7.2 Extended Nonlinear Susceptibility
165
Fig. 7.3 Definitions of
parameters for the SFG
emission from polarization at
z, P B (z), embedded in the
medium β. The subscripts
f = 1, 2, 3 denote the two
incident lights and SFG,
respectively, e.g.
k
α
1I = k
α
I (ω 1 ),
k
β
2T = k
β
T (ω 2 ), θ α
3R = θ α
R (()
k 1I
k 2I
k 1R
k 2R
k 3R
k 1T
k 3T
k 2T
k 3R
k 3T
3R
x
z
3T
bulk polarization
bulk polarization P (z)
B
and
χ pppp = χ 1 + χ 2 + χ 3 .
(7.39)
[Problem 7.1] Derive Eq. (7.39) from the three independent elements of Eq. (7.38)
in the isotropic condition.
Therefore, in isotropic bulk media, Eq. (7.36) is transformed into
P
B
G,p =
q,r
χ
B0
G,pqr ((, ω 1 , ω 2 )L I,q (ω 1 )L I,r (ω 2 )E
α
I,q (ω 1 )E
α
I,r (ω 2 ),
(7.40)
where
χ B0
G,pqr ((, ω 1 , ω 2 ) = l G
ζ
Q1,β
2
((, ω 1 , ω 2 )δ pq k
β
T ,r (ω 1 ) + ζ
Q2,β
1
((, ω 1 , ω 2 )δ pr k
β
T ,q (ω 2 )
+ζ
Q1,β
3
((, ω 1 , ω 2 )δ qr k
β
T ,p (ω 1 ) + ζ
Q2,β
3
((, ω 1 , ω 2 )δ qr k
β
T ,p (ω 2 )
f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )
(7.41)
and the ζ ’s are expressed using the notations of Eq. (7.38) by
ζ
Q2,β
1
((, ω 1 , ω 2 ) ≡ χ
D2,β
1
((, ω 1 , ω 2 ) − χ
Q,β
1 ((, ω 1 , ω 2 ),
(7.42)
ζ
Q1,β
2
((, ω 1 , ω 2 ) ≡ χ
D1,β
2
((, ω 1 , ω 2 ) − χ
Q,β
2 ((, ω 1 , ω 2 ),
(7.43)
ζ
Q1,β
3
((, ω 1 , ω 2 ) ≡ χ
D1,β
3
((, ω 1 , ω 2 ) − χ
Q,β
3 ((, ω 1 , ω 2 ),
(7.44)
ζ
Q2,β
3
((, ω 1 , ω 2 ) ≡ χ
D2,β
3
((, ω 1 , ω 2 ) − χ
Q,β
3 ((, ω 1 , ω 2 ).
(7.45)