7.2 Extended Nonlinear Susceptibility
169
l T
1
ε α (()k x (()
ζ
Q1β
3
ε
α (()k
β
T ,z (ω 1 )k x (() + ε
β (()k x (ω 1 )k
α
R,z (()
+ζ
Q2β
3
ε
α (()k
β
T ,z (ω 2 )k x (()+ε
β (()k x (ω 2 )k
α
R,z (()
f
β
z (()f
β
y (ω 1 )f
β
y (ω 2 ),
(7.57)
• PPP case:
χ
B
G,xxz = χ
B
G,yyz , χ
B
G,xzx = χ
B
G,yzy , χ
B
G,zxx = χ
B
G,zyy ,
(7.58)
χ
B
R,zzz = l R
⎧
⎪ ⎨
⎪ ⎩
ζ
Q1β
2
k
β
T (ω 1 ) × k
β
T (ω 2 )
y
k x (ω 2 )
+ ζ
Q2β
1
k
β
T (ω 2 ) × k
β
T (ω 1 )
y
k x (ω 1 )
+ζ
Q1β
3
k
β
T (ω 1 ) × k
β
R (()
y
k x (()
+ζ
Q2β
3
k
β
T (ω 2 ) × k
β
R (()
y
k x (()
⎫
⎪ ⎬
⎪ ⎭
f
β
z (()f
β
z (ω 1 )f
β
z (ω 2 ),
(7.59)
χ
B
T ,zzz = l T
⎧
⎪ ⎨
⎪ ⎩
ζ
Q1β
2
k
β
T (ω 1 ) × k
β
T (ω 2 )
y
k x (ω 2 )
+ ζ
Q2β
1
k
β
T (ω 2 ) × k
β
T (ω 1 )
y
k x (ω 1 )
+ ζ
Q1β
3
k
β
T ,z (ω 1 ) +
ε β (()
ε α (()
k α
R,z (()k x (ω 1 )
k x (()
+ζ
Q2β
3
k
β
T ,z (ω 2 ) +
ε β (()
ε α (()
k α
R,z (()k x (ω 2 )
k x (()
f
β
z (()f
β
z (ω 1 )f
β
z (ω 2 ),
(7.60)
where G = T or R, and we used f x (ω f ) = f y (ω f ) to derive Eq. (7.58).
The difference in the definitions for the bulk contribution, χ B0
G and χ B
G , are
attributed to different treatment of geometric factors related to the light wavevectors,
k
i
G (ω f ). The latter definition χ B
G allows us to use the apparently same expressions
for the effective susceptibilities in Eqs. (7.15), (7.16), (7.17), (7.18).
7.2.6 Summary of Derivation
To summarize the SSP, SPS, PSS, and PPP measurements of C ∞v systems, the relevant effective susceptibilities incorporating the dipole and quadrupole contributions
are given in the following forms,