Abstract

A quasi-discrete Hankel transform (QDHT) is presented as a new and efficient framework for numerical evaluation of the zero-order Hankel transform. A discrete form of Parseval’s theorem is obtained for the first time to the authors’ knowledge, and the transform matrix is discussed. It is shown that the S factor, defined as the products of a truncated radius, is critical to building the QDHT.

© 1998 Optical Society of America

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References

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  1. A. E. Siegman, Opt. Lett. 1, 13 (1977).
    [CrossRef]
  2. V. Magni and G. Gerullo, J. Opt. Soc. Am. A 9, 2031 (1992).
    [CrossRef]
  3. P. K. Murphy and N. C. Gallagher, J. Opt. Soc. Am. 73, 1130 (1983).
    [CrossRef]
  4. G. Agnesi and G. C. Reali, J. Opt. Soc. Am. A 10, 1872 (1993).
    [CrossRef]
  5. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  6. B. D. Gupta, Mathematical Physics (Vikas, Skylard, India, 1980).
  7. f1r1=2R12?n=1?f1nJ0jnr1R1J1-2jn,f1n=?0R1f1r1J0jnr1R1r1dr1=12?f2jn2?R1.
  8. I. N. Sneddon, Fourier Transform (University of Glasgow, Glasgow, Scotland, 1951).
  9. ?0?f2r22r2dr2=?0R2f2r2g2*r2r2dr2=1?R222?n=1??m=1?f1jn2?R22×J1-2jn×J1-2jm×SS.SS=?0R2J0jnr2R2J0jmr2R2r2dr2=1/2R22J1jn?mn.
  10. S. Wolfram, Mathematica:?A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1991).

1993 (1)

1992 (1)

1983 (1)

1977 (1)

Agnesi, G.

Brigham, E. O.

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

Gallagher, N. C.

Gerullo, G.

Gupta, B. D.

B. D. Gupta, Mathematical Physics (Vikas, Skylard, India, 1980).

Magni, V.

Murphy, P. K.

Reali, G. C.

Siegman, A. E.

Sneddon, I. N.

I. N. Sneddon, Fourier Transform (University of Glasgow, Glasgow, Scotland, 1951).

Wolfram, S.

S. Wolfram, Mathematica:?A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1991).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Other (6)

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

B. D. Gupta, Mathematical Physics (Vikas, Skylard, India, 1980).

f1r1=2R12?n=1?f1nJ0jnr1R1J1-2jn,f1n=?0R1f1r1J0jnr1R1r1dr1=12?f2jn2?R1.

I. N. Sneddon, Fourier Transform (University of Glasgow, Glasgow, Scotland, 1951).

?0?f2r22r2dr2=?0R2f2r2g2*r2r2dr2=1?R222?n=1??m=1?f1jn2?R22×J1-2jn×J1-2jm×SS.SS=?0R2J0jnr2R2J0jmr2R2r2dr2=1/2R22J1jn?mn.

S. Wolfram, Mathematica:?A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1991).

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Figures (1)

Fig. 1
Fig. 1

Results of Hankel transforming top-hat function F3r by a QDHT; N=40, S=128.02, R1=1.01, and R2=20.12. (a) Top-hat function transformed by the QDHT. The filled circles were acquired from the function J12πr/r. (b) Top-hat function reproduced after two QDHT’s.

Tables (1)

Tables Icon

Table 1 Comparison of QFHT and QDHT

Equations (24)

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f2r2=2π0f1r1J02πr1r2r1dr1,
f1r1=2π0f2r2J02πr1r2r2dr2,
f1r1f2r20.
f1r1=1πR12n=1f2jn2πR1J1-2jnJ0jnr1R1,
f2r2=1πR22n=1f1jn2πR2J1-2jnJ0jnr2R2.
f2jm2πR1=1πR22n=1f1jn2πR2J1-2jnJ0jnjmS,
f1jn2πR2=1πR12m=1f2jm2πR1J1-2jmJ0jnjmS,
S=2πR1R2.
F1n=f1jn2πR2J1-1jnR1,
F2n=f2jn2πR1J1-1jnR2,
F2m=n=1NCmnF1n,
F1n=m=1NCnmF2m,
Cmn=2SJ0jnjmSJ1-1jnJ1-1jm,
EkSCk-I,
[0.59200.67920.43380.6792-0.1308-0.72220.4338-0.72220.5388],
[0.71450.67560.16090.6756-0.6004-0.35910.1609-0.35910.5345],
[-0.350.84-1.280.84-2.123.49-1.283.49-7.06]×10-6,
[-0.00200.0053-0.01140.0053-0.01470.0341-0.01130.0341-0.1073].
S=jN+1,
Cmn=2jN+1J0jnjmjN+1J1-1jnJ1-1jm, n, m=1, N.
0f1r12r1dr1=0f2r22r2dr2.
0f2r22r2dr2=12π2R22n=1f1jn2πR22J1-2jn,
0f1r12r1dr1=12π2R12n=1f2jn2πR12J1-2jn.
2S2n=1NF1n2=2S2m=1NF2m2.

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