Origin of SFG Process

The induced dipole ( I) of a molecule placed in an electric field ( ) is given by 

Where a, P and y are the linear polarizability, the first- and second-hyper polarizabilities, respectively, and are represented by second, third and fourth rank tensors, respectively, and is a static polarizability. 

In condensed phases, it is more convenient to consider the dipole moment per unit volume or polarization (P) 

When the electric field ( ) is weak, as in a non-laser light source, the second and third terms can be neglected and polarization (P) can be expressed as below, corresponding to linear optics. 

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