Abstract

We describe a new method for the numerical calculation of the zero-order Hankel (Fourier–Bessel) transform that has a high computational efficiency and an accuracy that can be 2 orders of magnitude greater than that of the standard quasi-fast Hankel procedure. The new method offers particular advantages in calculating optical beam propagation and resonator modes at high Fresnel numbers.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 661.
  2. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [Crossref] [PubMed]
  3. J. D. Talman, “Numerical Fourier and Bessel transforms in logarithmic variables,”J. Comput. Phys. 29, 35–48 (1978).
    [Crossref]
  4. E. C. Cavanagh, B. D. Cook, “Numerical evaluation of Hankel transforms via Gaussian–Laguerre polynomial expansions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 361–366 (1979).
    [Crossref]
  5. A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
    [Crossref]
  6. G. P. Agrawal, M. Lax, “End correction in the quasi-fast Hankel transform for optical-propagation problems,” Opt. Lett. 6, 171–173 (1981).
    [Crossref] [PubMed]
  7. S. M. Candel, “Dual algorithms for fast calculation of the Fourier–Bessel transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 963–972 (1981).
    [Crossref]
  8. P. K. Murphy, N. C. Gallagher, “Fast algorithm for the computation of the zero-order Hankel transform,”J. Opt. Soc. Am. 73, 1130–1137 (1983).
    [Crossref]
  9. Ref. 1, pp. 727 and 805–806.
  10. K. E. Oughtstun, “Unstable resonators modes,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 366–373.
  11. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 198–223.
  12. Ref. 11, p. 99.

1983 (1)

1981 (2)

G. P. Agrawal, M. Lax, “End correction in the quasi-fast Hankel transform for optical-propagation problems,” Opt. Lett. 6, 171–173 (1981).
[Crossref] [PubMed]

S. M. Candel, “Dual algorithms for fast calculation of the Fourier–Bessel transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 963–972 (1981).
[Crossref]

1980 (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
[Crossref]

1979 (1)

E. C. Cavanagh, B. D. Cook, “Numerical evaluation of Hankel transforms via Gaussian–Laguerre polynomial expansions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 361–366 (1979).
[Crossref]

1978 (1)

J. D. Talman, “Numerical Fourier and Bessel transforms in logarithmic variables,”J. Comput. Phys. 29, 35–48 (1978).
[Crossref]

1977 (1)

Agrawal, G. P.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 198–223.

Candel, S. M.

S. M. Candel, “Dual algorithms for fast calculation of the Fourier–Bessel transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 963–972 (1981).
[Crossref]

Cavanagh, E. C.

E. C. Cavanagh, B. D. Cook, “Numerical evaluation of Hankel transforms via Gaussian–Laguerre polynomial expansions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 361–366 (1979).
[Crossref]

Cook, B. D.

E. C. Cavanagh, B. D. Cook, “Numerical evaluation of Hankel transforms via Gaussian–Laguerre polynomial expansions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 361–366 (1979).
[Crossref]

Frisk, G. V.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
[Crossref]

Gallagher, N. C.

Lax, M.

Martinez, D. R.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
[Crossref]

Murphy, P. K.

Oppenheim, A. V.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
[Crossref]

Oughtstun, K. E.

K. E. Oughtstun, “Unstable resonators modes,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 366–373.

Siegman, A. E.

Talman, J. D.

J. D. Talman, “Numerical Fourier and Bessel transforms in logarithmic variables,”J. Comput. Phys. 29, 35–48 (1978).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

E. C. Cavanagh, B. D. Cook, “Numerical evaluation of Hankel transforms via Gaussian–Laguerre polynomial expansions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 361–366 (1979).
[Crossref]

S. M. Candel, “Dual algorithms for fast calculation of the Fourier–Bessel transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 963–972 (1981).
[Crossref]

J. Acoust. Soc. Am. (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,”J. Acoust. Soc. Am. 68, 523–529 (1980).
[Crossref]

J. Comput. Phys. (1)

J. D. Talman, “Numerical Fourier and Bessel transforms in logarithmic variables,”J. Comput. Phys. 29, 35–48 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Other (5)

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 661.

Ref. 1, pp. 727 and 805–806.

K. E. Oughtstun, “Unstable resonators modes,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 366–373.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 198–223.

Ref. 11, p. 99.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

(a) Zero-order Hankel transform g(y) of f ( x ) = 5 / 2 π x 2 ( 0 x 1 ) for a Fresnel number Nf = 10; (b) difference (error) between the analytical transform and the results obtained with the numerical transformations FHATHA and QFHT.

Fig. 2
Fig. 2

Maximum error generated by the FHATHA and the QFHT algorithms in the numerical calculation of the Hankel transform of f ( x ) = 5 / 2 π x 2 for two Fresnel numbers, Nf = 10 and Nf = 200.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

g ( y ) = 2 π 0 1 f ( x ) J 0 ( 2 π N f y x ) x d x ,
2 π ξ n ξ n + 1 f ^ ( x ) J 0 ( 2 π N f y x ) x d x = f ( x n ) N f y [ ξ n + 1 J 1 ( 2 π N f y ξ n + 1 ) - ξ n J 1 ( 2 π N f y ξ n ) ] ,
g ^ ( y ) = 1 N f y n 0 N - 1 [ f ( x n ) - f ( x n + 1 ) ] ξ n + 1 J 1 ( 2 π N f y ξ n + 1 ) ,
ξ 0 = 0 , ξ n = exp [ α ( n - N ) ] for n = 1 , 2 , , N , x n = y n = x 0 exp ( α n ) for n = 0 , 1 , , N - 1 ,
x 0 = [ 1 + exp ( α ) ] exp ( - α N ) / 2.
g ^ ( y m ) = 1 N f y m n 0 N - 1 [ f ( x n ) - f ( x n + 1 ) ] k n × exp [ α ( n + 1 - N ) ] × J 1 { 2 π N f x 0 exp [ α ( n + m + 1 - N ) ] } ,
k 0 = 2 exp ( α ) + exp ( 2 α ) [ 1 + exp ( α ) ] 2 [ 1 - exp ( - 2 α ) ] 3 8 α + 1 2 + .
α = - ln [ 1 - exp ( α ) ] / ( N - 1 ) ,
g ^ ( y m ) = 1 / ( N f y m ) FFT [ FFT ( φ n ) IFFT ( j 1 n ) ] ,
φ n = { k 0 [ f ( x 0 ) - f ( x 1 ) ] exp [ α ( 1 - N ) ] for n = 0 [ f ( x n ) - f ( x n + 1 ) ] exp [ α ( n + 1 - N ) ] for n = 1 , 2 , , N - 1 0 for n = N , N + 1 , 2 N - 1 ,
g ˜ ( y m ) = 2 π α n 0 N - 1 f ( x n ) x n 2 J 0 { 2 π N f x 0 2 exp [ α ( n + m ) ] } + π f ( x 0 ) x 0 2 ,
n 0 N - 1 α x n f ( x n ) δ ( x - x n ) ,
g ( y ) = 10 π η - 4 [ 2 η 2 J 0 ( η ) + ( η 3 - 4 η ) J 1 ( η ) ] ,

Metrics