A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ｜z〉g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula （Huygens-Fresnel integration transformation describing optical diffraction）. The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz -- rz^＊ in the ｜Z〉g representation, while ｜z〉g in phase space is graphically denoted by an ellipse.

The two-mode quantum Fresnel operator and the multiplication rule of 2D Collins diffraction formula

By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional （2D） Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac＇s symbolic method

By virtue of the new technique of performing integration over Dirac＇s ket-bra operators, we ex- plore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel- Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, de- riving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel opera- tor （GFO） to correspond to the generalized Fresnel transform （GFT） in classical optics. We derive GFO＇s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum op- tics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product （IWOP） of operators are fully used. All these have confirmed Dirac＇s assertion： ＂...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory＂.