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Substitute these into (1.2.2) and (1.2.3). Let fmin and fmax be, respectively, the minimum and maximum values of the surface profile f(r~). Evaluating (1.2.2b) for z < fmin and (1.2.3a) for z > fmax, we obtain the equations of
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e-ik",-or'.L eik.f(r'.L)
[e( -kz)e( -k z ) + h( -kz)h( -k z )] . a(rjJ (1.2.7a)
+ [-h(-kz)e(-k z ) + e(-kz)h(-kz )] . b(r~JJ}
0= - 1
k_ dk et'-k - et'k 1 > Z 1
",- 1 ",-
k 1z
-, e- t'-k",- r",- e- tOk 1>'f(-') r",-
x {:l [el(klZ)el(klZ) +hl(klz)hl(klZ)]
+ [-hl(klZ)el(kIz)+el(klZ)hl(klZ)] .b(rjJ }
The above equations are the extended boundary conditions and can be used to solve for the surface fields along with the following equations, which are results of (1.2.4a) and (1.2.4b) ,
n(rjJ .a(rjJ = 0 n(T'1-) . b(rjJ = 0
(1.2.8a) (1.2.8b)
from (1.2.8a) and (1.2.8b) we have
a z (r 1- -
_' ) _ (A x
af(r'.L) + Aaf(r'.L)) _ (_' )
ax' Yay' . a1- r 11- r 1-
(1.2.10) (1.2.11)
b (-' ) z r 1-
= (A af (r'.L) + Aaf (r'.L)) . b (-' )
ax' Yay'
with a z and bz as the z-components of a and b, respectively. Once the surface fields are obtained, the scattered field in region 0 and the transmitted field in medium 1 can be derived by using (1.2.2a) and (1.2.3b). The scattered fields and transmitted fields are calculated by evaluating (1.2.2a) and (1.2.3b) for z > fmax and z < fmin, respectively. For
2 Scattering by Dielectric Rough Surface
z > f(r'.L)
E (r) = - _1_
dk ik.L.r.L eikzz ~
dr' e-ik.L 'r~ e-ikzf(r~)
{[e(kz)e(k z ) + h(kz)h(kz )] . a(r'.L)
+ [-h(kz)e(k z ) + e(kz)h(k z )] . b(r'.L) }
We note that the expression for the scattered field is similar to that of the extinction theorem. The property can be exploited later on in simplifying the analysis. Similarly for transmitted fields, we have
E (r) = _1_
eik.L r.L e-ik tz Z ~ k 1z
dr' e-ik.L r~ eiktz f (r~)
[e 1(-k1z)fh (-k 1z ) + h1(-k1z)h 1(-k 1z )] . a(r'.L)
+ [-h 1 ( -k1z)e1 (-k 1z ) + e1 (-k1z)h 1(-k 1z )] . b(r'.L) }
The objective is to solve for the surface fields using (1.2.7) and (1.2.8) and then to solve for the scattered and transmitted fields using (1.2.13). Equations (1.2.7) and (1.2.10) through (1.2.13) are exact. To solve for the surface fields, the perturbation method makes use of series expansions. Let
= 2:: a(m) (r'.L)
b(r'.L) =
2:: b(m) (r'.L)
where superscript (m) denotes mth order solution. We also have
(1.2.15a) (1.2.15b)
In SPM, f(1"..d and its derivatives are regarded as small parameters. Thus
not only the rms heights are small but also the slopes have to be small. The expansions of (1.2.14) and (1.2.15) are substituted into (1.2.7) to obtain the set of equations for the different-order solutions. From (1.2.10), (1.2.11) and (1.2.14) we obtain
a~O) (1"1J = b~O) (r'.d = 0
(m)(-' ) _
r..1 -
8f(1"1J 8x'
+' 8f(1"1J) y 8y' +'
. a..1
r ..1
(1.2.17a) (1.2.17b)
b(m)(-, ) = ( ' 8f(1"1J z r ..1 x 8x'
The assumptions are
8 f (1"1J ) . -b(m-l)(_, ) 8y' ..1 r ..1
_' (_' 8f 8f kzf (r ..i)' k1zf r ..i), 8x" 8y'
Note that the components of the surface fields are of a lower order than the horizontal components because of the small slope approximation. Substituting (1.2.14) and (1.2.15) into (1.2.7) and (1.2.10) through (1.2.11) and equating the same-order terms, we can calculate the surface fields to the zerothorder, the first-order, and so on. Then, from (1.2.13), the scattered fields can be obtained to different orders. In the following, we solve for the surface fields and scattered fields up to the second order. The zeroth-order solutions are just the reflected and transmitted fields of a flat surface. The first-order solution gives the lowest-order incoherent scattered intensities. However, the first-order solution does not give the depolarization effect in the backscattering direction. The second-order solution gives the lowest-order correction to the coherent reflection and transmission coefficients. Also, the depolarization of the backscattered power is manifested. As shown in previous section, the second order solution is needed to ensure energy conservation. This is particularly important for the calculation of emissivity.