  In this paper, we present a thorough investigation for a spontaneous
parametric four-wave mixing process in third-order nonlinear waveguides with
various continuous tapering patterns. It has been previously shown that these
devices can quasi-phase-match the four-wave-mixing process and enhance its
conversion efficiency by orders of magnitude. By altering the tapering profile
curve we found that these devices can enable single-photon sources with either
narrow or broadband spectral widths at on-demand frequencies. Using our model,
we were also able to identify the waveguide length at which the single-photon
spectral purity is maximised.
