Abstract

The duality between the well-known Zernike polynomial basis set and the Fourier–Bessel expansion of suitable functions on the radial unit interval is exploited to calculate Hankel transforms. In particular, the Hankel transform of simple truncated radial functions is observed to be exact, whereas more complicated functions may be evaluated with high numerical accuracy. The formulation also provides some general insight into the limitations of the Fourier–Bessel representation, especially for infinite-range Hankel transform pairs.

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References

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  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.
  2. J. Markham and J.-A. Conchello, "Numerical evaluation of Hankel transforms for oscillating functions," J. Opt. Soc. Am. A 20, 621-630 (2003).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), Section 9.2 and Appendix VII.
  4. F. Lado, "Equation of state of the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092-3096 (1968).
    [CrossRef]
  5. H. Fisk Johnson, "An improved method for computing the discrete Hankel transform," Comput. Phys. Commun. 43, 181-202 (1987).
    [CrossRef]
  6. D. Lemoine, "The discrete Bessel transform algorithm," J. Chem. Phys. 101, 3936-3984 (1994).
    [CrossRef]
  7. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, "Quasi-discrete Hankel transform," Opt. Lett. 23, 409-411 (1998).
    [CrossRef]
  8. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004).
    [CrossRef]
  9. A. Prata, Jr., and W. V. T. Rusch, "Algorithm for computation of Zernike polynomials expansion coefficients," Appl. Opt. 28, 749-754 (1989).
    [PubMed]
  10. A. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (U. S. Government Printing Office, 1972), Chap. 22.
  11. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966), Chap. XVIII.
  13. The computer package, MATHEMATICA (Wolfram Research, Inc.), conveniently includes an algorithm for the roots of the derivatives of the Bessel functions.
  14. F. S. Gibson and F. Lanni, "Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy," J. Opt. Soc. Am. A 8, 1601-1613 (1991).
  15. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1980), pp. 1037, 8.965.

2004 (1)

2003 (1)

1998 (1)

1994 (1)

D. Lemoine, "The discrete Bessel transform algorithm," J. Chem. Phys. 101, 3936-3984 (1994).
[CrossRef]

1991 (1)

1989 (1)

1987 (1)

H. Fisk Johnson, "An improved method for computing the discrete Hankel transform," Comput. Phys. Commun. 43, 181-202 (1987).
[CrossRef]

1968 (1)

F. Lado, "Equation of state of the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092-3096 (1968).
[CrossRef]

Abramowitz, A.

A. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (U. S. Government Printing Office, 1972), Chap. 22.

Andrews, G. E.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.

Askey, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), Section 9.2 and Appendix VII.

Chen, M.

Chen, W.

Conchello, J.-A.

Gibson, F. S.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1980), pp. 1037, 8.965.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Huang, M.

Huang, W.

Johnson, H. Fisk

H. Fisk Johnson, "An improved method for computing the discrete Hankel transform," Comput. Phys. Commun. 43, 181-202 (1987).
[CrossRef]

Lado, F.

F. Lado, "Equation of state of the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092-3096 (1968).
[CrossRef]

Lanni, F.

Lemoine, D.

D. Lemoine, "The discrete Bessel transform algorithm," J. Chem. Phys. 101, 3936-3984 (1994).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.

Markham, J.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.

Prata, A.

Roy, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.

Rusch, W. V. T.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1980), pp. 1037, 8.965.

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.

Stegun, I.

A. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (U. S. Government Printing Office, 1972), Chap. 22.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966), Chap. XVIII.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), Section 9.2 and Appendix VII.

Yu, L.

Zhu, Z.

Appl. Opt. (1)

Comput. Phys. Commun. (1)

H. Fisk Johnson, "An improved method for computing the discrete Hankel transform," Comput. Phys. Commun. 43, 181-202 (1987).
[CrossRef]

J. Chem. Phys. (2)

D. Lemoine, "The discrete Bessel transform algorithm," J. Chem. Phys. 101, 3936-3984 (1994).
[CrossRef]

F. Lado, "Equation of state of the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092-3096 (1968).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Other (8)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), Section 9.2 and Appendix VII.

A. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (U. S. Government Printing Office, 1972), Chap. 22.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966), Chap. XVIII.

The computer package, MATHEMATICA (Wolfram Research, Inc.), conveniently includes an algorithm for the roots of the derivatives of the Bessel functions.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1980), pp. 1037, 8.965.

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Figures (14)

Fig. 1
Fig. 1

Logarithm of the absolute difference between the exact result and the Fourier–Bessel series (upper curve) and the Dini series (lower curve) plotted near the unit interval end point.

Fig. 2
Fig. 2

Evaluation of the Hankel transform of the sweep signal in the Zernike–Bessel representation retaining 150 terms in the summation.

Fig. 3
Fig. 3

Logarithm of the absolute value of the relative error between the Gauss–Kronrod and the Zernike–Bessel evaluations for 80 terms for the Hankel transform of the sweep function.

Fig. 4
Fig. 4

Logarithm of the absolute value of the relative error between the Gauss–Kronrod and the Zernike–Bessel evaluations for 120 terms for the Hankel transform of the sweep function.

Fig. 5
Fig. 5

Logarithm of the absolute value of the relative error between the Gauss–Kronrod and the Zernike–Bessel evaluations for 150 terms for the Hankel transform of the sweep function.

Fig. 6
Fig. 6

Real part of the out-of-focus pupil function at a propagation distance of 0.01575 mm derived from Eq. (4) of Gibson and Lanni.[14]

Fig. 7
Fig. 7

Evaluation of the Hankel transform of the pupil function in the Zernike–Bessel representation retaining 120 terms in the summation.

Fig. 8
Fig. 8

Logarithm of the absolute value of the relative error between the Gauss–Kronrod and the Zernike–Bessel evaluations for 120 terms for the Hankel transform of the pupil function.

Fig. 9
Fig. 9

Quasi-discrete zeroth-order Hankel transform of the sweep function in Fig. 2 using 256 mesh points and a 120-term Zernike–Bessel representation plotted as a function of radial variable.

Fig. 10
Fig. 10

Logarithm of the absolute value of the relative error between the exact transform of the sweep function and the 256-point quasi-discrete zeroth-order Hankel transform as a function of radial variable.

Fig. 11
Fig. 11

Logarithm of the absolute value of the relative error between the exact transform of the sweep function and the 512-point quasi-discrete zeroth-order Hankel transform as a function of radial variable.

Fig. 12
Fig. 12

Logarithm of the absolute value of the difference between the sinc function and the 150-term Zernike–Bessel representation plotted as a function of radial variable.

Fig. 13
Fig. 13

Logarithm of the absolute value of the difference between the exact transform of the sinc function and the 256-point quasi-discrete first-order Hankel transform as a function of radial variable.

Fig. 14
Fig. 14

Logarithm of the absolute value of the difference between the exact transform of the sinc function and the 150-term Zernike–Bessel representation of the first-order Hankel transform plotted as a function of radial variable.

Equations (31)

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f ̂ ( u , v ) = p = i p f ̂ p ( ρ ) exp ( i p ϕ ) ,
f ̂ p ( ρ ) = 2 π 0 f p ( r ) J p ( 2 π r ρ ) r d r ,
f p ( r ) = 2 π 0 f ̂ p ( ρ ) J p ( 2 π ρ r ) ρ d ρ .
0 1 R n l ( ρ ) R n l ( ρ ) ρ d ρ = 1 2 ( n + 1 ) δ n n ,
P n ( α , β ) ( x ) = ( 1 ) n R n l ( ρ ) ρ α ,
f ( ρ ) = m = 1 2 c m J l + 1 2 ( α l m ) J l ( α l m ρ ) ,
c m = 0 1 f ( ρ ) J l ( α l m ρ ) ρ d ρ .
R n l ( ρ ) = 2 ( 1 ) ( n l ) 2 m = 1 J n + 1 ( α l m ) α l m J l + 1 2 ( α l m ) J l ( α l m ρ ) ,
0 1 R n l ( ρ ) J l ( k ρ ) ρ d ρ = ( 1 ) ( n l ) 2 J n + 1 ( k ) k .
f ( ρ ) = n = l 2 ( n + 1 ) b n R n l ( ρ ) ,
b n = 0 1 f ( ρ ) R n l ( ρ ) ρ d ρ .
f ( ρ ) = n = l 2 ( n + 1 ) b n R n l ( ρ ) = m = 1 2 c m J l + 1 2 ( α l m ) J 1 ( α l m ρ ) ,
b n = ( 1 ) ( n l ) 2 m = 1 c m J n + 1 ( α l m ) α l m ,
c m = J l + 1 2 ( α l m ) n = l ( 1 ) ( n l ) 2 2 ( n + 1 ) b n J n + 1 ( 2 π α l m ) 2 π α l m .
J l ( k ρ ) = n = l ( 1 ) ( n l ) 2 2 ( n + 1 ) J n + 1 ( k ) k R n ( ρ ) ,
J l ( α l m ρ ) = n = l ( 1 ) ( n l ) 2 2 ( n + 1 ) J n + 1 ( α l m ) α l m R n ( ρ ) .
ρ n = 2 m = 1 J n ( α n m ρ ) α n m J n + 1 ( α n m ) ,
δ n n 2 ( n + 1 ) = 2 m = 1 J n + 1 ( α l m ) J n + 1 ( α l m ) α l m 2 J l + 1 2 ( α l m )
1 4 ( n + 1 ) = m = 1 1 α n m 2 .
f ̂ p ( ρ ) = J p + 1 ( ρ ) ρ ,
c m = 0 1 J p + 1 ( ρ ) ρ J p ( α p m ρ ) ρ d ρ ,
c m J p + 1 ( α p m ) α p m
f p ( r ) 2 m = 1 J p ( α l m r ) α p m J p + 1 ( α l m ) ,
f ( ρ ) = m = 1 2 d m J l ( β l m ρ ) ,
d m = β l m 2 { β l m 2 [ J l ( β l m ) ] 2 + ( β l m 2 l 2 ) [ J l ( β l m ) ] 2 } 1 0 1 t J l ( β l m t ) f ( t ) d t ,
ρ n = 2 m = 1 β n m J n + 1 ( β n m ) J n ( β n m ρ ) ( β n m n 2 ) J n 2 ( β n m ) .
f ( r ) = sin ( π d b a { [ r ( b a ) d + a ] 2 a 2 } ) ,
f ̂ 0 ( r ) = 2 π n = 0 ( 1 ) n 2 J n + 1 ( 2 π r ) 2 π r 2 ( n + 1 ) 0 1 f ( ρ ) R n 0 ( ρ ) ρ d ρ ,
f ( r ) = sin ( 2 π γ r ) 2 π γ r
f ̂ ( ρ ) = { ρ p cos ( p π 2 ) 2 π γ γ 2 ρ 2 ( γ + γ 2 ρ 2 ) p if 0 ρ γ sin [ p arcsin ( γ ρ ) ] 2 π γ ρ 2 γ 2 if ρ > γ .
P n ( α , β ) ( cos θ ) cos [ ( n + α + β + 1 2 ) θ ( π 4 ) ( 1 + 2 α ) ] π n [ sin ( θ 2 ) ] 1 2 + α [ cos ( θ 2 ) ] 1 2 + β + O ( n 3 2 ) ,

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