Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation

Abstract

A formalism is introduced for the non-perturbative, purely numerical, solution of the reduced Rayleigh equation (RRE) for the scattering of light from two-dimensional penetrable rough surfaces. Implementation and performance issues of the method, and various consistency checks of it, are presented and discussed. The proposed method is found, within the validity of the Rayleigh hypothesis, to give reliable results. For a nonabsorbing metal surface the conservation of energy was explicitly checked, and found to be satisfied to within 0.03%, or better, for the parameters assumed. This testifies to the accuracy of the approach and a satisfactory discretization. As an illustration, we calculate the full angular distribution of the mean differential reflection coefficient for the scattering of p-or s-polarized light incident on two-dimensional dielectric or metallic randomly rough surfaces defined by (isotropic or anisotropic) Gaussian and cylindrical power spectra. Simulation results obtained by the proposed method agree well with experimentally measured scattering data taken from similar well-characterized, rough metal samples, or to results obtained by other numerical methods.

Numerical studies of the transmission of light through a two-dimensional randomly rough interface

“…The function I(γ|q ) was evaluated by expanding the integrand in Eq. (17) in powers of ζ(x ) and calculating the Fourier transform of ζ n (x ) by the fast Fourier transform [2]. For these expansions we used the first N = 18 terms.…”

“…In the theoretical and experimental studies of the interaction of an electromagnetic wave with a two-dimensional randomly rough dielectric surface, the great majority have been devoted to the reflection problem [1][2][3], and less attention has been paid to studies of the transmission of light through such surfaces. Greffet [4] obtained a reduced Rayleigh equation for the transmission amplitudes in the case where light incident from vacuum is transmitted through a two-dimensional randomly rough interface into a dielectric medium, and obtained a recursion relation for the successive terms in the expansions of the amplitudes in powers of the surface profile function.…”

Numerical solutions of the Rayleigh equations for the scattering of light from a two-dimensional randomly rough perfectly conducting surface

“…Equations (17)- (19) were solved by the method described in detail in [25]. A realization of the surface profile function was generated on a grid of N x × N x points within a square region of the x 1 x 2 plane of edge L. In evaluating the q ∥ integral in Eqs.…”

“…A realization of the surface profile function was generated on a grid of N x × N x points within a square region of the x 1 x 2 plane of edge L. In evaluating the q ∥ integral in Eqs. (17), the infinite limits of integration were replaced by finite limits jq ∥ j < Q∕2, and the integration was carried out by a two-dimensional version of the extended midpoint rule [26, p. 161] applied to a grid in the q 1 q 2 plane that is determined by the Nyquist sampling theorem [26, p. 605] and the properties of the discrete Fourier transform [25]. The function IγjQ ∥ was evaluated by expanding the integrand in Eq.…”

Dispersion of polarization coupling, localized and collective plasmon modes in a metallic photonic crystal mapped by Mueller Matrix Ellipsometry

“…As such, the Rigorous Coupled Wave Analysis (RCWA) [36] has been demonstrated to be suitable to model the specular Mueller matrix for various semiconductor cylinders [37], and recently fully rigorous boundary element or surface integral methods have been developed [38,39]. Moreover, the moderate aspect ratio of the hemispheroidal particles considered in this study may allow for using (approximate) numerical methods, for instance, the reduced Rayleigh equation technique [27,32,40]. This latter technique also has the additional advantage of being well suited for studying the optical effects of sample imperfections caused by e.g.…”