2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM in Java

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1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
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In view of (1.1.34), (1.1.35), and (1.1.22a)-(1.1.22d) and the fact that h(k z ) qi = -h( -k z ) . qi and h(k z ) . Pi = -h( -k z ) . Pi, the terms inside the two
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E~l)(r) =
dk-Lik-Lor-L+ikzz :z
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[e(kz)A~l)(k-L) +h(kz )(- ~A~l)(k-L))]
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(1. 1. 37b)
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. iF(k-L - ki-L)
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The incident power per unit area is
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Z = _ COS O i 27]
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(1.1.38)
The power per unit area associated with the first-order fields (which is also that of the incoherent wave) is,
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(5, . z = ~Re\ E(l) x H(l)*). z / s A s A) 2 s
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Using (1.1.37b), we obtain
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(1.1.39a)
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H~l)(r) =~J dk~ik~or-L+ik~z:~ [h(k~)A~l)(k~)+e(k~)~A~l)(k~)]
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Also (1.1.40) and we have 1 (5s . z) = 27]
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(1.1.39b)
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:z W(k-L - ku) [IA~1)(k-L)12+ IA~l)(k-L) k; n
kx = ky = kz = dk x dk y = k sin Os cos 1s k sin Os sin 1s k cos Os k 2 sin Os cos OsdOsd1s
(1.1.41)
where P stands for propagating waves with k p :::; k. This is because the real part is taken to get power in (1.1.39), and only the propagating waves of the spectrum in (1.1.41) have nonzero real part and contribute to power. Casting (1.1.41) in terms of directions in angular variables (Os, 1s), we have
(1.1.42a) (1.1.42b) (1.1.42c) (1.1.42d)
We can write (1.1.41) as
(Ss' z) = lSi'
.1 zl 1
n /2
dOsSlllOs
"((k s , ki ) d1s--'---'-----'---'-47f
(1.1.43)
1.2 Second-Order Solutions
where (1.1.44) To get 'Yhh and 'Yvh, we let incident wave be TE with ei = e( -k iz ) so that a~O) = 2kiz /k, a1 ) = O. Then 'Yhh corresponds to the IA~l) 12 in (1.1.44) while 1 2 'Yvh corresponds to the IA1 ) k /kl in (1.1.44). Here h stands for horizontal z polarization (TE) and v stands for vertical polarization (TM).
'Yhh 'Yvh
= 167fk W(ki- - kii-) cos (<Ps - <Pi) COSOi cos Os
(1.1.45a) (1.1.45b)
= 167fk W(ki- - kd) sm (<Ps - <Pi) COSOi
To get 'Yvv and 'Yhv, we let ei = h( -kiz ) so that a~O) = 0 and a1 ) = 2. Then 'Yhv corresponds to the IA~1)12 in (1.1.44) while 'Yvv corresponds to the - (1) . ) kz/k 12 m ( 1.1.44. Thus 1A p
'Yhv
'Yvv
167fk 4 2 2 --0- W(ki- - kd) cos Os sm (<Ps - <Pi) (1.1.45c) cos i 167fk 4 . . 2 = --0- W(ki- - kd)[sm Os smOi - cos(<Ps - <Pi)] (1.1.45d) cos i
In the backscattering direction (Os and
= Oi and <Ps = 7f+<Pi) one has ki- = -kd
(1.1.46)
so that
O"vv O"vh O"hv O"hh
4 2 2 = 167fk W( -2k d )(1 + sin Oi)
(1.1.47a) (1.1.47b) (1.1.47c)
= 0 = 167fk W( -2kd) cos Oi
(1.1.47d)
It is noteworthy that in the backscattering direction, there is no depolarization for a linearly polarized incident wave. Also, 0"vv is larger than O"hh. If Oi is close to grazing, so that Oi - t 90 0 , then O"vv is much larger than O"hh.
Second-Order Solutions
For the second-order solution, let
A(2\ki-) = A~2)(ki-)q(ki-)
+ A12)(ki-)p(ki-) + A~2)(ki-)z
(1.1.48)
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
Balancing (1.1.19) to the second order, we get two equations (one for e( -k z ) component, the other for h( -k z ) component):
A~2) (k-L) + ikz J dk~ [( e( -kz ) . q(k~)) A~l)(k~)
+ (e( -kz ) . P(k~)) A~l) (k~)] P(k-L - k~)
- [k~ (e(-k z )' qi)a~O)+~; (e(-k z )' Pi)a~O)] P(2)(k-L -kd) = 0
(1.1.49a)
~ A~2) (k-L) + ~ A~2) (k-L) + ikz J dk~ [ (h( -kz ) . q(k~)) A~l) (k~)
+ (h(-k z )' p(k~)) A~l)(k~) + ~ A~l)(k~)]P(k-L - k~)
- [~; (h(-k z )' qi)a~O) + ~; (h(-k z )' Pi)a~O)] P(2)(k-L The third equation for the three components is from (1.1.20)
ki-L) = 0 (1.1.49b)
A~2)(k-L)
=i J
dk~P(k-L - k~)(k-L - k~)
+ P(k~)A~l)(k~)}
(1.1.50)
. {q(k~)A~l)(k~)
We can solve the three equations (1.1.49a) , (1.1.49b) , and (1.1.50) for the three unknowns A~2), A~2), and A~2). However, it is easier to use (1.1.18) and make use of the similarity of the integrand of (1.1.18) to (1.1.49a) and (1.1.49b). From (1.1.18), the second-order scattered field is
E~2) = -~ dk~ik' 'i+ik :. {e(k.1( A\2)(k~)
- ikz J dk~ [e(k z ) . q(k~)A~l)(k~) + e(k z ) p(k~)A~l)(k~)] P(k-L -k~) - [k~ (e(k z ). qi)a~O) + ~; (e(k z ). Pi)a~O)] P(2)(k-L + k d ))