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Imagine a large bandgap material which is irradiated by an intense laser beam. If the photon energy is only high enough for 1/5 of the bandgap, is there a way to approximate the absorption by 5-photon excitation, i.e. the ratio of transmitted to initial Intensity?

All I found is related to 2 or 3 Photon absorption but not higher orders in the pertubation series ...

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On a practical level, I'm not sure that it matters. Did you notice how strongly the rate for those processes depended on intensity? It will get (much) worse for more photons and you need a pretty intense laser to make much use of the 3 photon process. – dmckee♦ Jan 16 at 16:13
Indeed, those effects are extremely small. I want to show that this process is irrelevant in comparison to any intensities that might be produced in a lab. I could go for a rough estimate and just say that if 2 or 3-photon absorption has very small values, then 5 will certainly be neglegible. – BandGap Jan 17 at 17:22
For a very rough, handwaving appraoch, the leading order difference between the two- and three-photon dependence on intensity gives you a idea of how adding a photon changes the scaling. – dmckee♦ Jan 17 at 17:34
Hmm, could you elaborate on what you mean by "leading order difference"? – BandGap Jan 17 at 18:02
The rate for each process will be expressible as a function of the intentist, $I$ in some form like $R(I) = aI^n + bI^{n+1} + \dots$. The exponent $n$ is the leading order. I believe I recall seing that $R_2(I) \propto I^3$ and $R_3(I) \propto I^5$ which suggests you get two powers of $I$ for each additional photon meaning that $R_5(I) \propto I^9$. Which is certainly not rigorous, but would be enough for me to put the issue on a back burner. – dmckee♦ Jan 17 at 19:05
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The best approximation I found so far is in Bloembergen: Nonlinear Optics. It is stated that successive orders of Polarisation are reduced by a factor

$|E|/|E_{at}|$,

with the applied electric field $E$ and the atomic electric field $E_{at}$. This will be sufficient for my case.

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