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_{B}= ± tan

^{-1}√

*E*, where

*E*is the dielectric constant of the dielectric medium and θ

_{B}is the Brewster angle. Nevertheless, we show in this paper that if the mean differential reflection coefficient is measured only in the angular range |θ

_{s}| < θ

_{B}, these data can be inverted to yield accurate results for the normalized surface height correlation function for weakly rough surfaces. Several parameterized forms of this correlation function, and the minimization of a cost function with respect to the parameters defining these representations, are used in the inversion scheme. This approach also yields the rms height of the surface roughness, and the dielectric constant of the scattering medium if it is not known in advance. The input data used in this minimization procedure consist of computer simulation results for surfaces defined by exponential and Gaussian surface height correlation functions, without and with the addition of multiplicative noise. The proposed inversion scheme is computationally efficient.

*s*polarization incident on a free-standing or supported film, both of whose surfaces are one-dimensional rough surfaces.

*p*- or

*s*-polarized light incident on it at an arbitrary angle of incidence

*θ*0 from one of them into the other at an arbitrary but specified angle of transmission

*θ*t that is not defined in terms of

*θ*0 by Snell's law. We call such transmission nonstandard refraction.

_{1}+Δk)θ(k

_{1}+Δk-Q)+θ(Q-k

_{2}+Δk-Q)θ (k

_{2}+Δk-Q)+θ(-Q-k

_{1}+Δk)θ (k

_{1}+Δk+Q)+θ(-Q-k

_{2}+Δk)θ (k

_{2}+Δk+Q)], where θ(z) is the Heaviside unit step function, k

_{1}= k

_{R}-k

_{0},k

_{2}=k

_{R}-k

_{0, k(subscript R}is the real part of the wavenumber of the surface plasmon polariton of frequency ω supported by the planar vacuum-metal interface, and k

_{0}is related to the angle of incidence measured clockwise from the x

_{3}-axis by k

_{0}=(ω/c)sinθ

_{0}. An explanation is provided for why a surface defined by this power spectrum produces enhanced backscattering at only the angle of incidence given by θ

_{s}=-θ

_{0}, and it is confirmed by numerical calculations of the angular dependence of the intensity of the light scattered diffusely from it.

_{1}in the region x

_{3}> H, a second dielectric medium characterized by a dielectric constant ε

_{2}in the region ζ(x

_{1}) < x < H, and vacuum in the region x < ζ(x

_{1}). The surface profile function ζ(x

_{1}) is assumed to be a single-valued function of x

_{1}that is differentiable and constitutes a random process. The structure is illuminated from the region x

_{3}> H by s-polarized light whose plane of incidence is the x

_{1}x

_{3}-plane. By the use of geometrical optics limit of phase perturbation theory we show how to design the surface profile function ζ(x

_{1}) in such a way that the mean differential transmission coefficient has a prescribed form within a specific range of the angles of transmission, and vanishes outside this range. In particular, we consider the case that the incident s-polarized light in incident normally on this structure, and the mean intensity of the transmitted light is constant within a specific range of the angle of transmission, and vanishes outside it. Numerical simulation calculations show that the transmitted intensity indeed has this property.

_{0}in the region x

_{3}$GTR D; a metal film characterized by a complex, frequency-dependent dielectric function(epsilon)

_{1}((omega) ) in the region 0 < x

_{3}< D; a dielectric film characterized by a dielectric constant (epsilon)

_{2}in the region (zetz) (x

_{1}) < x- 3) < 0; and vacuum ((epsilon)

_{3}equals 1) in the region x

_{3}< (zetz) (x

_{1}). The light whose plane of incidence is the x

_{1}x

_{3}- plane, in incident through the prism. For the surface profile function (zetz) (x

_{1}) we take the form (zetz) (x

_{1}) equals -d(theta) (x

_{1})(theta) (L-x(

_{1}), where (theta) (x

_{1}) is the Heaviside unit step function. Thus we have a dielectric film thickness d and dielectric constant (epsilon)

_{2}covering the half of the lower surface (x

_{3}equals 0) of the metal film defined by x

_{1}$GTR0, or a dielectric film of thickness d and dielectric constant (epsilon)

_{2}covering the part of the lower surface (x

_{3}equals 0) of the metal film defined by 0 < x

_{1}< L. The reduced Rayleigh equation for the amplitude of the light scattered back into the prism, R(qk), is obtained, and solved by the Wiener-Hopf method, and the result is used to calculate the intensity of the scattered field in the far field region as a function of x

_{1}for a fixed value of x

_{3}for several values of the wavelength of the incident light. The results provide information about the scattering of the surface plasmon polariton at the metal-vacuum interface, excited by the incident light, by an index step on that interface. A brief discussion of the transmission of light through this system is also given.

_{3}$GTR (zetz) (x

_{1}), and a dielectric medium characterized by a complex dielectric constant (epsilon) in the region x

_{3}< (zetz) (x

_{1}). The surface profile function (zetz) (x

_{1}) is assumed to be a single-valued function of x

_{1}, that is differentiable as many times as is necessary, and to constitute a zero-mean stationary, Gaussian random process. It has been recently been shown that a local relation can be written between L(x

_{1}(omega) ) equalsV [deltaH

_{2}$GTR(x

_{1},x

_{3}(omega) )/(delta) x

_{3}]

_{x3}equals0) and H(x

_{1}(omega) ) equalsV [H

_{2}$GTR(x

_{1},x

_{3}(omega) )]

_{x3equals0}, where H

_{2}$GTR(x

_{1},x

_{3}(omega) ) is the single nonzero component of the total magnetic field in the vacuum region, in the case of a p-polarized electromagnetic field whose plane of incidents is the x

_{1}x

_{3}-plane. This relation has the form L(x

_{1}(omega) ) equals I(x

_{1}(omega) )H(I(x

_{1}(omega) ), where the surface impedance I(I(x

_{1}(omega) ) depends on the surface profile function (zetz) (x

_{1}) and on the dielectric constant (epsilon) of the dielectric medium. A completely analogous relation exists when L(x

_{1}(omega) ) equalsV [(delta) E

_{2}$GTR(x

_{1},x

_{3}(omega) )/(delta) x

_{3}]

_{x3equals0}) and H(x

_{1}(omega) ) EQV [E

_{2}(x

_{1},x

_{3}(omega) )]

_{x3equals0}, where E

_{2}$GTR(x

_{1},x

_{3}(omega) ) is the single nonzero component of the electric field in the vacuum region, in the case of an s-polarized electromagnetic field whose plane of incidence is the x

_{1}x

_{3}-plane. Our goal in this work is to obtain the relation between the values of L(x

_{1}(omega) ) and H(x

_{1}(omega) ) averaged over the ensemble of realizations of the surface profile function (zetz) (x

_{1}). This we do by the use of projection operators and Green's second integral identity in the plane.

_{3}$GTR d

_{1}+(zetz)

_{1}(x

_{1}), a dielectric film in the region, d

_{2}+(zetz)

_{1}(x

_{1}), and a metal in the region x

_{3}< d

_{2}+ (zetz) (x

_{1}). This system is illuminated from the vacuum side by p-polarized light whose wavelength is allowed to vary from 0.2micrometers to 1.2micrometers . The film is assumed to have a dielectric function that is insensitive to the wavelength of the incident light. In obtaining the numerical results reported here the metal substrate is taken to be silver. The dielectric function of silver for a given wavelength is obtained by interpolation from experimental values. The surface profile functions,(zetz)

_{1,2}(x

_{1}), are assumed to be either zero or single-valued functions of x

_{1}that are differentiable as many times as is necessary, and to constitute zero-mean, stationary Gaussian random processes. Their surface height auto-correlation function is characterized by a Gaussian power spectrum. We study and discuss the wavelength dependence of R((lambda) ) and U((lambda) ) for several scattering systems obtained by turning on and off the surface profile functions (zetz)

_{1,2}(x

_{1}) and/or the correlation between these two surface profile functions.

_{3}$GTR(zetz) (x

_{1}); a dielectric film characterized by a real,positive, dielectric constant (epsilon) in the region -D < x

_{3}< (zetz) (x

_{1}); and a vacuum in the region x

_{3}<-D. The surface profile function (zetz) (x

_{1}) is assumed to be a single-valued funtion of x

_{1}, that is differentiable, and constitutes a random process. This structure is illuminated from the region x

_{3}$GTR (zetz) (x

_{1}) by s-polarized light whose plane of incidence is the x

_{1}x

_{3}-plane. By the use of the geometrical optics limit of phase perturbation theory we show how to design the surface profile function (zetz) (x

_{1}) in such a way that the mean differential transmission coefficient has a prescribed form within a specified range of the angle of transmission, and vanishes outside this range. In particular, we consider the case in which the transmitted intensity is constant within a specified range of the angle of transmission, and vanishes outsides it. Rigorous numerical simulation calculations show that the transmitted intensity indeed has this property.

^{(1}) and C

^{(10}) to the angular intensity correlation function for the scattering of s-polarized light from a one-dimensional random interface between two dielectric media. The calculations are carried out on the basis of a new approach that separates out explicitly the contributions C

^{(1}) and C

^{(10}) to the angular intensity correlation function. The contribution C

^{(1}) displays peaks associated with the memory effect and the reciprocal memory effect. In the case of a dielectric-dielectric interface, which does not support surface electromagnetic surface waves, these peaks arise from the coherent interference of multiply-scattered lateral waves supported by the interface. The contribution C

^{(10}) is a structureless function of its arguments.

^{(1}) plus C

^{(10}) plus C

^{(1.5}) where C

^{(1}), C

^{(10}) and C

^{(1.5}) are terms arising from three distinct classes of diagrams. The contribution C

^{(1}), which contains the memory and time-reversed memory effect terms, has been studied before and is proportional to (delta) (vector q minus vector k minus vector q' plus vector k'). C

^{(10}) is a new term (of the same order of magnitude as C

^{(1})) in the scattering of light from volume disorder which is found to be proportional to (delta) (vector q minus vector k plus vector q' minus vector k'). C

^{(1.5}) is a new term in the scattering of light from volume disorder which exhibits an unrestricted dependence on vector q, vector k, vector q', vector k' and a series of interesting intensity peaks related to the resonant scattering of light by the volume disorder. The contributions C

^{(10}) and C

^{(1.5}) have been considered in the study of the speckle correlation function for the scattering of light from rough surfaces, but this is the first consideration of these terms in the scattering of light from volume disorder.

^{(1}) plus C

^{(10}) plus C

^{(1.5}), where C

^{(1}), C

^{(10}) and C

^{(1.5}) are terms arising from three distinct scattering processes. The contribution C

^{(1}), which contains the memory and time-reversed memory effect terms, has been studied before, and is proportional to (delta) (vector q minus vector k minus vector q' plus vector k'). C

^{(10}) is a new term (of the same order of magnitude as C

^{(1})) in the scattering of light from volume disorder, that is found to be proportional to (delta) (vector q minus vector k plus vector q' minus vector k'). C

^{(1.5}) is a new term in the scattering of light from volume disorder that exhibits an unrestricted dependence on vector q, vector k, vector q', vector k' and a series of interesting intensity peaks related to the resonant scattering of light by the volume disorder. The contributions C

^{(10}) and C

^{(1.5}) were considered in the earlier study of the speckle correlation function for the scattering of light from randomly rough surfaces, but have only recently been considered in the scattering of light from volume disorder.

^{(1}); (2) an additional short range term of comparable magnitude C

^{(10}); (3) a long range term C

^{(2}); (4) an infinite range term C

^{(3}); (5) and a terms C

^{(1.5}) that along with C

^{(2}) displays peaks associated with the excitation of guided or surface waves.

_{min}less than k less than k

_{max}that includes the wave numbers q

_{1}((omega) ) and q

_{2}((omega) ) of the surface plasmon polaritons supported by the film at the frequency (omega) of the incident light. The existence of two surface electromagnetic waves leads to the appearance of two satellite peaks in the angular dependence of the intensity of the incoherent component of the light scattered from the film at scattering angles (theta)

_{s}given by sin (theta)

_{s}equals - sin (theta)

_{i}plus or minus (c/(omega) )[q

_{1}((omega) ) - q

_{2}((Omega) )], where (theta)

_{i}is the angle of incidence of the light, in addition to the enhanced backscattering peak in the retroreflection direction (theta)

_{s}equals -(theta)

_{i}. At the same time satellite peaks occur in the angular dependence of the intensity of the light transmitted incoherently through the film at angles of transmission (theta)

_{t}given by sin(theta)

_{t}equals - sin (theta)

_{i}plus or minus (c/(omega) )[q

_{1}((omega) ) - q

_{2}((omega) )], in addition to the enhanced transmission peak in the antispecular direction (theta)

_{t}equals -(theta)

_{i}. These results are compared with those for a metal film whose rough surface is characterized by a Gaussian power spectrum yielding the same rms height and rms slope as the West-O'Donnell power spectrum.

_{3}equals (zetz) (x), with x equals (x

_{1}, X

_{2}), where (zetz) (x) is a single-valued function of x that constitutes a zero- mean, stationary, isotropic, Gaussian random process defined by the properties <(zetz) (x)> equals 0, <(zetz) (x)(zetz) (x

^{1})> equals (sigma)

^{2}W(x - x

^{1}), and (sigma)

^{2}equals <(zetz)

^{2}(x)>. The angle brackets here denote an average over the ensemble of realizations of the surface profile function (zetz) (x). The results are used to compute the probability density P

_{1}(x)[P

_{2}(x)] that the nearest maximum (minimum) to a given maximum (minimum) is at a distance x from the latter; and the probability density P

_{3}(x) that the nearest minimum to a given maximum is at a distance x. Results are presented for random surfaces defined by surface height autocorrelation functions W(x) equals exp(-x

^{2}/a

^{2}), a

^{2}/(x

^{2}+ a

^{2}), and 2[(k

_{2}

^{2}- k

_{1}

^{2})x

^{2}]

^{-1}[k

_{2}xJ

_{1}(k

_{2}x) - k

_{1}xJ

_{1}(k

_{1}x)], where J

_{1}(z) is a Bessel function. Results are also presented for a novel type of one-dimensional random surface used in recent experimental studies of enhanced backscattering.

_{(alpha}). The photonic band structure of the average periodic system has been calculated, and the edges of the band gaps have been found by a careful study of the convergence of the calculation. Maxwell's equations are integrated numerically in space and time to yield the electromagnetic field radiated by a line source emitting light at frequency (omega) whose current amplitude has a Gaussian shape in time. By looking in the disordered system at the electromagnetic energy stored between consecutive pairs of cylinders of radii m(Delta) R and (m + 1)(Delta) R, where m equals 0,1,2..., centered on the line source we show that the system displays strong localization of light. The localization length is found to be much smaller for frequencies in the gaps of the photonic band structure of the average periodic medium than for frequencies outside the gaps.

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