2.4 Solutions to Problems
25
(c2) Incident angles (θ i1 (ω 1 ), θ i2 (ω 2 )) and incident media (i1, i2) of the
visible and infrared lights.
(c3) Direction of SFG.
The SFG signal is emitted in two directions in i = α and β. The
geometry of SFG detection is accordingly twofold, whether one observes
the reflected signal or transmitted signal.
Finally, we briefly discuss the heterodyne measurement of SFG, instead of
the conventional (homodyne) measurement of SFG signal intensity. The general
formula of Eq. (2.25) describes the intensity of SFG signal, I i (() in Eq. (2.24). The
light intensity is proportional to the square of the electric field, and thus Eq. (2.25)
includes the square of the effective χ (2) , |χ
(2)
eff | 2 . The result of Eq. (2.25) corresponds
to the homodyne measurement of SFG intensity.
Recent development of the phase-sensitive or heterodyne SFG measurement
allows for detecting the amplitude and phase of the SFG signal [4, 6]. The phase
information of χ (2) is useful to the interpretation of SFG signals, and allows
for detailed comparison between experimental and computational results of SFG
signals, as we discuss in subsequent chapters. The heterodyne SFG signal can detect
the output SFG field itself, that is ˆ
e
i (() · E i (() in Eq. (2.22), which includes χ
(2)
eff
(without taking the square). Accordingly, the heterodyne signal is also governed by
the same factors (A)–(C) mentioned above. The discussion in this section on the
mechanism of SFG emission holds for the heterodyne SFG spectroscopy as well,
while the difference from the conventional SFG lies in the detection method of SFG
signals.
2.4 Solutions to Problems
2.4.1 Boundary Condition at Interface (1)
[Problem 2.1] Explain the derivation of Eqs. (2.7) and (2.8), by taking account of
the fact that the interface polarization P (2) is singular at z = 0.
We also note that the solution of ∇ ·B(r) = 0 (Eq. (2.4)) is regular, as the system
contains no magnetic source at the interface.
Equation (2.7) We integrate the Maxwell equation of (c) (Eq. (2.4)) in the small
volume V (the left panel of Fig. 2.2) and apply the Gauss’ divergence theorem as
V
(∇ · B)dr =
σ
B · n dσ = {B z (z = +0) − B z (z = −0)} l
2
+
side
B · n dσ = 0.
25
(c2) Incident angles (θ i1 (ω 1 ), θ i2 (ω 2 )) and incident media (i1, i2) of the
visible and infrared lights.
(c3) Direction of SFG.
The SFG signal is emitted in two directions in i = α and β. The
geometry of SFG detection is accordingly twofold, whether one observes
the reflected signal or transmitted signal.
Finally, we briefly discuss the heterodyne measurement of SFG, instead of
the conventional (homodyne) measurement of SFG signal intensity. The general
formula of Eq. (2.25) describes the intensity of SFG signal, I i (() in Eq. (2.24). The
light intensity is proportional to the square of the electric field, and thus Eq. (2.25)
includes the square of the effective χ (2) , |χ
(2)
eff | 2 . The result of Eq. (2.25) corresponds
to the homodyne measurement of SFG intensity.
Recent development of the phase-sensitive or heterodyne SFG measurement
allows for detecting the amplitude and phase of the SFG signal [4, 6]. The phase
information of χ (2) is useful to the interpretation of SFG signals, and allows
for detailed comparison between experimental and computational results of SFG
signals, as we discuss in subsequent chapters. The heterodyne SFG signal can detect
the output SFG field itself, that is ˆ
e
i (() · E i (() in Eq. (2.22), which includes χ
(2)
eff
(without taking the square). Accordingly, the heterodyne signal is also governed by
the same factors (A)–(C) mentioned above. The discussion in this section on the
mechanism of SFG emission holds for the heterodyne SFG spectroscopy as well,
while the difference from the conventional SFG lies in the detection method of SFG
signals.
2.4 Solutions to Problems
2.4.1 Boundary Condition at Interface (1)
[Problem 2.1] Explain the derivation of Eqs. (2.7) and (2.8), by taking account of
the fact that the interface polarization P (2) is singular at z = 0.
We also note that the solution of ∇ ·B(r) = 0 (Eq. (2.4)) is regular, as the system
contains no magnetic source at the interface.
Equation (2.7) We integrate the Maxwell equation of (c) (Eq. (2.4)) in the small
volume V (the left panel of Fig. 2.2) and apply the Gauss’ divergence theorem as
V
(∇ · B)dr =
σ
B · n dσ = {B z (z = +0) − B z (z = −0)} l
2
+
side
B · n dσ = 0.