Optical Four-Wave Mixing Operator, Fresnel Operators and Three-Mode Entangled State Representation

We analyse the optical four-wave mixing operator S and relate it to the two-mode Fresnel operator. It is shown that the direct product of the two-mode Fresnel operator and the single-mode Fresnel operator has a natural representation on the basis of a three-mode entangled state, which is constructed by S and a beam splitter transform.