Quantum optical formulation of difference-frequency generation and optimal cloning of spatial modes

We present a quantum optical formulation of difference-frequency generation (DFG) that incorporates the spatial modes of light. It reproduces the well established result for classical light beams and establishes the relation of DFG to stimulated and spontaneous parametric downconversion. As shown here, these relations determine that stimulated parametric down-conversion can realise $N\xrightarrow{} M $ $d$-dimensional optimal quantum cloning.