I was reading about second-harmonic generation (SHG) crystals (or frequency-doubling crystals) used to produce green laser light from IR.

What low-level process in the crystal is actually driving the combination of two photons into one? I'm thinking of a process at the QED level (or at least motivating what happens at that level). The wikipedia entry on SHG doesn't really address this (it just mentions it's a non-linear phenomena..).

Or put another way - what's the simplest system possible (theoretically) that can show the phenomena even if at a very low efficiency?

Edit: A supplement to the question, is what entropy-related constraints are there on the incoming photons in these processes, in order to avoid for example turning heat into nicer higher-energy photons?

  • $\begingroup$ Excellent question - I should know the answer but I don't. There doesn't seem to be many people around who can describe optical processes like this at the fundamental level - I've tried for a long time to find someone to draw Feynman diagrams with only fundamental particles in them (you see a few "Feynman diagrams" in texts with 'atoms" or "systems' in them). I'm guessing that this won't be photons truly combining: it will be a perfectly elastic absorption and re-emission such that no energy, no momentum and no angular momentum lost to the environment. Secondly, it will likely .... $\endgroup$ Feb 12, 2014 at 22:22
  • $\begingroup$ ...involve "real" states, as opposed to the virtual state that leads to two photon fluorescence (in the latter case, the virtual state is not halfway between the two real states: some energy is lost to the medium and, for example, fluorescein has a two-photon absorption peak at roughly 780nm, whereas its fluorescence peak is 532nm ($532\ne 780/2$!). SHG, as you know, outputs a true second harmonic. $\endgroup$ Feb 12, 2014 at 22:26
  • $\begingroup$ Thanks, yeah I have a bunch of other processes I'd like to understand at this level once we're done with this one ;) What's also interesting is what the "entropic getaway" is for this situation - I assume there are strict demands on the incoming photon kinematic properties (since in general you can't make nice high-energy photons out of chaotic heat photons). $\endgroup$
    – BjornW
    Feb 12, 2014 at 22:49
  • $\begingroup$ Bjorn, I think you should also add your last comment to your question, or maybe ask it as a second question once you've got an answer to the present one. It too is an excellent question and in practice SHG is always done with lasers - but, as you say, is this simply because lasers are a way to get to the intensities needed for the nonlinearity, or is there an "entropic gateway"? $\endgroup$ Feb 12, 2014 at 22:52

1 Answer 1


One thing to note is that it is not just the crystal, but a crystal which is coherently pumped by a strong laser field in which one sees SHG. In this setting, the pump creates a periodic modulation of the index of refraction (thus the requirement for a non liner medium) which effectively acts as a phase grating.

One way of thinking about SHG is that it is the dynamical Casimir effect. Here, instead of a mirror oscillating at optical frequency, the optical path length of the crystal oscillates at this frequency due to the nonlinear response of the crystal, creating photons from the vacuum, at the expense of the photons modulating the crystal.

In terms of more traditional photon-photon processes, you can of course write out Feynman diagrams and calculate the scattering matrices for the annihilation of two photons in one mode and creation of another in a phase matched direction. An example is given (For the inverse process) here. (see page 100 or so).

Also, to address your last point, (which I'm not as familiar with). As I understand it, parametric processes such as SHG do not transfer energy to the medium and so there no reservoir to couple thermal energy to the photons. The process is entirely coherent.

Hope this helps.

  • $\begingroup$ Thanks, and +1 for the link to the non-linear optics lectures as well. When you write that the crystal is pumped by a strong laser field, would you say that the doubling process can't happen without a macroscopic amount of photons in the process? Or it's more like you use the strong laser field to get a usable signal in practice? $\endgroup$
    – BjornW
    Feb 13, 2014 at 22:36
  • $\begingroup$ Hi Bjorn, the $\chi^{(2)}$ coefficient dictating the probability of the process occurring is a function of the pump strength. In principle a single pump photon could drive SHG, but the process would be unlikely. As an order of magnitude estimate: a good conversion efficiency at 1W of pump is about 0.1. 1W of blue light corresponds to roughly $10^{20}$ photons per second, and the process is linear in the E-field and thus goes as the square-root of the power. You could thus expect an efficiency of about $10^{-11}$ at the single photon/s intensity level, i.e. one event every 3000-ish years. $\endgroup$
    – Orko
    Mar 9, 2014 at 22:41

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