Abstract

The method originally proposed by Siegman [ Opt. Lett. 1, 13 ( 1977)] for evaluating numerically the Hankel transform by fast-Fourier-transform techniques is reconsidered. A novel analytical form is found that permits a numerical computation of the Hankel transform with accuracy comparable with that of Siegman’s approach without lower-end corrections; but for simplicity and numerical efficiency the original approach remains unsurpassed.

© 1993 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. S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1980).
  3. 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]
  4. J.-L. Doumont, “Laser beam and resonator calculations on desktop computers,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1991).
  5. G. P. Agrawal, M. Lax, “End correction to the quasi-fast Hankel transform for optical propagation problems,” Opt. Lett. 6, 171–173 (1981).
    [CrossRef] [PubMed]
  6. K. E. Oughstun, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (Elsevier, New York, 1987), pp. 164–387.
  7. V. Magni, S. De Silvestri, “Fast Hankel transformation of high accuracy for optical beam propagation,” in Conference on Lasers and Electro-Optics 1992, Vol. 12 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 508–509.
  8. matlab Version 3.5, by MathWorks, Inc., Natick, Mass. 01760 (1991).
  9. 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]

1992 (1)

1981 (1)

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]

1977 (1)

Agrawal, G. P.

Cerullo, G.

De Silvestri, S.

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]

V. Magni, S. De Silvestri, “Fast Hankel transformation of high accuracy for optical beam propagation,” in Conference on Lasers and Electro-Optics 1992, Vol. 12 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 508–509.

Doumont, J.-L.

J.-L. Doumont, “Laser beam and resonator calculations on desktop computers,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1991).

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]

Lax, M.

Magni, V.

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]

V. Magni, S. De Silvestri, “Fast Hankel transformation of high accuracy for optical beam propagation,” in Conference on Lasers and Electro-Optics 1992, Vol. 12 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 508–509.

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]

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]

Oughstun, K. E.

K. E. Oughstun, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (Elsevier, New York, 1987), pp. 164–387.

Sheng, S.-C.

S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1980).

Siegman, A. E.

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. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Other (5)

K. E. Oughstun, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (Elsevier, New York, 1987), pp. 164–387.

V. Magni, S. De Silvestri, “Fast Hankel transformation of high accuracy for optical beam propagation,” in Conference on Lasers and Electro-Optics 1992, Vol. 12 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 508–509.

matlab Version 3.5, by MathWorks, Inc., Natick, Mass. 01760 (1991).

S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1980).

J.-L. Doumont, “Laser beam and resonator calculations on desktop computers,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1991).

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Equations (8)

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g ( ρ ) = 2 π 0 f ( r ) J l ( 2 π ρ r ) r d r ,
g ^ ( y ) = - f ^ ( x ) j ^ ( x + y ) d x .
LEC l = 2 π 0 r 0 f ( r ) J l ( 2 π ρ r ) r d r ,
g ( ρ ) = 2 π 0 b [ l f ( r ) - r d d r f ( r ) ] J l + 1 ( 2 π ρ r ) 2 π ρ d r + π b 2 f ( b ) 2 J l + 1 ( 2 π ρ b ) 2 π ρ b .
- 0 b d f d r r ρ J 1 ( 2 π ρ r ) d r + f ( b ) b ρ J 1 ( 2 π ρ b ) - j = 0 N - 1 ( d f d r ) j r j ρ J 1 ( 2 π ρ r j ) Δ r j + f ( b ) b ρ J 1 ( 2 π ρ b ) = - j = 1 N - 1 [ f ( r j + 1 ) - f ( r j ) ] r j ρ J 1 ( 2 π ρ r j ) + f ( b ) b ρ J 1 ( 2 π ρ b ) .
f ( r ) j = 0 N - 1 f ( r j + 1 ) [ θ ( r - r j ) - θ ( r - r j + 1 ) ] ,
LEC 0 = f ( 0 ) r 0 ρ J 1 ( 2 π ρ r 0 ) + ( d 2 f d r 2 ) 0 r 0 3 4 ρ [ J 1 ( 2 π ρ r 0 ) - J 3 ( 2 π ρ r 0 ) ] ,
LEC 1 = ( d f d r ) 0 r 0 2 ρ J 2 ( 2 π ρ r 0 ) + ( d 3 f d r 3 ) 0 r 0 4 12 ρ [ J 2 ( 2 π ρ r 0 ) - J 4 ( 2 π ρ r 0 ) ] .

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