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1.2 Dielectric Rough Surfaces
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with (2.1.98) Thus
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ht N h(O, 0) = [~ (1 -
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+ (1 + Rho) COSOt]
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cos(cPt - cPi) (2.1.99a)
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Vt Nh(O, 0) = [- ~ (1- Rho) COSOi cosOt - (1
. sin(cPt - cPi)
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(2.1.99b)
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ht Nv(O, 0)
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[~ (1 + R vo ) + (1 -
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R vo ) COSOi COSOt] sin(cPt - cPi) (2.1.99c)
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cos(cPt - cPi) (2.1.99d)
Vt Nv(O,O) =
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[~ (1 + R vo ) cosOt + (1- R vo ) COSOi ]
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COS Oli
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The coherent component only exists in the specular transmission direction.
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(k t , ki) = 47r---
cos Oi sm Ii x r5(Ot - Oli)r5(cPt - cPi)r5 ab
11 + R bo 2 9a exp [ -(k 1 COS Oli . 0
k cos 0d h
2 2J
where 9a = Snell's law
7]1/7]
for a
= V,
= 7]/7]l
for a
= h,
(2.1.100) is related to Oi by
(2.1.101)
Geometrical Optics Solution
Under the geometrical optics limit as k - t 00, the asymptotic solution to the Kirchhoff-approximated diffraction integrals can be derived using the method of stationary phase. The coherent component of the scattered fields will vanish in this limit, and only the incoherent component will remain. The bistatic scattering coefficients for the reflected and transmitted fields are proportional to the probability of the occurrence of the slopes which will specularly reflect or transmit the incident wave into the observation direction. The bistatic scattering coefficients satisfy reciprocity but violate energy conservation. This is due to the neglect of the effects of multiple scattering and shadowing . The scattering coefficients are next modified to incorporate the shadowing effects. The sum of reflected and transmitted intensities are then shown to be always less than the incident intensity since only single scattering solutions are used. From (2.1.54) the exponential phase factor is 1/J = kd .;p' = kdxx' + kdyY' + kdzf(x', y') (2.1.102)
2 KIRCHHOFF AND RELATED METHODS
To determine the stationary phase point, we differentiate (2.1.102) with respect to x' and set it equal to zero. f)'ljJ ax' = 0 = kdx + kdzao (2.1.103) At the stationary phase point we have
kdx ao = -kdz Similarly, by differentiating the phase term 'ljJ with respect to
(.l __
(2.1.104)
we get
kdy (2.1.105) kdz The slopes a o and f30 are such that the incident and scattered wave directions form a specular reflection. This can be seen from the fact that from (2.1.52) we have
fJo -
(2.1.106)
Replacing the surface slopes a and f3 by a o and f3o, we obtain, using (2.1.54),
(2.1.107)
where
(II*) =
/1 drJ..l df~eik'LL.(rl--rl,Jeikdz(f(r_d-f(r~)))
\ A" A"
(2.1.108)
The above integral can be solved by the asymptotic method. For large k, contributions of the integral come from regions where (x', y') is close to (x,y). Expanding f(x',y') about (x,y),
f(x', y') = f(x, y)
+ a(x' - x) + f3(y' - y) +...
u = k(x - x') v = k(y - y')
(2.1.109)
and replacing the integration variables by
(2.1.110a) (2.1.110b)
we obtain
(II*) = (:2Ao
where
JJ dudv exp [iU(qx + aqz) +iv(qy + f3qz) + 0 (l)])
(2.1.111) (2.1.112)
1.2 Dielectric Rough Surfaces
Ignoring the O(1jk) and higher order terms, we have
* 47f A o (II ) = j;2(o(qx
Therefore
(kl~~ II*) = 47f A
It follows that
+ aqz)o(qy + (3qz))
(2.1.113)
00 -00
dad{3 o(qx+ a qz)o(qy+{3qz) p(a, (3) (2.1.114)
where p(a, (3) is the probability density function for the slopes at the surface.
o d 47f p k d X . 1m I ( 1 11*\ _ - -A (- - -k Y ) k->oo kJz kdz ' k dz
(2.1.115)
For the Gaussian random rough surface
p(a, (3) = 27fh2
[a + (32 ] IC"(0)1 exp - 2h IC"(0)1
(2.1.116)
where C"(O) is the double derivative of the correlation function at p = O. Thus, h 2 IC"(0)1 is the mean square surface slope 8 2 , and for the Gaussian correlation function with correlation length l we have
2 = h2IC"(0)1 =
(2.1.117)
Substituting (2.1.116) into (2.1.115) gives
II* 27fA o e [_ kJx + kJv ] ( ) - kJzh2IC"(0)1 xp 2kJz h2 IC"(0)1
For an incident field with polarization polarization as is given by
(2.1.118)
b, i
the scattered intensity for
(lEs(r)1
where
~:~~: las' Fb(ao,(3o)1 2 (I 1*)
(2.1.119)
Fb(ao, (3o) = F(ao, (3o) IC;=b;
and using (2.1.55a), we find that
(2.1.120)
la s Fb(a o,{3o)1
with
Ikd l4
= k2lkiXksl4kJzfba
(2.1.121)
(2.1.122a) (2.1.122b)
2 KIRCHHOFF AND RELATED METHODS
(2.1.122c) (2.1.122d)
and R h are evaluated at
kdx/kdz X + kdy/kdzY
+Z Jk'jx/k'jz + k'jy/k'jz + 1
(2.1.123)
Then, the bistatic scattering coefficients for the reflected intensities are, in view of (2.1.88)
rab(k s , ki )
1 [ = cosBilki x ksl4kdz 2h2IC"(O)1 exp -
Ikdl 4 -
2 + k dy ] 2k~zh2IC"(O)1
k dx
(2.1.124) We note that from (2.1.122) the geometrical optics solution does not depend on whether the total or scattered field is used in the diffraction integral because F( 0', /3) is evaluated at the stationary phase point. The backscattering cross sections are defined to be
aab(ki ) = cos Bi/~b( -ki , ki )