Walk-Off and Phase-Matching Bandwidth in Nonlinear Crystals

The temporal walk-off results from a mismatch of the frequency-dependent group velocities in a crystal, i.e. from a group velocity mismatch. It is quite obvious that this effect can degrade the conversion efficiency of nonlinear processes, if originally copropagating pulses at different wavelengths lose their temporal overlap during the passage through a crystal. It is interesting to realize that this effect is directly related to the finite phase-matching bandwidth: the frequency dependence of the phase mismatch is proportional to the group velocity mismatch. In fact, the loss of overlap in the time domain and the limited bandwidth are just two different descriptions of the same fundamental physical phenomenon.

The following thought may actually cast doubts on that claim. Consider a chirped ultrashort pulse participating in a nonlinear interaction. Its optical spectrum is broader than that of a transform-limited pulse with the same pulse duration. Therefore, it can be more strongly affected by the finite phase-matching bandwidth, while it might not appear to be more vulnerable to temporal walk-off. So we have a situation where in the time domain it is less apparent that the conversion efficiency can be reduced. But it is in fact not unusual that a certain phenomenon can be more easily understood in the frequency domain than in the time domain (or vice versa). In that particular case, in the time domain it is necessary to consider how nonlinear amplitude contributions from different sections in the crystal add up.

Concerning spatial walk-off, there is a similar relation as above, but in that case to the angular phase-matching bandwidth. Spatial walk-off reduces the conversion efficiency for tightly focused laser beams, because the spatial overlap is then lost. Tightly focused beams also have a larger beam divergence, so that the problem is also apparent in the domain of propagation angles – which essentially is related to the spatial domain via a spatial Fourier transform.

Again, there is a case where the two domains at a first glance appear to lead to different consequences. Consider a beam with poor beam quality, having a large beam radius but nevertheless a relatively large beam divergence. The finite angular phase-matching bandwidth makes it clear that the conversion efficiency can suffer, while in the spatial domain it is not obvious. But again, the phenomenon can also be understood in the spatial domain – for example, by examining in detail a numerical simulation based on coupled differential equations for amplitudes with an x and y dependence.

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