Abstract

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.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef] [PubMed]
  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]
  3. D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
    [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. E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
    [CrossRef]
  6. E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
    [CrossRef]
  7. V. Magni, G. Cerullo, S. De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992).
    [CrossRef]
  8. J. A. Ferrari, “Fast Hankel transform of order zero,” J. Opt. Soc. Am. A 12, 1812–1813 (1995).
    [CrossRef]
  9. R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
    [CrossRef]
  10. Li Yu, M. Huang, M. Chen, W. Chen, W. Huang, Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
    [CrossRef]
  11. J. Markham, J. A. Conchello, “Numerical evaluation of Hankel transforms for oscillating functions,” J. Opt. Soc. Am. A 20, 621–630 (2003).
    [CrossRef]
  12. A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
    [CrossRef]
  13. A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
    [CrossRef]
  14. W. E. Higgins, D. C. Munson, “An algorithm for computing general integer-order Hankel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 86–97 (1987).
    [CrossRef]
  15. J. A. Ferrari, D. Perciante, A. Dubra, “Fast Hankel transform of nth order,” J. Opt. Soc. Am. A 16, 2581–2582 (1999).
    [CrossRef]
  16. 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]
  17. P. K. Murphy, N. C. Gallagher, “Fast algorithm for thecomputation of the zero-order Hankel transform,” J. Opt. Soc. Am. 73, 1130–1137 (1983).
    [CrossRef]
  18. B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
    [CrossRef]
  19. A. Agnesi, G. C. Reali, G. Patrini, A. Tomaselli, “Numerical evaluation of the Hankel transform: remarks,” J. Opt. Soc. Am. A 10, 1872–1874 (1993).
    [CrossRef]
  20. L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
    [CrossRef]
  21. B. W. Suter, R. A. Hedges, “Understanding fast Hankel transforms,” J. Opt. Soc. Am. A 18, 717–720 (2001).
    [CrossRef]
  22. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Harcourt–Academic, San Diego, Calif., 2001).
  23. A. L. Garcı́a, Numerical Methods for Physics (Prentice-Hall, Englewood Cliffs, N.J., 2000).
  24. N. M. Temme, “An algorithm with Algol60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives,” J. Comput. Phys. 32, 270–279 (1979).
    [CrossRef]
  25. The propagation of optical fields by applying the angular spectrum of plane waves is well documented in literature, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).
  26. J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
    [CrossRef]

2003 (2)

J. Markham, J. A. Conchello, “Numerical evaluation of Hankel transforms for oscillating functions,” J. Opt. Soc. Am. A 20, 621–630 (2003).
[CrossRef]

J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
[CrossRef]

2001 (1)

2000 (1)

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

1999 (1)

1998 (2)

1995 (1)

1993 (1)

1992 (1)

1991 (1)

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

1987 (1)

W. E. Higgins, D. C. Munson, “An algorithm for computing general integer-order Hankel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 86–97 (1987).
[CrossRef]

1986 (1)

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

1985 (1)

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

1983 (2)

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

P. K. Murphy, N. C. Gallagher, “Fast algorithm for thecomputation of the zero-order Hankel transform,” J. Opt. Soc. Am. 73, 1130–1137 (1983).
[CrossRef]

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. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

1979 (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]

N. M. Temme, “An algorithm with Algol60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives,” J. Comput. Phys. 32, 270–279 (1979).
[CrossRef]

1978 (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

1977 (1)

Agnesi, A.

Agrawal, G. P.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Harcourt–Academic, San Diego, Calif., 2001).

Barakat, R.

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]

Cerullo, G.

Chávez-Cerda, S.

J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
[CrossRef]

Chen, M.

Chen, W.

Conchello, J. A.

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]

De Silvestri, S.

Dubra, A.

Ferrari, J. A.

Frish, G. V.

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

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Gallagher, N. C.

Garci´a, A. L.

A. L. Garcı́a, Numerical Methods for Physics (Prentice-Hall, Englewood Cliffs, N.J., 2000).

Goodman, J. W.

The propagation of optical fields by applying the angular spectrum of plane waves is well documented in literature, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
[CrossRef]

Hansen, E. W.

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

Hedges, R. A.

Higgins, W. E.

W. E. Higgins, D. C. Munson, “An algorithm for computing general integer-order Hankel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 86–97 (1987).
[CrossRef]

Huang, M.

Huang, W.

Knockaert, L.

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

Lax, M.

Magni, V.

Markham, J.

Martinez, D. R.

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

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Mook, D. R.

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

Munson, D. C.

W. E. Higgins, D. C. Munson, “An algorithm for computing general integer-order Hankel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 86–97 (1987).
[CrossRef]

Murphy, P. K.

Oppenheim, A. V.

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

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Parshall, E.

Patrini, G.

Perciante, D.

Reali, G. C.

Rodri´guez-Masegosa, R.

J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
[CrossRef]

Sandler, B. H.

Siegman, A. E.

Suter, B. W.

B. W. Suter, R. A. Hedges, “Understanding fast Hankel transforms,” J. Opt. Soc. Am. A 18, 717–720 (2001).
[CrossRef]

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

Temme, N. M.

N. M. Temme, “An algorithm with Algol60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives,” J. Comput. Phys. 32, 270–279 (1979).
[CrossRef]

Tomaselli, A.

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Harcourt–Academic, San Diego, Calif., 2001).

Yu, Li

Zhu, Z.

IEEE Trans. Acoust. Speech Signal Process. (6)

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

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]

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

W. E. Higgins, D. C. Munson, “An algorithm for computing general integer-order Hankel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 86–97 (1987).
[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]

IEEE Trans. Signal Process. (2)

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Comput. Phys. (1)

N. M. Temme, “An algorithm with Algol60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives,” J. Comput. Phys. 32, 270–279 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Lett. (3)

Proc. IEEE (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Pure Appl. Opt. (1)

J. C. Gutiérrez-Vega, R. Rodrı́guez-Masegosa, S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” Pure Appl. Opt. 5, 276–282 (2003).
[CrossRef]

Other (3)

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Harcourt–Academic, San Diego, Calif., 2001).

A. L. Garcı́a, Numerical Methods for Physics (Prentice-Hall, Englewood Cliffs, N.J., 2000).

The propagation of optical fields by applying the angular spectrum of plane waves is well documented in literature, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).

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Figures (4)

Fig. 1
Fig. 1

Exact and computed Hankel transforms of the sinc function with γ=5 for (a) p=1 and (c) p=4. Solid curves denote exact transforms, and crosses denote the calculated transforms. Dynamic errors are shown in (b) and (d).

Fig. 2
Fig. 2

(a) Analytical [f2(ν)] and computed [f2*(ν)] transforms, (b) exact and retrieved functions f1*(r). Both plots correspond to N=1024 points sampled according to the criteria of the pQDHT and the QFHT.

Fig. 3
Fig. 3

Focusing evolution of a fourth-order Bessel beam along the plane (r, z). Physical parameters are chosen to produce a circular ring of radius 1 mm in the focal plane (at z=0.5 m).

Fig. 4
Fig. 4

Transverse radial intensity of the focused Bessel beam at the focal plane.

Tables (1)

Tables Icon

Table 1 Comparison of the pQDHT and the QFTT

Equations (23)

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f2(ν)=2π0f1(r)Jp(2πνr)rdr,
f1(r)=2π0f2(ν)Jp(2πrν)νdν,
f1(rR)=0,f2(νV)=0,
f(r)=m=1cpmJpαpm ra,0ra,
cpm=1a2Jp+12(apm) 0af(r)Jpαpm rardr.
f2αpm2πR=1πV2 n=1N f1(αpn/(2πV))Jp+12(αpn)JpαpnαpmS,
f1αpn2πV=1πR2 m=1N f2(αpm/(2πR))Jp+12(αpm)JpαpnαpmS,
F2(m)=f2αpm2πR|Jp+1(αpm)|-1V,
F1(n)=f1αpn2πV|Jp+1(αpn)|-1R;
F2(m)=n=1NTmnF1(n),
F1(n)=m=1NTnmF2(m),
Tmn=2Jp(αpnαpm/S)|Jp+1(αpn)||Jp+1(αpm)|S
0|f1(r)|2rdr=0|f2(ν)|2νdν.
m=1N |f2(αpm/(2πR))|22π2R2Jp+12(αpm)=n=1N |f1(αpn/(2πV))|22π2V2Jp+12(αpn),
2S2 n=1N|F1(n)|2=2S2 m=1N|F2(m)|2.
f2(ν)=νp cos(pπ/2)2πγγ2-v2(γ+γ2-v2)p,0ν<γ
=sin[p arcsin(γ/ν)]2πγν2-γ2,ν>γ.
e(ν)=20 log10|f2(ν)-f2*(ν)|max|f2*(ν)|.
1=1N j=1N|f1-f1*|.
f2(ν)=ap+1 Jp+1(2πaν)ν.
prop¯=exp(i2πΔzλ-2-ν¯2).
H¯=H¯prop¯.
u¯j=T * (H¯J¯)

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