4.1 Introduction

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dielectric. When this Green's function convolves with the surface fields on the dense grid, it will be just the product of a sparse matrix and a column vector. The second observation allows us, when using the free-space Green's function to convolve with the surface fields of dense grid, to first average the values of surface unknowns on the dense grid and then place them on the coarse grid. In 5, the PBTG method was implemented for 1-D surface (2-D scattering problem). In this section, we (i) extend the PBTG to. 2-D rough surface (3-D scattering problem), (ii) combine the PBTG method with the sparse matrix canonical grid method (SMCG) for improving CPU and memory requirements, and (iii) study bistatic scattering coefficients and emissivity for wave scattering from 2-D dielectric rough surface with high permittivity. The wave interaction in the rough surface is divided into: (1) very near field, (2) near field, and (3) non-near field. (1) Very near field is of distance of separation less than half a wavelength. (2) Near field separation is between half a wavelength and rd wavelengths. (3) Non-near field is beyond r d wavelengths. For very near field interactions, we use the usual product of sparse matrix and column vector. For near-field and non-near field interactions, the free space Green's function is slowly varying on the dense grid. We first average the fields on the dense grid to get fields on the coarse grid. For the non-near field interactions, we further expand free space Green's function on a canonical grid of a horizontal surface so that the fast Fourier Transform (FFT) can be applied. In the lower medium, the non-near field interactions are neglected because of lossy properties of the lower medium. The approach is denoted as PBTG/SMCG. The computational complexity and the memory requirements for the algorithm are 0 (Nscg log( N scg )) and O(Nscg ), respectively, where N scg is the number of grid points on the coarse grid. The second-order small perturbation method (SPM) will be studied in Volume III. Also, SPM agrees with the small slope approximation in emissivity calculation for half-space case [Irisov, 1997]. Monte Carlo simulations of emissivities are compared with those of the second-order SPM. In Section 4.2, the formulation of the problem of wave impinging upon a 2-D dielectric surface (3-D scattering problem) is described, and the surface integral equations are converted into a matrix equation using a single grid discretization. In Section 4.3, we describe the physics-based two-grid algorithm and combine it with the sparse matrix canonical grid method. The mathematical expressions of the bistatic scattering coefficients and the emissivity are given. In Sections 4.4 and 4.5, numerical results are illustrated.

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63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES

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Formulation and Single Grid Implementation

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Consider an electromagnetic wave, Ei(r) and Hi(r), impinging upon a 2-D dielectric rough surface with a random height profile z = f(x, y). It is tapered so that the illuminated rough surface can be confined to the surface area Lx x L y . The direction of incident wave is ki = sin ()i cos cPd:; + sin ()i sin cPill cos ()iZ. The incident fields for TE wave incidence are given as

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where

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e( -k z ) = :

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(xk y - fJk x ) (xkx + fJk y) +

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(6.4.3)

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h( -k z ) =

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(6.4.4)

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The spectrum of the incident wave, E(k x , k y ), is given as

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E(k x , k y ) = -2

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4'iT . exp [i (kixx

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1]00 dx ]00 dy exp( -ikxx - ikyY) -00 -00

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+ kiyY) (1 + w)] exp( -t)

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(6.4.7)

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where t = t x + t y = (x 2 + y 2)/g2 and

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tx = ty

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~--=-----:"~--=----:"'_---'--'-:::":

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(cos ()i cos cPiX + cos ()i sin cPiy)2 g2 cos2 (); (- sin cPiX

(6.4.8) (6.4.9)

= -'---------'-----g---,2,...------'----'-----

+ cos cPiY

4.3 Physics-Based Two-Grid Method

1 w - _ ( 2t x - 1 g2 cos2 ()i

- kr

2t - 1) + -----"-y---;:-_

(6.4.10)

The six component surface integral equations are given by (6.3.3)-(6.3.6). As in Section 3.1, we also take the x and y components of the tangential field equations and the normal component of the field equation. The method of moments (MoM) is used to discretize the integral equation. The resulting matrix equations are

Pl n '"' [Zmn 1(1) 6