92
S. Roy et al.
2.2 SFG Intensity and Vibrational Spectra
The SFG signal, generated at an interface, depends on the material parameter, χ
(2)
i jk .
In other words, sum frequency signal carries interfacial material response via χ
(2)
i jk
which is correlated with second-order molecular hyperpolarizability (β lmn ) and hence
with the vibrational/electronic transition of interfacial molecules. For vibrational
measurement, spatiotemporal overlap of a high power fixed frequency visible light
(E V I S , i.e., E ω 1 ) and a tuneable infrared frequency (E I R , i.e., E ω 2 ) on a material
surface (e.g., on the surface of water) will generate the SFG signal following Eq. 6.
When the VIS and IR fields are expressed in terms of their actual magnitudes, P
(2)
i,SFG
takes the form,
P
(2)
i,SFG = o χ
(2)
i jk (K j E V I S )(K k E I R )
(9)
K j and K k are the Fresnel factors which define the amplitude coefficients of the
reflected VIS and IR light, repectively. Using a similar Fresnel factor or ‘L-factor’,
P
(2)
i,SFG can be correlated to the sum frequency electric field (E i,SFG ) as follows [6,
10, 11].
E i,SFG = L i P
(2)
i,SFG = L i o χ
(2)
i jk (K j E V I S )(K k E I R )
(10)
Thus, the intensity of the emitted sum frequency light which is related to the
square of the electric field will be
I SFG ∝
E i,SFG
2 ∝
Li o χ
(2)
i jk (K j E V I S )(K k E I R )
2
(11)
Equation 11 is frequently expressed in the simplified form
I SFG ∝
χ
(2)
i jk
2
I V I S I I R
(12)
As discussed in Sect. 2, χ
(2)
i jk represents the ensemble averaged second-order
molecular hyperpolarizabilities (β lmn ) such that,
χ
(2)
i jk =
N s
o
β lmn
(13)
where N s is the number density of the surface molecules contributing to SFG signal,
indicates the ensemble average orientational distribution. (l, m, n) signify the
generic indices corresponding to molecular coordinate (a, b, c). At interface, orientation of a molecule is usually expressed by the angle of its principal axis with
the surface normal. It is very unlikely that the molecular symmetry axes (a, b, c)
will coincide with the laboratory-frame coordinate (x, y, z). Hence, to correlate χ
(2)
i jk
with the molecular property β lmn , a coordinate transformation is necessary which is
S. Roy et al.
2.2 SFG Intensity and Vibrational Spectra
The SFG signal, generated at an interface, depends on the material parameter, χ
(2)
i jk .
In other words, sum frequency signal carries interfacial material response via χ
(2)
i jk
which is correlated with second-order molecular hyperpolarizability (β lmn ) and hence
with the vibrational/electronic transition of interfacial molecules. For vibrational
measurement, spatiotemporal overlap of a high power fixed frequency visible light
(E V I S , i.e., E ω 1 ) and a tuneable infrared frequency (E I R , i.e., E ω 2 ) on a material
surface (e.g., on the surface of water) will generate the SFG signal following Eq. 6.
When the VIS and IR fields are expressed in terms of their actual magnitudes, P
(2)
i,SFG
takes the form,
P
(2)
i,SFG = o χ
(2)
i jk (K j E V I S )(K k E I R )
(9)
K j and K k are the Fresnel factors which define the amplitude coefficients of the
reflected VIS and IR light, repectively. Using a similar Fresnel factor or ‘L-factor’,
P
(2)
i,SFG can be correlated to the sum frequency electric field (E i,SFG ) as follows [6,
10, 11].
E i,SFG = L i P
(2)
i,SFG = L i o χ
(2)
i jk (K j E V I S )(K k E I R )
(10)
Thus, the intensity of the emitted sum frequency light which is related to the
square of the electric field will be
I SFG ∝
E i,SFG
2 ∝
Li o χ
(2)
i jk (K j E V I S )(K k E I R )
2
(11)
Equation 11 is frequently expressed in the simplified form
I SFG ∝
χ
(2)
i jk
2
I V I S I I R
(12)
As discussed in Sect. 2, χ
(2)
i jk represents the ensemble averaged second-order
molecular hyperpolarizabilities (β lmn ) such that,
χ
(2)
i jk =
N s
o
β lmn
(13)
where N s is the number density of the surface molecules contributing to SFG signal,
indicates the ensemble average orientational distribution. (l, m, n) signify the
generic indices corresponding to molecular coordinate (a, b, c). At interface, orientation of a molecule is usually expressed by the angle of its principal axis with
the surface normal. It is very unlikely that the molecular symmetry axes (a, b, c)
will coincide with the laboratory-frame coordinate (x, y, z). Hence, to correlate χ
(2)
i jk
with the molecular property β lmn , a coordinate transformation is necessary which is