152
7 Quadrupole Contributions from Interface and Bulk
The quadrupole contributions were recognized at an early stage of studies on the
second-order optical processes [1, 16, 20]. Bloembergen et al. proposed a theory of
SHG involving the quadrupole contribution in centrosymmetric media [2, 3], and the
theory of SHG has been sophisticated by subsequent studies[6–8, 18, 19]. Here we
formulate the quadrupole contributions in SFG as well as in SHG. While the theory
of SFG and SHG is mostly common, the SFG spectroscopy has some different
sensitivity to the quadrupole contributions from SHG since the two incident fields
can propagate with different wavevectors. Once the quadrupole effect on SFG is
understood, SHG is regarded as a special case of SFG.
In the present chapter, some arguments given in preceding chapters have to
be modified in order to incorporate the quadrupole contributions. First, the threelayer model of SFG in Chap. 2 is generalized to take account of the structure
within the interface in Sect. 7.1. Then the second-order polarization is extended to
include the quadrupole originating from both interface and bulk in Sect. 7.2. As
a consequence, the second-order susceptibility effectively consists of four terms
of different characters, namely χ ID , χ IQ , χ IQB , and χ B . Section 7.3 presents the
microscopic formulas of those terms in both the energy representation and timedependent representation, after the discussion of χ (2) in Chap. 3. Section 7.4 argues
the invariance of the quadrupole formulas with respect to the molecular origin, a
fundamental requirement for the sound theory. Summary is then given in Sect. 7.5.
7.1 Beyond the Three-Layer Model
We extend the electrodynamic theory of SFG and generalize the three-layer model
in Chap. 2 by taking account of depth- (z-) dependent dielectric properties of the
interface. The phenomenological three-layer model is not capable of describing
internal structure within the interfacial layer. In order to evaluate the quadrupolar
polarizations at the surface, we need to account for the structure of electric field
gradient over the surface region in a molecular scale.
Notations for Surface SFG Figure 7.1 summarizes the optical geometry and
related notations for the SFG measurement. This geometry is identical to that in
Fig. 2.1, though the interface region around z ≈ 0 is not treated as a phenomenological third layer in Fig. 7.1. While the microscopic definition of the origin z = 0
is somewhat arbitrary due to finite thickness of the interface, this arbitrariness does
not affect the following discussion toward treating the surface and bulk contributions
from a unified view. This is because the thickness of the interface is generally much
shorter than the wavelengths of the light fields. (This condition is usually satisfied
in the surface nonlinear spectroscopy.)
Figure 7.1 shows all the wavevectors k G with the subscripts G = I, R and T ,
which distinguish the incident, reflected and transmitted fields, respectively. In the
following discussion the medium α is assumed to be vacuum for simplicity, and
hence ε α = 1, though extension to other situations is straightforward. The absolute
7 Quadrupole Contributions from Interface and Bulk
The quadrupole contributions were recognized at an early stage of studies on the
second-order optical processes [1, 16, 20]. Bloembergen et al. proposed a theory of
SHG involving the quadrupole contribution in centrosymmetric media [2, 3], and the
theory of SHG has been sophisticated by subsequent studies[6–8, 18, 19]. Here we
formulate the quadrupole contributions in SFG as well as in SHG. While the theory
of SFG and SHG is mostly common, the SFG spectroscopy has some different
sensitivity to the quadrupole contributions from SHG since the two incident fields
can propagate with different wavevectors. Once the quadrupole effect on SFG is
understood, SHG is regarded as a special case of SFG.
In the present chapter, some arguments given in preceding chapters have to
be modified in order to incorporate the quadrupole contributions. First, the threelayer model of SFG in Chap. 2 is generalized to take account of the structure
within the interface in Sect. 7.1. Then the second-order polarization is extended to
include the quadrupole originating from both interface and bulk in Sect. 7.2. As
a consequence, the second-order susceptibility effectively consists of four terms
of different characters, namely χ ID , χ IQ , χ IQB , and χ B . Section 7.3 presents the
microscopic formulas of those terms in both the energy representation and timedependent representation, after the discussion of χ (2) in Chap. 3. Section 7.4 argues
the invariance of the quadrupole formulas with respect to the molecular origin, a
fundamental requirement for the sound theory. Summary is then given in Sect. 7.5.
7.1 Beyond the Three-Layer Model
We extend the electrodynamic theory of SFG and generalize the three-layer model
in Chap. 2 by taking account of depth- (z-) dependent dielectric properties of the
interface. The phenomenological three-layer model is not capable of describing
internal structure within the interfacial layer. In order to evaluate the quadrupolar
polarizations at the surface, we need to account for the structure of electric field
gradient over the surface region in a molecular scale.
Notations for Surface SFG Figure 7.1 summarizes the optical geometry and
related notations for the SFG measurement. This geometry is identical to that in
Fig. 2.1, though the interface region around z ≈ 0 is not treated as a phenomenological third layer in Fig. 7.1. While the microscopic definition of the origin z = 0
is somewhat arbitrary due to finite thickness of the interface, this arbitrariness does
not affect the following discussion toward treating the surface and bulk contributions
from a unified view. This is because the thickness of the interface is generally much
shorter than the wavelengths of the light fields. (This condition is usually satisfied
in the surface nonlinear spectroscopy.)
Figure 7.1 shows all the wavevectors k G with the subscripts G = I, R and T ,
which distinguish the incident, reflected and transmitted fields, respectively. In the
following discussion the medium α is assumed to be vacuum for simplicity, and
hence ε α = 1, though extension to other situations is straightforward. The absolute