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# Second Harmonic Generation and Absorption Spectra

Light of frequency shining on a nonlinear material can generate, in addition to the fundamental frequency , higher harmonics of frequencies , , , and so on. If the material is noncentrosymmetric, the dipoles produced by the valence electronic motion driven by the nonlinear polarization can produce even-order harmonic generation. The simplest even harmonic generation has double the fundamental frequency of and is known as second harmonic generation (SHG).
As in the linear optic case, the nonlinear susceptibility can be investigated classically using an anharmonic Lorentz model. From the model, one can calculate the resonance SHG intensity (from the absolute value of the susceptibility) as well as the SHG absorption spectrum (from the imaginary value of the susceptibility). This Demonstration shows that there are two resonance and absorption peaks at the resonance and half-resonance frequencies.

### 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.

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