I-D RANDOM ROUGH SURFACE SCATTERING in .NET

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9 I-D RANDOM ROUGH SURFACE SCATTERING
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Thus the incoherent power flow exactly cancels the second term of (9.3.26), giving the relation cos Oi (5 s . z) = (5s . z)coh + (5s . z)incoh = ~
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which exactly obeys energy conservation. Thus if we define incoherent wave
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'Ps = 1/;s - (1/;s)
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From (9.3.30), we define the power flow per unit area of the incoherent wave as (9.3.34) Casting in terms of angular integration, we let
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kx =ksinOs kz =kcosOs
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dOs cos 2 OsI(k x = k sin Os)
(9.3.37)
Thus if we divide (9.3.37) by the incident power per unit area of (9.3.22), we can define the incoherent bistatic scattering coefficients a(Os) as
a (Os) =
. kcos 2 0s 0 I (k x = k sm Os) cos i
(9.3.38)
Note that (9.3.38) is defined in such a manner that the integration of a(Os) over Os will combine with the reflected power of the coherent wave to give an answer that obeys energy conservation. For first-order scattering, 'P~l)(kx) = 1/;~l)(kx), so that from (9.3.33) and (9.3.30) we obtain (9.3.39)
3.2 Neumann Problem for l-D Surface
and (9.3.38) assumes the form
0'( Os) = 4k 3 cos 2 Os cos Oi W( k sin Os - k sin Oi)
(9.3.40)
The results for Gaussian and exponential correlation functions can be readily obtained by using the spectral densities as listed in (9.2.14) and (9.2.15). The backscattering coefficient is, for Os = -Oi,
0'( -Oi)
4k 3 cos3 Oi W( -2k sin Oi)
3.2 Neumann Problem for One-Dimensional Surface
Let the incident wave be
nl,.
'f/tnc _ eikiXX-iki.Z -
(9.3.41)
where kix = k sin Oi, k iz = k cos Oi. The scattered wave is written as a perturbation series:
'ljJs = 1/J~O)
+ 1/J~1) + 'IjJ~2) + ...
(9.3.42)
The wave function is 1/J = 1/Jinc + 1/Js. The boundary condition at z = f(x) is
8n (1/Jinc + 1/Js) = 0
Since the surface normal is
(9.3.43)
(9.3.44)
the boundary condition at z
= f(x) becomes
_ 8f (81/Jinc 8x 8x
+ 81/J s ) + (81/Jinc + 81/J s ) = 0
8x 8z 8z
(9345) . .
We let the scattered field be represented by
1/Js(r) =
dkxeikxx+ik,z1/Js(kx)
(9.3.46)
To calculate 1/Js(kx ), we match boundary condition of (9.3.45):
- 1:
_ ~~ (ikiXeikiXX-iki.!(X)) _ ikizeikiXX-iki.!(X) dkxikxeikxx+ik,!(x)1/Js(kx)
dkxikzeikxx+ik,!(x)1/Js(kx)
(9.3.47)
9 l-D RANDOM ROUGH SURFACE SCATTERING
In a spectral domain, we let
'l/Js(k x ) =
) 'l/JiO) (k x ) + 'l/Ji1)(k x ) + 'l/Ji 2 (k x ) + ...
(9.3.48)
Also we expand exp(ikzJ(x)) in a power series
eik.f(x) = 1 + ikzJ(x) -
~k;f2 + ...
'k ize iki",x
(9.3.49)
Putting (9.3.48) and (9.3.49) into (9.3.47) gives
df 'l - dx ['k ix eiki",x
-h1:
'k - Z iz f
2" kiz f2)]
2 - 'l'k iz f - 2" k iz f2)
dkxikxeikxx
(1 +
ikzf(x) -
~k;f2)
. ('l/JiO) (k x ) + 'l/Ji1)(k x ) + 'l/Ji 2 (k x ) + ...) )
dkxikzeik",x
(1 +
ikzJ(x) -
~k;f2)
(9.3.50)
. ('l/J~O)(kx)
+ 'l/Ji 1)(k x ) + 'l/J~2)(kx) + ...) = 0
-ikizeiki"'X
Balance to zeroth-order gives
dkxikzeik",x 'l/J~O) (k x ) = 0
Thus
'l/JiO)(k x ) = 8(kx - kix )
Balancing (9.3.50) to first order gives
(9.3.51)
2 _ df ik' eiki"'x _ k tZ feiki"'x _ d df d X tX X
dkxikzeik"'Xikzf(x)'l/JiO) (k x ) +
dk xikx eik"'Xnl,(O)(kx ) 'f/ s dkxikzeik",x'l/Jil) (k x ) = 0 (9.3.52)
Putting (9.3.51) into (9.3.52) gives
-2 ~f ikixeiki"'X - 2k;zeiki "'x f(x) x Let
dkxikzeikxx'l/Jil) (k x ) = 0
(9.3.53)
(9.3.54) (9.3.55)
3.2 Neumann Problem for l-D Surface
Substituting (9.3.54) and (9.3.55) in (9.3.53) gives
00 / -00
dk x2k xk.1X F(k X)ei(kx+kiX)x
2 dk X2k1Z F(k X)ei(kx+kiX)x
(9.3.56)
dkxikzeikxx'l/J~l)(kx)
In the first two integrals, we next change dummy variable of kx + kix This gives
dk x [2(k x - kix)kix - 2klz] F(k x - kiX)eikxx +
dkxikzeikxx'l/J~l) (k x ) =
(9.3.57)
Taking the inverse Fourier transfer of (9.3.57) gives
'l/J~l)(kx)
[(k x - kix)k ix - klz] F(k x - kix )
(9.3.58)
z Next we balance (9.3.50) to second order:
2i 2 = k [kxk ix - k ]F(k x - kix )
(9.3.59) Putting (9.3.51) and (9.3.58) into (9.3.59) gives, on canceling some of the terms,
Putting (9.3.58) in (9.3.60) then gives
- i: i: -:i: i: i:
_ df k. k. eikiXXf(x) dx 1X 1Z
+ ~k3 f2( X )eikiXX 2 1Z
dkxikxeikxx
['l/J~l)(kx) +ikzf(x)'l/J~O)(kx)]
dkxikzeikxx
['l/J~2)(kx) +ikzf(x)'l/J~l)(kx) - ~k;f2(x)'l/J~O)(kx)]
dkxikxeikxx'l/J~l)(kx)
- f(x)
dkxk;eikxx'l/J~l)(kx)
(9.3.60)
dkxikzeikxx'l/J~2) (k x ) =
dkxikzeikxx'l/Ji2) (k x ) dk'x [-k'xX k 2 + kz ]F(k'x )ei(kx+k~)x k 2i
dk x
[k X1X - k 2]F(k x - k ) k 1X
(9.3.61)
9 l-D RANDOM ROUGH SURFACE SCATTERING
On the right hand side of (9.3.61), let k~ k~ ~ k x . We get
+ kx
= k~
and then kx ~ k~,
Taking the inverse Fourier transfer of (9.3.62) gives
1/Ji2)(kx) =
f: f: f: :z f:
dkxeikxx
dkxeikxxikz1/Ji2) (k x )
dk~ [-k~kx + k 2]F(kx - k~) ~~ [k~kiX dk~ :'z (k 2 - k~kx)(k~kix -
2 k ] F(k~ - kix )
(9.3.62)
2 k )F(kx -
k~)F(k~ -
kix )
(9.3.63)
Coherent Wave Reflection
Taking the average of (9.3.58) gives
(1/J~l)(kx)) = 0
(9.3.64)
Taking the average of (9.3.63) and using (9.2.10) gives 00 2 2 (1/Ji 2)(kx )) = - k dk~ (k - :'~kiX)2 8(kx - kix)W(k~ - kix ) (9.3.65)
From (9.3.47), (9.3.65), and (9.3.51) we have
(1/Js("r)) =
dkxeikxx+ikzz
[(1/J~O)(kx)) + (1/J~2)(kx))]
= eikiXX+ikizZ [1 _
~ k.
Joo dk'x (k
k~kix)2 W(k'x z
k )] (9366) ~x ..
The coherent reflected wave is in the specular direction and is less than that of the flat surface for the case of Neumann boundary condition.