Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

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Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

Based on the Collins formula, the relationship between the coordinate transform matrix (WCTM) of the Wigner distribution function (WDF) and the ray transfer matrix (RTM) of an arbitrary first order optical system has been derived. By using this relation and the definition of fractional Fourier transform (FRT) in terms of WDF rotation, it is concluded that an arbitrary first order optical system can be generally decomposed into a thin lens and a FRT sub system whose order is not unique and depends on two concrete decomposing operations on the system. And when the system is reciprocally symmetric, a FRT can be implemented by it. In addition, the composition, that is also the decomposition condition of the complicated FRT optical system by cascading a series of FRT subsystems has also been derived by using the operations of RTM.