166
7 Quadrupole Contributions from Interface and Bulk
We note that Eq. (7.41) is derived using the orthogonal condition between the
electric field and the wave vector in the medium β,
p
(f
β
p (ω f )L I,p (ω f )E
α
I,p (ω f )) · k
β
T ,p (ω f ) = 0
(f = 1, 2).
(7.46)
In summary, the effective polarization P eff,G = P l +P B
G in Eq. (7.25) consists of
Eqs. (7.28) and (7.40). As a consequence, the effective second-order susceptibility
including the surface and bulk contribution, χ
(2)
eff,G in Eq. (7.24), is expressed as
χ
(2)
eff,G ((, ω 1 , ω 2 ) =
ˆ
e
i
G (()L G (()
χ
(2)
q0 G ((, ω 1 , ω 2 ) :
L I (ω 1 ) ˆ
e
α
I (ω 1 )
L I (ω 2 ) ˆ
e
α
I (ω 2 )
,
(7.47)
where
χ
(2)
q0 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
B0
G,pqr ((, ω 1 , ω 2 ).
(7.48)
Equation (7.47) is an extended form of Eq. (7.14) to incorporate the dipole and
quadrupole contributions. If the quadrupole terms of χ IQ , χ IQB and χ B0
G were
neglected in Eq. (7.48), Eq. (7.47) would coincide with Eq. (7.14).
7.2.5 Expression of Bulk Term χ B
For an interface of azimuthal C ∞v symmetry, the (achiral) SFG signal is detected
only in the SSP, SPS, PSS, or PPP combination for symmetry reasons, even though
the bulk contribution is taken into account. These polarization combinations are
related to specific tensor elements of χ ID in Eqs. (7.15), (7.16), (7.17), (7.18),
and their relations have been already discussed in Eqs. (3.49), (3.50), (3.51),
(3.52) in Chap. 3. However, the bulk contribution in Eq. (7.48) would break the
relation of Eqs. (7.15), (7.16), (7.17), (7.18) between the effective susceptibilities
and nonvanishing tensor elements. Here we discuss this relation when the bulk
contribution is taken into account.
In the original expressions of χ
(2)
eff,SSP , χ
(2)
eff,SPS , χ
(2)
eff,PSS and χ
(2)
eff,PPP in Eqs. (7.15),
(7.16), (7.17), (7.18), the tensor elements of χ ID cannot be simply replaced with
those of χ
(2)
q0G in Eq. (7.48) to incorporate the quadrupole contributions, because
several extra tensor elements of χ B0
G,pqr in Eq. (7.41) are not necessarily zero.
Equation (7.41) indicates that non-zero elements of χ B0
G,pqr are (yyx), (yxy), (xyy),
7 Quadrupole Contributions from Interface and Bulk
We note that Eq. (7.41) is derived using the orthogonal condition between the
electric field and the wave vector in the medium β,
p
(f
β
p (ω f )L I,p (ω f )E
α
I,p (ω f )) · k
β
T ,p (ω f ) = 0
(f = 1, 2).
(7.46)
In summary, the effective polarization P eff,G = P l +P B
G in Eq. (7.25) consists of
Eqs. (7.28) and (7.40). As a consequence, the effective second-order susceptibility
including the surface and bulk contribution, χ
(2)
eff,G in Eq. (7.24), is expressed as
χ
(2)
eff,G ((, ω 1 , ω 2 ) =
ˆ
e
i
G (()L G (()
χ
(2)
q0 G ((, ω 1 , ω 2 ) :
L I (ω 1 ) ˆ
e
α
I (ω 1 )
L I (ω 2 ) ˆ
e
α
I (ω 2 )
,
(7.47)
where
χ
(2)
q0 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
B0
G,pqr ((, ω 1 , ω 2 ).
(7.48)
Equation (7.47) is an extended form of Eq. (7.14) to incorporate the dipole and
quadrupole contributions. If the quadrupole terms of χ IQ , χ IQB and χ B0
G were
neglected in Eq. (7.48), Eq. (7.47) would coincide with Eq. (7.14).
7.2.5 Expression of Bulk Term χ B
For an interface of azimuthal C ∞v symmetry, the (achiral) SFG signal is detected
only in the SSP, SPS, PSS, or PPP combination for symmetry reasons, even though
the bulk contribution is taken into account. These polarization combinations are
related to specific tensor elements of χ ID in Eqs. (7.15), (7.16), (7.17), (7.18),
and their relations have been already discussed in Eqs. (3.49), (3.50), (3.51),
(3.52) in Chap. 3. However, the bulk contribution in Eq. (7.48) would break the
relation of Eqs. (7.15), (7.16), (7.17), (7.18) between the effective susceptibilities
and nonvanishing tensor elements. Here we discuss this relation when the bulk
contribution is taken into account.
In the original expressions of χ
(2)
eff,SSP , χ
(2)
eff,SPS , χ
(2)
eff,PSS and χ
(2)
eff,PPP in Eqs. (7.15),
(7.16), (7.17), (7.18), the tensor elements of χ ID cannot be simply replaced with
those of χ
(2)
q0G in Eq. (7.48) to incorporate the quadrupole contributions, because
several extra tensor elements of χ B0
G,pqr in Eq. (7.41) are not necessarily zero.
Equation (7.41) indicates that non-zero elements of χ B0
G,pqr are (yyx), (yxy), (xyy),