Abstract

We show that, at any Fresnel number, a suitable one-dimensional Fourier transform relates the complex-amplitude distribution along the optical axis with the zero-order circular harmonic of the amplitude transmittance of a two-dimensional diffracting screen. First, our general result is applied to recognize that any rationally nonsymmetric screen generates an axial-irradiance distribution that exhibits focal shift. In this way we identify a wide set of two-dimensional screens that produce the same focal shift as that produced by the clear circular aperture. Second, we identify several apodizers for shaping the axial-amplitude distribution. We discuss some examples for achieving high-precision focusing, axial hyperresolution, or high focal depth.

© 1994 Optical Society of America

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References

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  1. M. De, L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
    [Crossref]
  2. M. Plight, “The rapid calculation of the optical transfer function for on-axis systems using the orthogonal properties of the Chebyshev polynomials,” Opt. Acta 25, 849–860 (1978).
    [Crossref]
  3. V. N. Mahajan, “Zernike polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [Crossref]
  4. J. Ojeda-Castaneda, L. R. Berriel-Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
    [Crossref] [PubMed]
  5. J. Ojeda-Castaneda, P. Andrés, M. Martínez-Corral, “Zone plates with cells apodized by Legendre profiles,” Appl. Opt. 29, 1299–1303 (1990).
    [Crossref] [PubMed]
  6. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [Crossref]
  7. 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]
  8. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye–Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
    [Crossref]
  9. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [Crossref]
  10. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [Crossref] [PubMed]
  11. Y. Li, “Focusing nontruncated elliptical Gaussian beams,” Opt. Commun. 68, 317–323 (1988).
    [Crossref]
  12. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [Crossref]
  13. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [Crossref]
  14. J. Ojeda-Castaneda, P. Andrés, M. Martinez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31, 4600–4602 (1992).
    [Crossref] [PubMed]
  15. J. Ojeda-Castaneda, L. R. Berriel-Valdós, E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26, 2770–2772 (1987).
    [Crossref] [PubMed]
  16. B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
    [Crossref]
  17. J. Ojeda-Castaneda, G. Ramírez, “Zone plates for zero-axial irradiance,” Opt. Lett. 18, 87–89 (1993).
    [Crossref] [PubMed]

1993 (1)

1992 (1)

1990 (1)

1988 (2)

1987 (1)

1983 (1)

1982 (2)

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

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

1981 (4)

V. N. Mahajan, “Zernike polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
[Crossref]

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

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

1978 (1)

M. Plight, “The rapid calculation of the optical transfer function for on-axis systems using the orthogonal properties of the Chebyshev polynomials,” Opt. Acta 25, 849–860 (1978).
[Crossref]

1977 (1)

M. De, L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[Crossref]

1969 (1)

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[Crossref]

1964 (1)

Andrés, P.

Berriel-Valdós, L. R.

De, M.

M. De, L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[Crossref]

Erkkila, J. H.

Frieden, B. R.

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[Crossref]

Givens, M. P.

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

Hazra, L. N.

M. De, L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[Crossref]

Li, Y.

Y. Li, “Focusing nontruncated elliptical Gaussian beams,” Opt. Commun. 68, 317–323 (1988).
[Crossref]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

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

Mahajan, V. N.

Martinez-Corral, M.

Martínez-Corral, M.

McCutchen, C. W.

Montes, E.

Ojeda-Castaneda, J.

Plight, M.

M. Plight, “The rapid calculation of the optical transfer function for on-axis systems using the orthogonal properties of the Chebyshev polynomials,” Opt. Acta 25, 849–860 (1978).
[Crossref]

Ramírez, G.

Rogers, M. E.

Spjelkavik, B.

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

Stamnes, J. J.

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

Wolf, E.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

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

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

Opt. Acta (3)

M. De, L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[Crossref]

M. Plight, “The rapid calculation of the optical transfer function for on-axis systems using the orthogonal properties of the Chebyshev polynomials,” Opt. Acta 25, 849–860 (1978).
[Crossref]

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[Crossref]

Opt. Commun. (5)

Y. Li, “Focusing nontruncated elliptical Gaussian beams,” Opt. Commun. 68, 317–323 (1988).
[Crossref]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

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

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

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

Opt. Lett. (2)

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram: (a) the diffracting screen, (b) the optical setup under consideration.

Fig. 2
Fig. 2

Azimuthal average for obtaining p 0(r) from the pupil function along any centered ring with radius r.

Fig. 3
Fig. 3

Generation of a family of apodized pupils with the same transmittance profile but with different central obscuration ratios: (a) generating function q 0(ζ) = 1 − ζ2, (b) some members of the family of 2-D annular pupils generated with the above q 0(ζ).

Fig. 4
Fig. 4

Focal-shift effect produced by an absorbing screen whose azimuthally averaged amplitude transmittance is p 0(r) = 0.5: (a) gray-scale representation of the annular aperture, (b) normalized axial-irradiance distribution for N = 2.5. The normalized axial-irradiance distribution for the clear annular pupil is the same.

Fig. 5
Fig. 5

Normalized axial-irradiance distribution for the pupil functions in Fig. 3(b) (solid curve) and for the clear annular pupil (dashed curve).

Fig. 6
Fig. 6

Axial hyperresolving effect: (a) apodizers whose azimuthally averaged amplitude transmitance is mapped into the 1-D function q 0(ζ) = ζ2, (b) normalized axial-irradiance distribution for these pupil functions (solid curve) and for the clear annular pupil (dashed curve).

Fig. 7
Fig. 7

First apodizer for high-precision focusing: (a) annular amplitude transmittance, (b) normalized axial-irradiance distribution.

Fig. 8
Fig. 8

Second apodizer for high-precision focusing: (a) amplitude transmittance, (b) normalized axial-irradiance distribution.

Equations (21)

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u ( z ) = [ - i / λ f ( f + z ) ] exp ( i k z ) 0 2 π a a p ( r , θ ) × exp [ - i k z r 2 / 2 f ( f + z ) ] r d r d θ ,
p ( r , θ ) = m = - + p m ( r ) exp ( i m θ ) ,
p m ( r ) = ( 1 / 2 π ) 0 2 π p ( r , θ ) exp ( - i m θ ) d θ .
u ( z ) = [ - i k / f ( f + z ) ] exp ( i k z ) a a p 0 ( r ) × exp { - i k [ a 2 z / 2 f ( f + z ) ] ( r / a ) 2 } r d r ,
ζ = 2 ( r / a ) 2 - 2 1 - 2 - 1 ,             q 0 ( ζ ) = p 0 ( r ) .
u ( z ) = [ π a 2 ( 1 - 2 ) / 2 λ f ( f + z ) ] - 1 1 q 0 ( ζ ) × exp { - i 2 π [ a 2 ( 1 - 2 ) z / 4 λ f ( f + z ) ] ζ } d ζ .
N = a 2 ( 1 - 2 ) / λ f ,             W 20 = N z / 2 ( f + z ) ,             h ( W 20 ) = u ( z ) ,
h ( W 20 ) = [ π ( N - 2 W 20 ) / 2 f ] - 1 1 q 0 ( ζ ) exp ( - i π W 20 ζ ) d ζ ,
p 0 ( r ) = k ,             a r a ,
q 0 ( ζ ) = n = 0 + a n P n ( ζ ) .
h ( W 20 ) = ( π N / f ) n = 0 + ( - i ) n a n j n ( π W 20 ) .
p 0 ( r ) = 2 ( r / a ) 2 - 2 1 - 2 - 1             if a r a
h ( W 20 ) 2 = ( π N / f ) 2 [ j 1 ( π W 20 ) ] 2 .
h ( W 20 ) = 1 - δ ( W 20 ) ,
q 0 ( ζ ) = δ ( ζ ) - 0.5 P 0 ( ζ ) .
δ ( ζ ) = n = - + ϕ n * ( 0 ) ϕ n ( ζ ) .
δ ( ζ ) = n = 0 + 2 n + 1 2 P n ( 0 ) P n ( ζ )
δ ( ζ ) = lim M m = 0 M [ ( - 1 ) m ( 2 m ) ! ( 4 m + 1 ) / 2 2 m + 1 ( m ! ) 2 ] P 2 m ( ζ ) .
q 0 ( ζ ) = lim M m = 1 M [ ( - 1 ) m ( 2 m ) ! ( 4 m + 1 ) / 2 2 m + 1 ( m ! ) 2 ] P 2 m ( ζ ) ,
h ( W 20 ) = lim M m = 1 M [ ( 2 m ) ! ( 4 m + 1 ) / 2 2 m + 1 ( m ! ) 2 ] j 2 m ( π W 20 ) ,
h ( W 20 ) = 1 - [ sin ( π W 20 ) / π W 20 ] .

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