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-I) - -I] } (1.2.67)

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AF)(k.l) = J

dk~F(k.l - k~)i(k.l - k~) . A~)(k~)

dk~ F(k.l - k~)F(k~ -

ku )

. [(k.l - k~) . (Aq(k~)q(k~)

+ Ap(k~)])(k~))] (1.2.68)

Substituting the zeroth- and the first-order solutions in (1.2.67) and using (1.2.68), we have

E~2)(r) =

J dk.lik.L.1-.L+ikzz :z

. {e(k,){ k; (kj, - k,) [(e(-k,) . qi HOI + (c(-k,) . Pi) a~OI] F(21(k~

+ (k1z - k z ) J

-I -I

- kn )

dk~ [(h( -k z ) . q(k~)) Bq(k~) + (h( -kz ) . p(k~)) Bp(k~)]

. F(k.l - ku)F(k.l - k.l)

k~z (k 1z -

k z ) [(h(-k z )' qi)b~O) + (h(-k z ). Pi)

2 , { k p kz(kr - k ) + h(k z ) -k (krkz + k2k1z)

00 1 -I -00

b~O)]FC2)(k.l

- k u )}

dk.l(k.l - k.l) F(k.l - k.l)

- -I

. F(k~ - ki.l) (Aq(k~)q(k~) + Ap(k~)p(k~))

k~(k2 - kr) J - I -I - krkz + k 2k dk.lF(k.l-k.l) 1z

. [(h( -k z ) . q(k~)) Aq(k~)

+ (h( -k z ) . p(k~)) Ap(k~)] F(k~ -

ku )

2.2 Second-Order Solutions

E~2)(r) =

dk.lik~.r~+ikzz

. { e(k z ) [f~;) (k.l, ki.l) (e( -kiz ) . ei)

+ f~i) (k.l, ki.l) (h( -kiz ) . ei)]

( h( -k A

A A) (2) + h(kz ) [(2) (k.l' -: (A -kiz ) . ei + f hh (k.l' -ki.l) fhe ki.l) e(

A iz ) . ei )]}

(1.2.70) Then comparing (1.2.69) and (1.2.70), we obtain,

(1.2.71a)

1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM

(2) fhe (k.i,kilJ = -

~ kz

(q(k~)A~(k~) + p(k~)A;(k~) )

i: i:

k (kr - k 2 ) k z (krkz + k2k 1z )

dk~F(k.i - k~)F(k~ -

ku)(k.i -

dk~F(k.i - k~)F(k~ -

h( -kz) . q(k~)) A~(k~) + (h( -kz ) . p(k~)) A;(k~)]

- k;k r F(2) (k.i - ki.i)

(e( -kz ) . Pi) (1 + Rho)

2 k kl z ' , k iz - -2- F (2)(- - -ki.i) (h( -k z ) qi ) T(1- Rho) k.i + kzkl z

Jdk~F(k.i

k~)F(k~ 2

ki.i)

. [(e( -k z ) . q(k~)) B~(k~)

+ (e( -kz ) P(k~)) B;(k~)]}

(1.2071c)

(2) k --:::. hh ( .i,kt.t)

__ ~

{ - ; kz

(q(k~)A~(k~) + p(k~)A;(k~))

i: i:

(kr - k ) k z (krkz + k2k 1z )

dk~F(k.i - k~)F(k~ -

ki.i)(k.i -

dk~ F(k.i - k~)F(k~ -

ki.i)

. [(h( -kz ) . q(k~) )A~(k~) + (h( -kz ) . p(k~) )A;(k~)]

kzkr , - -2- F (2) (k.i - -ki.i) (, -kz ) qi ) e(

k iZ -T )

(1 - R vo )

k k 1z ' - -2- F (2) (k.i - -ku) (h( -k z ) . Pi ) (1 + R vo )

+ kzk 1z

Jdk~F(k.i

k~)F(k~ -

ku )

.[(e( -kz ) . q(k~)) B~(k~)+ (e( -kz) p(k~)) B;(k~)] } (1.2071d)

Note that in the above there are wavevectors, k.i, k.i' and ki.io The wavevec-/

2.2 Second-Order Solutions

tor kl.. denotes scattered direction. The wavevector kil.. denotes incident direction. Since this is the second-order scattering, there is an intermediate direction k~, of which an integration is required in (1.2.71). Substituting (1.2.48) into (1.2.71a) and also using k1z - kz = (kr - k2)j(k 1z + kz), we obtain