  We derive frequency correlation and exit probability expressions for photons
generated via spontaneous parametric downconversion (SPDC) in nonlinear
waveguides that exhibit linear scattering loss. Such loss is included within a
general Hamiltonian formalism by connecting waveguide modes to reservoir modes
with a phenomenological coupling Hamiltonian, the parameters of which are later
related to the usual loss coefficients. In the limit of a low probability of
SPDC pair production, the presence of loss requires that we write the usual
lossless generated pair state as a reduced density operator, and we find that
this density operator is naturally composed of two photon, one photon, and zero
photon contributions. The biphoton probability density, or joint spectral
intensity (JSI), associated with the two-photon contribution is determined not
only by a phase matching term, but also by a loss matching term. The relative
size of the loss coefficients within this term lead to three qualitatively
different regimes of SPDC JSIs. If either the pump or generated photon loss is
much higher than the other, the side lobes of the phase matching squared sinc
function are washed out. On the other hand, if pump and generated photon loss
are appropriately balanced, the lossy JSI is identical to the lossless JSI.
Finally, if the generated photon loss is frequency dependent, the shape of the
JSI can be altered more severely, potentially leading to generated photons that
are less frequency correlated though also produced less efficiently when
compared to photons generated in low-loss waveguides.
