1. What is Phase?

Phase is one of those terms that is bandied about in science-fiction TV shows because it sounds vaguely technical and difficult to understand. Most of the time they get it wrong, or at best semi-close, but still a fair distance off.¹



So what is phase? Well, let's go back to the mathematical description. Imagine a simple sine wave. This has the mathematical equation

y = sin(x)

A wave in phase with this wave has the equation

y = sin(x)

and one which is out of phase has the equation

y = sin(x+π)

where the π refers to the two waves having a phase difference of π radians, or 180°.


This is one problem that there is when defining phase. There is no such thing as absolute phase in a wave. Only the phase difference, strictly speaking, can be defined.


The waves above have the same frequency and so relative phase is fairly straightforward to calculate. However, this need no be the case. It is possible for the two waves to be at different frequencies. This means that in space, one wave seems to 'slip' past the other, and when the two are added strange things happen (as an example of this, try adding together sin(x) and sin(1.25x) and see what happens - if you have a graphic calculator or even at a push Excel). At one point in space the two waves wil be in phase, at another they will be out of phase.

2. How does phase work in optics?

Well, in optics our laser beam is an electromagnetic wave.² This wave, believe it or not, is a sine wave³.If two light waves, both at the same frequency, are present, one can describe the phase difference between them exactly as was done in the abstract mathematical case in section 1. However, if the frequencies are different then problems occur, especially if the two waves are travelling through a medium, and not a vacuum, because dispersion begins to play a part too.

3. Matching phase - how and why?

All of this comes together when frequency conversion experiments in nonlinear optics are considered. The example here is second harmonic generation, but a similar procedure will cover parametric generation and sum and difference frequency mixing.



In second harmonic generation, a laser beam with a frequency ω (the fundamental) is sent into a crystal, and it generates light at a frequency of 2ω (the harmonic). There are coupled differential equations that cover this, but for un-phasematched light the upshot is that over a distance called the coherence length light is converted from the fundamental to the harmonic and back-converted to the fundamental, so at this distance there is no harmonic at all. It then starts to convert back and forth over further coherence lengths. The coherence length can be thought of as the distance over which the two waves have a phase difference of less than 90°. This phase slippage occurs because the fundamental travels faster through the cyrstal than the harmonic, because of dispersion.



There is a way around this, though. Most nonlinear crystals are birefringent. If the angle of the crytal is chosen to be just right, and the harmonic and fundamental are in different polarisations⁴ then the fundamental can have a higher refractive index than the harmonic, resulting in a (slight) reduction in speed. This, if the maths is right, will give the harmonic the same speed as the fundamental, and so they remain in phase. There will be no back conversion and the harmonic will continue to grow.

The question of phasematching gets a little more complicated in other frequency conversion situations, but essentially a similar process will be followed.

4. Quasiphasematching, or cheating

There is another way to get round the back-conversion problem. The conversion direction is dependent on the exact crystal structure in the direction of propagation. If the crystal structure is flipped, back-conversion can be prevented and gain in the harmonic can be seen all the way through. This is called quasiphasematching. This can be applied down the entire length of such a crystal. The process of creating such a crystal structure is called poling, as the polarity of the crystal is changed. The manufacture of such crystals requires a voltage of around 100kV to be applied across a crystal of around 1cm width and so really should not be tried at home. To distinguish these crystals from normal ones (which will of necessity be made of the same material) the name of the crystal is prefixed by PP. So, if you have a crystal of RTA⁶, and periodically pole it⁷, the resulting crystal is called PPRTA.