  We analytically treat the scattering of two counter-propagating photons on a
two-level emitter embedded in an optical waveguide. We find that the
non-linearity of the emitter can give rise to significant pulse-dependent
directional correlations in the scattered photonic state, which could be
quantified via a reduction in coincident clicks in a Hong-Ou-Mandel measurement
setup, analogous to a linear beam splitter. Changes to the spectra and phase of
the scattered photons, however, would lead to reduced interference with other
photons when implemented in a larger optical circuit. We introduce suitable
fidelity measures which account for these changes, and find that high values
can still be achieved even when accounting for all properties of the scattered
photonic state.
