  By the newly developed technique of integration within an ordered product
(IWOP) of operators, we explore 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
can not only find some new quantum mechanical unitary operators which
correspond to the known optical transformations, deriving a new theorem for
calculating quantum tomogram of density operators, but also can reveal some new
classical optical transformations. 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 to finding new classical optical
transformations. The complex wavelet transform and the condition of mother
wavelet are studied in the context of quantum optics too. Throughout our
discussions, the coherent state, the entangled state representation of the
two-mode squeezing operators and the IWOP technique are fully used. All these
confirms 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".
