IR-visible sum frequency generation spectroscopy uses two laser beams (an infrared probe, and a visible pump) that spatially and temporally overlap at a surface of a material or the interface between two media. An output beam is generated at a frequency of the sum of the two input beams. The two input beams must be able to access the surface with sufficiently high intensities, and the output beam must be able to reflect off (or transmit through) the surface in order to be detected. Broadly speaking, most sum frequency spectrometers can be considered as one of two types,
scanning systems (those with narrow bandwidth probe beams) and
broadband systems (those with broad bandwidth probe beams). For the former type of spectrometer, the pump beam is a visible wavelength laser held at a constant frequency, and the other (the probe beam) is a tunable infrared laser — by tuning the IR laser, the system can scan across molecular resonances and obtain a vibrational spectrum of the interfacial region in a piecewise fashion.
Nonlinear susceptibility For a given nonlinear optical process, the polarization \overrightarrow{P} which generates the output is given by :\overrightarrow{P} = \epsilon_0\left(\chi^{(1)}\overrightarrow{E} + \chi^{(2)}\overrightarrow{E}^2 + \chi^{(3)}\overrightarrow{E}^3 + \dots + \chi^{(n)}\overrightarrow{E}^n\right) = \epsilon_0 \sum_{i=1}^n \chi^{(i)}\overrightarrow{E}^{i} where \chi^{(i)} is the ith order nonlinear susceptibility, for i \in [1,2,3,\dots,n]. It is worth noting that all the even order susceptibilities become zero in
centrosymmetric media. A proof of this is as follows. Let I_{inv} be the inversion operator, defined by I_{inv} \overrightarrow{L} = -\overrightarrow{L} for some arbitrary vector \overrightarrow{L}. Then applying I_{inv} to the left and right hand side of the polarization equation above gives :I_{inv}\overrightarrow{P} = -\overrightarrow{P} = I_{inv}\left(\epsilon_0 \sum_{i=1}^n \chi^{(i)}\overrightarrow{E}^{i}\right) = \epsilon_0\sum_{i=1}^n \chi^{(i)}\left(I_{inv}\overrightarrow{E}\right)^{i} = \epsilon_0\sum_{i=1}^n (-1)^i\chi^{(i)}\overrightarrow{E}^{i}. Adding together this equation with the original polarization equation then gives :\overrightarrow{P}-\overrightarrow{P} = \overrightarrow{0} = \epsilon_0\sum_{i=1}^n \left(1 + (-1)^i\right)\chi^{(i)}\overrightarrow{E}^{i} = 2\epsilon_0\sum_{i=1}^{n/2} \chi^{(2i)}\overrightarrow{E}^{(2i)} which implies \chi^{(2i)} = 0 for i \in [1,2,3,\dots,n/2] in centrosymmetric media.
Q.E.D. [Note 1: The final equality can be proven by
mathematical induction, by considering two cases in the inductive step; where k is odd and k is even.] [Note 2: This proof holds for the case where n is even. Setting m = n - 1 gives the odd case and the remainder of the proof is the same.] As a second-order nonlinear process, SFG is dependent on the 2nd order susceptibility \chi^{(2)}, which is a third rank tensor. This limits what samples are accessible for SFG. Centrosymmetric media include the bulk of gases, liquids, and most solids under the assumption of the electric-dipole approximation, which neglects the signal generated by multipoles and magnetic moments.
SFG intensity The output beam is collected by a detector and its intensity I is calculated using :I(\omega_3;\omega_1,\omega_2)\propto|\chi^{(2)}|^2I_1(\omega_1)I_2(\omega_2) where \omega_1 is the visible frequency, \omega_2 is the IR frequency and \omega_3 = \omega_1 + \omega_2 is the SFG frequency. The constant of proportionality varies across literature, many of them including the product of the square of the output frequency, \omega_2 and the squared secant of the reflection angle, \sec^2 \beta. Other factors include index of refractions for the three beams. Because it is not currently known how to adequately correct for nonresonant interferences, it is very important to experimentally isolate the resonant contributions from any nonresonant interference, often done using the technique of nonresonant suppression. The resonating contribution is from the vibrational modes and shows changes in resonance. It can be expressed as a sum of a series of Lorentz oscillators :\sum_q \frac{A_q}{\omega_2-\omega_{0_q}+i\Gamma_q} where A is the strength or amplitude, \omega_0 is the resonant frequency, \Gamma is the damping or linewidth coefficient (FWHM), and each q > 1 indexes the normal (resonant vibrational) mode. The amplitude is a product of \mu, the induced dipole moment, and \alpha, the polarizability.
Orientation information From the second order susceptibility, it is possible to ascertain information about the orientation of molecules at the surface. \chi^{(2)} describes how the molecules at the interface respond to the input beam. A change in the net orientation of the polar molecules results in a change of sign of \chi^{(2)}. As a rank 3 tensor, the individual elements provide information about the orientation. For a surface that has
azimuthal symmetry, i.e. assuming C_{\infty} rod symmetry, only seven of the twenty seven tensor elements are nonzero (with four being linearly independent), which are :\chi^{(2)}_{zzz}, :\chi^{(2)}_{xxz} = \chi^{(2)}_{yyz}, :\chi^{(2)}_{xzx} = \chi^{(2)}_{yzy}, and :\chi^{(2)}_{zxx} = \chi^{(2)}_{zyy}. The tensor elements can be determined by using two different polarizers, one for the
electric field vector perpendicular to the
plane of incidence, labeled S, and one for the electric field vector parallel to the plane of incidence, labeled P. Four combinations are sufficient: PPP, SSP, SPS, PSS, with the letters listed in decreasing frequency, so the first is for the sum frequency, the second is for the visible beam, and the last is for the infrared beam. The four combinations give rise to four different intensities given by :I_{PPP}=|f'_zf_zf_z\chi_{zzz}^{(2)}+f'_zf_if_i\chi_{zii}^{(2)}+f'_if_zf_i\chi_{zii}^{(2)}+f'_if_if_z\chi_{iiz}^{(2)}|^2, :I_{SSP}=|f'_if_if_z\chi_{iiz}^{(2)}|^2, :I_{SPS}=|f'_if_zf_i\chi_{zii}^{(2)}|^2, and :I_{PSS}=|f'_zf_if_i\chi_{zii}^{(2)}|^2 where index i is of the interfacial xy-plane, and f and f' are the linear and nonlinear Fresnel factors. By taking the tensor elements and applying the correct transformations, the orientation of the molecules on the surface can be found. == Experimental setup ==