Abstract

Improved expressions are given for the performance parameters for transverse and axial gains for complex pupil filters. These expressions can be used to predict the behavior of filters that give a small axial shift in the focal intensity maximum and also predict the changes in gain for different observation planes.

© 2007 Optical Society of America

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References

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  1. C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. A 5, 643 (1988).
    [CrossRef]
  2. D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, Opt. Lett. 28, 607 (2003).
    [CrossRef] [PubMed]
  3. S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, Opt. Lett. 29, 932 (2004).
    [CrossRef] [PubMed]
  4. C. J. R. Sheppard, Optik (Stuttgart) 99, 32 (1995).
  5. P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

2005 (1)

P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
[CrossRef]

2004 (1)

2003 (1)

1995 (1)

C. J. R. Sheppard, Optik (Stuttgart) 99, 32 (1995).

1988 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Cagigal, M. P.

Campos, J.

Canales, V. F.

de Juana, D. M.

Escalera, J. C.

Gundu, P. N.

P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
[CrossRef]

Hack, E.

P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
[CrossRef]

Hegedus, Z. S.

Ledesma, S.

Oti, J. E.

Rastogi, P.

P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, Optik (Stuttgart) 99, 32 (1995).

C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. A 5, 643 (1988).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Yzuel, M. J.

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

Opt. Commun. (1)

P. N. Gundu, E. Hack, and P. Rastogi, Opt. Commun. 249, 101 (2005).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

C. J. R. Sheppard, Optik (Stuttgart) 99, 32 (1995).

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

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Figures (2)

Fig. 1
Fig. 1

(a) Axial gain factors G A along the axis calculated from a fourth-order expansion of intensity with a defocus filter u 0 = 1 . The exact behavior and that predicted using second- and third-order expansions are also shown. (b) Axial gain factors G A at u = u 0 corresponding to the true focus position calculated from Eq. (6) (fourth order). The exact behavior and that predicted using second- and third-order expansions are also shown.

Fig. 2
Fig. 2

(a) Transverse gain factors G T along the axis calculated from a fourth-order expansion of intensity with a defocus filter u 0 = 1 . The behavior predicted using second- and third-order expansions is also shown. (b) Axial gain factors G T at u = u 0 corresponding to the true focus position calculated from Eq. (7) (fourth order). The behavior predicted using second- and third-order expansions is also shown.

Equations (12)

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U ( v , u ) = 2 0 1 P ( ρ ) J 0 ( v ρ ) exp ( 1 2 i u ρ 2 ) ρ d ρ ,
I ( v , u ) = I 0 2 u Im ( I 0 I 1 * ) v 2 2 Re ( I 0 I 1 * ) u 2 4 [ Re ( I 0 I 2 * ) I 1 2 ] + u v 2 4 Im ( I 0 I 2 * ) + u 3 24 Im ( I 0 I 3 * 3 I 1 I 2 * ) + v 4 32 Re ( I 0 I 2 * + 2 I 1 2 ) + u 2 v 2 16 Re ( I 0 I 3 * I 1 I 2 * ) + u 4 192 Re ( I 0 I 4 * 4 I 1 I 3 * + 3 I 2 2 ) .
u F = 2 Im ( I 0 I 1 * ) Re ( I 0 I 2 * ) I 1 2 ,
S = I 0 2 u F 2 Im ( I 0 I 1 * )
G A ( u ) = 24 2 I ( v , u ) u 2 I ( v , u ) v = 0 ,
G T ( u ) = 2 2 I ( v , u ) v 2 I ( v , u ) v = 0 .
G A = 12 Re ( I 0 I 2 * ) I 1 2 u F 2 Im ( I 0 I 3 * 3 I 1 I 2 * ) u F 2 8 Re ( I 0 I 4 * 4 I 1 I 3 * + 3 I 2 2 ) I 0 2 u F 2 Im ( I 0 I 1 * ) ,
G T = 2 Re ( I 0 I 1 * ) u F 2 Im ( I 0 I 2 * ) u F 2 8 Re ( I 0 I 3 * I 1 I 2 * ) I 0 2 u F 2 Im ( I 0 I 1 * ) ,
G A ( u ) = 12 I 0 I 2 I 1 2 u 2 8 ( I 0 I 4 4 I 1 I 3 + 3 I 2 2 ) I 0 2 u 2 4 ( I 0 I 2 I 1 2 ) ,
G T ( u ) = 2 I 0 I 1 u 2 8 ( I 0 I 3 I 1 I 2 ) I 0 2 u 2 4 ( I 0 I 2 I 1 2 ) ,
I ( v , u ) = 1 v 2 4 u 2 48 + 5 v 4 192 + u 2 v 2 192 + u 4 5760
F = I 1 ( I 0 I 2 I 1 2 ) I 0 ( I 0 I 3 I 1 I 2 ) ,

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