Quantum correspondence of the mixed Lenz–Fresnel transform in classical optics

We find the quantum correspondence (a four-parameter squeezing operator U(r,s,μ)U(r,s,μ)) of the mixed optical Lenz–Fresnel transform, i.e. that the kernel of Lenz–Fresnel transform is just the matrix element of U(r,s,μ)U(r,s,μ) in the entangled states. The group multiplication rule of U(r,s,μ)U(r,s,μ) is proved by virtue of its coherent entangled state representation which is essential to this correspondence.