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Current understanding:

Nonlinear optics is classified by the effect of an electric field applied to a medium. In nonlinear crystals, when a field is applied the dielectric polarization does not respond proportionally to it.

Nonlinear mediums are employed to change the wavelength of light. When two rays of the same frequency are phase matched incident to such a medium, they are absorbed and remitted from the surface as a single ray with double the frequency.

On an individual photon basis, two photons enter the material and are destroyed, while another is created with twice the frequency due to energy and momentum conservation. (Two are absorbed then a single re-emitted.)

My source of confusion:

Frequency mixing seems to occur if and only if photon absorption takes place. However, non-linear crystals are transparent, so most photons are not absorbed, and are instead scattered by the electric fields of surrounding atoms.

The question:

If very little absorption takes place, by what mechanism does frequency mixing take place?

Bonus question: If frequency mixing does take place, are all photons mixed? Or does unmixed light leave the crystal as well?

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Frequency mixing is the mechanism. Mental pictures of the form "two photons are absorbed and one bigger photon is emitted" are useful but they're ultimately just not that accurate, unless you're willing to consider any interaction of light with materials (including the dispersion of light by glass) as continuous absorption-and-emission loops.

That said, in frequency mixing there generally is a transfer of energy from one beam to the other. If the regime is right, the amount of energy that's actually deposited into the material can be negligible, so any energy that makes its way to the $2\omega$ mode is taken out directly from the $\omega$ driver. The physical mechanism is the excitation of a polarization $\mathbf P(t)$ in the medium that produces an electric field that reinforces the $2\omega$ and interferes destructively with the $\omega$. In the ideal case, the polarization only mediates the exchange and it only keeps a negligible amount of energy itself (though the nonlinear equivalent of the Kramers-Kronig relations places restrictions on how much this is possible).

If the process is perfectly phase-matched, then it is possible to achieve a $100\%$ conversion efficiency, though this will normally take a good bit of finesse in the lab to achieve. Achieving this also requires the length of the medium to be precisely correct: if the medium gets too long, the energy flow will reverse (through stimulated parametric down-conversion) until all the energy is back in the fundamental.

In most real-world applications, though, only a fraction of the energy of the driver is utilized. This is an issue with e.g. green laser pointers, which are actually frequency-doubled IR lasers and need a filter to kill the (dangerous and invisible) IR beam; in low-quality pointers this is missing, causing a safety hazard that can be diagnosed with a simple home experiment.

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  • $\begingroup$ I actually do consider any interaction of light with materials as continuous absorption and emission loops. With the added consideration of lights path integral which takes into account all possible outcomes as opposed to a single path. Is something wrong with this line of thinking? If so, is it incorrect or just inefficient? $\endgroup$ – user400188 Dec 5 '17 at 23:30
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    $\begingroup$ No, there's nothing wrong with it, as such. It does, however, hide the fact that all of those virtual loops are best seen as a single quasi-particle that's being looked at from an inconvenient basis. That's not fatal in this case, but choosing the wrong basis can cause you to miss out on important quasiparticles - see e.g. Robert Laughlin's Nobel lecture for some examples where the quasiparticle really is king. $\endgroup$ – Emilio Pisanty Dec 5 '17 at 23:33
  • $\begingroup$ One comment on your answer, writing: " If the regime is right, the amount of energy that's actually deposited into the material can be negligible, so any energy that makes its way to the 2ω mode is taken out directly from the ω driver. " Makes it sound like energy can be taken out of the material. That is, it can be supplied from someplace other than the ω driver. From the rest of the content, it is perfectly clear that this is not the case. I am just pointing this out to improve your answer. $\endgroup$ – user400188 Dec 5 '17 at 23:48

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