Abstract

A family of generalized Jinc functions is defined and analyzed. The zero-order one is just the traditional Jinc function. In terms of these functions, series-form expressions are presented for the Fresnel diffraction of a circular aperture illuminated by converging spherical waves or plane waves. The leading term is nothing but the Airy formula for the Fraunhofer diffraction of circular apertures, and those high-order terms are directly related to those high-order Jinc functions. The truncation error of the retained terms is also analytically investigated. We show that, for the illumination of a converging spherical wave, the first 19 terms are sufficient for describing the three-dimensional field distribution in the whole focal region.

© 2003 Optical Society of America

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References

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  1. M Born, E Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 8.
  2. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2.
  4. A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Sec. 18.4.
  5. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in system of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  6. Y. Li, “Three-dimensional intensity distribution in low-Fresnel-number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
    [CrossRef]
  7. P. Wang, Y. Xu, W. Wang, Z. Wang, “Analytical expression for Fresnel diffraction,” J. Opt. Soc. Am. A 15, 684–688 (1998).
    [CrossRef]
  8. J. C. Heurtley, “Analytical expression for Fresnel diffraction: comment,” J. Opt. Soc. Am. A 15, 2929–2930 (1998).
    [CrossRef]
  9. P. L. Overfelt, D. J. White, “Analytical expression for Fresnel diffraction: comment,” J. Opt. Soc. Am. A 16, 613–615 (1999).
    [CrossRef]
  10. Y.-T. Wang, Y. C. Pati, T. Kailath, “Depth of focus and the moment expansion,” Opt. Lett. 20, 1841–1843 (1995).
    [CrossRef] [PubMed]
  11. M Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972).
  12. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  13. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  14. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  15. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  16. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [CrossRef]
  17. Q. Cao, J. Jahns, “Focusing analysis of the pinhole photon sieve: individual far-field model,” J. Opt. Soc. Am. A 19, 2387–2393 (2002).
    [CrossRef]
  18. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
    [CrossRef]
  19. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).The Parseval theorem is the special case f(x, y)= g(x, y) of Theorem IV of this reference.
    [CrossRef]
  20. W. H. Southwell, “Asymptotic solution of the Huygens–Fresnel integral in circular coordinates,” Opt. Lett. 3, 100–102 (1978).
    [CrossRef]
  21. A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [CrossRef]
  22. J. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
    [CrossRef]

2002 (3)

2001 (1)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

1999 (1)

1998 (2)

1995 (1)

1987 (1)

1984 (1)

1982 (1)

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

1981 (4)

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
[CrossRef]

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

1978 (1)

1968 (1)

Adelung, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Berndt, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Born, M

M Born, E Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 8.

Braat, J.

Cao, Q.

Dirksen, P.

Erkkila, J. H.

Givens, M. P.

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2.

Harm, S.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Heurtley, J. C.

Jahns, J.

Janssen, A. J. E. M.

Johnson, R. L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Kailath, T.

Kipp, L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Lalor, E.

Li, Y.

Overfelt, P. L.

Pati, Y. C.

Rogers, M. E.

Seemann, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Sec. 18.4.

Skibowski, M.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Southwell, W. H.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Wang, P.

Wang, W.

Wang, Y.-T.

Wang, Z.

White, D. J.

Wolf, E

M Born, E Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 8.

Wolf, E.

Xu, Y.

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

Nature (London) (1)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Opt. Commun. (3)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Opt. Lett. (2)

Other (5)

M Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972).

M Born, E Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 8.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Sec. 18.4.

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Figures (5)

Fig. 1
Fig. 1

Functional curves of Jinc 0 ( u ) , Jinc 1 ( u ) , and Jinc 2 ( u ) .

Fig. 2
Fig. 2

Asymptotic behavior of Jinc n ( u ) for large n. For comparison, Jinc 10 ( u ) and Jinc 30 ( u ) have been amplified 22 and 62 times, respectively.

Fig. 3
Fig. 3

Schematic view of the system configuration.

Fig. 4
Fig. 4

Relation between the number M 0 of the terms needed and N 2 .

Fig. 5
Fig. 5

Transverse field distribution at the plane corresponding to N 2 = 2 : (a) real part, (b) imaginary part. The Fresnel number is chosen such that N = 5 . The solid lines are the exact results, and the stars are the analytical results of the first 19 terms.

Equations (36)

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Jinc ( u ) = J 1 ( u ) u ,
Jinc ( u ) = 1 u 2 0 u vJ 0 ( v ) d v ,
Jinc n ( u ) = 1 u 2 n + 2 0 u v 2 n + 1 J 0 ( v ) d v ,
Jinc n ( u ) = m = 0 n ( - 2 ) m n ! ( n - m ) ! J m + 1 ( u ) u m + 1 ,
Jinc 1 ( u ) = J 1 ( u ) u - 2   J 2 ( u ) u 2 ,
Jinc 2 ( u ) = J 1 ( u ) u - 4   J 2 ( u ) u 2 + 8   J 3 ( u ) u 3 ,
Jinc 3 ( u ) = J 1 ( u ) u - 6   J 2 ( u ) u 2 + 24   J 3 ( u ) u 3 - 48   J 4 ( u ) u 4 .
J n ( u ) = 2 π u cos u - n π 2 - π 4
lim u Jinc n ( u ) = Jinc 0 ( u ) = J 1 ( u ) u .
Jinc n ( 0 ) = 1 2 ( n + 1 )
lim n Jinc n ( u ) = 1 2 ( n + 1 )   J 0 ( u ) .
U ( R ,   z ) = π λ z 0 a U 0 ( r ) exp jk   r 2 2 z J 0 kRr z r d r ,
U ( ρ ,   z ) = π N 1 0 1 exp ( j π N 2 ρ 1 2 ) J 0 ( 2 π N 1 ρ ρ 1 ) ρ 1 d ρ 1 ,
N 2 = a 2 λ 1 z - 1 f .
exp ( j π N 2 ρ 1 2 ) = n = 0 j n n !   ( π N 2 ) n ρ 1 2 n .
U ( ρ ,   z ) = n = 0 U n ( ρ ,   z ) ,
U n ( ρ ,   z ) = π j n n !   ( π N 2 ) n N 1 Jinc n ( 2 π N 1 ρ ) .
U ( 0 ,   z ) = N 1 j 2 N 2   [ exp ( j π N 2 ) - 1 ] .
I ( 0 ,   z ) = | U ( 0 ,   z ) | 2 = ( N 2 + N ) 2 N 2 2 sin 2 π N 2 2 ,
S M = 0 | U ( ρ ,   z ) - V M ( ρ ,   z ) | 2 ρ d ρ 0 | U ( ρ ,   z ) | 2 ρ d ρ ,
U ( ρ ,   z ) - V M ( ρ ,   z ) U M ( ρ ,   z )
U ( ρ ,   z ) = 2 π N 1 0 W ( ρ 1 ) J 0 ( 2 π N 1 ρ ρ 1 ) ρ 1 d ρ 1 ,
S M 0 | W M ( ρ 1 ) | 2 ρ 1 d ρ 1 0 | W ( ρ 1 ) | 2 ρ 1 d ρ 1 = ( π N 2 ) 2 M ( 2 M + 1 ) ( M ! ) 2 .
0 u v 2 n + 1 J 0 ( v ) d v = u 2 n + 1 J 1 ( u ) + 2 nu 2 n J 0 ( u ) - 4 n 2 0 u v 2 n - 1 J 0 ( v ) d v ,
0 u v 2 n + 1 J 0 ( v ) d v = u 2 n + 1 J 1 ( u ) - 2 n 0 u v 2 n J 1 ( v ) d v .
Jinc n - 1 ( u ) = m = 0 n - 1 ( - 2 ) m ( n - 1 ) ! ( n - 1 - m ) ! J m + 1 ( u ) u m + 1 ,
Jinc n ( u ) = J 1 ( u ) u + 2 n u 2   J 0 ( u ) + L ( u ) ,
L ( u ) = - 4 n 2 u 2 Jinc n - 1 ( u ) .
L ( u ) = m = 0 n - 1 2 n ( - 2 ) m + 1 n ! ( n - m - 1 ) ! J m + 1 ( u ) u m + 3 .
L ( u ) = ( - 2 ) n n !   J n + 1 ( u ) u n + 1 + ( - 2 ) n n !   J n - 1 ( u ) u n + 1 + m = 0 n - 2 2 n ( - 2 ) m + 1 n ! ( n - m - 1 ) ! J m + 1 ( u ) u m + 3 .
L ( u ) = m = n - 1 n ( - 2 ) m n ! ( n - m ) ! J m + 1 ( u ) u m + 1 + ( - 2 ) n - 1 n ! [ n - ( n - 1 ) ] ! J n - 2 ( u ) u n + m = 0 n - 3 2 n ( - 2 ) m + 1 n ! ( n - m - 1 ) ! J m + 1 ( u ) ( u ) m + 3 ,
( - 2 ) i n ! ( n - i ) ! J i + 1 ( u ) u i + 1 + ( - 2 ) i n ! ( n - i ) ! J i - 1 ( u ) u i + 1 .
L ( u ) = m = 1 n ( - 2 ) m n ! ( n - m ) ! J m + 1 ( u ) u m + 1 - 2 n u 2   J 0 ( u ) .
Jinc n ( u ) = 0 1 v 1 2 n + 1 J 0 ( uv 1 ) d v 1 .
Jinc n ( u ) = 1 2 ( n + 1 ) - g ( v 1 ) J 0 ( uv 1 ) d v 1 ,
lim n Jinc n ( u ) = 1 2 ( n + 1 ) - δ ( v 1 - 1 ) J 0 ( uv 1 ) d v 1 ,

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