Zeroth- and First-Order Solutions in Java

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1.1 Zeroth- and First-Order Solutions
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k z kxky kz . 1 h kz . q (-I ) = =F kk k ( ' - kyk 1 ) = k sm ( cPk - cPk ) k..L x p ' p
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(1.1.22g) (1.1.22h)
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(1.1.22i) (1.1.22))
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h( kz) . p(k~) = =F k:zk l
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(kxk~ + kyk~) = =F ~ COS(cPk - cP~)
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1.1 Zeroth- and First-Order Solutions
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Zeroth-Order Solution
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Balancing (1.1.19) and (1.1.20) to the zeroth order, we obtain A~O)Oc..L) = 0 (1.1.23)
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[e( -kz)e( -k z ) + h( -kz)h( -k z )] . A~) (k..L)
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(1.1.24)
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Solution of (1.1.24) can easily be calculated in the (1.1.21) and (1.1.22). Let
((J,P, z) system defined
q(ki..L) Pi = p(ki..L)
be the
eli =
(1.1.25) (1.1.26)
q and p for the incident direction.
-(0) (0)
Then the solution of (1.1.24) is
A..L (k..L) = (qi aq + Piap )8(k..L - ki..L)
(1.1.27) (1.1.28a) (1.1.28b)
where
a~O)
= 2e( -kiz ) . ei k~z
ap = 2h( -k iz ) . ei
The Dirac delta function in (1.1.27) indicates that the zeroth-order surface field consists of only a single spectral component that corresponds to specular reflection. Substituting (1.1.27) in (1.1.18) gives the zeroth-order solution of the scattered field as
E~O)(r)
{-e(kiZ)(e i ' e(-kiZ )) + h(kiZ)(e i ' h(-kiZ ))} /ki"-:T"-+ikiZz
(1.1.29)
which is the response of a flat surface.
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
First-Order Solution
Balancing (1.1.19) to the first order gives 0= { e( -kz)e( -kz) + h( -kz)h( -kz)}
. [A (1) (7el-)
+ ikz dk~ A (O\k~)F(kl- - k~)]
(1.1.30)
Note that (1.1.30) has only two independent components since it states that the projection of the vector in the square bracket onto the two polarization directions are equal to zero. The third component can be obtained by balancing (1.1.20) to the first order that gives
A~l)(kl-) =
Jdk~F(kl-
k~)(kl- - k~). A~)(k~)
(1.1.31)
From (1.1.27) and (1.1.31), we have
A~l)(kl-) = iF(kl- - kil-)(kl- - kd)' ((lia~O) + Pia1 )
= iF(kl- - k d ){ -kpsin(<pk -
<Pi)a~O)
(1.1.32)
+ [k pCOS(<pk To solve (1.1.30), let
<Pi) - kip] a1 ) }
A (1) (kl-)
= A~l) (kl-)q(kl-) + A11) (kl-)p(kl-) + A~l) (kl-)z
(1.1.33)
Note that in here we use q(kl-) and p(kl-) as basis vectors. Previously [Shin, 1984; Tsang et al. 1985], the representation of A (1) (kl-) was made differently using qi and Pi as basis vectors. The present set of basis vectors simplifies subsequent calculations. We find it more convenient to use q and Pas defined by the scattered directions. Substituting (1.1.33) and (1.1.27) into (1.1.30) gives two equations in which the dot product of e( -kz ) with the squarebracketed terms in (1.1.30) gives zero and the dot product of h( -k z ) with the square-bracketed terms in (1.1.30) also gives zero. From (1.1.33) and (1.1.30) the dot product of e( -k z ) with the squarebracketed terms gives
A~l)(kl-)
-ikze( -k z) . [a~O)qi
+ a10)Pi] F(kl- -
kil-)
(1.1.34)
The dot product of h( -kz ) with the square-bracketed terms in (1.1.30) gives
~ A11) (kl-) =
~ A~l)(kl-) -
ikzh( -k z)
[a~O)qi + a10)Pi] F(kl- -
kd ) (1.1.35)
101 Zeroth- and First-Order Solutions
Thus the two components A~1) and A11) depend on the projection of the polarizations e( -k z ) and h( -k z ) on the incident polarizations as projected on the x-y plane. Substituting (1.1.35) into (1.1.34) and using (1.1.22) gives 1 the explicit expressions for A~1) (/e l-) and A1 \kl-) :
A~1) (kl-) = iF(kl- - ka)A~1) (kl-)
(1.1.36a) (1.1.36b) (1.1.36c) (1.1.36d)
A11)(kl-)
where
iF(kl- - ka)A~1)(kl-)
A~1) (kl-)
-kz [a~O) cos(<Pk - <Pi)
+ a1 ) sin( <Pk - <pd]
k + k~zipa~O)
A11)(kl-) = k k
a~O) sm(<pk - <Pi) - a1 ) COS(<pk - <Pi) ]
For TE excitation, a~O)
= 2kiz /k, a~O) = 0
(1.1.36e)
(1.1.361)
- (1) 2k zkiz A q (- = - - k - cos (<Pk - <Pi ) kl-) - (1) A p (kl-)
2kkiz . ----r;- sm(<pk -
<Pi )
For TM excitation, a~O)
= 0, a1 ) = 2 we have
(1. 1. 36g)
A~1)(kl-) = -2k z sin(<pk - <Pi) -(1) 2 2 A p (kl-) = k {kpk ip - k COS(<pk - <Pi)}
(1.1.36h)
Substituting into (1.1.18) gives the first-order scattered field as
E~1) (r)
dkl- i k1- 01'1- +ik,z ~ [e(k z )e(kz ) + h(kz)h(k z )]
. [A~1)(kl-)q(kl-)
+ A11)(kl-)p(kl-) + zA11) (kl-)
- ikzF(kl- - k a ) ( qia~O)
1 = _ _ 2
+ Pia~O)) ]
- t k dkl- etO-k1- 0101- +"k ,z_
. { e(kz ) [A~1) (kl-) - ikzF(kl- - ka)e(k z ) .
(qia~O) + Pia1 ))]
+ h(k z ) [- :z A11)(kl-) + ~ A11)(kl-)
- ikzF(kl- - ka)h(k z ) .
(qia~O) + Pia1 ))] }
(1.1.37a)