7.2 Extended Nonlinear Susceptibility
163
∂
∂s
exp
ik
β
T (ω 1 ) · r−iω 1 t
f
β
r (ω 2 )L I,r (ω 2 )E
α
I,r (ω 2 ) exp
ik
β
T (ω 2 ) · r−iω 2 t
+
q,r,s
f
β
p (()χ
D2,β
pqrs ((, ω 1 , ω 2 ) f
β
q (ω 1 )L I,q (ω 1 )E
α
I,q (ω 1 ) exp
ik
β
T (ω 1 ) · r−iω 1 t
f
β
r (ω 2 )L I,r (ω 2 )E
α
I,r (ω 2 )
∂
∂s
exp
ik
β
T (ω 2 ) · r − iω 2 t
−
q,r,s
f
β
p (()χ
Q,β
pqrs ((, ω 1 , ω 2 )f
β
q (ω 1 )L I,q (ω 1 )E
α
I,q (ω 1 )f
β
r (ω 2 )L I,r (ω 2 )E
α
I,r (ω 2 )
·
∂
∂s
exp
ik
β
T (ω 1 ) · r − iω 1 t
exp
ik
β
T (ω 2 ) · r − iω 2 t
.
(7.31)
P (2),B is expressed in a similar manner as in Eq. (7.26) by separating the phase
factor along x,
P
(2),B (r, ,, t) = P
B (z) exp (ik x (()x − iit) ,
(7.32)
where P B (z) is a function of z in the bulk region. By performing the derivative of
phase factor by
∂
∂s
exp
ik
β
G (ω f ) · r − iω f t
= ik
β
G,s exp
ik
β
G (ω f ) · r − iω f t
,
the bulk polarization of Eq. (7.31) or (7.32) is written as
P
(2),B
p
(r, ,, t) = P
B
p (z) exp (ik x (()x − iit)
= i
q,r,s
f
β
p (()
χ
D1,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 1 ) + χ
D2,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 2 )
−χ
Q,β
pqrs ((, ω 1 , ω 2 )
k
β
T ,s (ω 1 ) + k
β
T ,s (ω 2 )
· f
β
q (ω 1 )L I,q (ω 1 )E
α
I,q (ω 1 ) exp
ik
β
T (ω 1 ) · r − iω 1 t
· f
β
r (ω 2 )L I,r (ω 2 )E
α
I,r (ω 2 ) exp
ik
β
T (ω 2 ) · r − iω 2 t
(7.33)
and thus
P
B
p (z) = i
q,r,s
χ
D1,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 1 ) + χ
D2,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 2 )
−χ
Q,β
pqrs ((, ω 1 , ω 2 )
k
β
T ,s (ω 1 ) + k
β
T ,s (ω 2 )
· f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )L I,q (ω 1 )L I,r (ω 2 )E
α
I,q (ω 1 )E
α
I,r (ω 2 )
· exp
i
k
β
T ,z (ω 1 ) + k
β
T ,z (ω 2 )
z
.
(7.34)