kxm = kix

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(4.3.10) (4.3.11)

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be the wave vector components of the mth Floquet mode. The integral equation becomes

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e-ikx",x+iknn!(x) k u(x)

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(4.3.12)

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where u(x) is as given by (4.3.6). To solve (4.3.12), a Fourier series expansion of w(x) can be made

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4 RANDOM ROUGH SURFACE SIMULATIONS

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to denote the coupling between the mth and nth Floquet modes, then the matrix equation is

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In actual numerical implementation, the number of modes are truncated by keeping all the propagating modes Ikx171l < k and a reasonable number of evanescent modes. Suppose we keep modes from - N to M so that we have a total of N + M + 1 modes, we then have the standard matrix equation of dimension M + N + 1.

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(4.3.16)

m = -N, -N + 1, ... ,-1,0, 1, ... ,M. Suppose P = 25.6 wavelengths. Then 21f 21f 21f P (25.6)>. (25.6)>' Thus

To illustrate, let k ix =

k x171 = k ix

+-25.6

(normal incidence). Then

k _ mk

x171 -

The propagating modes are m = -25, -24, ... , -1,0,1, ... ,25, a total of 51 propagating modes. Suppose we put 5 evanescent modes on both sides giving us 61 modes. Thus we have Pn, n = -30, -29, -28, -27, ... , -1,0,1, ... ,30. The larger the value of P, the smaller is the mode spacing, and the more modes we have. For example, for P = 25.6, we have 60/25.6 = 2.3 modes per wavelength. This method of mode expansion has the advantage over the method of subsectional basis function, which needs 10 points (i.e., 10 subsectional basis functions) per wavelength. The method of subsectional basis function needs 256 unknowns for the case P = 25.6>'. On the other hand, the periodic boundary condition method uses 51 modes. However, the disadvantage of the mode expansion method is that the diagonal elements of the matrix equation may not be dominant. As can be seen from (4.3.14), A 171n depends on f (x). If f (x) is small, then A mn has large diagonal elements. For the special case when f(x) = 0, we have A mn = ~6mn and all off-diagonal elements are zero. However, for large values of f(x), the diagonal elements

3.1 Periodic Boundary Condition

may not be large compared with the off-diagonal elements. Also for k zm imaginary, depending on how large is f(x), the integrand in (4.3.14) can be exponentially large or exponentially small. Thus some off-diagonal elements can be exponentially large or exponential small, giving an ill-conditioned matrix. Nevertheless, the periodic boundary condition method is quite useful in the small height limit. After the 'Wn's are solved, we can calculate the scattered field as follows:

1/Js(x', z')

where

2::=

m=-(XJ

bmeikxmx'+ikzmz'

(4.3.17)

bm =

2::=

n=-(XJ

B mn W n

(4.3.18)

(4.3.19) The incident power per unit area is 1 Sine' Z = - - cosBi 217 The power per unit area outflowing from the surface is

(4.3.20)

(-Ss . z/ = Re 2Wf-L'l/J s EN: ) 8z

(4.3.21)

If we integrate over a period, then

~ jf dx (Ss' z) = Re ( P

_~ 2

m==-oo

zm Ibm l2 2Wf-L ) k

2::=

m. propagating

modes only

Ibm 12kzm 2k17

(4.3.22) We only have propagating modes in (4.3.22) because k zm of evanescent modes are imaginary. Next we need to convert the result to bistatic scattering coefficients as a function of scattered angle. Let

k xm = ksine s (4.3.23) For each discrete m, there corresponds a scattered angle Bs . Thus we have discrete scattered angles. The spacing of k xm is 27r I P, so that in the limit of large P the discrete angles almost form a continuum. To convert the summation to integration, we note that since the kxm's are spaced 27r I P apart we have

(4.3.24)

4 RANDOM ROUGH SURFACE SIMULATIONS

Thus

= -P

L 21T

f}.k xm = -P

21T -k

dk xm = -P

_2!:

dOs k cos Os

(4.3.25)

propagf.lt.

prop.<tgi'lt.

From (4.3.22) and (4.3.25) and using k zm

A -dx (Ss . z) = P I

21T 271

= k cos Os we obtain

dOs kcos 2 Oslbml 2

(4.3.26)

_2!:

Since the incident power per unit area is ficient is

CO;:i, the bistatic scattering coef-

Pk 2 2 0 cos Oslbml (4.3.27) 21T cos i In 5, we shall compare the numerical results of using periodic boundary condition with that of using a large surface length L of many wavelengths.

a-(Os) =

3.2 MFIE for TE Case of PEC

For scattering by PEe, we have m,ed EFIE as given by (4.1.11) for TE case and MFIE as given by (4.1.151) for TM case. For EFIE of the TE case, the self patch is of order O(f}.x lnf}.x) as given by (4.1.28) and the non-self patch is of order O( f}.x). On the other hand, for MFIE of the TM case, the self patch is equal to 1/2 as represented by (4.1.156). Non-self patch is of order O(f}.x). Thus MFIE gives larger diagonal elements for the impedance matrix and that gives a better condition number. In this section, the MFIE for the TE case is derived which results in larger diagonal elements than the EFIE for TE case. We use (4.1.10) and take normal derivative with respect to r' and let r' approach the surface iL' . \7'~inc(r')-

~ ds[iL' . \7' g(r, r')][iL \7~(r)] = { ~' . \7'~(r')

(4.3.28)

The self patch analysis can be performed as in Section 1.5 for the Neumann case. The integral equation becomes, for rand r' on surface S, 0,' . \J'7jJinc(r')