Abstract

We discuss from the viewpoint of the Strehl ratio versus defocus, or the normalized axial-irradiance distribution, the influence of decentering the dark mask of an annular pupil. Our treatment, which is valid for pupil apertures with any Fresnel number, permits us to infer that the axial behavior of a noncentrally obscured pupil is equivalent to that of an apodizer with continuous amplitude variations. Hence the Strehl ratio versus defocus of an optical system can be shaped by use of noncentered dark masks that act as continuous gray apodizers. Several numerically evaluated examples are presented.

© 1994 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  4. J. Ojeda-Castañeda, L. R. Berriel-Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett.13, 183–185 (1988).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [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]
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    [CrossRef]

1994 (1)

1992 (2)

1988 (3)

1981 (1)

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

1978 (1)

1976 (1)

1964 (1)

Andrés, P.

Berriel-Valdós, L. R.

J. Ojeda-Castañeda, L. R. Berriel-Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett.13, 183–185 (1988).

Díaz, A.

Friedman, E.

Li, Y.

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

Mahajan, V. N.

Mani, S. A.

Martínez-Corral, M.

McCutchen, C. W.

Ojeda-Castañeda, J.

Pons, A.

Sutton, G. W.

Tepichin, E.

Weiner, M. M.

Wolf, E.

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

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

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

Opt. Lett. (1)

J. Ojeda-Castañeda, L. R. Berriel-Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett.13, 183–185 (1988).

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the optical setup: (a) coordinate system, (b) lateral displacement ηa of the dark mask.

Fig. 2
Fig. 2

Factor in Eq. (6) as a function of W20 for various values of N.

Fig. 3
Fig. 3

Apodization effect created by lateral displacement of a circular dark mask into a clear circular pupil: (a) azimuthally averaged pupil functions for three different values of the displacement parameter η, (b) Strehl ratio versus defocus for N = 100, (c) Strehl ratio versus defocus for N = 2.

Fig. 4
Fig. 4

Normalized light throughput versus lateral displacement for the clear aperture (dashed line) and for a window with Gaussian transmittance (solid curve).

Fig. 5
Fig. 5

Same as in Fig. (3), but here the pupil function has a window with Gaussian transmittance.

Fig. 6
Fig. 6

Shaping the Strehl ratio versus defocus: (a) 1-D pupil amplitude transmittance q0(ζ), (b) Strehl ratio versus defocus for N = 100, (c) Strehl ratio versus defocus for N = 2. Dotted curves correspond to η = 0 and ∊ = 0, dashed curves correspond to η = 0 and ∊ = 1 − (2)1/2/2, and solid curves correspond to η = (2)1/2/2 and ∊ = 1 − (2)1/2/2.

Equations (9)

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T ( z ) = i λ f ( f + z ) exp ( ikz ) 0 2 π 0 a t ( r , θ ) × exp [ i 2 π z 2 λ f ( f + z ) r 2 ] r d r d θ ,
T ( z ) = i 2 π λ f ( f + z ) exp ( ikz ) 0 a t 0 ( r ) × exp [ i 2 π z a 2 2 λ f ( f + z ) ( r a ) 2 ] r d r ,
ζ = ( r a ) 2 0.5 , q 0 ( ζ ) = t 0 ( r ) ,
T ( z ) = Q ( W 20 ) = π ( N 2 W 20 ) f 0.5 0.5 q 0 ( ζ ) × exp ( i 2 π W 20 ζ ) d ζ ,
N = a 2 λ f , W 20 = N z 2 ( f + z ) ,
S ( W 20 ) = | Q ( W 20 ) | 2 | Q ( 0 ) | 2 = ( N 2 W 20 ) 2 N 2 × | 0.5 0.5 q 0 ( ζ ) exp ( i 2 π W 20 ζ ) d ζ | 2 | 0.5 0.5 q 0 ( ζ ) d ζ | 2 = ( N 2 W 20 ) 2 N 2 S 0 ( W 20 ) .
p ( r , θ ) = R ( r ) t ( r , θ ) ,
p 0 ( r ) = R ( r ) t 0 ( r ) .
R ( r ) = exp [ π ( r ω ) 2 ] .

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