Details

The linear and second-harmonic susceptibility can be calculated from the one-dimensional anharmonic Lorentz model:

,

where is the displacement of the electron, is the fundamental frequency of the system, is the electronic charge, is the damping constant, is the electron mass, is the driving field, and is the nonlinear coefficient given by

,

where is the lattice constant. Solving the model using an approximation method similar to the Rayleigh–Schrödinger perturbation theory gives the linear susceptibility

and the second harmonic susceptibility

,

where

is the linear susceptibility for the frequency .

For a step-by-step derivation of the formulas, see [1].

Here

,

, the vacuum permittivity,

,

,

,

,

,

.

The absorption curve can be obtained from . Although not shown here, the real SHG susceptibility can simply be obtained using . The real linear and SHG susceptibility does not violate causality because it is consistent with the Kramers–Kronig relation. This is, however, not always the case in nonlinear optics, especially in time-resolved spectroscopy involving pump-probe input (e.g. four-wave mixing), where when coming into the nonlinear material, the pump signal precedes the probe signal, or in processes where the probe produces self-induced absorption on the material. More on this can be found in [2].

References

[1] R. W. Boyd, Nonlinear Optics, 2nd ed., San Diego: Academic Press, 2003.

[2] V. Lucarini, J. J. Saarinen, K.-E. Peiponen, E.-M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research, New York: Springer, 2005.