Classical- and Heterodyne-Detected Vibrational Sum …
89
constant, known as molecular polarizability. μ ind is not necessarily in the direction
of E and hence α is vectorial in nature. For high intensity light (intensity ∝ |E|
2 ),
specifically when the amplitude of the light electric field is comparable to the atomic
field - the electric field with which the electrons are held by the atomic nuclei (typically, 5 × 10
11 V/m), μ ind deviates from the linear dependence of E; it becomes a
higher-order polynomial of E, as expressed by Taylor series.
μ ind = α E + β E
2
+ γ E
3
+ . . .
(1)
where, the coefficients β, γ are the corresponding molecular hyperpolarizabilities.
For a macroscopic system, the induced dipole moment is expressed as the induced
dipole moment per unit volume, known as polarization ( P) such that
P = N μ ind = o χ
(1) E + o χ
(2) E
2
+ o χ
(3) E
3
+ . . .
= P
(1)
+ P
(2)
+ P
(3)
+ . . .
(2)
where, N is the number of molecules per unit volume (or per unit area for surface);
indicates the ensemble average over orientational distribution; o is the electric
constant in vacuum; χ
(1)
, χ
(2)
, χ
(3) are electric susceptibilities corresponding to the
molecular polarizabilities α, β and γ , respectively. Thus, χ
(n) provides a macroscopic
description of the susceptibility of a material towards polarization via the effective
sum of the corresponding molecular (hyper) polarizabilities. Similar to molecular
(hyper) polarizability, χ
(n) is also vectorial in nature. The higher order polarization
terms, P
(2)
, P
(3) , etc., become increasingly important for light from a pulsed laser.
For instance, Light from a femtosecond (fs) pulsed laser (1.0 W, 1.0 kHz, 50 fs pulse
duration) produces an energy density of the order of ~10
19 W/m
2 (i.e., a field strength
of ~8.5 × 10
10 V/m) when focused on a material with a lens of 50 cm focal length. In
general, the spectroscopy associated with the higher order polarizations ( P
(2)
, P
(3) ,
etc.) are known as nonlinear spectroscopy.
Sum frequency is generated by the oscillating dipole corresponding to the secondorder polarization, P
(2) . Assuming two intense lights of angular frequencies ω 1 and
ω 2 , falling on a material, P
(2) can be expressed as,
P
(2)
(t) = o χ
(2) E(t)
2
(3)
where, E(t) = (E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
) +
E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
.
By successive expansion, P
(2)
(t) takes the form,
P
(2)
(t) = o χ
(2)
2E 1 E
∗
1 + 2E 2 E
∗
2
+ (E
2
1 e
−2iω 1 t
+ E
2
2 e
−2iω 2 t
+ E
∗
2
1 e
2iω 1 t
+ E
∗
2
2 e
2iω 2 t
)
+
2E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+ 2E
∗
1 E 2 e
i(ω 1 −ω 2 )t
+
2E 1 E 2 e
−i(ω 1 +ω 2 )t
+ 2E
∗
1 E
∗
2 e
i(ω 1 +ω 2 )t
(4)
89
constant, known as molecular polarizability. μ ind is not necessarily in the direction
of E and hence α is vectorial in nature. For high intensity light (intensity ∝ |E|
2 ),
specifically when the amplitude of the light electric field is comparable to the atomic
field - the electric field with which the electrons are held by the atomic nuclei (typically, 5 × 10
11 V/m), μ ind deviates from the linear dependence of E; it becomes a
higher-order polynomial of E, as expressed by Taylor series.
μ ind = α E + β E
2
+ γ E
3
+ . . .
(1)
where, the coefficients β, γ are the corresponding molecular hyperpolarizabilities.
For a macroscopic system, the induced dipole moment is expressed as the induced
dipole moment per unit volume, known as polarization ( P) such that
P = N μ ind = o χ
(1) E + o χ
(2) E
2
+ o χ
(3) E
3
+ . . .
= P
(1)
+ P
(2)
+ P
(3)
+ . . .
(2)
where, N is the number of molecules per unit volume (or per unit area for surface);
indicates the ensemble average over orientational distribution; o is the electric
constant in vacuum; χ
(1)
, χ
(2)
, χ
(3) are electric susceptibilities corresponding to the
molecular polarizabilities α, β and γ , respectively. Thus, χ
(n) provides a macroscopic
description of the susceptibility of a material towards polarization via the effective
sum of the corresponding molecular (hyper) polarizabilities. Similar to molecular
(hyper) polarizability, χ
(n) is also vectorial in nature. The higher order polarization
terms, P
(2)
, P
(3) , etc., become increasingly important for light from a pulsed laser.
For instance, Light from a femtosecond (fs) pulsed laser (1.0 W, 1.0 kHz, 50 fs pulse
duration) produces an energy density of the order of ~10
19 W/m
2 (i.e., a field strength
of ~8.5 × 10
10 V/m) when focused on a material with a lens of 50 cm focal length. In
general, the spectroscopy associated with the higher order polarizations ( P
(2)
, P
(3) ,
etc.) are known as nonlinear spectroscopy.
Sum frequency is generated by the oscillating dipole corresponding to the secondorder polarization, P
(2) . Assuming two intense lights of angular frequencies ω 1 and
ω 2 , falling on a material, P
(2) can be expressed as,
P
(2)
(t) = o χ
(2) E(t)
2
(3)
where, E(t) = (E 1 e
−iω 1 t
+ E
∗
1 e
iω 1 t
) +
E 2 e
−iω 2 t
+ E
∗
2 e
iω 2 t
.
By successive expansion, P
(2)
(t) takes the form,
P
(2)
(t) = o χ
(2)
2E 1 E
∗
1 + 2E 2 E
∗
2
+ (E
2
1 e
−2iω 1 t
+ E
2
2 e
−2iω 2 t
+ E
∗
2
1 e
2iω 1 t
+ E
∗
2
2 e
2iω 2 t
)
+
2E 1 E
∗
2 e
−i(ω 1 −ω 2 )t
+ 2E
∗
1 E 2 e
i(ω 1 −ω 2 )t
+
2E 1 E 2 e
−i(ω 1 +ω 2 )t
+ 2E
∗
1 E
∗
2 e
i(ω 1 +ω 2 )t
(4)