186
7 Quadrupole Contributions from Interface and Bulk
7.5 Summary
In this section we have presented a unified treatment of the dipole and quadrupole
contributions to the second-order optical responses. The concept of effective susceptibility χ (2) is expanded to incorporate both the surface and bulk contributions.
This treatment allows us to quantitatively evaluate both surface and bulk contributions in experimental SFG/SHG spectra, and to calculate them from microscopic
expressions of general hyperpolarizabilities.
The source of the SFG/SHG signals consists of four terms, i.e. χ ID , χ IQ , χ IQB
and χ B in Eq. (7.50). These four terms can be characterized in various ways as
follows, which help us understand the physical meanings of these terms.
Dipole vs. Quadrupole χ ID originates from the induced electric dipole, while
the other three terms, χ IQ , χ IQB , and χ B originate from the quadrupole (electric
quadrupole and magnetic dipole).
The conventional SFG/SHG theory within the dipole approximation considers
only the χ ID term. The present theory shows that the conventional theory can be
straightforwardly extended by replacing χ ID with χ q = χ ID + χ IQ + χ IQB + χ B if
we properly define the latter three terms.
Interface vs. Bulk χ ID and χ IQ reflect properties of interface, while χ IQB and χ B
are determined solely by bulk properties.
It is worth noting that χ IQB reflects no interface properties, as seen in Eq. (7.30),
since it is derived from the lower bound of the integral over the interface region in
Eq. (7.27).
Dependence on Optical Geometry χ ID , χ IQ and χ IQB are independent of the
optical geometry of measurement as they involve no wavevector k
i
G (ω f ) or related
quantities. On the other hand, χ B depends on the optical geometry as described in
Sects. 7.2.4 and 7.2.5.
This difference allows us to distinguish χ B from the other terms by changing the
optical geometry of measurement.
The characteristics of the four terms, χ ID , χ IQ , χ IQB and χ B , are schematically
summarized in Fig. 7.5.
7.6 Solutions to Problems
7.6.1 Isotropic Tensor Components
[Problem 7.1] Derive Eq. (7.39) from the three independent elements of Eq. (7.38)
in the isotropic condition.
7 Quadrupole Contributions from Interface and Bulk
7.5 Summary
In this section we have presented a unified treatment of the dipole and quadrupole
contributions to the second-order optical responses. The concept of effective susceptibility χ (2) is expanded to incorporate both the surface and bulk contributions.
This treatment allows us to quantitatively evaluate both surface and bulk contributions in experimental SFG/SHG spectra, and to calculate them from microscopic
expressions of general hyperpolarizabilities.
The source of the SFG/SHG signals consists of four terms, i.e. χ ID , χ IQ , χ IQB
and χ B in Eq. (7.50). These four terms can be characterized in various ways as
follows, which help us understand the physical meanings of these terms.
Dipole vs. Quadrupole χ ID originates from the induced electric dipole, while
the other three terms, χ IQ , χ IQB , and χ B originate from the quadrupole (electric
quadrupole and magnetic dipole).
The conventional SFG/SHG theory within the dipole approximation considers
only the χ ID term. The present theory shows that the conventional theory can be
straightforwardly extended by replacing χ ID with χ q = χ ID + χ IQ + χ IQB + χ B if
we properly define the latter three terms.
Interface vs. Bulk χ ID and χ IQ reflect properties of interface, while χ IQB and χ B
are determined solely by bulk properties.
It is worth noting that χ IQB reflects no interface properties, as seen in Eq. (7.30),
since it is derived from the lower bound of the integral over the interface region in
Eq. (7.27).
Dependence on Optical Geometry χ ID , χ IQ and χ IQB are independent of the
optical geometry of measurement as they involve no wavevector k
i
G (ω f ) or related
quantities. On the other hand, χ B depends on the optical geometry as described in
Sects. 7.2.4 and 7.2.5.
This difference allows us to distinguish χ B from the other terms by changing the
optical geometry of measurement.
The characteristics of the four terms, χ ID , χ IQ , χ IQB and χ B , are schematically
summarized in Fig. 7.5.
7.6 Solutions to Problems
7.6.1 Isotropic Tensor Components
[Problem 7.1] Derive Eq. (7.39) from the three independent elements of Eq. (7.38)
in the isotropic condition.