Understanding fast Hankel transforms

Suter, Bruce W.; Hedges, Robert A.

doi:10.1364/josaa.18.000717

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…Since the seminal work by Siegman in 1977, 1 a number of algorithms for the numerical evaluation of the HT have been reported in the literature, for both zero-order HTs [2][3][4][5][6][7][8][9][10][11] and high-order HTs. [12][13][14][15][16][17][18][19][20][21] Unfortunately, the performance of a method for computing the HT is highly dependent on the function to be transformed, and thus it is difficult to determine the optimal algorithm for a given function. In optics, we often deal with problems where the HT and the IHT need to be computed thousands of times starting from a known analytical expression for the input function.…”

Section: Introductionmentioning

confidence: 99%

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

2004

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The method originally proposed by Yu et al. [Opt. Lett. 23, 409 (1998)] for evaluating the zero-order Hankel transform is generalized to high-order Hankel transforms. Since the method preserves the discrete form of the Parseval theorem, it is particularly suitable for field propagation. A general algorithm for propagating an input field through axially symmetric systems using the generalized method is given. The advantages and the disadvantages of the method with respect to other typical methods are discussed.

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Section: Introductionmentioning

confidence: 99%

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

2004

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“…In the most general case the heating function H(r,z), and hence initial pressure p 0 (r,z)ϭ⌫H(r,z), will be given as values on a grid. In this case numerical Fourier and Hankel transforms 25,26 may be used to calculate p 0 (k r ,) from Eqs. ͑35͒ and ͑36͒.…”

Section: A Cylindrically Symmetric Heating Functionsmentioning

confidence: 99%

Fast calculation of pulsed photoacoustic fields in fluids usingk-space methods

Cox

Beard

2005

The Journal of the Acoustical Society of America

183

121

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Two related numerical models that calculate the time-dependent pressure field radiated by an arbitrary photoacoustic source in a fluid, such as that generated by the absorption of a short laser pulse, are presented. Frequency-wavenumber (k-space) implementations have been used to produce fast and accurate predictions. Model I calculates the field everywhere at any instant of time, and is useful for visualizing the three-dimensional evolution of the wave field. Model II calculates pressure time series for points on a straight line or plane and is therefore useful for simulating array measurements. By mapping the vertical wavenumber spectrum directly to frequency, this model can calculate time series up to 50 times faster than current numerical models of photoacoustic propagation. As the propagating and evanescent parts of the field are calculated separately, model II can be used to calculate far- and near-field radiation patterns. Also, it can readily be adapted to calculate the velocity potential and thus particle velocity and acoustic intensity vectors. Both models exploit the efficiency of the fast Fourier transform, and can include the frequency-dependent directional response of an acoustic detector straightforwardly. The models were verified by comparison with a known analytic solution and a slower, but well-understood, numerical model.

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“…Analytical evaluations of (1.2) and (1.3) are rare and their numerical computations are difficult because of the oscillatory behavior of the Bessel function and the infinite length of the interval. Since seminal work by Siegman [19] in 1977, a number of algorithms for the numerical evaluation of the Hankel transform have been published for both zero-order [5,6,[9][10][11][12][20][21][22][23][24] and high-order [25][26][27][28][29][30][31][32][33] Hankel transform. Unfortunately, the efficiency of a method for computing Hankel transform is highly dependent on the function to be transformed, and thus it is difficult to choose the optimal algorithm for given function.…”

Section: Introductionmentioning

confidence: 99%

A New Stable Algorithm to Compute Hankel Transform Using Chebyshev Wavelets

Pandey¹

2010

CICP

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A new stable numerical method, based on Chebyshev wavelets for numerical evaluation of Hankel transform, is proposed in this paper. The Chebyshev wavelets are used as a basis to expand a part of the integrand, r f (r), appearing in the Hankel transform integral. This transforms the Hankel transform integral into a Fourier-Bessel series. By truncating the series, an efficient and stable algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν > −1. The method is quite accurate and stable, as illustrated by given numerical examples with varying degree of random noise terms εθ i added to the data function f (r), where θ i is a uniform random variable with values in [−1,1]. Finally, an application of the proposed method is given for solving the heat equation in an infinite cylinder with a radiation condition.

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Understanding fast Hankel transforms

Cited by 8 publications

References 9 publications

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

Fast calculation of pulsed photoacoustic fields in fluids usingk-space methods

A New Stable Algorithm to Compute Hankel Transform Using Chebyshev Wavelets

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