Phase-matching

Before starting to explain NLO effects it is probably useful to remember some interference basics. Phase-matching - an interference phenomenon - is important for NLO effects. Here's why: In order to get light out of something one must provide that it doesn't cancel out by interference. That is, the phase of the light wave must match with the optical length in the medium, or else it will cancel out at the medium/outside-boundary. One can adjust the optical length of a light beam inside a medium, e.g. a cubic crystal, by tilting it. At certain angles the optical length is a natural multiple of the wavelength of the light one wants to let out.

[X⁽²⁾ effects

X⁽²⁾ is used in frequency conversion experiments. Lasers can generally operate over a] certain [frequency band].

// Solid state lasers can be tuned, the Ti:Sapp laser for example can be tuned from 700-1080 nm, a quite impressive bandwidth IMO. Dye-lasers operate on virtually ANY wavelength, depending only on the used laser dye... that's why I would re-write the next section as follows: //

Even though there are lasers available for virtually any wavelength, it is often simpler to manipulate the existing laser radiation instead of buying another - often expensive - laser source. [If] one [wants top perform an experiment at a] different [frequency] than the one that is available, one [can generally still access these] using some X⁽²⁾-tricks.

Frequency doubling or second harmonic generation (SHG)

In certain materials it is possible to double the frequency of light. Very specific crystals, also called non-linear optic crystals, or NLO-crystals are such materials¹. To see what happens in such a material it is best to look at the power series again:

Here we should note that E is the oscillating field - a function of time - with the fequency f:

The square of a wave function yields another wave function with the double frequency:

Note the 2f in the first term of the last equation. That's where the light with the double frequency is coming from; or in other words, light with half the wavelength - or: The first overtone, the second harmonic, of the original light wave. [In effect, you can take (say) two red photons, and out the other end comes one blue photon. Energy is conserved.] In order to get the second harmonic out of the NLO crystal all one has to do is provide phase-matching conditions (eg. by tilting the crystal).

Sum- and difference- frequency mixing

The frequency mixing is another X⁽²⁾ process. It is mathematically equivalent to the SHG, but instead of adding two equal frequencies, different frequencies are added (or subtracted).

Where the first sine in the expression has the difference and the second sine the sum frequency. Normally one of the frequencies is the one from the laser. in that way either E₁ or E₂ are high enough to compensate for the tiny X⁽²⁾ value. The frequency to be added or subtracted could stem from any other light source. One can also select which one of the operations should take place. That is, if one wants the addition or the subtraction. To do that one can tilt the crystal, so that the undesired operation is interferd away - phase-matching.

Optical parametric generation, or downconversion

This is yet another X⁽²⁾ process. It works just like the sum frequency generation but the other way round. Instead of making one photon out of two, two photons (with lower frequency) are made out of one. The resulting frequencies add to form the original frequency (again, energy is conserved). Using the optic parametric downconversion, one can generate a huge number of lower energy colours from one high energy frequency (that is red out of blue, but normally infrared out of red). One of the nice things about the downconversion is that a huge range of frequencies can be generated from one original frequency (or wavelength)