We consider the effect of self-phase modulation and cross-phase modulation on
the joint spectral amplitude of photon pairs generated by spontaneous four-wave
mixing. In particular, the purity of a heralded photon from a pair is
considered, in the context of schemes that aim to maximise the purity and
minimise correlation in the joint spectral amplitude using birefringent
phase-matching and short pump pulses. We find that non-linear phase modulation
effects will be detrimental, and will limit the quantum interference visibility
that can be achieved at a given generation rate. An approximate expression for
the joint spectral amplitude with phase modulation is found by considering the
group velocity walk-off between each photon and the pump, but neglecting the
group-velocity dispersion at each wavelength. The group-velocity dispersion can
also be included with a numerical calculation, and it is shown that it only has
a small effect on the purity for the realistic parameters considered.Comment: 11 pages, 10 figure
