Abstract

Unlike the FFT, the Quasi Discrete Hankel Transform (QDHT) is not sampled on a uniform grid; in particular the field may no longer be sampled on axis. We demonstrate how the generalised sampling theorem may be applied to optical problems, evaluated with the QDHT, to efficiently and accurately reconstruct the optical field at any point. Without sacrificing numerical accuracy this is demonstrated to be typically 50× faster than using an equivalent 2D FFT.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

A. W. Norfolk and E. J. Grace, “New beat length for writing periodic structures using Bessel beams,” Opt. Commun. 283(3), 447–450 (2010).
[CrossRef]

2009

2005

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

2004

1999

1998

1978

1977

1975

A. J. Jerri and D. W. Kreisler, “Sampling Expansions with Derivatives for Finite Hankel and Other Transforms,” SIAM J. Math. Anal. 6(2), 262–267 (1975).
[CrossRef]

1964

L. L. Campbell, “A comparison of the sampling theorems of Kramer and Whittaker,” J. Soc. Indust. Appl. Math. 12, 117–130 (1964).
[CrossRef]

1959

H. P. Kramer, “A Generalized Sampling Theorem,” J. Math. & Phys. 38, 68–72 (1959).

Campbell, L. L.

L. L. Campbell, “A comparison of the sampling theorems of Kramer and Whittaker,” J. Soc. Indust. Appl. Math. 12, 117–130 (1964).
[CrossRef]

Chen, M.

Chen, W.

Ding, D.

Ersoy, O. K.

Feit, M. D.

Fleck, J. A.

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

Grace, E. J.

A. W. Norfolk and E. J. Grace, “New beat length for writing periodic structures using Bessel beams,” Opt. Commun. 283(3), 447–450 (2010).
[CrossRef]

Guizar-Sicairos, M.

Guti`errez-Vega, J. C.

Huang, M.

Huang, W.

Jerri, A. J.

A. J. Jerri and D. W. Kreisler, “Sampling Expansions with Derivatives for Finite Hankel and Other Transforms,” SIAM J. Math. Anal. 6(2), 262–267 (1975).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

Kramer, H. P.

H. P. Kramer, “A Generalized Sampling Theorem,” J. Math. & Phys. 38, 68–72 (1959).

Kreisler, D. W.

A. J. Jerri and D. W. Kreisler, “Sampling Expansions with Derivatives for Finite Hankel and Other Transforms,” SIAM J. Math. Anal. 6(2), 262–267 (1975).
[CrossRef]

Liu, X.

Norfolk, A. W.

A. W. Norfolk and E. J. Grace, “New beat length for writing periodic structures using Bessel beams,” Opt. Commun. 283(3), 447–450 (2010).
[CrossRef]

Siegman, A. E.

Srisungsitthisunti, P.

Xu, X.

Yu, L.

Zhu, Z.

Appl. Opt.

J. Math. & Phys.

H. P. Kramer, “A Generalized Sampling Theorem,” J. Math. & Phys. 38, 68–72 (1959).

J. Opt. Soc. Am. A

J. Soc. Indust. Appl. Math.

L. L. Campbell, “A comparison of the sampling theorems of Kramer and Whittaker,” J. Soc. Indust. Appl. Math. 12, 117–130 (1964).
[CrossRef]

Opt. Commun.

A. W. Norfolk and E. J. Grace, “New beat length for writing periodic structures using Bessel beams,” Opt. Commun. 283(3), 447–450 (2010).
[CrossRef]

Opt. Lett.

Proc. IEEE

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

SIAM J. Math. Anal.

A. J. Jerri and D. W. Kreisler, “Sampling Expansions with Derivatives for Finite Hankel and Other Transforms,” SIAM J. Math. Anal. 6(2), 262–267 (1975).
[CrossRef]

Other

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Sample points for the FFT (o) and zeroth order QDHT (x). Separation of QDHT points has been exaggerated for illustrative purposes.

Fig. 2.
Fig. 2.

Intensity comparison for axial point (r = 0) and first numerical point, α 1 = .3α 2.

Fig. 3.
Fig. 3.

Reconstruction error for test functions shown in Table 1.

Tables (2)

Tables Icon

Table 1. Typical beam-like functions.

Tables Icon

Table 2. Accuracy of field on axis and relative speed for each method.

Equations (8)

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

u(r,z=0)=[J0(α1r)+J0(α2r)] exp (r2) ,
f(r)=0Kmax Jn (krr) F (kr) kr d kr ,
KR=jn,N+1 .
jn,N<KmaxRmaxjn,N+1;
f0(r)=p=1f(rn,p)Sk(r) ,
Sk(r)=2rn,pJn(Kr)K(rn,p2r2)Jn+1(Krn,p) .
ε=f0(r)f(r).
εz=f0(0,z)f(0,z).

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