Optical Transformation:From Quantum Optics To Classical Optics

By virtue of the technique of integration within an ordered product (IWOP) of operators we find a one-to-one correspondence between the unitary operators in quantum optics and the diffraction integral transforms in classical optics. This is achieved by setting up new quantum mechanical representations that represent various photon field distributions and constructing new unitary operators that describe the transformations between those vectors. By this means, not only can the theory of representation and transformation be efficiently applied to the study of classical optics, but also the optical transform theory can be adopted to develop the quantum optical theory. The main content includes:The IWOP technique efficiently generalize the Newton-Leibniz integration formula which only applies to commuting functions of continuum variables to the noncommutative operators made of Dirac's symbols and greatly simplifies the calculation of operators. By using the IWOP technique we set up several new quantum mechanical representations, including deduced entangled state representation, intermediate coordinate-momentum state representation, intermediate entangled state representation, etc. We prove that the transformation between two conjugate deduced entangled state representations just corresponds to the famous Hankel transform in classical optics. The Hatikel transform is a most fundamental optical transform and is as important as the Fourier transform, however, people only know the transformation between the coordinate eigenstate and the momentum eigenstate corresponds to the Fourier transform before. The discover of the above-mentioned relation reveal the more general relation between two research regions, and now the optical transformations which abide by cylindrical symmetiy (including the Hankel transformation) can be studied by means of the deduced entangled representations. In addition, we find the results of measuring quantum states in the intermediate states correspond to the Radon transformations of Wigner functions of corresponding classical optical fields, which manifestly shows that the...