Generation of photon pairs through parametric processes in nonlinear waveguides with the account of losses

We present an analytical description of the process of spontaneous four-wave mixing in a cubic nonlinear fiber with linear losses. We consider the generation of photon pairs in the fiber when in the input of fiber is fed the pumping wave and single signal photon. The focus of attention is on three cases: when the signal photon propagates in the fiber without generating of biphotons; when the photon pair is generated; and when the photon is lost in the fiber. We also consider the cascade processes, but do not give them an analytical description because of their smallness. Description of the biphotons generation process we provide using the Schrodinger-type equation, and take into account the losses in the fiber through the introduction of the virtual beam splitters. We demonstrate the effectiveness of the generation of photon pairs through parametric processes.