170
7 Quadrupole Contributions from Interface and Bulk
χ
(2)
eff,G,SSP ((, ω 1 , ω 2 ) = L G,y (() L I,y (ω 1 ) L I,z (ω 2 ) sin θ α
I (ω 2 ) χ
(2)
q G,yyz ((, ω 1 , ω 2 ),
(7.61)
χ
(2)
eff,G,SPS ((, ω 1 , ω 2 ) = L G,y (() L I,z (ω 1 ) L I,y (ω 2 ) sin θ α
I (ω 1 ) χ
(2)
q G,yzy ((, ω 1 , ω 2 ),
(7.62)
χ
(2)
eff,G,PSS ((, ω 1 , ω 2 ) = L G,z (() L I,y (ω 1 ) L I,y (ω 2 ) sin θ i
G (() χ
(2)
q G,zyy ((, ω 1 , ω 2 ),
(7.63)
χ
(2)
eff,G,PPP ((, ω 1 , ω 2 ) =
− L G,x (() L I,x (ω 1 ) L I,z (ω 2 ) cos θ i
G (() cos θ α
I (ω 1 ) sin θ α
I (ω 2 ) χ
(2)
q G,xxz ((, ω 1 , ω 2 )
− L G,x (() L I,z (ω 1 ) L I,x (ω 2 ) cos θ i
G (() sin θ α
I (ω 1 ) cos θ α
I (ω 2 ) χ
(2)
q G,xzx ((, ω 1 , ω 2 )
+ L G,z (() L I,x (ω 1 ) L I,x (ω 2 ) sin θ i
G (() cos θ α
I (ω 1 ) cos θ α
I (ω 2 ) χ
(2)
q G,zxx ((, ω 1 , ω 2 )
+ L G,z (() L I,z (ω 1 ) L I,z (ω 2 ) sin θ i
G (() sin θ α
I (ω 1 ) sin θ α
I (ω 2 ) χ
(2)
q G,zzz ((, ω 1 , ω 2 ).
(7.64)
Equations (7.61), (7.62), (7.63), (7.64) are straightforward extension of Eqs. (7.15),
(7.16), (7.17), (7.18) by substituting χ ID with χ
(2)
qG ,
χ
(2)
q G,pqr ((, ω 1 , ω 2 ) =
χ
ID
pqr ((, ω 1 , ω 2 ) + χ
IQ
pqr ((, ω 1 , ω 2 ) + χ
IQB
pqr ((, ω 1 , ω 2 ) + χ
B
G,pqr ((, ω 1 , ω 2 ).
(7.50)
The four constituent terms of Eq. (7.50) are given by
χ
ID
pqr ((, ω 1 , ω 2 ) =
∞
z b
dz χ
D0
pqr (z, ,, ω 1 , ω 2 )f p (z, ,)f q (z, ω 1 )f r (z, ω 2 ),
(7.12)
χ
IQ
pqr ((, ω 1 , ω 2 ) =
∞
z b
dz
χ
D1
pqrz (z, ,, ω 1 , ω 2 )f p (z, ,)
∂f q (z, ω 1 )
∂z
f r (z, ω 2 )
+ χ
D2
pqrz (z, ,, ω 1 , ω 2 )f p (z, ,)f q (z, ω 1 )
∂f r (z, ω 2 )
∂z
+χ
Q
pqrz (z, ,, ω 1 , ω 2 )
∂f p (z, ,)
∂z
f q (z, ω 1 )f r (z, ω 2 )
,
(7.29)
χ
IQB
pqr ((, ω 1 , ω 2 ) = χ
Q,β
pqrz ((, ω 1 , ω 2 )f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 ),
(7.30)
and χ B
G, ((, ω 1 , ω 2 ) is given in Eqs. (7.54), (7.55), (7.56), (7.57), (7.58), (7.59),
(7.60).