Abstract

It is shown for the first time we believe, that when a spherical wave illuminates a certain type of diffracting screen, in addition to the expected focal-shift effect, depending on the value of the Fresnel number of the focusing system, a focal switch effect can appear, i.e., an increase in the height of the lateral lobe of the axial-intensity distribution over that of the central lobe.

© 1996 Optical Society of America

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References

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  1. D. A. Holmes, J. E. Korka, P. Avizonis, “Parametric study of apertured Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  2. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [CrossRef]
  3. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  4. V. N. Mahajan, “Axial irradiance and optimal focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  5. J. Ojeda-Castañeda, M. Martínez-Corral, P. Andrés, A. Pons, “Strehl ratio versus defocus for noncentrally obscured pupils,” Appl. Opt. 33, 7611–7616 (1994).
    [CrossRef] [PubMed]
  6. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  7. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  8. Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
    [CrossRef]
  9. M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef] [PubMed]
  10. P. Andrés, M. Martínez-Corral, J. Ojeda-Castañeda, “Off-axis focal shift for rotationally nonsymmetric screens,” Opt. Lett. 18, 1290–1292 (1993).
    [CrossRef] [PubMed]

1994 (2)

1993 (1)

1992 (1)

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

1983 (1)

1982 (2)

1981 (1)

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

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

1972 (1)

Andrés, P.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Avizonis, P.

Carter, W. H.

Holmes, D. A.

Korka, J. E.

Li, Y.

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[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.

Martínez-Corral, M.

Ojeda-Castañeda, J.

Pons, A.

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. (5)

J. Mod. Opt. (1)

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

Opt. Acta (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Opt. Commun. (2)

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

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

Opt. Lett. (1)

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Figures (4)

Fig. 1
Fig. 1

Geometry of the diffraction problem.

Fig. 2
Fig. 2

For the N = 10 case (a) the normalized version of the function I′(z) = |u′(z)|2 corresponding to the diffracting screens with transmittance q(ζ) = 4ζ2 (solid curve) and q(ζ) = 1 – 4ζ 2 (dashed curve), and to the corresponding circular aperture (dashed curve); (b) normalized axial-intensity distribution for the same screens as in (a).

Fig. 3
Fig. 3

Normalized intensity distribution corresponding to the axially superresolving screen, q(ζ) = 4ζ2, and N = 5.

Fig. 4
Fig. 4

Relative excess, ΔI = (IL IC )/I(0), of the maximum intensity IL of the lateral lobe over the maximum intensity IC of the central lobe for systems with different Fresnel numbers.

Equations (4)

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u ( z ) = exp ( ikz ) 2 π i λ f ( f + z ) 0 r 0 t ( r ) × exp [ i 2 π z 2 λ f ( f + z ) r 2 ] r d r ,
ζ = ( r r o ) 2 0.5 , q ( ζ ) = t ( r ) ,
u ( z ) = π N ( f + z ) 0.5 0.5 q ( ζ ) exp [ i 2 π N z 2 ( f + z ) ζ ] d ζ = π N ( f + z ) u ( z ) ,
I ( z ) = [ π N ( f + z ) ] 2 | 0.5 0.5 q ( ζ ) exp [ i 2 π N z 2 ( f + z ) ζ ] d ζ | 2 = ( π N ( f + z ) ) 2 I ( z ) .

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