Abstract

The generalization of zone plates and Fresnel lenses from the usual flat surfaces to curved surfaces is discussed. It is shown that, although Abbe’s sine law applies to zone plates but not to Fresnel lenses, both can yield single-element aplanats if formed on a curved surface. Third-order aberration formulas are given.

© 1977 Optical Society of America

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References

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  1. M. V. R. K. Murty, J. Opt. Soc. Am. 50, 923 (1960).
    [CrossRef]
  2. M. Young, J. Opt. Soc. Am. 62, 972 (1972).
    [CrossRef]
  3. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [CrossRef]
  4. E. Delano, J. Opt. Soc. Am. 64, 459 (1974).
    [CrossRef]
  5. J. Focke, “Higher Order Aberration Theory,” in Progress in OpticsE. Wolf, Ed. (North-Holland, Amsterdam, 1965) Vol. 4 p. 29.
    [CrossRef]

1974 (1)

1972 (1)

1960 (1)

1951 (1)

Delano, E.

Feder, D. P.

Focke, J.

J. Focke, “Higher Order Aberration Theory,” in Progress in OpticsE. Wolf, Ed. (North-Holland, Amsterdam, 1965) Vol. 4 p. 29.
[CrossRef]

Murty, M. V. R. K.

Young, M.

J. Opt. Soc. Am. (4)

Other (1)

J. Focke, “Higher Order Aberration Theory,” in Progress in OpticsE. Wolf, Ed. (North-Holland, Amsterdam, 1965) Vol. 4 p. 29.
[CrossRef]

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Equations (8)

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B = - y 2 ϕ [ ( u - 1 2 + u - 1 u + u 2 ) + 2 y c ( u - 1 + u ) ] + 8 m λ G y 4 , F = - y 2 ϕ u ¯ ( u - 1 + u + y c ) , C = - y 2 ϕ u ¯ 2 , E = 0 P = 0 a = - y 2 ϕ Δ λ / λ , b = 0.
B = - 1 4 y 4 ϕ 3 ( 4 S T + 3 T 2 + 1 ) + 8 m λ G y 4 , F = - 1 4 y 3 y ¯ ϕ 3 [ ( 3 T + T ¯ ) S + T 2 + 2 T T ¯ + 1 ] + 8 m λ G y 3 y ¯ , C = - 1 4 y 2 y ¯ 2 ϕ 3 [ 2 ( T + T ¯ ) S + 2 T T ¯ + T ¯ 2 + 1 ] + 8 m λ G y 2 y ¯ 2 , E = - 1 4 y y ¯ 3 ϕ 3 [ ( T + 3 T ¯ ) S + 3 T ¯ 2 + 1 ] + 8 m λ G y y ¯ 3 , P = 0 , a = - y 2 ϕ Δ λ / λ , b = - y y ¯ ϕ Δ λ / λ ,
B = - 1 4 y 2 ϕ ( 8 y c U + 3 U 2 + y 2 ϕ 2 ) + 8 m λ G y 4 , F = - 1 4 y ϕ ( 6 y y ¯ c U + 2 y 2 c U ¯ + y U 2 + 2 y U U ¯ + y 2 y ¯ ϕ ) + 8 m λ G y 3 y ¯ , C = - 1 4 y ϕ ( 4 y ¯ 2 c U + 4 y y ¯ c U ¯ + 2 y ¯ U U ¯ + y U ¯ 2 + y y ¯ 2 ϕ 2 ) + 8 m λ G y 3 y ¯ , E = - 1 4 y ¯ ϕ ( 2 y ¯ 2 c U + 6 y y ¯ c U ¯ + 3 y U ¯ 2 + y ¯ 2 y ϕ 2 ) + 8 m λ G y y ¯ 3 , P = 0 , a = - y 2 ϕ Δ λ / λ , b = - y y ¯ ϕ Δ λ / λ ,
Δ ζ 1 = 1 2 ( c - c ) ( 1 - n - 1 n ) r 2 ( V - 1 + c r ) ,
Δ ζ 2 = - 1 2 ( c - c ) ( 1 - n - 1 n ) l c V - 1 r 2 ,
- 2 n u ( Δ ζ 1 + Δ ζ 2 ) = ρ 2 ( Δ B ρ + F L · σ ) .
Δ B = - Δ c Δ n y 2 ( u i + u - 1 y c ) , F L = - Δ c Δ n y 2 u ¯ i ,
Δ B = - Δ c Δ n y 2 ( u i + u - 1 y c ) , F L = - Δ c Δ n y i ( y u ¯ - y ¯ u ) , Δ F = - Δ c Δ n y y ¯ ( u i + u - 1 y c ) , Δ C = - Δ c Δ n y y ¯ ( u ¯ i + u - 1 y ¯ c ) , Δ E = - Δ c Δ n y ¯ 2 ( u ¯ i + u - 1 y ¯ c ) , Δ P = Δ c Δ n y ¯ i ( y u ¯ - y ¯ u ) , Δ a = 0 , Δ b = 0.

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