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New Eigenmodes of Propagation in Quadratic Graded Index Media and complex fractional fourier transform

Communications in Theoretical Physics 2003年第1期39卷 97-100页

作者： FANHong－Yi FANYue DepartmentofPhysics ShanghaiJiaoTongUniversityShangahai200030China DepartmentofMaterialScienceandEngineering UniversityofScienceandTechnologyofChinaHefei230026China Department

By introducing a convenient complex form of the α-th 2-dimensional fractional fourier transform (CFFT) operation we find that it possesses new eigenmodes which are two-mode Hermite polynomials. We prove the eigenvalues of propagation in quadratic graded-index medium over a definite distance are the same as the eigenvalues of the α-th CFFT, which means that our definition of the α-th CFFT is physically meaningful.

关键词： complex fractional fourier transform two-mode Hermite polynomials

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

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Frontiers of physics 2012年第3期7卷 261-310页

作者： Hong-yi Fan Li-yun Hu Department of Physics Shanghai Jiao Tong UniversityShanghai 200030China College of Physics&Communication Electronics Jiangxi Normal UniversityNanchang 330022China Department of Material Science and Engineering University of Science and Technology of China Hefei 230026 China

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＂.

关键词： Dirac's symbolic method IWOP technique entangled state of continuum variables entangled Fresnel transform Collins formula Generalized Fresnel operator complex wavelet trans-form complex Wigner transform complex fractional fourier transform symplectic wavelet trans-form entangled symplectic wavelet transform Symplectic-dilation mixed wavelet transform frac-tional Radon transform new eigenmodes of fractional fourier transform

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