Abstract

An explicit evaluation is made of a simple end correction to the quasi-fast Hankel-transform algorithm. Application to a Gaussian beam shows that for a given accuracy, the use of this end correction permits a reduction of a factor of 8 in storage as well as a factor of 8 in running time.

© 1981 Optical Society of America

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References

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  1. R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 17, 95–113 (1965).
    [CrossRef]
  2. R. C. Le Bail, “Use of fast Fourier transform for solving partial differential equations,” J. Comput. Phys. 9, 440–465 (1972).
    [CrossRef]
  3. A. E. Siegman, E. A. Sziklas, “Mode calculations in unstable resonator with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  4. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  5. M. Lax, G. P. Agrawal, W. H. Louisell, “Continuous Fourier-transform spline solution of unstable resonator field-distribution,” Opt. Lett. 4, 303–305 (1979).
    [CrossRef] [PubMed]
  6. M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. (in press); G. P. Agrawal, M. Lax, J. H. Batteh, “Laser-induced channel for atmospheric transmission,” Opt. News 6(3), 37 (1980); G. P. Agrawal, M. Lax, J. H. Batteh, presented at the Optical Society of America Annual Meeting, October 1980.
  7. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  8. R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AE-17, 93–103 (1969).
    [CrossRef]
  9. A. E. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef] [PubMed]
  10. The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
    [CrossRef]
  11. W. D. Murphy, M. L. Bernabe, “Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators,” Appl. Opt. 17, 2358–2365 (1978).
    [CrossRef] [PubMed]
  12. W. P. Latham, T. C. Salvi, “Resonator studies with the Gardner–Fresnel–Kirchhoff propagator,” Opt. Lett. 5, 219–221 (1980).
    [CrossRef] [PubMed]

1980 (1)

1979 (1)

1978 (1)

1977 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975 (1)

1972 (1)

R. C. Le Bail, “Use of fast Fourier transform for solving partial differential equations,” J. Comput. Phys. 9, 440–465 (1972).
[CrossRef]

1969 (1)

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AE-17, 93–103 (1969).
[CrossRef]

1965 (2)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 17, 95–113 (1965).
[CrossRef]

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1959 (1)

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

Agrawal, G. P.

M. Lax, G. P. Agrawal, W. H. Louisell, “Continuous Fourier-transform spline solution of unstable resonator field-distribution,” Opt. Lett. 4, 303–305 (1979).
[CrossRef] [PubMed]

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. (in press); G. P. Agrawal, M. Lax, J. H. Batteh, “Laser-induced channel for atmospheric transmission,” Opt. News 6(3), 37 (1980); G. P. Agrawal, M. Lax, J. H. Batteh, presented at the Optical Society of America Annual Meeting, October 1980.

Batteh, J. H.

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. (in press); G. P. Agrawal, M. Lax, J. H. Batteh, “Laser-induced channel for atmospheric transmission,” Opt. News 6(3), 37 (1980); G. P. Agrawal, M. Lax, J. H. Batteh, presented at the Optical Society of America Annual Meeting, October 1980.

Bernabe, M. L.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Gardner, D. G.

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

Gardner, J. C.

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

Hockney, R. W.

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 17, 95–113 (1965).
[CrossRef]

Latham, W. P.

Lausch, G.

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

Lax, M.

M. Lax, G. P. Agrawal, W. H. Louisell, “Continuous Fourier-transform spline solution of unstable resonator field-distribution,” Opt. Lett. 4, 303–305 (1979).
[CrossRef] [PubMed]

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. (in press); G. P. Agrawal, M. Lax, J. H. Batteh, “Laser-induced channel for atmospheric transmission,” Opt. News 6(3), 37 (1980); G. P. Agrawal, M. Lax, J. H. Batteh, presented at the Optical Society of America Annual Meeting, October 1980.

Le Bail, R. C.

R. C. Le Bail, “Use of fast Fourier transform for solving partial differential equations,” J. Comput. Phys. 9, 440–465 (1972).
[CrossRef]

Louisell, W. H.

Meinke, W. W.

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Murphy, W. D.

Salvi, T. C.

Siegman, A. E.

Singleton, R. C.

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AE-17, 93–103 (1969).
[CrossRef]

Sziklas, E. A.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AE-17, 93–103 (1969).
[CrossRef]

J. Chem. Phys. (1)

The exponential transformation r = r0eαx is attributed to Gardner by Siegman in Ref. 9: D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959). However, its usefulness in connection with the solution of the radial Sehrödinger equation was recognized much earlier by Langer: see R. E. Langer, “On the connection formulas and the solution of the wave equation,” Phys. Rev. 51, 669–676 (1937).
[CrossRef]

J. Comput. Phys. (1)

R. C. Le Bail, “Use of fast Fourier transform for solving partial differential equations,” J. Comput. Phys. 9, 440–465 (1972).
[CrossRef]

J. Assoc. Comput. Mach. (1)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 17, 95–113 (1965).
[CrossRef]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Lett. (3)

Other (1)

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. (in press); G. P. Agrawal, M. Lax, J. H. Batteh, “Laser-induced channel for atmospheric transmission,” Opt. News 6(3), 37 (1980); G. P. Agrawal, M. Lax, J. H. Batteh, presented at the Optical Society of America Annual Meeting, October 1980.

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Tables (1)

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Table 1 Numerical Values of ϕ(ρ) = exp(−ρ2/2) with and without End Correction

Equations (6)

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ϕ ( ρ ) = 0 ψ ( r ) J 0 ( ) r d r ,
r = r 0 e αx , ρ = ρ 0 e αy ,
g ( y ) = j ( x + y ) f ( x ) d x ,
ϕ ( ρ ) = ϕ c ( ρ ) + C ( ρ ) + C ( ρ ) ,
C ( ρ ) = 0 r 0 ψ ( r ) J 0 ( ) r d r
C ( ρ ) ψ ( r 0 ) ( r 0 / ρ ) J 1 ( ρ r 0 ) ψ ( r 0 ) 1 / 2 r 0 2 ( 1 1 / 8 ρ 2 r 0 2 + ) ,

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