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How does the nonlinear absorption coefficient depend on the band gap? How can that coefficient be calculated theoretically? (Preferably with an example)

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Two photon absorption is the nonlinear mechanism most sensitive to band gap, and it depends on the band gap being twice $\hbar \omega$ or smaller. It can be estimated using standard second order perturbation theory. See for example http://aristotle.sri.com/srini/73-JAP.pdf

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