Relation between quantum tomography and optical Fresnel transform

Corresponding to optical Fresnel transformation characteristic of ray transfer matrix elements (A;B;C;D); AD-BC = 1, there exists Fresnel operator F(A;B;C;D) in quantum optics, we show that under the Fresnel transformation the pure position density |x>_rs,rs__rs,rs_ is just the wave function of its Fresnel transformed state F|phi>, i.e. rs_= . Similarly, we find F|p>_rs,rs_