Introduction

In a classical sense Optics is the science that deals with the paths of light. In a more modern sense optics will have to deal with the interactions between light and matter, because the paths are strongly dependent on this interaction For a long time it was thought that interactions between light and matter were linear - that is, the amount of some interaction was directly proportional to the amount of light involved. If one were to observe otherwise, some nonlinear effect is taking place.

Nonlinear optics is a relatively new field. While the first nonlinear effects were seen in the 1960's with the invention of the laser, the effects themselves had been postulated decades beforehand, even in the 1930's. It is only with the invention of the laser that it has been possible to attain the high intensities required to experiment in this field. Nonlinear optics is now a well-developed science, and is used for many technical applications.

A. Linear optics.

To explain nonlinear optics (NLO) it is best to recap linear optics. Linear optics includes the well-known phenomena of absorption, emission, reflection, refraction and diffraction.

The formal description of linear optics and indeed nonlinear optics requires quantum mechanics. Instead of starting with the Schrodinger equation, one can imagine the processes by looking at a commonly used analogy.

The most common analogy goes like this: imagine a spring with a mass on the end. If a force is applied to this, the spring will extend and there will be a restoring force generated by the spring to oppose the motion. Upon releasing the mass after extending the spring simple harmonic motion¹ begins. The mass will then oscillate forever.

If other forces are applied to the mass, in particular a damping force which causes the oscillation to slow and stop, and a driving force (with some frequency) then it is called damped, driven harmonic motion. Although the calculations are tedious, it can be shown that the oscillation of the mass has a frequency that exactly matches that of the driving force. This is the principle of linear optics. We replace the driving force with an electric field (like in a laser beam) and the displacement of the mass by a quantity known as the electric polarisation. The mass itself becomes the electrons in the atoms of the matter we put the light through, and these exhibit a restoring force and damping force when displaced.

The polarisation can then be defined in terms of the incident optical (electric) field, and there is a relationship between them, which is that the polarisation equals the electric field multiplied by some term called the electric susceptibility, which is a complex term. The real part of this gives the refractive index of the material and the imaginary part gives the absorption or emission characteristics. At low intensity the relationship is linear, so that the polarisation is proportional to the field applied, and hence (when the medium 'relaxes') the emitted/transmitted radiation has the same frequency.

Linear optics is responsible for, or is the domain of, effects like refractive index, dispersion², birefringence³, absorption and emission.

B. Nonlinear Optics

In our spring analogy, when the mass is displaced a long way from its equilibrium position, the relationship becomes more complicated. In optics, a similar thing happens. The exact relationship between polarisation and electric field is a power series; thus polarisation is an addition of a linear component, plus a term with field squared, plus a term with field cubed and so on to higher powers.

P(E)=X⁽¹⁾E + X⁽²⁾E² + X⁽³⁾E³ + ...

Thus polarisation is an addition of a linear component, plus a term with field squared, plus a term with field cubed, and so on. Each term has its own susceptibility⁴ factor, X⁽¹⁾, X⁽²⁾ etc. The susceptibilities from X⁽²⁾ onwards are known as nonlinear susceptibilities. The symbol for susceptibility is the Greek character chi, which looks very similar to an X. For the squared term, known as a second-order nonlinearity, one talks about X⁽²⁾; for the cubed term, known as a third-order nonlinearity, one talks about X⁽³⁾ and so on.

In a strict sense X⁽ⁿ⁾ is not a number, but a tensor⁵, which is why symmetry plays such an important role in certain nonlinear processes. Roughly speaking, X⁽ⁿ⁾ will give a system of numbers which become very small the higher the order n. Or putting it another way, the effects beyond X⁽²⁾ are very small. For that reason, under normal light conditions one will not observe second- or higher-order effects, because the numbers are thousands or millions of times smaller than the first-order effects. Nevertheless, these factors become important when high intensity fields are used, as the very large values of E² and E³ and so on compensate for the low values of X. Such intensities can be achieved using lasers, and especially pulsed lasers.

Each order of nonlinearity gives rise to different effects, and within each order there can be several slightly different effects.

X⁽²⁾ effects.

X⁽²⁾ is used in frequency conversion experiments. Lasers can generally operate over a small frequency band, and there are a limited number of frequencies that lasers exist for. If an experiment needs a frequency that does not exist as a simple laser frequency, one can generally still perform the experiment, but by being a little more cunning. There are different ways of doing this, but most of these are based around some X⁽²⁾ process.

1. Frequency doubling.

In certain materials it is possible to double the frequency of the light. Crystals are commonly used (very specific ones) and when a high intensity laser beam is sent though it, some of the photons are 'glued' together. In effect, one can take (say) two red photons, and out the other end comes one blue photon. Energy is conserved.

2. Sum- and difference- frequency mixing.

Another X⁽²⁾ process. Here two different frequency laser beams are sent into the crystal, and the photons can mix together and form either the sum of the two frequencies, or the difference.

3. Optical parametric generation.

Yet another X⁽²⁾ process. Here, a single frequency of light is sent into a crystal, where the original photon is split into two photons, whose energies add up to give the original energy of the first. We call the higher energy photon the signal and the lower energy photon the idler.

X⁽³⁾ effects.

These are based on the cube of the electric field, and can be used in a number of different applications. X⁽³⁾ is especially useful in fibre optic communications.

1. Self-focusing.

The optical Kerr effect is an effect where the higher the intensity of the light, the higher the refractive index. This can lead to problems, since it means that in any medium, a high intensity beam will focus itself down - it is possible to create a lens by having a high refractive index in the centre and lower to the edges, rather than the traditional curved surfaces. This can cause catastrophic damage to components if the focusing occurs within them.

2. Self-phase modulation

This is an awkward one to explain in layman's terms. A travelling wave has a property called phase. The optical Kerr effect can modify the refractive index around an intense laser beam, altering the phase. The rate of change of phase with time is the optical frequency, hence if phase is modulated we can generate new frequencies close to the frequency of the original wave. In materials with very high nonlinearity, this means that we can generate white light from only a small frequency width input.

3. Multiphoton absorption

When a beam intersects with a medium, radiation will be absorbed if the frequency of light matches an absorption band or transition within the material. However, if there is a very intense beam, absorption can occur where the medium removes two photons at half the frequency, or three at one-third, and reacts as if they add up to a single photon at the multiple of frequency required. The photons must interact within a very short timescale (around one femtosecond or so).

All of these optical effects can be useful. Some can be a pain in the neck, especially if they are unwanted, but much useful science can be done using them. There are other effects too, but these are the main ones.