Phase matching and acceptance parameters

2.5.1 Phase matching

Phase matching is the term used to describe processes by which the momentum or phase mismatch term, Ak, can be made equal to zero. The name phase matching arises because in this way the free electromagnetic waves propagate with the same phase

velocity as the non-linear polarisation waves which give rise to them. This is necessary to allow the waves to grow continuously. A number of ways of achieving phase matching have been achieved, of which the most common is the use of the birefringence of the material.

As given previously in section 2.4. the phase match relation is defined as

(2.4.8) Using the relation ki = 27tniAi, where ni are the refractive indices and X[ are the free

space wavelengths (i =p,s,i), we can re-express the phase matching condition in the more useful form of

Xp Xi (2.5.1.1)

In normally dispersive materials, the refractive index decreases with increasing wavelength, and the condition given by eqn 2.4.8 cannot generally be met. However, in

s,k

s,k

X

Ch. 2 : Theory o f Parametric Interactions

anisotropic crystals, which most materials used in second order non-linear experiments are, the refractive index depends additionally on the direction of propagation with respect to the dielectric axes and on the polarisation of the wave [25]. This behaviour is shown schematically in fig. 2 . 1 which shows an example of the optical indicatrix, or

index ellipsoid, and the axial convention for describing it.

Theory shows that two waves, with orthogonal polarisations, can propagate independently in the crystal. These waves must satisfy the Fresnel equation which determines the dependence of refractive index with propagation direction, and can be written as [26]

sin^ Qcos^ (p , sin^ 6sin^ tp , cos^ ^ _ n ^ ^ The two orthogonal polarisations experience different refractive indices, except in the case where the direction of propagation is along an optic axis. Thus by propagating through the crystal at a specific angle with a specific polarisation we can control the refractive index, and hence the k values, and in certain circumstances we can satisfy eqn 2.4.8, the phase matching condition. Because eqn 2.4.8 is a vector relation, it is possible to achieve phase matching for waves which are either collinearly or noncollinearly propagating.

Angular phase matching, as described above, is not the only means by which phase matching can be achieved. The refractive indices can also be changed by controlling the temperature of the crystal, though the dependence on temperature is stronger in some crystals than in others. If adjustment of the temperature allows phase matching to be achieved for propagation along one of the principal axes of the optical indicatrix, then this has advantages in terms of angular and spectral properties which will shortly be described.

Another technique which has seen success is the use of guided wave and periodic structures. As the phase velocity of a guided wave can differ significantly from that of the same wave in the bulk medium, it is possible to find combinations of modes in which the phase matching conditions are satisfied. Phase matching can also be achieved by introducing a periodicity into the structure (notably for guided waves) that modifies either the propagation constants, or the sign or magnitude of the non-linearity. A reversal of the sign of the non-linearity with a periodicity equal to an odd integer multiple of the coherence length (defined by Akz = n) is referred to as quasi-phase matching and was in fact one of the original techniques suggested by Armstrong et al [5]. Ignoring the difficulty of producing these structures, they have the advantage that they allow phase matching where it was otherwise unobtainable, and they also allow the use of large non-linearities which may also not have been accessible with phase

Ch.2 ; Theory of Parametric Interactions matching by biréfringent techniques.

2.5.2 Angular acceptance

As mentioned previously, the gain that can be obtained depends on the intensity of the interacting beams. Thus it is obviously advantageous to have as high an intensity as possible in the crystal to maximise the gain. The limit to this being that at some point the crystal will start to damage [27]. If the pump source is of limited energy then it is likely that to approach the damage threshold, the beam will have to be focused. Focusing of the beam inherently means that different parts of the beam will propagate through the crystal at different angles, and it will therefore become impossible to satisfy the phase match condition for the whole beam simultaneously. It is therefore appropriate to define a degree of focusing that can be utilised before the deviation away from phase matching negates the effect of the focusing. This is commonly referred to as the angular acceptance of the crystal, and it is defined to be the angular deviation from perfect phase matching at which the magnitude of the mismatch is no greater than tc/1

[28].

Generally the direction through the crystal at which phase matching occurs is specified by two angles, 9, the angle between the k vector and the z-axis and, (|), the angle between the projection of the k vector in the xy plane and the x-axis. This means that an acceptance angle can be defined with respect to both these planes. The derivation applies to both. If we expand the k-vector, to second order, about one of the phase matching angles, say 0, this can be represented by

+ (2.5.2.1)

Inserting this in eqn. 2.3.4 and applying the condition that the centre wavelengths are matched, kp^ - k^^ - = 0, we get

Ak - 2n

dQ A. dQ <50+ 7T

1 1 1 d% Se^

(2.5.2.2) where the derivatives are evaluated at 9.

For phase match angles greater than 5°, expansion to first order is sufficiently accurate [29]. If propagation is close to one of the principle dielectric axes of the crystal, i.e. 9 =

0 or 9Q0, then the first term vanishes and the remaining second order term means that

the effect of the angle on the phase mismatch is much reduced. This results in an increase in the acceptance angle of the crystal for this direction, which, because of the relaxation of alignment sensitivity, is termed 'non-critical phase matching' (NCPM). In the case where the first term dominates, the tighter alignment sensitivity leads to

Ch. 2 : Theory of Parametric Interactions 'critical phase matching’.

If the condition on the maximum magnitude of the allowable angular phase mismatch, Ak = 7t/l, is applied then an acceptance angle can be calculated from the resulting linear

(critical) or quadratic (non-critical) equation in 6 6, which is normally quoted in

mrad.cm for critical and mrad.cm^ for NCPM.

2.5.3 Poynting vector walk-off

Critical phase matching does have a major disadvantage. In order to achieve phase matching, at least one of the waves must be of extra-ordinary or e- polarisation [25]. Because of the angular dependence of the refractive index, as described earlier, e-waves propagate with elliptical rather than spherical wavefronts. This results in the direction of energy flow, referred to as the Poynting vector, differing from the direction of the wave front normals. This means that although an e- and an o-wave may start off in the same direction, the e-wave will follow the Poynting vector and 'walk off from the direction of the o-wave. This effect is seen to have a dramatic effect on amplification and oscillation thresholds as it results in a decoupling of the waves.

This effect is closely related to angular acceptance in that focusing of the interacting beams beyond a certain point reduces the coupling between the beams despite the fact that higher intensities are available due to the effects of focusing. In the case of angular acceptance, larger spot sizes, and hence less divergent beams, results in less phase mismatch across the wavefront. Larger spot sizes also reduce the effects of walk-off as for a given amount of walk-off, specified by the walk-off angle p, the fraction of the beams still overlapping after a distance, Ic, will be larger for larger spot sizes. This is perhaps more easily explained with the use of a diagram, see fig. 2.2, in which the rule

of thumb of w = pic, as a minimum spot size is illustrated.

I

fig. 2.2 Effect of walk off on the interaction of two beams

Ch. 2 : Theory of Parametric Interactions

match geometry where all wavefronts approximate to circular and the wavevector and poynting vector are collinear for all three waves (assuming collinear phase matching).

2.5.4 Spectral Acceptance

Obviously the phase mismatch condition can only be satisfied exactly for one set of pump, signal and idler wavelengths. It is of interest to know by how much these can vary before the gain is reduced by the resulting phase mismatch. This is completely analogous to the calculation of angular acceptance previously carried out. An equation similar to eqn S.2.2 . 2 can be derived where the k-vector is expanded in terms of ScD or

dX. The pump spectral acceptance is calculated by holding the signal and idler frequencies fixed, and equally the bandwidth of signal and idler can be calculated by holding the other two frequencies constant, remembering that the additional relation

5co^ = -ÔÛ). is required for energy conservation. The actual expression for the signal

bandwidth for the case of o-oe phase matching in KTP is given in section 5.8.1.