[Paper Review] Numerical simulations of scattering of light from two-dimensional surfaces using the Reduced Rayleigh Equation

Tor Nordam, Paul Anton Letnes+1 more|arXiv (Cornell University)|Apr 23, 2012

Optical Polarization and Ellipsometry11 references4 citations

TL;DR

This paper presents a non-perturbative, purely numerical method based on the Reduced Rayleigh Equation (RRE) to simulate light scattering from two-dimensional penetrable rough surfaces with arbitrary polarization and dielectric properties. The approach achieves high accuracy, with energy conservation verified to within 0.03% for non-absorbing metallic surfaces, enabling reliable simulation of full angular and polarized scattering distributions for isotropic and anisotropic Gaussian and cylindrical power spectra surfaces.

ABSTRACT

A formalism is introduced for the non-perturbative, purely numerical, solution of the reduced Rayleigh equation for the scattering of light from two-dimensional penetrable rough surfaces. As an example, we apply this formalism to study the scattering of p- or s-polarized light from two- dimensional dielectric or metallic randomly rough surfaces by calculating the full angular distribution of the co- and cross-polarized intensity of the scattered light. In particular, we present calculations of the mean differential reflection coefficient for glass and silver surfaces characterized by (isotropic or anisotropic) Gaussian and cylindrical power spectra. The proposed method is found, within the validity of the Rayleigh hypothesis, to give reliable results. For a non-absorbing 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.

Motivation & Objective

• To develop a non-perturbative, purely numerical method for solving the Reduced Rayleigh Equation (RRE) for light scattering from two-dimensional penetrable rough surfaces.
• To enable accurate simulation of the full angular and polarized distribution of scattered light, including co- and cross-polarized intensities.
• To validate the method through energy conservation checks and systematic variation of surface roughness parameters.
• To demonstrate feasibility of running such simulations on standard desktop hardware with GPU acceleration, reducing reliance on costly supercomputing resources.
• To extend the applicability of the RRE formalism to complex, randomly rough dielectric and metallic surfaces with isotropic and anisotropic power spectra.

Proposed method

• The Reduced Rayleigh Equation (RRE) is solved numerically for reflection, with the unknown being the scattering amplitude, using a boundary integral formulation.
• The surface is discretized into a grid of Nq elements, and the system matrix is assembled from Green’s function integrals over the surface elements.
• The resulting linear system is solved via LU decomposition, with performance optimized using the MAGMA library for GPU-accelerated computation.
• The mean differential reflection coefficient is computed from the scattering amplitudes to obtain the full angular distribution of scattered intensity with polarization resolution.
• The method is applied to dielectric (glass) and metallic (silver) surfaces with isotropic and anisotropic Gaussian and cylindrical power spectra.
• Energy conservation is used as a key validation metric, computed by integrating the total scattered power over all angles.

Experimental results

Research questions

• RQ1 Can the Reduced Rayleigh Equation be reliably solved in a purely numerical, non-perturbative manner for two-dimensional randomly rough surfaces?
• RQ2 Does the proposed numerical method preserve energy conservation to a high degree of accuracy for non-absorbing metallic surfaces?
• RQ3 How does the angular distribution of scattered light, including polarization, depend on surface roughness and power spectrum anisotropy?
• RQ4 To what extent can GPU-accelerated LU decomposition reduce computational cost and enable simulations on standard desktop hardware?
• RQ5 What is the upper limit of surface roughness (δ/a) for which the RRE formalism remains valid and energy-conserving?

Key findings

• The method achieves energy conservation to within 0.03% or better for a non-absorbing metallic surface, confirming high numerical accuracy and adequate discretization.
• The mean differential reflection coefficient exhibits a well-defined retro-reflection peak, consistent with the enhanced backscattering phenomenon, for weakly rough metallic surfaces.
• The angular distributions of scattered intensity exhibit symmetry properties previously observed in other simulation methods for strongly rough surfaces.
• The simulation remains energy-conserving for surface roughness ratios δ/a ≤ 0.12, indicating a practical upper limit for the validity of the RRE under the assumed parameters.
• GPU-accelerated LU decomposition shows comparable performance to traditional supercomputing nodes for moderate system sizes, suggesting feasibility for desktop-based simulations.
• The method is applicable to a wide range of systems, including clean and multilayered structures, with potential for use in designing novel optical materials.