Abstract

Our purpose is to bring to the attention of the optical community our recent work on the numerical evaluation of zero-order Hankel transforms; such techniques have direct application in optical diffraction theory and in optical beam propagation. The two algorithms we discuss (Filon–Simpson and Filon-trapezoidal) are reasonably fast and very accurate; furthermore, the errors incurred are essentially independent of the magnitude of the independent variable. Both algorithms are then compared with the recent (fast-Fourier-transform-based Hankel transform algorithm developed by Magni, Cerullo, and Silvestri (MCS algorithm) [J. Opt. Soc. Am. A 9, 2031 (1992)] and are shown to be superior. The basic assumption of these algorithms is that the term in the integrand multiplying the Bessel function is relatively smooth compared with the oscillations of the Bessel function. This condition is violated when the inverse Hankel transform has to be computed, and the Filon scheme requires a very large number of quadrature points to achieve even moderate accuracy. To overcome this deficiency, we employ the sampling expansion (Whittaker’s cardinal function) to evaluate numerically the inverse Hankel transform.

© 1998 Optical Society of America

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References

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  1. R. Barakat, “The numerical evaluation of diffraction integrals,” in The Computer in Optical Research, R. Frieden, ed. (Springer, New York, 1980), Chap. 2.
  2. L. Bingham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N. J., 1988).
  3. D. Elliott, K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and 5.
  4. A. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef]
  5. P. Murphy, N. Gallagher, “Fast algorithm for the computation of the zero-order Hankel transform,” J. Opt. Soc. Am. 73, 1130–1137 (1983). Contains references to other FFT-based Hankel transform algorithms.
    [CrossRef]
  6. G. Agrawal, M. Lax, “End correction in the quasi-fast Hankel transform for optical propagation problems,” Opt. Lett. 6, 171–173 (1981).
    [CrossRef] [PubMed]
  7. A. Oppenheim, G. Frisk, D. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
    [CrossRef]
  8. S. Candel, “An algorithm for the Fourier–Bessel transform,” Comput. Phys. Commun. 23, 343–353 (1981).
    [CrossRef]
  9. V. Magni, V. Cerullo, S. Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992).
    [CrossRef]
  10. D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).
  11. R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
    [CrossRef]
  12. R. Barakat, B. Sandler, “Filon trapezoidal schemes for Hankel transforms of orders zero and one,” Appl. Math. Lett. (to be published).
  13. R. Barakat, B. Sandler, “Numerical evaluation for first-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. (to be published).
  14. L. Filon, “On a quadrature formula for trigonometric integrals,” Proc. R. Soc. Edin. 49, 38–47 (1928).
  15. C. Trantner, Integral Transforms in Mathematical Physics (Methuen, London, 1966), Chap. 6. This is the only book that contains full details of Filon’s work.
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  17. I. Gradshteyn, I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
  18. I. Sneddon, Fourier Transforms (Dover, New York, 1995), Chap. 3.
  19. G. Watson, Theory of Bessel Functions (Cambridge, London, 1944).
  20. F. Oliver, ed., Royal Society Mathematical Tables: Vol. 7, Bessel Functions, Part III, Zeros and Associated Values (Cambridge University Press, Cambridge, 1960).
  21. R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
    [CrossRef]
  22. R. Barakat, “Solution to an Abel integral equation for bandlimited functions by means of sampling theorems,” J. Math. Phys. (Cambridge, Mass.) 43, 332–335 (1964).
  23. A. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [CrossRef]

1996 (1)

R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

1992 (1)

1991 (1)

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

1983 (1)

1981 (2)

1978 (1)

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

1977 (2)

A. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
[CrossRef]

A. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

1964 (2)

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
[CrossRef]

R. Barakat, “Solution to an Abel integral equation for bandlimited functions by means of sampling theorems,” J. Math. Phys. (Cambridge, Mass.) 43, 332–335 (1964).

1928 (1)

L. Filon, “On a quadrature formula for trigonometric integrals,” Proc. R. Soc. Edin. 49, 38–47 (1928).

Agrawal, G.

Barakat, R.

R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
[CrossRef]

R. Barakat, “Solution to an Abel integral equation for bandlimited functions by means of sampling theorems,” J. Math. Phys. (Cambridge, Mass.) 43, 332–335 (1964).

R. Barakat, B. Sandler, “Filon trapezoidal schemes for Hankel transforms of orders zero and one,” Appl. Math. Lett. (to be published).

R. Barakat, B. Sandler, “Numerical evaluation for first-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. (to be published).

R. Barakat, “The numerical evaluation of diffraction integrals,” in The Computer in Optical Research, R. Frieden, ed. (Springer, New York, 1980), Chap. 2.

Berger, D.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Bingham, L.

L. Bingham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N. J., 1988).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Candel, S.

S. Candel, “An algorithm for the Fourier–Bessel transform,” Comput. Phys. Commun. 23, 343–353 (1981).
[CrossRef]

Cerullo, V.

Chamaly, S.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Elliott, D.

D. Elliott, K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and 5.

Filon, L.

L. Filon, “On a quadrature formula for trigonometric integrals,” Proc. R. Soc. Edin. 49, 38–47 (1928).

Frisk, G.

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

Gallagher, N.

Gradshteyn, I.

I. Gradshteyn, I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Jerri, A.

A. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Lax, M.

Levy, J.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Magni, V.

Martinez, D.

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

Mercier, D.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Monceau, P.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Murphy, P.

Oppenheim, A.

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

Parshall, E.

R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

Perreau, M.

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Rao, K.

D. Elliott, K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and 5.

Ryzhk, I.

I. Gradshteyn, I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Sandler, B.

R. Barakat, B. Sandler, “Numerical evaluation for first-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. (to be published).

R. Barakat, B. Sandler, “Filon trapezoidal schemes for Hankel transforms of orders zero and one,” Appl. Math. Lett. (to be published).

Siegman, A.

Silvestri, S.

Sneddon, I.

I. Sneddon, Fourier Transforms (Dover, New York, 1995), Chap. 3.

Trantner, C.

C. Trantner, Integral Transforms in Mathematical Physics (Methuen, London, 1966), Chap. 6. This is the only book that contains full details of Filon’s work.

Watson, G.

G. Watson, Theory of Bessel Functions (Cambridge, London, 1944).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Math. Lett. (1)

R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

Comput. Phys. Commun. (1)

S. Candel, “An algorithm for the Fourier–Bessel transform,” Comput. Phys. Commun. 23, 343–353 (1981).
[CrossRef]

J. Math. Phys. (Cambridge, Mass.) (1)

R. Barakat, “Solution to an Abel integral equation for bandlimited functions by means of sampling theorems,” J. Math. Phys. (Cambridge, Mass.) 43, 332–335 (1964).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Phys. I (Paris) (1)

D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).

Opt. Lett. (2)

Proc. IEEE (2)

A. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

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

Proc. R. Soc. Edin. (1)

L. Filon, “On a quadrature formula for trigonometric integrals,” Proc. R. Soc. Edin. 49, 38–47 (1928).

Other (11)

C. Trantner, Integral Transforms in Mathematical Physics (Methuen, London, 1966), Chap. 6. This is the only book that contains full details of Filon’s work.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

I. Gradshteyn, I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

I. Sneddon, Fourier Transforms (Dover, New York, 1995), Chap. 3.

G. Watson, Theory of Bessel Functions (Cambridge, London, 1944).

F. Oliver, ed., Royal Society Mathematical Tables: Vol. 7, Bessel Functions, Part III, Zeros and Associated Values (Cambridge University Press, Cambridge, 1960).

R. Barakat, “The numerical evaluation of diffraction integrals,” in The Computer in Optical Research, R. Frieden, ed. (Springer, New York, 1980), Chap. 2.

L. Bingham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N. J., 1988).

D. Elliott, K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and 5.

R. Barakat, B. Sandler, “Filon trapezoidal schemes for Hankel transforms of orders zero and one,” Appl. Math. Lett. (to be published).

R. Barakat, B. Sandler, “Numerical evaluation for first-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Plot of exact solution given by Eq. (3.6) with μ=1.5 (dashed curve) versus direct numerical calculation of Eq. (5.1) with Filon–Simpson with N=200 points (solid curve).

Fig. 2
Fig. 2

Same data as for Fig. 1, but with N=400 points.

Fig. 3
Fig. 3

Same data as for Fig. 1, but with N=800 points.

Tables (6)

Tables Icon

Table 1 Averaged Absolute Error (×10-8) over Blocks of Ten Values between Exact Results [Eq. (3.2)] and the Numerical Computations (Filon–Simpson Algorithm)

Tables Icon

Table 2 Averaged Absolute Error (×10-9) over Blocks of Ten Values between Exact Results [Eq. (3.6)] and the Numerical Computations (Filon–Simpson Algorithm)

Tables Icon

Table 3 Averaged Absolute Error (×10-7) over Blocks of Ten Values between Exact Results [Eq. (3.6)] and the Numerical Computations (Filon-Trapezoidal Algorithm)

Tables Icon

Table 4 Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Exact Value of b=1 for N=50 and 30

Tables Icon

Table 5 Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Estimated Value of b>1 (b=1.1) for N=50 and 30

Tables Icon

Table 6 Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Estimated Value of b<1 (b=0.9) for N=50 and 30

Equations (43)

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

H(r)=0h(p)J0(rp)pdp.
H(r)=abh(p)J0(rp)pdp,
F(x)=abf(p)exp(ixp)dp.
h(p)=c1+c2(p-p2k+1)+c3(p-p2k+1)+c3(p-p2k+1)2,
HK(r)=p2kp2k+2h(p)J0(rp)pdp,
(p2k+2-p2k+1)=(p2k+1-p2k)=δ.
xyJ0(y)dy=xJ1(x),
H(r)=k=0(N-2)/2Hk(r).
H(r)=1r[hNJ1(rpN)pN-h0J1(rp0)p0]-12δr3[(hN-2-4hN-1+3hN)$0(rpN)]-12δr3k=0(N-2)/2Qk$0(rp2k)+1δ2r4k=0(N-2)/2(h2k+2-2h2k+1+h2k)$1(rp2k, rp2k+2),
Q0=h2-4h1+3h0,
Qk=h2k+2-4h2k+1+6h2k-4h2k-1+h2k-2.
$0(x)0xJ1(y)ydy,
$1(x1, x2)x1x2$0(y)dy.
h(p)=A+Bp,
A=1δ(pk+1hk-pkhk+1),
B=1δ(hk+1-hk),
Hk(r)pkpk+1h(p)J0(rp)pdp,
Hk(r)=1r[hk+1J1(rpk+1)-hkJ1(rpk)]-1δr3(hk+1-hk)[$0(rpk+1)-$0(rp2k)],
H(r)=k=1NHk(r)=1r[hNJ1(rpN)-h0J1(rp0)]-1δr3k=0N(hk+1-hk)[$0(rpk+1)-$0(rpk)].
H(0)=abh(p)pdp,
h(p)=2π[arccos p-p(1-p2)1/2],0p1,
H(r)=2J1(r)r2,0r<,
maxerror=5.485×10-7(atr=2),
minerror=-1.227×10-8(atr=64)
maxerror=9.554×10-8(atr=2)
minerror=-2.092×10-9(atr=76)
|error|=110m=MM+9|erroratrm|,
maxerror=1.785×10-5(atr=2),
minerror=-2.452×10-8(atr=96)
maxerror=4.536×10-6(atr=2)
minerror=1.209×10-8(atr=96)
maxerror=2.030×10-6(atr=2)
minerror=4.446×10-9(atr=85)
h(p)=(1-p2)μ,0p1.
(1-p2)μexp(-μp2).
H(r)=2μΓ(μ+1)Jμ+1(r)rμ+1.
h(p)=p2,
H(r)=1r3[(r2-4)J1(r)+2rJ0(r)].
r=(2πNf)y,
h(p)=pΔ,
h(p)=0H(r)J0(pr)rdr.
h(p)=2b2n=1Hαnb [J1(αn)]-2J0αnpb.
J0(αn)=0.

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