  Single-photon sources are necessary components for many prospective quantum
technologies. One candidate for a single-photon source is spontaneous
parametric down conversion combined with a heralding photon detection. The
heralded light pulse from such a source, is typically treated as single-mode,
this treatment, however, is incomplete. We develop a full multimode description
based on the exact Bogoliubov treatment of the down conversion process. We then
provide a perturbative and effective treatment, which illustrates the most
important physical mechanisms and permits analytical estimates of the success
probability and purity of single-photon states under practical heralding
conditions, both without relying on the precise detection time of the heralding
photon and when accepting photons only in a narrow window around the time of
the detection. This permits us to characterize the emitted light under three
different assumptions for the pump pulse. For spontaneous parametric down
conversion with a very short pump pulse, we find the single-mode description to
be accurate, while for longer pump pulses and continuous pumping, a multimode
description is necessary. Our findings can be used to guide the design of
quantum information protocols based on heralded single-photon sources, as their
performance may depend on the multimode nature of the sources.
