Optical scaled Fresnel--Fourier transform obtained via intermediate coordinate-- momentum representation

Using the intermediate coordinate--momentum representation |x\rangle_s,r, we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F(r,s) and the Fourier transform operator \mathcalF by decomposing U. We also find that the matrix element _s,r\langle x|U|f\rangle just corresponds to an optical scaled Fresnel--Fourier transform.