

n¼1

¼ ð2pÞ dðp2  q2 Þ

dx1 expðiðp1  q1 Þx1 þ ðbðpk Þ  b0 ðqk ÞÞfðx1 ÞÞ

 2pm d p1  q1  Im ðbðpk Þ  b0 ðqk ÞÞ; a m¼1 1 X

ð2:7Þ

where Im ðcÞ ¼

1 a

Z

1a 2

dx1 ei

2pmx a 1

ecfðx1 Þ :

ð2:8Þ

12a

At the same time, in order to satisfy the Bloch–Floquet theorem, we set  1 X 2pn 2pd q1  k1  a ¼ p; s: Aa ðqk Þ ¼ 2pdðq2  k2 ÞaðnÞ a ðkk Þ; a n¼1

ð2:9Þ

Thus, kk ¼ ðk1 ; k2 ; 0Þ is the two-dimensional wave vector of the surface polariton. When the results given by Eqs. (2.7) and (2.9) are substituted into Eq. (2.3), we obtain as the equation satisfied by apðnÞ ðkk Þ and asðnÞ ðkk Þ 1 1 X X X m¼1 n¼1 b¼p;s

2

ðnÞ

ð2pÞ dðp1  Km Þdðp2  k2 ÞMðabÞ mn ðkk Þab ðkk Þ ¼ 0;

a ¼ p; s;

ð2:10Þ

208

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

where $

Mm;n ðkk Þ ¼

1 Imn ðbðKm ; k2 Þ  b0 ðKn ; k2 ÞÞ 2 ðKm þ k22 Þ2 bðKm ; k2 Þ  b0 ðKn ; k2 Þ 0 bðKm ; k2 Þb0 ðKn ; k2 Þ½Km Kn þ k22  1  B ðKm2 þ k22 ÞðKn2 þ k22 Þ 1B

ðKn2 þ k22 Þ2 B @ x ðKm  Kn Þk2 b0 ðKn ; k2 Þ i c ðKm2 þ k22 ÞðKn2 þ k22 Þ

1 x bðKm ; k2 ÞðKm  Kn Þk2 c ðKm2 þ k22 ÞðKn2 þ k22 Þ C C C

x 2 2 A K K þk

i

m

c

n

ð2:11Þ

2

ðKm2 þ k22 ÞðKn2 þ k22 Þ

and where, to simplify the notation, we have defined Km ¼ k1 þ ð2pm=aÞ, so that b0 ðKm ; k2 Þ  2 1 2 1 ½Km2 þ k22  ðx=cÞ 2 and bðKm ; k2 Þ  ½Km2 þ k22  ðxÞðx=cÞ 2 . On equating to zero the coefficient of each delta function, we obtain the equation for apðnÞ ðkk Þ and aðnÞ s ðkk Þ in the form 1 X X ðnÞ MðabÞ a ¼ p; s; m ¼ 0; 1; 2; . . . ð2:12Þ mn ðkk Þab ðkk Þ ¼ 0; b¼p;s n¼1

The dispersion relation for surface polaritons is obtained by equating to zero the determinant of the coefficients in this system of equations. 3. Results In carrying out the numerical solution of Eq. (2.12) the dielectric function of the substrate, ðxÞ, will be assumed to be real and to have the simple free electron metal form 2

ðxÞ ¼ 1  ðxp =xÞ ;

ð3:1Þ

where xp is the plasma frequency of the conduction electrons in the metal. The surface polaritons that we will be concerned with are therefore surface plasmon polaritons. In our numerical calculations a value of hxp ¼ 3:78 eV appropriate for silver [26] is assumed. We will be interested in the frequency range 0 6 x < xp in which ðxÞ is negative, which is the region in which surface plasmon polaritons exist. We assumed that the dielectric function of the metal is real because initial calculations (whose results we do not report) showed that within graphical accuracy using a complex dielectric function does not change the real part of the resulting complex frequency of a surface plasmon polariton from the real value obtained by the use of a dielectric function of the form (3.1) with the same xp . This had already been pointed out earlier by Barnes et al. [16]. The solutions xðkk Þ of the dispersion relation obtained from Eq. (2.12) possess three general properties. The first is that, for a fixed value of k2 , xðkk Þ is a periodic function of k1 with a period 2p=a. This can be seen if we replace k1 by k1 þ ð2p=aÞ in Eq. (2.12), replace the summation variable n by s  1, relabel m by r  1, and define aaðs1Þ ðk1 þ ð2p=aÞ; k2 Þ ¼ bðsÞ a ðk1 ; k2 Þ: In this way we obtain the same set of homogeneous equations for the b-coefficients as we began with for the a-coefficients, so that xðk1 þ ð2p=aÞ; k2 Þ is a solution of the same equation that xðk1 ; k2 Þ satisfies. The second property of xðkk Þ is that it is an even function of kk : xðk1 ; k2 Þ ¼ xðk1 ; k2 Þ. For, if we replace k1 by k1 , and k2 by k2 in Eq. (2.12), change the summation variable n into s, replace m by r, and define aðsÞ ðkk Þ ¼ bðsÞ a a ðkk Þ, we obtain the same set of equations for the b-coefficients as we began with for the a-coefficients, except that in place of Irs ð Þ we have Irþs ð Þ. However, if the surface profile function fðx1 Þ is an even function of x1 , as it will be in the examples studied here, then, from Eq. (2.8) we see that Im ðcÞ ¼ Im ðcÞ:

ð3:2Þ

It then follows that xðkk Þ is a solution of the same equation that xðkk Þ satisfies. In the more general case that the surface profile function fðx1 Þ is not an even function of x1 , a more complicated argument [27] yields the same result.

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

209

The third property of xðkk Þ is that for a given value of k2 it is an even function of k1 : xðk1 ; k2 Þ ¼ xðk1 ; k2 Þ. To see this we replace k1 by k1 in Eq. (2.12), change the summation variable n into s, replace m by r, and ðsÞ ðsÞ define aðsÞ ðk1 ; k2 Þ ¼ bðsÞ p p ðkk Þ, as ðk1 ; k2 Þ ¼ bs ðkk Þ. The resulting set of equations for the b-coefficients is the set of equations with which we began for the a-coefficients, except that in place of Irs ð Þ we have Irþs ð Þ. However, in the case that the surface profile function fðx1 Þ is an even-function of x1 , it follows from Eq. (3.2) that xðk1 ; k2 Þ is a solution of the same equation that xðk1 ; k2 Þ satisfies. These three properties of xðkk Þ have the consequence that all of the distinct solutions of the dispersion relation are obtained if we restrict the wavenumber k1 to the interval 0 6 k1 6 p=a, i.e. to the right-hand half of the first Brillouin zone for the grating structure. In 1addition, it is straightforward to deduce from Eqs. (2.2a) and (2.2b) that if k1 and k2 are real, and ðk12 þ k22 Þ2 is larger than x=c, b0 ðKn ; k2 Þ is purely real when x is also real. From Eq. (2.1), we see that the electric field in the vacuum region in this case tends to zero as x3 ! 1. At the same time, since ðxÞ is assumed to be real and negative, bðKn ; k2 Þ is purely real when x is also real. This1 means that the electric field inside the metal also tends to zero as x3 ! 1. For k1 and k2 real and ðk12 þ k22 Þ2 smaller than x=c, b0 ðKn ; k2 Þ can become purely imaginary for x real, for some values of n, and the electric field (2.1) describes waves that radiate energy into the vacuum as they propagate along the surface, and are attenuated thereby. In this case the solutions of the dispersion relation for real k1 and k2 are complex, xðkk Þ ¼ xR ðkk Þ  ixI ðkk Þ, where the imaginary part of the frequency is related to the lifetime of 1 the wave, sðkk Þ, by sðkk Þ ¼ ½2xI ðkk Þ . Thus, the Bloch-type surface plasmon polaritons that are true eigenmodes of the corrugated structure, i.e. possess an infinite lifetime, for a given value of k2 are found only in the region of the ðx; k1 Þ-plane bounded from the left by the dispersion curve of volume electromagnetic 1 waves in the vacuum, x ¼ cðk12 þ k22 Þ2 , and from the right by the boundary of the first Brillouin zone, k1 ¼ p=a. This region is called the nonradiative region of the ðx; k1 Þ-plane. We will restrict our attention in this paper to solutions of the dispersion relation that lie inside the nonradiative region, i.e. to pure surface electromagnetic waves. The limiting case of surface plasmon polaritons propagating parallel to the grooves and ridges of the grating will also be considered. This situation corresponds to the conditions k1 ¼ 0; k2 P ðx=cÞ. In the numerical calculations of the dispersion curves the infinite determinant formed from the coefficients in Eq. (2.12) was replaced by the determinant of the 2ð2N þ 1Þ 2ð2N þ 1Þ matrix obtained by restricting m and n to run from N to N . The zeros of this determinant were found by fixing k1 and k2 and increasing x from 0 to xp in small increments Dx. To simplify the calculation we sought sign changes in the real part of this determinant as we successively increased x by Dx. If a sign change was found in going from x ¼ nDx to x ¼ ðn þ 1ÞDx, the real part of the determinant was calculated at x ¼ ðn þ 12ÞDx, and it was determined whether the zero occurred in the interval ðnDx; ðn þ 12ÞDxÞ or in the interval ððn þ 12ÞDx; ðn þ 1ÞDxÞ. The real part of the determinant was then calculated at the midpoint of the interval in which the zero occurred, and the process was repeated for a total of six times. In this way the zero was found with an error smaller than Dx=26 . It was then verified that the imaginary part of the determinant also vanished at the frequency found in this way. The search was stopped once the three lowest frequency solutions were found. The convergence of the solutions found in this way was tested by increasing N and seeing if they approached stable limiting values. Our criterion for convergence was a change of smaller than 0.1% as N was increased to N þ 1. In all our calculations a value of N ¼ 5 (a 22 22 matrix) proved to be sufficient to yield convergent solutions. As we are interested in the conical propagation of surface plasmon polaritons across the grating surface, in the numerical calculations we set k1 ¼ kk cos h, k2 ¼ kk sin h, where h is the angle between the surface plasmon polariton wave vector and the x1 -axis (the normal to the grooves and ridges of the grating). The angle h was varied in the interval 0 6 h 6 p=2, and kk was restricted by the condition 0 6 kk 6 p=ða cos hÞ (i.e. 0 6 k1 6 p=a). The vacuum light line is given by x ¼ ckk . To explore the sensitivity of the surface plasmon polariton dispersion curves in conical propagation to the surface profile function fðx1 Þ, we have calculated these curves for three different choices for this

210

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

function. We consider them and the corresponding results in turn. The period of each grating profile function has been chosen to be a ¼ 0:2 lm, so that we will be working in a frequency and wavevector range where the effects of retardation are important. However, in this range the surface plasmon polariton dispersion curves still deviate considerably from the vacuum light line. Finally, we will investigate the dependence of the width of the lowest gap in the surface plasmon polariton spectrum on the angle of propagation for sinusoidal gratings of different periods. 3.1. Sinusoidal profile The first surface profile function we consider is the sinusoidal profile fðx1 Þ ¼ f0 cosð2px1 =aÞ:

ð3:3Þ

In our numerical calculations we have assumed the values f0 ¼ 0:02 lm and a ¼ 0:2 lm. The Fourier coefficient Im ðcÞ for this surface profile function, obtained from Eq. (2.8), is Im ðcÞ ¼ Im ðcf0 Þ;

ð3:4Þ

where Im ðxÞ is a modified Bessel function. In Figs. 1(a)–(d) we have plotted the dispersion curves as functions of kk for four different, increasing, values of h. The shaded areas show the absolute band gaps, i.e. gaps for all values of kk along a given direction of propagation. The case h ¼ 0 corresponds to propagation of the surface plasmon polariton normal to the grooves and ridges of the grating. We see that the frequencies of the three branches increase monotonically as h increases. Thus, the position of the center of the gap between the first and second branches at the boundary of the first Brillouin zone, kk ¼ p=ða cos hÞ, increases with increasing h. However, the width of the gap decreases with increasing h, and vanishes at h ¼ p=2. At this value of h the surface plasmon polariton is propagating parallel to the grooves and ridges of the surface, and the boundary of the first Brillouin zone has moved off to infinity. We have therefore plotted the corresponding dispersion curves for kk out to only 3p=a. We note, however, pffiffiffi that for h ¼ p=2, as kk tends to infinity all branches of the dispersion curve tend to the frequency xp = 2, which is the limiting frequency of a surface plasmon polariton at a planar vacuum–metal interface as kk ! 1. This is because as kk tends to infinity and the wavelength of the surface plasmon polariton becomes smaller than any linear dimension characterizing the surface profile function, the surface plasmon polariton ‘‘sees’’ a surface that is locally planar, and all branches of its dispersion curve tend to the limiting frequency it has on such a surface. These properties of the gap are depicted explicitly in Fig. 2. In Fig. 2(a) we have plotted the width of the gap, dx, as a function of h, and in Fig. 2(b) we have plotted the frequency of the center of the gap, x0 , as a function of h. The former function decreases to zero as h increases from 0 to p=2, while the latter function increases with increasing h. 3.2. Symmetric sawtooth profile The second surface profile function we consider is the symmetric sawtooth profile defined by 8 4H > > x1 ; a=2 6 x1 6 0; 4H > :H  x1 ; 0 6 x1 6 a=2 a for jx1 j 6 a=2. This is an example of a surface profile function whose Fourier series representation has an infinite number of terms. In our calculations we assumed the values H ¼ 0:005 lm and a ¼ 0:2 lm. The Fourier coefficient Im ðcÞ for this surface profile function is given by

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

211

Fig. 1. Dispersion curves for the three lowest frequency branches as functions of kk for surface plasmon polaritons propagating across The the sinusoidal grating defined by Eq. (3.3) with f0 ¼ 0:02 lm and a ¼ 0:2 lm, for four values of h. p ffiffiffi inset to Fig. 1(a) shows three periods of the surface profile function. The horizontal dashed curve is at the frequency x ¼ xp = 2, the maximum frequency of a surface plasmon polariton at a planar vacuum–metal interface, and the dotted line is the vacuum light line x ¼ ckk .

Im ðcÞ ¼

8 4H c sinhðH cÞ > > > < 4H 2 c2 þ p2 m2 ;

m even;

> > 4H c coshðH cÞ > : ; 4H 2 c2 þ p2 m2

m odd:

ð3:6Þ

Strictly speaking, according to a theorem of Millar [23], the Rayleigh hypothesis is not valid for this nonanalytic surface profile. Nevertheless, it has been shown in earlier calculations of the dispersion curves of surface plasmon polaritons propagating normal to the grooves and ridges of a grating defined by a symmetric sawtooth profile function [3] that convergent results can be obtained if H =a is not too large ð 6 0:2Þ and if N is also not too large. The latter restriction is due to the fact that if a finite number of terms is kept in the Fourier series for fðx1 Þ we are dealing with an analytic surface profile function for which the Rayleigh hypothesis is valid for sufficiently small H =a. However, as the number of terms in the Fourier series for fðx1 Þ is increased, we come closer and closer to a nonanalytic surface profile function, the range of

212

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

Fig. 2. (a) The width of the gap between the first and second branches at the boundary of the first Brillouin zone for surface plasmon polaritons propagating across the sinusoidal grating defined by Eq. (3.3) with f0 ¼ 0:02 lm and a ¼ 0:2 lm, as a function of h. (b) The frequency of the center of this gap as a function of h.

values of H =a for which the Rayleigh hypothesis is valid becomes smaller and smaller [28], and in the limit as N goes to infinity it vanishes. With these considerations in mind we have carried out numerical calculations of the dispersion curves for the conical propagation of surface plasmon polaritons across a grating defined by a symmetric sawtooth profile function. In Figs. 3(a)–(d) we have plotted the dispersion curves as functions of kk for four increasing values of h. As in Fig. 1 the shaded areas show the absolute band gaps, i.e. gaps for all values of kk along a given direction of propagation. In contrast with the results plotted in Figs. 1(a)–(d), where the group velocity of the surface plasmon polaritons on the second branch is positive, it is negative for surface plasmon polaritons on the second branch for this choice for the surface profile function and its parameters. However, as in the preceding example, the frequencies of the three branches plotted increase with increasing h, the frequency of the center of the gap at the boundary of the first Brillouin zone between the first two branches decreases, and the width of the gap decreases. In Figs. 4(a) and (b) we have plotted the dependence of the width of the gap and the frequency of the center of the gap on h, respectively, which plots confirm the trends seen in Figs. 3(a)–(d). 3.3. A superposition of Gaussians The third surface profile function we consider is defined by " # 2 1 X 4ðx1  ‘aÞ fðx1 Þ ¼ H exp  : R2 ‘¼1

ð3:7Þ

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

213

Fig. 3. The same as Fig. 1, except that the surface plasmon polariton is propagating across the symmetric sawtooth grating defined by Eq. (3.5) with H ¼ 0:005 lm and a ¼ 0:2 lm: The circles show the dispersion curves calculated in the framework of small-amplitude perturbation theory.

This is another example of a surface profile function whose Fourier series representation has an infinite number of terms. In our numerical calculations we used the values H ¼ 0:03 lm, R ¼ 0:075 lm, and a ¼ 0:2 lm. In calculating the Fourier coefficient Im ðcÞ for this surface profile function, the function fðx1 Þ was calculated very accurately numerically from Eq. (3.7) for 0 6 x1 6 a=2, and Im ðcÞ was then calculated from  n Z 12a 1 2X ðcH Þ 2pmx1 n Im ðcÞ ¼ dx1 cos ð3:8Þ f ðx1 Þ: a n¼0 n! a 0 The fact that fðx1 Þ is an even function of x1 was used in obtaining Eq. (3.8).

214

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

0.05

δω/ωp

0.04 0.03 0.02 0.01 0.00 (a)

0

π/6

0

π/6

θ

π/3

π/2

π/3

π/2

0.75

ω0/ωp

0.70 0.65 0.60 0.55 (b)

θ

Fig. 4. The same as Fig. 2, except that the surface plasmon polariton is propagating across the symmetric sawtooth grating defined by Eq. (3.5) with H ¼ 0:005 lm and a ¼ 0:2 lm. The dashed curve in Fig. 4(a) is the result obtained in the framework of small-amplitude perturbation theory.

The additional degree of freedom provided by the existence of a second lateral length scale R in addition to the period a, means that the surface profile function defined by Eq. (3.7) can describe a grating surface whose peaks are sharper than those of a sinusoidal profile, and whose valleys are shallower, for example. The dispersion curves as functions of kk for four increasing values of h are plotted in Figs. 5(a)–(d) for surface plasmon polaritons propagating across this surface. The shaded areas show the absolute band gaps, i.e. gaps for all values of kk along a given direction of propagation. The group velocity of the surface plasmon polaritons on the second branch is positive, as it is in the results for the sinusoidal surface profile function plotted in Figs. 1(a)–(d). The frequencies of the three branches plotted increase monotonically as h increases. The position of the center of the gap between the first and second branches at the Brillouin zone boundary increases with increasing h, and the width of this gap decreases with increasing h, and vanishes at h ¼ p=2. These features of the gap are clearly revealed in the results plotted in Fig. 6. 3.4. The angular dependence of the gap width All the results presented in Sections 3.1–3.3 show a similar dependence of the width of the lowest gap on the angle of propagation of surface plasmon polaritons. It decreases with increasing h, and vanishes at h ¼ p=2. However, at large angles of propagation h a weak structure in the angular dependence of the gap width appears, which will be shown below to be related to the interaction of the second and third branches in the surface polariton spectrum.

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

215

Fig. 5. The same as Fig. 1, except that the surface plasmon polariton is propagating across the grating surface defined by Eq. (3.7) with A ¼ 0:03 lm R ¼ 0:075 lm, and a ¼ 0:2 lm.

However, such angular dependences of the width of the lowest gap are typical only for gratings of sufficiently small periods. In Fig. 7 we present plots of the width of the lowest gap in the surface polariton spectrum as a function of the angle of propagation h for different values of the periods of the sinusoidal grating define by Eq. (3.3). In the calculations whose results are illustrated in Fig. 7 the height of each grating was chosen to be the same, namely f0 ¼ 20 nm, as that of the grating used in obtaining the results presented in Figs. 1 and 2. All plots presented in Fig. 7 are far from a cos2 h dependence predicted by Mills [19]. The gap widths increase as the angle of propagation increases, reach a maximum, and drop to zero at h ¼ 90°. The angular position of the maximum of the gap width moves to smaller angles of propagation with the decrease of the period of the grating, while the maximum itself becomes less and less pronounced. The magnitudes of the gap widths are, of course, much smaller, since the gratings are considerably shallower than those used in obtaining the results plotted in Figs. 1 and 2.

216

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

Fig. 6. The same as Fig. 2, except that the surface plasmon polariton is propagating across the grating surface defined by Eq. (3.7) with A ¼ 0:03 lm, R ¼ 0:075 lm, and a ¼ 0:2 lm.

0.30 0.25 0.20 δω/ωp

×30

×10

0.15 0.10 0.05 0.00

0

π/6

θ

π/3

π/2

Fig. 7. The angular depedence of the width of the gap between the first and second branches at the boundary of the first Brillouin zone for surface plasmon polaritons propagating across the sinusoidal grating defined by Eq. (3.3) with f0 ¼ 20 nm and different periods a ¼ 200 nm, solid line; a ¼ 600 nm, dotted line; and a ¼ 1000 nm, dashed line.

4. Discussion The results of our nonperturbative calculations presented in Sections 3.1–3.3 display several peculiar features. First of all, as has already been mentioned, the group velocity of the surface plasmon polaritons of the second branch is positive over the entire nonradiative region in all cases considered here except the case

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

217

where surface polaritons propagate across a symmetric sawtooth grating at an angle h smaller than p=6. (see Figs. 3(a) and (b)). This means that the lowest gap opened up in the surface polariton spectrum due to the Bragg scattering by the grating is sufficiently large to change the frequency dependence of the second branch so that the group velocity changes its sign. Second, the angular dependence of the gap width is far from the cos2 h dependence predicted in both [19,20]. What is more, the dependence is not quite monotonic at large angles of propagation h (see Figs. 2(a), 4(a) and 6(a)). The most pronounced structure is displayed by the width of the gap produced by the symmetric sawtooth profile, Fig. 4(a). The parameters used in our numerical calculations in this case are such that we can apply small-amplitude perturbation theory to analyze the results obtained for the symmetric sawtooth grating, while those obtained for the sinusoidal grating and the grating formed by the superposition of Gaussians cannot be described by perturbation theory. However, the processes occurring at the latter surfaces are the same and can be interpreted on the basis of perturbation theory. The starting point of the perturbation theory is Eq. (2.12) in which the functions Im ðcÞ defined by Eq. (2.8) are expanded in powers of the surface profile function fðx1 Þ 1 ð4:1Þ Im ðcÞ ¼ dm0 þ cbf ð1Þ ðmÞ þ c2 bf ð2Þ ðmÞ þ ; 2 where dm0 is the Kronecker symbol, and Z 12a 2pm bf ðpÞ ðmÞ ¼ 1 dx1 ei a x1 fp ðx1 Þ: ð4:2Þ a 12a We will first analyze the angular dependence of the width of the lowest gap that opens up in the surface polariton spectrum. To obtain the expression for the width in the same approximation as used in [19,20] it is enough to keep only the two lowest branches in Eq. (2.12), i.e. to restrict ourselves to the first Bragg scattering of the surface plasmon polaritons. In this case, to second order in the surface profile function, the dispersion relation for surface plasmon polaritons takes the form h i2 ðppÞ ðppÞ ð4:3Þ DðK0 ; k2 ; xÞDðK1 ; k2 ; xÞ ¼ bf ð1Þ ð1Þ M0;1 ðkk ÞM1;0 ðkk Þ; where DðKm ; k2 ; xÞ ¼

ðxÞb0 ðKm ; k2 Þ þ bðKm ; k2 Þ 1 bð2Þ ðppÞ  f ð0ÞMm;m ðkk ÞðbðKm ; k2 Þ  b0 ðKm ; k2 ÞÞ ðxÞ  1 2 ðspÞ h i2 M ðpsÞ m;ðmþ1Þ ðkk ÞMðmþ1Þ;m ðkk Þ ð1Þ b  f ð2m þ 1Þ : ðssÞ Mðmþ1Þ;ðmþ1Þ ðkk Þ

ð4:4Þ

Eq. (4.3) is known to describe well the two lowest branches of the dispersion relation for surface plasmon polaritons propagating on a shallow grating perpendicular to the grooves (k2 ¼ 0) [4]. The width of the gap at the boundary of the first Brillouin zone can be obtained by solving Eq. (4.3) for k1 ¼ p=a. As a result we obtain a simple expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxðhÞ c p 2 ðxsp Þ 1  ðxsp Þ ð1Þ b ; ð4:5Þ ¼ 4j f ð1Þj xp xp a 2 ðxsp Þ þ 1 where the dielectric function of the metal is evaluated at the frequency of the unperturbed surface polariton xsp at the boundary of the first Brillouin zone, i.e xsp is the solution of the dispersion relation  1=2 p xsp ðxsp Þ ¼ : ð4:6Þ a cos h c ðxsp Þ þ 1

218

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

In obtaining Eq. (4.5) we have neglected the influences of the p- to s- and s- to p-scattering corrections to DðKm ; k2 ; xÞ since they turned out to be negligibly small. Thus, after solving Eq. (4.6) for the frequency xsp and expressing the dielectric function through the angle of propagation 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  2 2 2 x2p a2 xp a ð4:7Þ ðxsp Þ ¼ 4 2 2 cos2 h þ cos4 h þ 1 5; 2p c 2p2 c2 we can rewrite Eq. (4.5) in a more transparent form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u 2

 ðx a=2pcÞ cos h  1 þ ðxp a=2pcÞ cos4 h þ 1 dxðhÞ c p 2t p ð1Þ b : ¼ 2j f ð1Þj 2 xp xp a ðxp a=2pcÞ cos4 h þ 1

ð4:8Þ

The angular dependence of the gap width calculated with the use of Eq. (4.8) is shown in Fig. 4(a) by a dashed line together with the results of a numerical solution of Eq. (2.12). It should be noted that Eq. (4.5) coincides with the expressions for the gap width obtained in [19,20] (when differences in notation and a misprint in [19] are taken into account). However, in both papers cited the frequency and the wavenumber of an unperturbed surface polariton at the boundary of the first Brillouin zone was taken to be independent of the angle of propagation. This assumption led to the cos2 h dependence of the gap width and to much smaller values of the gap width at large angles of propagation. As seen from Fig. 4(a) the results of the perturbative approach for the gap width agree well with the results of numerical calculations up to large values of the angle h, where structure in the results of the numerical calculations occurs. We note that in the case of the sinusoidal grating and the one formed by a superposition of Gaussians the results of the perturbation theory underestimate the real gap width. Simple and quite crude estimates already show the origin of the structure observed in Fig. 4(a). For the parameters of the symmetric sawtooth grating assumed the frequency of the second branch at the boundary of the first Brillouin zone reaches the frequency of the third, unperturbed by the grating, branch (xsp ðp=a; k2 Þ þ dx=2 ¼ xsp ð3p=a; k2 Þ, where xsp ðk1 ; k2 Þ is the frequency of a surface plasmon polariton on a planar surface), when the angle of incidence is hc ¼ 69:8°. The gap opened between the third and the fourth branches makes this angle even smaller. From our numerical results it follows that hc ¼ 66:5°. At larger angles of propagation the second and third branches would cross, when noninteracting, within the first Brillouin zone. This would lead to a seemingly smaller gap width calculated as the difference of frequencies between the lowest and the second lowest branch (as was done in the numerical calculations), which in this case is the third, rather than second branch, while in the perturbative approach used here it is still the difference between the frequencies of the first and second branches. Analogous, simple, estimates show that the frequency of the second branch at the boundary of the first Brillouin zone becomes higher than its frequency at k1 ¼ 0 at angles of propagation larger than 54°. Thus, for h > 54° the group velocity of the surface polaritons of the second branch should be expected to be positive, while it is negative for smaller angles of propagation. This trend is clearly seen in Figs. 3(a)–(d). To describe the effect of the branch crossing on the basis of perturbation theory we can proceed as follows. In the vicinity of the boundary of the first Brillouin zone the dispersion relation for the second branch can be derived from Eq. (2.12) by taking into account only the first and the second branches, as was done in derivation of Eq. (4.4), with the result that ðppÞ

b ðK1 ; k2 Þ  DðK1 ; k2 Þ  bf ð1Þ ð1Þbf ð1Þ ð1Þ D

ðppÞ

M1;0 ðkk ÞM0;1 ðkk Þ ¼ 0; DðK0 ; k2 Þ

ð4:9Þ

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

219

where DðKm ; k2 Þ is given by Eq. (4.4). In a similar fashion the dispersion relation for the third branch in the vicinity of the Brillouin zone boundary can be derived from Eq. (2.12) by taking into account only the third and the fourth branches, with the result that ðppÞ

b ðK1 ; k2 Þ  DðK1 ; k2 Þ  bf ð1Þ ð3Þbf ð1Þ ð3Þ D

ðppÞ

M1;2 ðkk ÞM2;1 ðkk Þ ¼ 0: DðK1 ; k2 Þ

ð4:10Þ

When the angle of propagation is quite small Eqs. (4.9) and (4.10) describe well the dispersion of surface plasmon polaritons of both branches. However, with increasing h the branches merge and at some critical angle they cross. In the vicinity of the crossing point it is no longer possible to consider the branches as noninteracting, and both Eqs. (4.9) and (4.10) are no longer valid. When the interaction between the second and third branch is taken into account the dispersion relation for the two branches takes the form ðppÞ ðppÞ b ðK1 ; k2 Þ D b ðK1 ; k2 Þ ¼ bf ð1Þ ð2Þbf ð1Þ ð2ÞM1;1 D ðkk ÞM1;1 ðkk Þ:

ð4:11Þ

A typical picture of the branch interaction is shown in Fig. 8. The calculations are done on the basis of Eq. (4.11) for the symmetric sawtooth profile characterized by the parameters used in obtaining the results shown in Figs. 3 and 4. The angle of propagation is h ¼ 70°. The dashed lines show the dispersion curves obtained from Eqs. (4.9) and (4.10), i.e. without any interaction between the branches. In Fig. 9 we present the dispersion curves for surface plasmon polaritons propagating at large angles h across the three types of gratings considered in the Section 3, obtained by numerical solution of the exact dispersion relation that follows from Eq. (2.12). The angles h were chosen so that the repulsion of the second and third branches can be clearly seen. Finally, the dispersion curves of all three branches of interest can be obtained from Eq. (2.12) by keeping higher-order Bragg scatterings. To second order in the surface profile function the dispersion relation has the form 2 Y n¼2

DðKn ; k2 Þ ¼

2 X

ðppÞ ðppÞ DðKi ; k2 ÞDðKj ; k2 ÞDðKl ; k2 Þbf ð1Þ ðm  nÞbf ð1Þ ðn  mÞMm;n ðkk ÞMn;m ðkk Þ;

ð4:12Þ

i;j;l;m;n¼2

ω/ωp

0.695

0.690

0.685 1.8

2.0

2.2 k||a/π

2.4

2.6

Fig. 8. An example of the interaction of the second and third branches of the surface plasmon polariton propagating across the symmetric sawtooth grating with H ¼ 0:005 lm and a ¼ 0:2 lm at an angle h ¼ 70°.

220

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

Fig. 9. Dispersion curves for the three lowest frequency branches as functions of kk for surface plasmon polaritons propagating across the sinusoidal (a) and symmetric sawtooth (b) gratings and the grating formed by a superposition of Gaussians (c) at an angle h sufficiently large that the repulsion of the second and third branches occurs inside the first Brillouin zone. The inset to each figure shows the second and the third branches in the region of their repulsion. The angles of propagation are: (a) h ¼ 57°, (b) h ¼ 69°, and (c) h ¼ 82:5°.

where DðKm ; k2 ; xÞ ¼

ðxÞb0 ðKm ; k2 Þ þ bðKm ; k2 Þ 1 bð2Þ ðppÞ  f ð0ÞMm;m ðkk ÞðbðKm ; k2 Þ  b0 ðKm ; k2 ÞÞ ðxÞ  1 2 

2 X n¼2 ðn6¼mÞ

bf ð1Þ ðm  nÞbf ð1Þ ðn  mÞ

ðpsÞ ðspÞ Mm;n ðkk ÞMn;m ðkk Þ ðssÞ

Mn;n ðkk Þ

ð4:13Þ

and in the sum on the right-hand side of Eq. (4.12) only contributions with i 6¼ j 6¼ l 6¼ n 6¼ m are kept. The solution of Eq. (4.12) for the three lowest branches is presented in Figs. 3(a)–(c) together with the results of

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

221

the numerical calculations. Thus, in this case the perturbative solution indeed describes well the dispersion curves of surface polaritons on the symmetric sawtooth grating considered here. In a recent experiment [29] the spectrum of surface plasmon polaritons propagating at an angle to the normal to the grooves and ridges of a grating ruled on a silver surface has been studied experimentally. The main goal of the experiment was to measure the width of the lowest gap opened in the surface plasmon polariton spectrum at the boundary of the first Brillouin zone as a function of the angle of propagation. Since this gap is in the nonradiative region, to measure it a double grating technique was used. In this technique, the corrugation was produced by an interference pattern on the surface of photoresist which, after development, was covered by a thick silver layer. Due to the nonlinearities of the processes the grating profile contained several harmonics in addition to the fundamental grating (of period a). However, only the lowest harmonic (of period a=2) is usually strong enough to be taken into account in experiments. In the case where the fundamental grating is sufficiently weak, the gap in the surface polariton spectrum that opens up at the boundary of the second Brillouin zone can be regarded as the gap opened solely due to the harmonic grating at the boundary of its first Brillouin zone. Thus, in this case, the fundamental grating only transfers the gap into the radiative region. The surface profile function of the system studied in [29] has the form  fðx1 Þ ¼ d1 sin

2p x1 a



 þ d2 sin

4p p x1  ; a 2

ð4:14Þ

where d1 ¼ 17:3 nm, d2 ¼ 2:8 nm, and a ¼ 633 nm. In Fig. 10(a) we plot the angular dependence of the width of the gap opened at the boundary of the second Brilluoin zone in the spectrum of surface plasmon polaritons propagating across the double grating ruled on a silver surface, and in Fig. 10(b) we plot the frequency of the center of the gap as a function of h. The results plotted in Fig. 10 were obtained by a numerical, nonperturbative, solution of Eq. (2.12). In obtaining the plots in Fig. 10 we used the value of the plasma frequency hxp ¼ 8:4 eV, which was estimated from the experimental data, presented in [29], that at h ¼ 0° the wavelength of the center of the gap was measured to be k0 ¼ 651 nm. This allows estimating the real part of the dielectric function as ð2pc=k0 Þ ¼ 18:4, which agrees reasonably well with the experimental values given in [30]. In the same plot the crosses show the experimental values of the gap width [29]. As is seen, the results of the numerical calculations underestimate the gap width by about 45%, but describe well the angular dependence for small h. It should be noted that the value of the gap width at h ¼ 0° presented in Fig. 8(b) of [29] is about 28% higher than that stated in the text of [29], dk ¼ 12 nm, (with k ¼ 645 nm and kþ ¼ 657 nm, where k are the wavelengths of the lower and upper branches), which leads to hdxðhÞ ¼ 0:035 eV. We note that the parameters of the harmonic grating used in [29] are such that perturbation theory can be applied to study the spectrum of surface plasmon polaritons. The estimates of the value of the gap width at h ¼ 0 with the use of the results of [3,4,19,20], as well as the expressions presented in [29] and in the present paper, give the same value  hdx  0:025 eV. Thus, at small angles of propagation the angular dependence of the gap width is in good qualitative agreement with the experimental results. At large angles of propagation the gap width decreases rapidly, almost vanishes at h0 ¼ 76°, then grows, and after passing through a maximum tends to zero as h ! p=2. Due to the phase shift p=2 between the gratings, their contributions to the ‘‘efficiency’’ of the scattering are of opposite sign and, as a result, cancel each other at some particular angle h0 . The surface plasmon polaritons propagating at this angle with respect to the grooves and ridges of the gratings do not undergo Bragg scattering and do not feel the presence of the gratings. The additional structure that is seen at h ¼ 60° is due to the interaction of the second and third branches. It should be noted , that although the parameters of the harmonic grating used in [29] are well suited for the use of perturbation theory, the fundamental grating was extremely strong, so the perturbative approach breaks down.

222

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

Fig. 10. (a) The width of the gap between the second and third branches at the center of the Brillouin zone for surface plasmon polaritons propagating across the double grating defined by Eq. (4.14) as a function of h. (b) The frequency of the center of this gap as a function of h. The crosses show the results presented in Fig. 8 of [29].

5. Conclusions In this paper we have carried out nonperturbative calculations of the dispersion curves of surface plasmon polaritons propagating at an angle to the grooves and ridges of a weakly corrugated classical metallic grating. The theory of these calculations, which is based on the reduced Rayleigh equation for the electric field above and on a two-dimensional rough surface [21], however, is applicable more generally to a grating surface bounding any medium that is characterized by a frequency-dependent dielectric function that is negative in some frequency range or ranges. It is only necessary to use the appropriate form of the dielectric function ðxÞ in solving the dispersion equation obtained from Eq. (2.12). If we take as our reference situation the propagation of surface plasmon polaritons normal to the grooves and ridges of the grating ðh ¼ 0°Þ we are led to the following conclusions about the behavior of the dispersion curves as functions of kk in the interval 0 6 kk 6 p=ða cos hÞ ð0 6 k1 6 p=aÞ in the case of gratings of a comparatively small period. In the first place, the frequencies of the three lowest frequency branches of the dispersion curve in the nonradiative region of the ðx; kk Þ-plane, which are all that were studied here, increase with increasing h. As a consequence the frequency of the center of the gap at the boundary of the first Brillouin zone between the first and second branches, increases with increasing h. In the second place, the width of this gap decreases with increasing h, and vanishes at h ¼ p=2. At large angles of propagation h the second and third branches interact, and their repulsion leads to the appearance of structure in the angular dependences of both the gap width and frequency of the center of the gap at the boundary of the first Brillouin zone. Finally, although there are some quantitative differences between the surface plasmon

M. Kretschmann et al. / Optics Communications 215 (2003) 205–223

223

polariton dispersion curves associated with the different surface profile functions employed in our numerical calculations, their qualitative features listed above are largely independent of the particular surface profile function assumed. The width of the gap at the boundary of the first Brillouin zone and its dependence of the angle of propagation of surface plasmon polaritons depends strongly on the period of the grating. For large periods the gap width increases with increasing h, reaches a maximum, and vanishes at h ¼ p=2. As the period of the grating decreases, the width of the gap at h ¼ 0 increases, so that its angular dependence gradually transforms into one that decreases monotonically with an increase of h. Finally, we have investigated the angular dependence of the width of the gap that opens up in the spectrum of a surface plasmon polariton propagating across the double grating used in the experiments reported in [29]. The particular gap of interest is the one opened at the boundary of the second Brillouin zone of the double grating. The results of our calculations agree qualitatively and semi-quantitatively with the experimental results of [29]. Acknowledgements This research was supported in part by NSF Grant INT-0084423. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

P.M. van den Berg, J.C.M. Borburgh, Appl. Phys. 3 (1974) 55. F. Toigo, A. Marvin, V. Celli, N.R. Hill, Phys. Rev. B 15 (1977) 5618. B. Laks, D.L. Mills, A.A. Maradudin, Phys. Rev. B 23 (1981) 4965. A.A. Maradudin, in: V.M. Agranovich, D.L. Mills (Eds.), Surface Polaritons, North-Holland, Amsterdam, 1982, p. 405. N.E. Glass, M. Weber, D.L. Mills, Phys. Rev. B 29 (1984) 6548. S.R. Seshadri, J. Appl. Phys. 57 (1985) 4874. E.P. DaSilva, G.A. Farias, A.A. Maradudin, J. Opt. Soc. Am. A 4 (1987) 2022. W.L. Barnes, T.W. Preist, S.C. Kitson, J.R. Sambles, Phys. Rev. B 54 (1996) 6227. R.H. Ritchie, E.T. Arakawa, J.J. Cowan, Phys. Rev. Lett. 21 (1968) 1530. W. Knoll, M.R. Philpott, J.D. Swalen, A. Girlando, J. Chem. Phys. 77 (1982) 2254. Y.J. Chen, E.S. Koteles, R.J. Seymour, G.J. Sonek, I.M. Ballantyne, Solid. State Commun. 46 (1983) 95. D. Heitmann, N. Kroo, C. Schulz, Z. Szentirmay, Phys. Rev. B 35 (1987) 2660. H. Raether, in: Surface Plasmons, Springer-Verlag, Berlin, 1988, p. 113. B. Fischer, T.M. Fischer, W. Knoll, J. Appl. Phys. 75 (1994) 1577. H. Lochbihler, Phys. Rev. B 50 (1994) 4795. W.L. Barnes, T.W. Preist, S.C. Kitson, J.R. Sambles, N.P.K. Cotter, D.J. Nash, Phys. Rev. B 51 (1995) 11164. S.C. Kitson, W.L. Barnes, G.W. Bradberry, J.R. Sambles, J. Appl. Phys. 79 (1996) 7383. D. Maystre, in: E. Wolf (Ed.), Progress in Optics, vol. XXI, North-Holland, Amsterdam, 1984, pp. 3–67. D.L. Mills, Phys. Rev. B 15 (1977) 3097. S.R. Seshadri, J. Appl. Phys. 58 (1985) 1733. G.C. Brown, V. Celli, M. Haller, A. Marvin, Surf. Sci. 136 (1984) 381. R. Petit, M. Cadilhac, C. R. Acad. Sci. Paris B 262 (1966) 468. R.F. Millar, Proc. Camb. Phil. Soc. 65 (1969) 773. R.F. Millar, Proc. Camb. Phil. Soc. 69 (1971) 217. R.F. Millar, Radio Sci. 8 (1973) 785. F. Abeles, T. Lopez-Rios, in: V.M. Agranovich, D.L. Mills (Eds.), Surface Polaritons, North-Holland, Amsterdam, 1982, p. 260. A.A. Maradudin, W. Zierau, Geophys. J. Int. 118 (1994) 325. P.M. van den Berg, J.T. Fokkema, J. Opt. Soc. Am. 69 (1979) 27. W.L. Barnes, S.C. Kitson, T.W. Preist, J.R. Sambles, J. Opt. Soc. Am. A 14 (1997) 1654. E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, New York, 1985.