An accurate and efficient numerical simulation approach to electromagnetic wave scattering from two-dimensional, randomly rough, penetrable surfaces is presented. The use of the M\"uller equations and an impedance boundary condition for a two-dimensional rough surface yields a pair of coupled two-dimensional integral equations for the sources on the surface in terms of which the scattered field is expressed through the Franz formulas. By this approach, we calculate the full angular intensity distribution of the scattered field that is due to a finite incident beam of $p$-polarized light. We specifically check the energy conservation (unitarity) of our simulations (for the non-absorbing case). Only after a detailed numerical treatment of {\em both} diagonal and close-to-diagonal matrix elements is the unitarity condition found to be well-satisfied for the non-absorbing case (${\mathcal U}>0.995$), a result that testifies to the accuracy of our approach.
Deep Dive into The Scattering of Electromagnetic Waves from Two-Dimensional Randomly Rough Penetrable Surfaces.
An accurate and efficient numerical simulation approach to electromagnetic wave scattering from two-dimensional, randomly rough, penetrable surfaces is presented. The use of the M"uller equations and an impedance boundary condition for a two-dimensional rough surface yields a pair of coupled two-dimensional integral equations for the sources on the surface in terms of which the scattered field is expressed through the Franz formulas. By this approach, we calculate the full angular intensity distribution of the scattered field that is due to a finite incident beam of $p$-polarized light. We specifically check the energy conservation (unitarity) of our simulations (for the non-absorbing case). Only after a detailed numerical treatment of {\em both} diagonal and close-to-diagonal matrix elements is the unitarity condition found to be well-satisfied for the non-absorbing case (${\mathcal U}>0.995$), a result that testifies to the accuracy of our approach.
The scattering of electromagnetic waves from twodimensional randomly rough penetrable surfaces has been studied theoretically for more than five decades. The methods used in these studies in recent years, where attention has been directed toward multiple-scattering phenomena, have been either analytical in nature or computational. Chief among the former methods has been the small-amplitude perturbation theory [1][2][3], while several different computational methods have been used in solving the scattering problem. In the earliest calculation of this type [4] the system of coupled inhomogeneous integral equations for the tangential components of the total electric and magnetic fields on the rough surface obtained from scattering theory, was converted into a system of inhomogeneous matrix equations by the use of the method of moments [5], which was then solved by Neumann-Liouville iteration [6]. This is a formally exact approach but one that is computationally (and memory) intensive.
In subsequent work on this problem approximate solutions of the exact integral equations have been sought. In the sparse-matrix flat-surface iterative approach [7,8] the matrix elements for two close points on the surface are treated exactly, while those connecting two widely separated points are treated approximately, in an iterative solution of the matrix equations. Soriano and Saillard [9] have combined the sparsematrix flat-surface iterative approach with an impedance approximation [10] to calculate co-polarized and crosspolarized bistatic scattering coefficients of aluminum randomly rough surfaces for comparison with results obtained from perfectly conducting surfaces.
An approach that combines the fast multipole method [11] and the sparse-matrix flat-surface iterative approach has been developed by Jandhyala et. al [12]. Despite these advances, the calculation of the scattering of electromagnetic waves from two-dimensional, penetrable, randomly rough surfaces, remains a computationally intensive problem, in need of further improvements in the methods used to solve it.
In this paper we use the Franz formulas of electromagnetic scattering theory [13,14] to obtain expressions for the amplitude of the electromagnetic field scattered from a two-dimensional, rough, metallic or dielectric surface in terms of the tangential components of the total electric and magnetic fields on the surface. The independent elements of these tangential field components satisfy a system of four coupled inhomogeneous two-dimensional integral equations -the Müller equations [15,16] -derived from Franz formulas. This system of four integral equations is reduced to a system of two integral equations by the use of an impedance boundary condition at a two-dimensional rough metallic surface [17], and its solution is used to calculate the mean differential reflection coefficient.
The approach to the scattering of an electromagnetic field from a rough metallic or dielectric surface outlined here is similar to the approach of Soriano and Saillard [5] in its use of an impedance boundary condition to reduce the number of coupled integral equations that have to be solved from four to two. However, there are still important differences between these two approaches. The first is that we do not use the sparse-matrix flat-surface iterative approach: the matrix elements connecting any two points on the surface are calculated accurately. Moreover, those connecting two nearby points are calculated more accurately than in the work of Soriano and Saillard. The second is that we calculate the full angular intensity distribution of the scattered field, which allows us to check the unitarity (energy conservation) of the scat- tered field (for the non-absorbing case). This enabled the identification of critical aspects of the numerical implementation that, if not handled properly, could lead to erroneous results and/or a significant drop in accuracy. This important point seems to have been overlooked in previous publications. The third is that although the occurrence of hyper-singular kernels is avoided in both approaches by the use of the Müller equations [15,16], some differences are found between our resulting matrix elements and those of Soriano and Saillard that appear to affect the unitarity of the scattered field [24]. Moreover, contrary to what was reported in Ref. [15], we find that matrix element terms containing the Green’s function of the metal also have to be taken into account for some offdiagonal elements in order to produce accurate results. The fourth is that we do not use the beam decomposition method [19] for the incident beam in which a wide beam is represented by a weighted sum of shifted narrow beams. Instead we use a single wide incident beam.
The physical system that we consider consists of vacuum [ε 0 = 1] in the region x 3 > ζ(x ) [where x = (x 1 , x 2 , 0)] and a non-magnetic metal in the region x 3 < ζ(x ) that is characterized by a complex,
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