Abstract

Radially symmetric pupil plane phase retardation functions are derived that extend focal depth and alleviate third-order spherical aberration (SA) effects. The radial symmetry of these functions means that they can be more conveniently manufactured by use of traditional techniques such as diamond machining than previously reported filters with rectangular symmetry. The method employs minimization of the variation of Strehl ratio with defocus, W20, and SA, W40. The performance of the derived phase filters is illustrated by comparison with standard optical systems and with previously reported phase filters.

© 2003 Optical Society of America

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References

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2002 (2)

2001 (2)

W. Chi and N. George, Opt. Lett. 26, 875 (2001).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, Proc. SPIE 4471, 272 (2001).
[CrossRef]

2000 (1)

1998 (1)

1995 (1)

1992 (1)

1991 (2)

1988 (1)

1986 (1)

1985 (1)

1971 (1)

1960 (1)

Andrés, P.

Bara, S.

Berriel Valdos, L. R.

Berriel-Valdos, L. R.

Borovikov, V. A.

V. A. Borovikov, Uniform Stationary Phase Method, Vol. 40 of IEE Electromagnetic Waves Series (Institution of Electrical Engineers, London, 1994).

Cathey, W. T.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, Proc. SPIE 4471, 272 (2001).
[CrossRef]

E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995).
[CrossRef]

Chi, W.

Colautti, C.

Davidson, N.

Díaz, A.

Dowski, E. R.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, Proc. SPIE 4471, 272 (2001).
[CrossRef]

E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995).
[CrossRef]

Friberg, A. T.

Friesem, A. A.

George, N.

Harvey, A. R.

S. Mezouari and A. R. Harvey, Proc. SPIE 4768, 21 (2002).
[CrossRef]

Hasman, E.

Jaroszewicz, Z.

Kolodziejczyk, A.

Mezouari, S.

S. Mezouari and A. R. Harvey, Proc. SPIE 4768, 21 (2002).
[CrossRef]

Mino, M.

Montes, E. L.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

Okano, Y.

Popov, S. Y.

Sherif, S. S.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, Proc. SPIE 4471, 272 (2001).
[CrossRef]

Sicre, E. E.

Sochacki, J.

Thaning, A.

Welford, W. T.

Zalvidea, D.

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Figures (4)

Fig. 1
Fig. 1

Axial intensity as a function of SA of W 40 , with zero-defocus aberration. The solid curve represents an ideal circular lens, the dark dashed plot corresponds to the use of an additional LF ( β = 5.6 π and β 0 / β = 0.401 ), and the light dotted plot corresponds to the QF ( α = 0.75 π ).

Fig. 2
Fig. 2

Axial intensity as a function of defocus of W 20 . The curves are as defined in Fig. 1.

Fig. 3
Fig. 3

Computed MTFs obtained with optical systems suffering from defocus W 20 and spherical aberration W 40 of 0, λ / 2 , λ , and 2 λ . The rectangularly separable filter has a 4 π peak-to-valley optical path difference.

Fig. 4
Fig. 4

Variation of the intensity of the focal point as a function of the SA parameter, W 40 , and β . When β = 0 , the pupil function is identical to a circular aperture, and the maximum intensity (aberration free, W 40 = 0 ) is normalized to 1. The relative intensity decreases drastically as β increases.

Equations (9)

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I W 20 , W 40 = 4 π 2 0 p ˜ ρ exp i 2 π W 20 ρ / ρ 0 2 + W 40 ρ / ρ 0 4 ρ d ρ 2 ,
p ˜ ρ = exp i 2 π θ ρ 0 ρ ρ 0 0 ρ > ρ 0 ,
I W 20 , W 40 = π 2 ρ 0 4 - exp i 2 π Φ ξ + W 40 ξ 2 + W 20 + W 40 ξ d ξ 2 ,
d d ξ Φ ξ + W 40 ξ 2 + W 20 + W 40 ξ | ξ = ξ s = 0 ,
I W 20 , W 40 2 π 3 ρ 0 4 1 2 W 40 + Φ ξ s ,
d d W 20 1 2 W 40 + Φ ξ s = 0 ,
d d W 40 1 2 W 40 + Φ ξ s = 0 .
θ ρ = β 0 ρ / ρ 0 4 + β ρ / ρ 0 4 log ρ / ρ 0 ,
θ ρ = α ρ / ρ 0 2 - 1 / 2 2 ,

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