Surface plasmon polariton mechanism for enhanced ... - OSA Publishing

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J. Opt. Soc. Am. A / Vol. 12, No. 11 / November 1995

Maradudin et al.

Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces A. A. Maradudin Department of Physics and Institute for Surface and Interface Science, University of California, Irvine, Irvine, California 92717

A. R. McGurn Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008

E. R. Méndez ´ de F´ısica Aplicada, Centro de Investigacion ´ Cient´ıfica y de Educacion ´ Superior de Ensenada, Division Apartado Postal 2732, Ensenada, Baja California, 22800, Mexico Received February 27, 1995; accepted May 4, 1995 Recent experimental results of West and O’Donnell [ J. Opt. Soc. Am. A 12, 390 (1995)] for the enhanced backscattering of p-polarized light from weakly rough, one-dimensional, random gold surfaces are compared with the predictions of two perturbative calculations of such scattering. The experimental surfaces were fabricated to possess power spectra that are nonzero in only a narrow range of wave numbers about the wave number of the surface plasmon polariton supported by them at the frequency of the incident light. As a consequence, enhanced backscattering that is due to the coherent interference of time-reversed scattering sequences involving counterpropagating surface plasmon polaritons is possible for only a limited range of values for the angles of incidence and scattering. The perturbative calculations used in the comparisons with experiment are the infinite-order calculation of McGurn et al. [ Phys. Rev. B 31, 4866 (1985)] in the small-roughness approximation and the small-amplitude perturbation theory of Maradudin and Méndez [ Appl. Opt. 32, 3335 (1993)] that is exact to fourth order in the surface-profile function. In both calculations the origin of enhanced backscattering is the coherent interference of multiply scattered surface plasmon polaritons with their time-reversed partners. The good quantitative and qualitative agreement between the theoretical and experimental results, with no fitting parameters, supports the conclusion of West and O’Donnell that their data demonstrate the existence of enhanced backscattering caused by the excitation of surface plasmon polaritons on a weakly rough random metal surface.  1995 Optical Society of America

In a paper published in 19851 it was predicted that if p-polarized light is incident on a one-dimensional, randomly rough metal surface, with the plane of incidence normal to the grooves and the ridges of the surface, the angular dependence of the intensity of the light that is scattered diffusely will display a sharp peak in the retroreflection direction. This peak has come to be called enhanced backscattering in reflection. The prediction of this effect was based on an infinite-order perturbation expansion of the scattered intensity in powers of the surface-profile function, and the results were valid only for small rms height, small rms slope, random surfaces. In addition, the surface-profile function was assumed to be a stationary, zero-mean, Gaussian random process, characterized by a Gaussian form for the surface height correlation function and hence by a Gaussian form for the power spectrum of the surface roughness. The mechanism responsible for this phenomenon can be described as follows. Let us consider a sequence of scattering events in which the incident light excites a surface electromagnetic wave, a surface plasmon polariton, through the surface roughness; the surface plasmon polariton propagates along the surface and is scattered 0740-3232/95/112500-07$06.00

s 2 1 more times by the roughness; at the final scattering event it is converted back into volume electromagnetic waves in the vacuum, propagating away from the surface. It is assumed that all such s-fold scattering sequences are uncorrelated because of the random nature of the surface. However, any given sequence and its time-reversed partner, in which the light and the surface plasmon polariton are scattered from the same points on the surface but in the reverse order, interfere constructively if the wave vectors of the initial and final waves are oppositely directed. These two waves have the same amplitude and phase and add coherently in forming the intensity of the scattered light. For scattering into directions other than the retroreflection direction the different partial waves have a nonzero phase difference and rapidly become incoherent, so that only their intensities add. Consequently, the intensity of scattering into the retroreflection direction is a factor of 2 larger than the intensity of scattering into other directions because of the cross terms that appear in the expression for the intensity in the former case. This enhancement occurs within a region of angular width Dus lyl about the retroreflection direction at normal incidence, where l is the wavelength  1995 Optical Society of America

Maradudin et al.

Vol. 12, No. 11 / November 1995 / J. Opt. Soc. Am. A

of the incident light and l is the mean free path of the surface plasmon polaritons propagating on the randomly rough surface.1 One must subtract the contribution of the single-scattering processes in obtaining this factorof-2 enhancement because it is not subject to coherent backscattering. In subsequent work2 it was shown that most of the enhanced backscattering arises from the double scattered (s 2) optical paths. Although the enhanced backscattering of light from large rms height, large rms slope, randomly rough, highly reflecting surfaces has been observed experimentally3 and its origin in the coherent interference of multiply scattered light waves has been proposed,3 unambiguous experimental evidence for the surface plasmon polariton mechanism for the enhanced backscattering of light from smooth, randomly rough surfaces has been lacking. Because of the weakness of the roughness of the surfaces to which the theory of Ref. 1 applies, the diffuse component of the scattered light is a very small fraction of the total scattered intensity, which is dominated by the specular component. In addition, the mean free path l of the surface plasmon polariton on such a smooth random surface at the wavelength l of the incident light is long compared with l, which results in a narrow enhanced backscattering peak. Both of these factors make the experimental observation of enhanced backscattering from weakly rough random surfaces difficult to achieve. An early effort to detect this effect4 was not definitive because the surfaces used in these measurements were not characterized independently of the scattering measurements, and the rms height and the transverse correlation length of the surface roughness were in fact obtained from a fit of theoretical expressions for the intensity of the scattered light to the experimental data. In recent, elegant experiments West and O’Donnell5 have demonstrated that smooth, one-dimensional, randomly rough metal surfaces can display enhanced backscattering when illuminated by p-polarized light with its plane of incidence normal to the generators of the surface and have shown unambiguously that the surface polariton mechanism for this effect proposed in Ref. 1 is responsible for their results. In this paper we offer theoretical support for their experimental results and conclusions, based on the results of Refs. 1 and 2. In their work West and O’Donnell fabricated onedimensional, random, gold surfaces that were stationary, Gaussian random processes by an extension of holographic grating fabrication methods.6 The power spectrum of the roughness of these surfaces, however, did not have the Gaussian form assumed in earlier theoretical and experimental studies but rather had the form gsjkjd

p fusk 2 kmin duskmax 2 kd kmax 2 kmin 1 us2kmin 2 kdusk 1 kmax dg .

(1)

The definition of gsjkjd used in Refs. 1 and 2 has been adopted, and usxd is the Heaviside unit step function. The choice of a power spectrum of this form was based on the following considerations. Light of frequency v, whose angle of incidence is u0 , can excite forward- and backward-propagating surface plasmon polaritons with

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wave numbers 6ksp svd, respectively, through the surface roughness, where ksp svd

" #1/2 v esvd . Re c esvd 1 1

(2)

In this equation esvd is the complex dielectric function of the metal. The conditions for this excitation may be described by the following pair of grating equations, in which the surface plasmon polaritons with wave numbers 6ksp svd correspond to the 61st diffracted orders, respectively: 1ksp svd

v sin u0 1 k1 , c

2ksp svd

v sin u0 2 k2 , c (3)

where k1 and k2 are wave numbers that are among those available in the power spectrum of the surface roughness. The surface plasmon polaritons, once excited, can then couple to volume electromagnetic waves in the vacuum above the surface and traveling away from it at a scattering angle us . This outward coupling may be described by a second pair of grating equations, v sin us 1ksp svd 2 k3 , c

v sin us 2ksp svd 1 k4 , c (4)

where k3 and k4 are again among the wave numbers contained in the power spectrum. In the retroreflection direction us 2u0 , Eqs. (3) and (4) are identical and require that k1 k4 and k2 k3 . These equations then describe the pair of time-reversed scatting processes whose coherent interference gives rise to enhanced backscattering. If it is desired to excite both forward- and backwardpropagating surface plasmon polaritons of wave numbers 6ks svd for all angles of incidence u0 in the range 2umax # u0 # 1umax , only a limited portion of the power spectrum plays a role. At the limiting angle u0 umax the range of wave numbers of the surface roughness necessary for the excitation of both surface plasmon polaritons follows from Eqs. (3) as kmin , k , kmax , where kmin ksp svd 2

v sin umax , c

kmax ksp svd 1

v sin umax . c

(5)

Thus, if the power spectrum is nonzero only for k between kmin and kmax (and for 2k between 2kmax and 2kmin ), it is readily verified from Eqs. (3) that values of k in these ranges permit the excitation of both surface plasmon polaritons for all u0 between 2umax and 1umax. Similarly, it follows from Eqs. (4) that the excited surface plasmon polaritons will be coupled to scattered volume waves corresponding to scattering angles us distributed continuously between 2umax and 1umax. The values of kmin and kmax entering the power spectrum (1) are determined by Eqs. (5). With this form of the power spectrum the incident light couples strongly to the surface plasmon polaritons over a limited range of angles of incidence rather than weakly over a large range of this angle, as is the case

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when a power spectrum gsjkjd peaked about k 0, e.g., a Gaussian, is used. This is because in the latter case the wave number of the surface plasmon polariton lies in the wings of the power spectrum (especially at normal incidence), where it is very small. The surfaces fabricated by West and O’Donnell were intended to possess the power spectrum (1) with kmin and kmax determined from Eqs. (5) with umax 13.5± and ksp svd as the wave number of surface plasmon polaritons at the frequency v corresponding to a wavelength of l 612 nm. The surfaces fabricated in this way were characterized by a contact profilometer. From the resulting surface profiles a power spectrum was obtained that possessed flat regions with distinct edges at kmin and kmax. From their results that 4pyskmin 1 kmax d > 578 nm and 2pyskmax 2 kmin d > 1315 nm we obtain the values kmin 8.4815 3 1023 nm21 and kmax 13.260 3 1023 nm21 . From these values and the result that kmax 2 kmin 2vyc sin umax , which follows from Eqs. (5), we find that umax 13.456±, in good agreement with the value of 13.5± assumed by West and O’Donnell. Results for two surfaces characterized by these values of kmin and kmax were presented by West and O’Donnell: surface A, with a rms height d 10.9 nm; and surface B, with a rms height d 8.3 nm. For a one-dimensional random surface defined by the equation x3 z sx1 d, where z sx1 d is a stationary, zeromean, Gaussian random process, the rms slope is given in terms of the power spectrum by " #1/2 Z ` dk 0 2 1/2 2 k gsjkjd s ksz sx1 dd l d . (6) 2` 2p For the power spectrum given by Eq. (1) this result becomes d s p skmin 2 1 kmin kmax 1 kmax 2 d1/2 . 3

(7)

In this way we find that s has the values 0.1194 and 0.09092 for surfaces A and B, respectively. The wavelength of light used in the measurements of West and O’Donnell ranged from l 543 nm to l 633 nm. Thus the conditions dyl ,, 1 and s ,, 1 for the validity of perturbation theory7 are well satisfied for these surfaces. We can also estimate the transverse correlation length of the surface roughness. On the basis of heuristic arguments and computer simulation results it has been suggested8 that a good estimate for this length is kd l, the average distance between consecutive peaks and valleys on the surface. In turn, a good estimate of kd l is provided by the reciprocal of the density of zeros of the random function z 0 sx1 d and is given in terms of the power spectrum by 8 2 31/2 Z ` 2 6 7 6 0 dkk gskd 7 7 k dl > p 6 (8) Z ` 6 7 4 4 dkk gskd 5 s

0

skmax3 2 kmin 3 d1/2 , 5 p 3 skmax5 2 kmin 5 d1/2

(9)

where the second form is obtained by the use of the power

spectrum (1). When the values of the wave numbers kmin and kmax given above are used in this expression, we find that k dl 278.13 nm. Compared with the values of the rms height d quoted above, this value justifies the description of surfaces A and B as weakly rough. We have calculated k≠Rpy≠us lincoh , the contribution to the mean differential reflection coefficient from the diffuse component of the scattered light on the basis of two theoretical results for this contribution. The first is the result of the infinite-order perturbation calculation obtained in Ref. 1, with an error of a factor of 2p present in the result of that study9 corrected. The second is the result of the small-amplitude perturbation calculation of Maradudin and Méndez,2 in which this contribution to the mean differential reflection coefficient was calculated exactly through terms of fourth order in the surface-profile function. In Fig. 1 we have plotted the experimental data of West and O’Donnell for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light, for their surface A (d 10.9 nm) for three different angles of incidence ( Fig. 3 of Ref. 5), together with the corresponding theoretical results calculated on the basis of both the infinite-order perturbation theory of Ref. 1 and the fourth-order perturbation theory of Ref. 2. The wavelength of the incident light is l 612 nm, and the dielectric constant of gold at this wavelength is esvd 29.00 1 i1.29, a value that was obtained10 by interpolation from published values of optical constants.11 It is seen from these results that the two sets of theoretical curves agree well with each other and are in both qualitative and quantitative agreement with the experimental data. An enhanced backscattering peak is seen in the results for angles of incidence u0 0± and 10±, whereas none is seen in the results for u0 18±. Since the angle umax 13.456±, the absence of an enhanced backscattering peak for u0 18± is due to the inability of the incident light to excite surface plasmon polaritons for u0 . umax on account of the nature of the power spectrum of the surface roughness. The height of the enhanced backscattering peak in both the experimental and theoretical results is nearly a factor of 2 greater than the height of the background at its position, as is expected on the basis of our introductory remarks. The nonzero contributions to kRpy≠us lincoh at large values of jus j are associated with single-scattering processes, as discussed by West and O’Donnell.5 An important consequence of all the higher-order terms in the perturbation expansion of k≠Rpy≠us lincoh that were included in Ref. 1 but were omitted in the calculations of Ref. 2 is to replace the decay rate of a surface plasmon polariton, which in the latter calculation is the decay rate De svd that is due to Ohmic losses, i.e., due to the imaginary part of the dielectric function, by a larger decay rate that adds to De svd an additional contribution Dsp svd, which is due to the roughness-induced conversion of the surface plasmon polariton into other surface plasmon polaritons and into volume electromagnetic waves in the vacuum above the surface. The decreased mean free path of the surface plasmon polariton used in the calculations of Ref. 1 leads to a larger angular width of the enhanced backscattering peak than that obtained in the calculations of Ref. 2. This larger angular width is seen

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Fig. 1. Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light when p-polarized light of wavelength l 612 nm is incident on a one-dimensional random gold surface characterized by the power spectrum given by Eq. (1). d 10.9 nm and esvd 29.00 1 i1.29. The angles of incidence u0 are as noted. The key to the symbols is as follows: experimental data of Ref. 5 (s), results of the infinite-order perturbation theory of Ref. 1 (n), results of the fourth-order perturbation theory of Ref. 2 (—).

in the theoretical results presented in Fig. 1, although the theoretical widths are still somewhat narrower than the experimental ones. The results of the infinite-order perturbation theory match the experimental results quantitatively somewhat more closely than do those of the fourth-order perturbation theory. However, the results of the fourth-order perturbation theory follow the shapes of the experimental curves more closely than do those of the infinite-order perturbation theory. The shapes of the mean differential reflection coefficient obtained by the infinite-order perturbation theory of Ref. 1 are sharper than those obtained by the theory of Ref. 2 and sharper than the experimental curves: the cutoffs at us 613.456± are more nearly vertical in the former results than in the latter. This is due to the fact that the perturbation theory of Ref. 1 was based on the small-roughness approximation, in which the expansion of the scattering potential in powers of the surface-profile function is truncated at the linear term. Consequently, the lowest-order irreducible vertex function used in the calculation of the ladder diagram contribution to the reducible vertex function, from which the contribution to the irreducible vertex function from the maximally crossed diagrams is obtained, is proportional to the power spectrum of the surface roughness. The end result of the calculation for k≠Rpy≠us lincoh is that it

is proportional to sums of products of two power spectra with different arguments. The vertical sides of the power spectrum defined by Eq. (1) are therefore reflected in the shapes of the results for k≠Rpy≠us lincoh . In contrast, the calculation in Ref. 2, although carried out in a different fashion, is equivalent to using a scattering potential that contains higher order terms in the scattering potential than the first, and consequently a lowest-order irreducible vertex function that is not simply proportional to the power spectrum, and that smooths out the vertical sides possessed by the latter. Nevertheless, it is gratifying that despite these differences the results of the two different theoretical approaches are in good agreement with each other and with the experimental data. The small quantitative differences between the theoretical and experimental results could be due to the fact that the dielectric function of gold in the frequency range used in the experiments of West and O’Donnell is not known definitively. The results of different investigators11,12 display some differences. Although we have found that the agreement between the theoretical and experimental results can be improved somewhat by adjustment of the values of the dielectric function used in our calculations, in obtaining the results presented here we have used the same values of esvd that were used by West and O’Donnell

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in planning the fabrication of their surfaces.10 Another possible source of the differences between the theoretical and experimental results is the sensitivity of the theoretical results to the details of the statistical description of the surface-profile function.13 As noted above, the theories presented in Refs. 1 and 2 were based on the assumption that the surface-profile function is a Gaussian random process. It has been demonstrated in Ref. 13 that even if the probability density function of heights is a Gaussian but the probability density function of slopes, and especially the probability density function of curvatures, depart from a Gaussian form, the dependence of k≠Rpy≠us lincoh on us can differ qualitatively and quantitatively from the dependence obtained on the assumption that all of these probability density functions are Gaussian. In the absence of experimental results for these probability density functions we cannot judge whether the differences between the theoretical and experimental results, at least in part, have this origin. In Fig. 2 we have plotted the theoretical and experimental ( Fig. 6 of Ref. 5) results for k≠Rpy≠us lincoh for surface A (d 10.9 nm) for four different wavelengths of the incident light, viz., l 633, 612, 594, and 543 nm, and a fixed angle of incidence, u0 4±. The dielectric function of gold at these wavelengths is esvd 29.93 1 i1.053,12 29.00 1 i1.29, 28.75 1 i1.41, and 26.59 1 i1.2014, respectively, with the last three values determined by interpolation from the data in Ref. 11.10 By changing the wavelength of the incident light, we also change the wave number of the surface plasmon polariton excited by it, ksp svd, according to Eq. (2). There should therefore be angular shifts in k≠Rpy≠us lincoh created by these changes in 1ksp svd and 2ksp svd. This may

Maradudin et al.

be seen if we determine the limits on the scattering angle us for converting the surface plasmon polariton with wave number 1ksp svd into volume electromagnetic waves in the vacuum. From the first of Eqs. (4) we find that v sin us max 1ksp svd 2 kmin , c v sin us min 1ksp svd 2 kmax . c

(10)

From the second of Eqs. (4) it follows that the conversion of the surface plasmon polariton with wave number 2ksp svd into volume waves occurs for scattering angles in the range s2us max, 2us min d. If we recall that the values of kmin and kmax were determined on the basis of the assumption that us min 2us max for a value of ksp svd corresponding to a wavelength of light of l 612 nm, we see from Eqs. (10) that us max and us min will shift from their values of 613.456±, respectively, as ksp svd is changed by a change in l. In a similar fashion it follows from the first of Eqs. (3) that the limits on the angle of incidence u0 for exciting a surface plasmon polariton with wave number 1ksp svd are v sin u0 max 1ksp svd 2 kmin , c v sin u0 min 1ksp svd 2 kmax . c

(11)

From the second of Eqs. (3) we find that the excitation of the surface plasmon polariton with wave number 2ksp svd occurs for angles of incidence in the range s2u0 max , 2u0 min d.

Fig. 2. Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light when p-polarized light is incident at an angle of 4± on a one-dimensional random gold surface characterized by the power spectrum given by Eq. (1) for the four wavelengths shown. d 10.9 nm. The symbols have the same meanings as those in Fig. 1.

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Fig. 3. Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light when p-polarized light of wavelength l 612 nm is incident on a one-dimensional random gold surface characterized by the power spectrum given by Eq. (1). d 8.3 nm, esvd 29.00 1 i1.29, and u0 10±. The symbols have the same meanings as those in Fig. 1.

The shifts in k≠Rpy≠us lincoh predicted by Eqs. (10) are seen in the results plotted in Fig. 2, where the values of us min and us max are indicated by thick vertical lines and the values of 2us max and 2us min are indicated by thin vertical lines. These limiting values coincide for the case l 612 nm. Enhanced backscattering is present in the experimental and theoretical results for all the wavelengths except l 543 nm. The reason for this is that it is found from Eqs. (11) that the incident light of this wavelength excites surface plasmon polaritons of wave number 1ksp svd for angles of incidence u0 only in the interval (23.9±, 20.1±), whereas it excites surface plasmon polaritons of wave number 2ksp svd for u0 only in the interval (220.1±, 3.9±). Thus, for u0 4±, surface plasmon polaritons with wave number 1ksp svd are strongly excited, but those with wave number ksp svd are not excited. The coherent interference between time-reversed scattering processes needed for enhanced backscattering is therefore absent, which leads to the suppression of this effect. The results of the fourth-order perturbation theory of Ref. 2 presented in Fig. 2 are in quantitative agreement with those of the infinite-order perturbation theory of Ref. 1 in the immediate vicinity of the retroreflection direction but are in poorer agreement away from it. As in the results presented in Fig. 1, the results of the latter theory display much sharper discontinuities at the scattering angles 6us min and 6us max , at which it is no longer possible for the surface plasmon polaritons to be converted into volume waves in the vacuum, than do the results of the former theory. In fact, the two-step form of the results of the infinite-order perturbation theory calculations, associated with the fact that, at wavelengths different from 612 nm, us min fi 2us max , is more pronounced than it is in the results of the fourth-order perturbation calculations and in the experimental data at the wavelengths l 633 nm and 594 nm. Such differences as exist between the two theoretical results and between the theoretical and experimental results are most likely due to the factors mentioned above in the discussion of Fig. 1. In Fig. 3 we have plotted the theoretical and experimental ( Fig. 7 of Ref. 5) results for k≠Rpy≠us lincoh for West and O’Donnell’s surface B (d 8.3 nm). The wavelength of the incident light is l 612 nm, and the angle of incidence is u0 10±. Again, the agreement between the theoretical and experimental results is quantitatively and

qualitatively quite good, except in the vicinity of us > 13.5±. If we compare the results presented in this figure with the corresponding results for surface A in Fig. 1, we find that in the single-scattering regime us , 241± the ratio of the data is approximately 1.64, which is close to the ratio of the squares of the rms heights of the two surfaces, s10.9y8.3d2 1.72. Similarly, the ratio of the data in the multiple-scattering regime (jus j , 13.45±) is approximately 2.80, which is comparable with the ratio of the fourth powers of the rms heights of these surfaces, s10.9y8.3d4 2.97. These results are in agreement with the predictions in Ref. 2 that the single-scattering contribution to k≠Rpy≠us lincoh scales as d 2 , whereas the dominant contribution to enhanced backscattering scales as d 4 . It should also be noted that West and O’Donnell found no evidence of enhanced backscattering in their results for the scattering of s-polarized light from the same surfaces used in the experiments with p-polarized light. This is in agreement with the conclusions of the theoretical work presented in Refs. 1 and 2. It is well known that a planar metal –vacuum interface does not support s-polarized surface plasmon polaritons.14 Consequently, weakly rough random metal surfaces possess no light /surface plasmon polariton multiple-scattering processes that can interfere coherently with their time-reversed partners to produce enhanced backscattering. In conclusion, the theoretical results presented here, which have been obtained with no fitting parameters, support the conclusion of West and O’Donnell that their data demonstrate the enhanced backscattering of light caused by the excitation of surface plasmon polaritons on a weakly rough random metal surface that was predicted in Ref. 1.

ACKNOWLEDGMENTS The authors are grateful to K. A. O’Donnell for communicating Ref. 5 to them before its publication and for sharing with them his experimental data. The work of A. A. Maradudin was supported in part by U.S. Army Research Office grant DAAL-03-92-G-0239. The work of A. R. McGurn was supported in part by National Science Foundation grant DMR 92-13793. The work of E. R. Méndez was supported in part by the Consejo Nacional de Ciencia y Technologia through grant F96-I9205.

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REFERENCES 1. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866 – 4871 (1985). 2. A. A. Maradudin and E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335 – 3343 (1993). 3. E. R. Méndez and K. A. O’Donnell, “Observation of depolarisation and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91 – 95 (1987); K. A. O’Donnell and E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194 – 1205 (1987). 4. Zu-Han Gu, R. S. Dummer, A. A. Maradudin, and A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537 – 543 (1989). 5. C. S. West and K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390 – 397 (1995). 6. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 4; E. K. Popov, L. V. Tsonev, and M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168 – 2173 (1993).

Maradudin et al. 7. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces ( Hilger, Bristol, UK, 1991), p. 39. 8. A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485 – 501 (1990). 9. V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225 – 2239 (1985). 10. K. A. O’Donnell, The School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, November 1994 ( personal communication). 11. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 294. 12. P. W. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 – 4379 (1972). 13. M. E. Knotts, T. R. Michel, and K. A. O’Donnell, “Comparison of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928 – 941 (1993). 14. E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface polaritons — electromagnetic waves at interfaces,” in Polaritons, E. Burstein and F. de Martini, eds. ( Pergamon, London, 1974), pp. 89 – 108.