156
7 Quadrupole Contributions from Interface and Bulk
(Here we have assumed both the incident phases i1 and i2 to be α in Fig. 7.1.)
As shown in Table 2.1, e(() is given as the result of the Fresnel transformation in
the interface, e(() = F i→j (() ˆ
e
i ((), and hence the left side is written by e(() ·
P S (() = ˆ
e
i (() · F i→j (()P S ((). The Fresnel factor F i→j is a symmetric tensor
and is given in Eq. (2.18), which involves the interfacial dielectric constant ε .
In the present chapter, however, we obviate the use of phenomenological
parameter ε , and incorporate the effect into the microscopic local field correction
f . Accordingly, χ
(2)
eff,G is defined here so as to satisfy the following condition,
ˆ
e
i
G (() · L G (()P
S (() = χ
(2)
eff,G E
α
I (ω 1 )E
α
I (ω 2 ),
(7.13)
where the subscript G (= R or T ) specifies reflected or transmitted SFG signal.
In Eq. (7.13) the optical factor L G (() is used instead of F i→j , and the local field
f at the frequency is incorporated in the definition of P S (() (see Eqs. (7.11)
and (7.12)). χ
(2)
eff,G thus introduced will be expanded to incorporate the quadrupole
contributions in the next Sect. 7.2.
By inserting Eq. (7.11) into (7.13), we get the χ
(2)
eff,G formula for the electric
dipole contribution,
χ
(2)
eff,G ((, ω 1 , ω 2 )
= ˆ
e
i
G (() · L G (()
χ
ID ((, ω 1 , ω 2 ) :
L I (ω 1 ) ˆ
e
α
I (ω 1 )
L I (ω 2 ) ˆ
e
α
I (ω 2 )
.
(7.14)
For an interface of azimuthal C ∞v symmetry, the effective susceptibilities for
possible polarization combinations are expressed as follows:
χ
(2)
eff,G, SSP ((, ω 1 , ω 2 ) = L G,y (() L I,y (ω 1 ) L I,z (ω 2 ) sin θ
α
I (ω 2 ) χ
ID
yyz ((, ω 1 , ω 2 ),
(7.15)
χ
(2)
eff,G, SPS ((, ω 1 , ω 2 ) = L G,y (() L I,z (ω 1 ) L I,y (ω 2 ) sin θ
α
I (ω 1 ) χ
ID
yzy ((, ω 1 , ω 2 ),
(7.16)
χ
(2)
eff,G, PSS ((, ω 1 , ω 2 ) = L G,z (() L I,y (ω 1 ) L I,y (ω 2 ) sin θ
i
G (() χ
ID
zyy ((, ω 1 , ω 2 ),
(7.17)
χ
(2)
eff,G, PPP ((, ω 1 , ω 2 ) =
− L G,x (() L I,x (ω 1 ) L I,z (ω 2 ) cos θ
i
G (() cos θ
α
I (ω 1 ) sin θ
α
I (ω 2 ) χ
ID
xxz ((, ω 1 , ω 2 )
− L G,x (() L I,z (ω 1 ) L I,x (ω 2 ) cos θ
i
G (() sin θ
α
I (ω 1 ) cos θ
α
I (ω 2 ) χ
ID
xzx ((, ω 1 , ω 2 )
+ L G,z (() L I,x (ω 1 ) L I,x (ω 2 ) sin θ
i
G (() cos θ
α
I (ω 1 ) cos θ
α
I (ω 2 ) χ
ID
zxx ((, ω 1 , ω 2 )
+ L G,z (() L I,z (ω 1 ) L I,z (ω 2 ) sin θ
i
G (() sin θ
α
I (ω 1 ) sin θ
α
I (ω 2 ) χ
ID
zzz ((, ω 1 , ω 2 ),
(7.18)
7 Quadrupole Contributions from Interface and Bulk
(Here we have assumed both the incident phases i1 and i2 to be α in Fig. 7.1.)
As shown in Table 2.1, e(() is given as the result of the Fresnel transformation in
the interface, e(() = F i→j (() ˆ
e
i ((), and hence the left side is written by e(() ·
P S (() = ˆ
e
i (() · F i→j (()P S ((). The Fresnel factor F i→j is a symmetric tensor
and is given in Eq. (2.18), which involves the interfacial dielectric constant ε .
In the present chapter, however, we obviate the use of phenomenological
parameter ε , and incorporate the effect into the microscopic local field correction
f . Accordingly, χ
(2)
eff,G is defined here so as to satisfy the following condition,
ˆ
e
i
G (() · L G (()P
S (() = χ
(2)
eff,G E
α
I (ω 1 )E
α
I (ω 2 ),
(7.13)
where the subscript G (= R or T ) specifies reflected or transmitted SFG signal.
In Eq. (7.13) the optical factor L G (() is used instead of F i→j , and the local field
f at the frequency is incorporated in the definition of P S (() (see Eqs. (7.11)
and (7.12)). χ
(2)
eff,G thus introduced will be expanded to incorporate the quadrupole
contributions in the next Sect. 7.2.
By inserting Eq. (7.11) into (7.13), we get the χ
(2)
eff,G formula for the electric
dipole contribution,
χ
(2)
eff,G ((, ω 1 , ω 2 )
= ˆ
e
i
G (() · L G (()
χ
ID ((, ω 1 , ω 2 ) :
L I (ω 1 ) ˆ
e
α
I (ω 1 )
L I (ω 2 ) ˆ
e
α
I (ω 2 )
.
(7.14)
For an interface of azimuthal C ∞v symmetry, the effective susceptibilities for
possible polarization combinations are expressed as follows:
χ
(2)
eff,G, SSP ((, ω 1 , ω 2 ) = L G,y (() L I,y (ω 1 ) L I,z (ω 2 ) sin θ
α
I (ω 2 ) χ
ID
yyz ((, ω 1 , ω 2 ),
(7.15)
χ
(2)
eff,G, SPS ((, ω 1 , ω 2 ) = L G,y (() L I,z (ω 1 ) L I,y (ω 2 ) sin θ
α
I (ω 1 ) χ
ID
yzy ((, ω 1 , ω 2 ),
(7.16)
χ
(2)
eff,G, PSS ((, ω 1 , ω 2 ) = L G,z (() L I,y (ω 1 ) L I,y (ω 2 ) sin θ
i
G (() χ
ID
zyy ((, ω 1 , ω 2 ),
(7.17)
χ
(2)
eff,G, PPP ((, ω 1 , ω 2 ) =
− L G,x (() L I,x (ω 1 ) L I,z (ω 2 ) cos θ
i
G (() cos θ
α
I (ω 1 ) sin θ
α
I (ω 2 ) χ
ID
xxz ((, ω 1 , ω 2 )
− L G,x (() L I,z (ω 1 ) L I,x (ω 2 ) cos θ
i
G (() sin θ
α
I (ω 1 ) cos θ
α
I (ω 2 ) χ
ID
xzx ((, ω 1 , ω 2 )
+ L G,z (() L I,x (ω 1 ) L I,x (ω 2 ) sin θ
i
G (() cos θ
α
I (ω 1 ) cos θ
α
I (ω 2 ) χ
ID
zxx ((, ω 1 , ω 2 )
+ L G,z (() L I,z (ω 1 ) L I,z (ω 2 ) sin θ
i
G (() sin θ
α
I (ω 1 ) sin θ
α
I (ω 2 ) χ
ID
zzz ((, ω 1 , ω 2 ),
(7.18)