2-mode Fresnel operator and entangled Fresnel transform

For 2-dimensional (2D) generalized Fresnel transform (GFT) in Fourier optics theory we find 2-mode quantum-optics Fresnel operator, which possesses the property of group multiplication as GFT does. Projecting the Fresnel operator onto the usual 2-mode coordinate representation leads to a new kind of entangled Fresnel transforms, which is quite different from the usual 2D GFT (the direct product of two 1D GFTs).