'" :E c.

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Surface position (wavelength)

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Figure 5.4.2 Comparison of t,he surface currents by matrix inversion and tenth-order MOMI for TE case at (Ji = 30 0 with h = 0.4..\ and l = 0.2..\, 80 points/..\ discretization and near-field integration are used in the simulation. L = 25.6..\.

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PBTG Combined with the Multilevel Fast Multipole Method

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Physics-Based Two-Grid Method Combined with the Multilevel Fast Multipole Method

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For scattering by lossy dielectric rough surfaces with large permittivities, we have introduced the PBTG method in Section 2. The PBTG method was further combined with BMIA/CAG in Section 2. In this section, the PBTG is used in conjunction with the steepest descent multilevel fast multipole method (SDFMM) to solve wave scattering from one-dimensional random lossy dielectric rough surfaces. The proposed algorithm has the computational complexities of O(Ndg ) for near-field interactions and O(Neg ) for nonnear field interactions, where N dg and Neg are the number of sampling points on the dense and coarse grid, respectively. Using the proposed algorithm, wave scattering from Gaussian and non-Gaussian rough surfaces is investigated and illustrated. Special emphasis is put on checking the accuracy of the algorithm and energy conservation. We illustrate the single grid method and the PBTG in Section 5.l. We discuss the computational complexity of the combined algorithm of the PBTG with the SDFMM in Section 5.2. In Section 5.3, we apply the algorithm to Gaussian rough surfaces and discuss the accuracy of the algorithm. In Section 5.4, we apply the method to study scattering by the modified power-law spectrum.

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5.1 Single Grid and PBTG

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Consider a tapered plane wave 1f;ineCr) incident on a random dielectric rough surface defined by z = f (x). The surface fields satisfy the dual surface integral equations. let G(r, r') and G1 (r, r') be the 2-dimensional Green's functions of free space and the medium, respectively. Let p be equal to f-Ld f-L and E1 / E for TE and TM polarization, respectively. Using the method of moments (MoM), the integral equations are cast into the matrix equations:

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L aiju(Xj) + L bij 1f;(xj) = 1f;ine(Xi)

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(5.5.1)

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L a~~) pU(Xj) + L b~~)1f;(xj) = 0

j=l j=l

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(5.5.2)

where u(x) =

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Jl + [f'(X)]2~~. Expressions aij, bij , a~~), and bg)

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are as in

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(5.2.6)-(5.2.9).

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5 FAST METHODS FOR ROUGH SURFACE SCATTERING

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The quantity N is the number of sampling points on the surfaces. The matrix elements aij, bij , ag-), and bg-) are determined by the Green's functions. We let Roman numeral subscripts i, j denote indexing with the dense with the coarse grid. Assume that the upper medium is free grid and '2, space and the lower medium is lossy with a relative complex permittivity E1. We can define a distance limit T)., as determined by the complex permittivity of the lower medium. Outside this limit the field interaction between the ith and the jth point is vanishingly small, and the lower medium Green's function can be set equal to zero. Therefore we can approximate

aij -

_(1) _

{a(l) 0 2)

Tij::; Tz > Tij _ TZ T"2) > TZ _

(5.5.3) (5.5.4)

b(l)

~ [;(1) = {bg-) Tij::; TZ

where Tij is the distance between the ith point and the jth point on the dense grid. Thus ag-) and bg-) are banded matrices and Equation (5.5.2) becomes

(5.5.5) Lag)pu(Xj) + L[;g)1/;(Xj) = 0 j=l j=l Based on the observation that the upper medium Green's function is slowly varying on the dense grid, we decompose the upper medium Green's function into near field and non-near field interactions

N N N

LaijU(Xj) = Lafjuj j=l j=l

+ Lai/uj

(5.5.6)

L bij1/;(Xj) = L bfj1/;j + L bi/1/;j j=l j=l j=l where afj' bfj' ail, and bil are determined by as, = {a ij 2) 0 bS , = {bij 2) 0 ns aij = Tij::; Tf Tij > Tf Tij::; Tf Tij > Tf Tij::;Tf Tij > Tf Tij::;Tf Tij > Tf

(5.5.7)

(5.5.8) (5.5.9) (5.5.10) (5.5.11)