New decomposition of the Fresnel operator corresponding to optical transformation in ABCD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property, we obtain new decomposition of the Fresnel operator as the product of quadratic phase operator, squeezing operator, and fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems. The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.