Classical- and Heterodyne-Detected Vibrational Sum …
93
carried by Eulers’ angle transformation [10, 12, 13]. Non-zero components of β lmn
are associated with the vibrational modes of the molecules. The expression for β lmn ,
obtained from quantum mechanical perturbative treatment, [1, 14] is as follows,
β lmn =
A v
ω v − ω I R − iΓ v
(14)
where, ω v and Γ v are the resonant frequency and the natural linewidth of the vth vibrational transition. ω I R is the tuneable IR frequency. A v is the amplitude coefficient
proportional to the product of Raman and IR transition moments [1, 15].
Therefore,
χ
(2)
i jk =
N s
o
A v
(ω v − ω I R − iΓ v )
=
N s A v
o
(ω v − ω I R )
(ω v − ω I R ) 2 + Γ 2
v
+ i
N s A v
o
Γ v
(ω v − ω I R ) 2 + Γ 2
v
= Re
χ
(2)
i jk
+ I m
χ
(2)
i jk
(15)
where,
Re[χ
(2)
i jk ] =
N s A v
o
(ω v − ω I R )
(ω v − ω I R ) 2 + Γ 2
v
and
Im[χ
(2)
i jk ] =
N s A v
o
Γ v
(ω v − ω I R ) 2 + Γ 2
v
In the above equation, χ
(2)
i jk realizes the resonance condition of VSFG. When the
incident IR frequency (ω I R ) is equal to the vth vibrational transition frequency (ω v )
of interfacial molecules (i.e., ω I R = ω v or ω v − ω I R = 0), the imaginary component,
Im
χ
(2)
i jk
becomes maximum and shows an absorptive band shape with the variation
of ω I R . The real component, Re
χ
(2)
i jk
= 0 at ω I R = ω v , and shows a dispersive
band shape with ω I R . The sum frequency intensity (I SFG ),
I SFG ∝
χ
(2)
i jk
2 ∝
N s A v
o
2
1
(ω v − ω I R ) 2 + Γ 2
v
(16)
also shows an absorptive band shape similar to the Im
χ
(2)
i jk
, which means the intensity of the sum frequency signal is maximum at vibrational resonance (ω I R = ω v ).
This is known as ‘resonantly enhanced’ VSFG (Fig. 3). Moreover, to have non-zero
value of χ
(2)
i jk , A v must be non-zero, i.e., the vibrational transition should be both
Raman and IR active. In other words, the vibrational modes of interfacial molecules
must be IR and Raman active to generate VSFG signal.