Generalized Fresnel operators corresponding to optical Fresnel diffraction and the squeezed states

Corresponding to optical Fresnel diffraction, we show that the exponential quadratic operator exp{−i[αP 2+ βQ 2+γ(PQ+QP)]/2} is actually a generalized single-mode Fresnel operator (GFO) in compact form, where [Q,P]=iħ. We also demonstrate that exp{iα[(Q 1+Q 2)2+(P 1−P 2)2]+iβ[(Q 1−Q 2)2+(P 1+P 2)2]+iγ(Q 1 P 2+Q 2 P 1)} is a two-mode GFO. Their disentangling formula and normal ordering form are derived with the use of technique of integration within an ordered product (IWOP) of operators and the coherent state representation. The squeezed states generated by these two GFOs are obtained.