14 May 2010 Numerical approximation of scalar diffraction through first order optical systems
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Luneburg's first order optical systems consist of sections of free space, lenses, and all possible combinations of these. The linear canonical transform (LCT), a parameterised linear integral transform, may be used to model the paraxial propagation of scalar optical fields through such systems. We consider the propagation of quasi-monochromatic, coherent wave fields, though more general calculations are possible. Numerical approximation of such systems is an active area of research, of interest for system design and analysis. We consider methods for the determination of the sampling requirements for the wave fields at the input and output of such calculations, in conjunction with the discretisation of the transform. We illustrate these considerations using phase space diagrams (PSDs), making use of the LCT's simple co-ordinate transforming effect on such diagrams. We discuss the implications of the cross-terms present in the Wigner distribution function, which are ignored in such PSD-based analyses, for the accuracy of the simulations and for the selection of sampling schemes. We examine the available algorithms for performing the transformations in O(N log N) time. In particular, we consider the relative merits of algorithms which decompose the optical system into special cases for which fast algorithms are better developed and also algorithms which decompose the calculations into smaller ones iteratively.
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John J. Healy, John J. Healy, John T. Sheridan, John T. Sheridan, } "Numerical approximation of scalar diffraction through first order optical systems", Proc. SPIE 7717, Optical Modelling and Design, 77171A (14 May 2010); doi: 10.1117/12.854053; https://doi.org/10.1117/12.854053

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