- Healy, John J.

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# Healy, John J.

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Healy, John J.

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Healy, John J.

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- PublicationReevaluation of the direct method of calculating Fresnel and other linear canonical transformsThe linear canonical transform may be used to simulate the effect of paraxial optical systems on wave fields. Using a recent definition of the discrete linear canonical transform, phase space diagram analyses of the sampling requirements of the direct method of calculating the Fresnel and other linear canonical transforms are more favorable than previously thought. Thus the direct method of calculating these Transforms may be used with fewer samples than previously reported simply by making use of an appropriate reconstruction filter on the samples output by the algorithm.
375Scopus© Citations 39 - PublicationDigital computation of the complex linear canonical transform(Optical Society of America, 2011-07-01)
; ; ; ; ; An efficient algorithm for the accurate computation of the linear canonical transform with complex transform parameters and with complex output variable is presented. Sampling issues are discussed and the requirements for different cases given. Simulations are provided to validate the results.506Scopus© Citations 11 - PublicationCross terms of the Wigner distribution function and aliasing in numerical simulations of paraxial optical systemsSampling a function periodically replicates its spectrum. As a bilinear function of the signal, the associated Wigner distribution function contains cross terms between the replicas. Often neglected, these cross terms affect numerical simulations of paraxial optical systems. We develop expressions for these cross terms and show their effect on an example calculation
452Scopus© Citations 11 - PublicationCases where the linear canonical transform of a signal has compact support or is band-limitedA signal may have compact support, be band-limited (i.e., its Fourier transform has compact support), or neither (“unbounded”). We determine conditions for the linear canonical transform of a signal having these properties. We examine the significance of these conditions for special cases of the linear canonical transform and consider the physical significance of our results
366Scopus© Citations 52 - PublicationFast linear canonical transformsThe linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande–Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions.
1051Scopus© Citations 78 - PublicationAdditional sampling criterion for the linear canonical transformThe linear canonical transform describes the effect of ﬁrst-order quadratic phase optical systems on a wave ﬁeld. Several recent papers have developed sampling rules for the numerical approximation of the transform. However, sampling an analog function according to existing rules will not generally permit the reconstruction of the analog linear canonical transform of that function from its samples. To achieve this, an additional sampling criterion has been developed for sampling both the input and the output wave ﬁelds.
376Scopus© Citations 43 - PublicationSpace-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithmsThe product of the spatial and spatial frequency extents of a wave field has proven useful in the analysis of the sampling requirements of numerical simulations. We propose that the ratio of these quantities is also illuminating. We have shown that the distance at which the so-called “direct method” becomes more efficient than the so-called “spectral method” for simulations of Fresnel transforms may be written in terms of this space-bandwidth ratio. We have proposed generalizations of these algorithms for numerical simulations of general ABCD systems and derived expressions for the “transition space-bandwidth ratio” above which the generalization of the spectral method is the more efficient algorithm and below which the generalization of the direct method is preferable.
485Scopus© Citations 31 - PublicationWigner cross-terms in sampled and other periodic signalsIf we sample a scalar wave field, it becomes periodic in frequency. We examine the cross-terms which occur between these periodic replicas in the Wigner-Ville distribution function of such a signal. We present analytic results for Gaussian signals. The results also have implications for physical systems which contain periodic gratings.
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