7.2 Extended Nonlinear Susceptibility
171
Fig. 7.4 Scheme of
derivation for the dipole and
quadrupole nonlinear
susceptibility terms from
(χ D0 , χ D1 , χ D2 , χ Q ) in
Eq. (7.20) to
(χ ID , χ IQ , χ IQB , χ B ) in
Eq. (7.50)
dipole
quadrupole
Q
D2
D1
D0
B
IQB
IQ
ID
(7.20)
(7.50)
(7.12)
(7.29)
(7.30)
(7.54)-(7.60)
These four constituent terms include various tensor elements of χ D0 , χ D1 , χ D2 ,
and χ Q . The nonvanishing and equivalent tensor elements of χ D0 , χ D1 , χ D2 , χ Q
for a system of C ∞v symmetry are summarized as follows [4].
χ
D0
pqr : (xxz) = (yyz), (xzx) = (yzy), (zxx) = (zyy), (zzz),
χ
D1
pqrs , χ
D2
pqrs , χ
Q
pqrs :
(zzzz), (yzyz) = (xzxz), (yyzz) = (xxzz), (zyyz) = (zxxz),
and the χ Q elements in isotropic bulk β are given on the basis of Eqs. (7.38)
and (7.39) by
χ
Q,β
pqrs : (xzxz) = (yzyz) = χ
Q,β
1 ,
(xxzz) = (yyzz) = χ
Q,β
2 ,
(zxxz) = (zyyz) = χ
Q,β
3 ,
(zzzz) = χ
Q,β
1
+ χ
Q,β
2
+ χ
Q,β
3 .
Finally, the extended effective susceptibility beyond the dipole approximation is
schematically summarized in Fig. 7.4. The second-order nonlinear optical response
to the electric field and its gradient is represented in Eq. (7.20), which includes χ D0 ,
χ D1 , χ D2 and χ Q . Those second-order response including both the interface and
bulk contributions are represented with the extended susceptibility,
χ
(2)
qG = χ
ID
+ χ
IQ
+ χ
IQB
+ χ
B
G .
(7.50)
The first term χ ID originates from the response of electric dipole, while the last three
terms χ IQ + χ IQB + χ B
G stem from the quadrupole.
The effective susceptibility χ
(2)
eff in Eqs. (3.49), (3.50), (3.51), (3.52) in Chap. 2
is accordingly extended to Eqs. (7.61), (7.62), (7.63), (7.64) in the present chapter.
By replacing the original χ (2) (=χ ID ) with χ
(2)
qG in Eq. (7.50), the apparently same
formula for the effective susceptibility is applied to describe the SFG signal.