Biréfringent phasematching

The most common method of phasematching makes use of the fact that many materials are biréfringent, that is to say, light of orthogonal polarisations propagating in an arbitraiy direction through such a material experiences different values of refractive index. Given that refractive index also varies with wavelength, Equation 2.18 can be satisfied under certain circumstances. In general, nonlinear biréfringent crystals have three optical axes, defined relative to the crystallographic structure,and designated x, y and z. Each axis has an associated refractive index, %, ny and n^. A wave propagating with its polarisation parallel to one of those axes will experience the index of that axis, and is referred to as an ordinary wave, or o-wave. Light propagating with the orthogonal polarisation, at an angle to an axis, will experience a value of refractive index which varies with angle, and is called an extraordinary wave, or e-wave. This situation is illustrated by the index ellipsoid, which is a suiface defined by the equation

n n n

An example of this is depicted in Fig. 2.1. This shows that the refractive indices seen by an o-wave and an e-wave can be represented by the axes of an ellipse normal to a wave propagating at an angle 0. For the situation illustrated, «e varies between nx for 6 = 0°, and n^ for 0 = 90°, while «o stays fixed at «y. Hence, phasematching can be satisfied by choosing suitable polarisations and propagation angles.

value of n seen by e-wave value of n seen by ^ o-wave direction of propagation b-wave polarisation e-wave polarisation

Figure 2.1 The index ellipsoid, illustrating the values of refractive index seen by an o-wave and an e-wave propagating at an angle in the biréfringent crystal.

Biréfringent crystals are divided into two categories: uniaxial and biaxial. Uniaxial crystals have two values of refractive index equal, so the three indices reduce to only two, known as «o and «e- A uniaxial crystal may be positive, where «e > /%o, or negative, where «o > «e- A biaxial crystal has ^ and either nx> r i y >

or Uz> Uy> Hx.

Different combinations of ordinary and extraordinary waves can be used to achieve the required effect; these combinations are referred to as Type I and Type II phasematching, and for parametric generation in positive and negative crystals are:

Positive Negative Type I o->e + e e->o + o Type II o e(o) + o(e) e o(e) + e(o)

The variations of refractive index with wavelength for crystals ai e described by Seilmeier equations, which are determined experimentally. Examples of Seilmeier equations for various materials can be found in Chapters 4, 5 and 6. After the selection of an appropriate nonlinear material, the Seilmeier equations can be used in the phasematching condition of Equation 2.19. By taking the angulai' variation of refractive index into account as well, the propagation angle required to generate specific signal and idler wavelengths from a given pump wavelength can be determined. For the configuration shown in Fig. 2.1, the angular variation of «e is given by

Icos^ 0 sin^0

* ^ ^ (2.22)

Tunability of the output of an OPO is frequently desirable, and this angular variation of index leads to the possibility of so-called angle tuning. The crystal in an oscillating OPO is slowly rotated, so that the incident pump beam experiences a vaiying refractive index, and the signal and idler wavelengths must alter to compensate for this and maintain phasematching. Other tuning methods include pump tuning, where the crystal is kept stationaiy while the pump wavelength is varied, and temperature tuning, which exploits the fact that some biréfringent materials exhibit a significant variation of refractive index with temperature.

Biréfringent phasematching can be further divided into the techniques of critical and noncritical phasematching. The latter method is configured such that the pump, signal and idler waves all propagate along a crystal optical axis, say the x-axis, and consequently see, in this case, either Uy or Under these conditions, the beams remain collinear throughout the crystal and can therefore interact over the whole crystal length. It is evident fi’om Equation 2.15 that this is desirable, because of the dependence of gain on the square of the interaction length. This arrangement obviously precludes the use of angle tuning, so pump and temperature tuning must be relied upon. However, often these techniques are unable to generate the desired

wavelengths, and it becomes necessary to propagate the beams at an angle to one of the optical axes. This arrangement is called critical phasematching, and is unfortunately prey to a problem known as Poynting vector walk-off, or spatial walk-off, which is illustrated in Fig. 2.2. An ordinary wave propagates through a crystal with its momentum vector, parallel to its Poynting vector, S', with the wavefronts normal to both vectors. However, an extraordinary wave suffers walk- off, where the Poynting vector moves away from the A:-vector by an angle p, the walk-off angle. The wave fronts remain normal to the A^-vector. Thus an ordinary and extraordinary beam will overlap with each other for a short distance only, and the interaction between them will be greatly reduced, leading to low gain and poor efficiency. Since birefnngent phasematching requires at least one extraordinary wave, Poynting vector walk-off affects a critically phasematched OPO, but not a noncritical OPO, where all the waves propagate along an optical axis in the crystal.

optic axis propagation direction interaction length D e-wave -► ^o o-wave_k r\

Figure 2.2 Poynting vector walk-off, whereby the Poynting vector of an e-wave moves away from the k-vector when the wave propagates at an angle to the optic axis, whereas those of an o-wave remain collinear.

One way of overcoming the problems of walk-off is to use noncollinear phasematching. The ordinary and extraordinary beams are propagated with their k~ vectors at an angle to each other in such a way that the extraordinary ^-vector walks onto the ordinary ^-vector, increasing the interaction volume. This subject is dealt with in more detail in Section 4.3, with specific reference to a noncollinearly phasematched OPO based on KTi0As0 4.

The walk-off angle for an extraordinary wave can be calculated from the dot product of the unit vectors of the electric field vector E, which is normal to the Poynting vector, and the displacement vector D, which is normal to the A-vector (Fig. 2.2):

p — arccos( jO) 23)

A detailed calculation of the walk-off of an e-polarised signal wave in KTi0As04 is given as an example in Appendix A.