Abstract

The relation between the variation of power and astigmatism in a progressive power optical element is considered. The classical Minkwitz identity is revisited. Then, it is shown how to extend this identity, which applies to the geometry of surfaces, to optical parameters that are determined by the geometry of reflected or refracted wavefronts.

© 2011 Optical Society of America

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References

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  1. G. Minkwitz, “Uber den Flachenastigmatismus Bei Gewissen Symmetrischen Aspharen,” Opt. Acta 10, 223–227 (1963).
    [CrossRef]
  2. J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
    [CrossRef]
  3. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (21 February 1967).
  4. R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
    [CrossRef]
  5. B. Bourdoncle, J. O. Chauveau, and J. L. Mercier, “Traps in displaying optical performances of a progressive addition lens,” Appl. Opt. 31, 3586–3593 (1992).
    [CrossRef] [PubMed]
  6. J. Rubinstein and G. Wolansky, “A class of elliptic equations related to optical design,” Math. Res. Lett. 9, 537–548 (2002).

2006 (1)

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
[CrossRef]

2005 (1)

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

2002 (1)

J. Rubinstein and G. Wolansky, “A class of elliptic equations related to optical design,” Math. Res. Lett. 9, 537–548 (2002).

1992 (1)

1963 (1)

G. Minkwitz, “Uber den Flachenastigmatismus Bei Gewissen Symmetrischen Aspharen,” Opt. Acta 10, 223–227 (1963).
[CrossRef]

Alvarez, L. W.

L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (21 February 1967).

Artal, P.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
[CrossRef]

Blendowske, R.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
[CrossRef]

Bourdoncle, B.

Campbell, C.

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

Chauveau, J. O.

Hayes, J. R.

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

King-Smith, E.

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

Mercier, J. L.

Minkwitz, G.

G. Minkwitz, “Uber den Flachenastigmatismus Bei Gewissen Symmetrischen Aspharen,” Opt. Acta 10, 223–227 (1963).
[CrossRef]

Rubinstein, J.

J. Rubinstein and G. Wolansky, “A class of elliptic equations related to optical design,” Math. Res. Lett. 9, 537–548 (2002).

Sheedy, J. E.

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

Villegas, E. A.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
[CrossRef]

Wolansky, G.

J. Rubinstein and G. Wolansky, “A class of elliptic equations related to optical design,” Math. Res. Lett. 9, 537–548 (2002).

Appl. Opt. (1)

Math. Res. Lett. (1)

J. Rubinstein and G. Wolansky, “A class of elliptic equations related to optical design,” Math. Res. Lett. 9, 537–548 (2002).

Opt. Acta (1)

G. Minkwitz, “Uber den Flachenastigmatismus Bei Gewissen Symmetrischen Aspharen,” Opt. Acta 10, 223–227 (1963).
[CrossRef]

Opt. Vis. Sci. (2)

J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Opt. Vis. Sci. 82, 916–924 (2005).
[CrossRef]

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).
[CrossRef]

Other (1)

L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (21 February 1967).

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Equations (19)

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H s = 1 2 · ( u ( 1 + | u | 2 ) 1 / 2 ) , K s = u x x u y y u x y 2 ( 1 + | u | 2 ) 2 .
C s ( 0 , Y ) X = 2 H s ( 0 , Y ) Y .
C s ( 0 , y ) x = 2 ( 1 + h 2 ( y ) ) 1 / 2 H s ( 0 , y ) y ,
u ( x , y ) = A ( y 3 / 3 + y x 2 ) .
C = 2 ( H 2 K ) 1 / 2 .
C s = 2 A 3 y 5 / ( 1 + A 2 y 4 ) 3 / 2 .
H m = Δ u 1 + | u | 2 , K m = 4 K s .
H m = 4 A y / ( 1 + A 2 y 4 ) , K m = 16 A 2 y 2 / ( 1 + A 2 y 4 ) 2 .
C m = 8 A x / ( 1 + A 2 y 4 ) , H m = 4 A y / ( 1 + A 2 y 4 ) .
2 u x 2 = 2 u y 2 , 2 u x y = 0.
u ( x , y ) = h ( y ) + 1 2 h ( y ) x 2 + O ( x 4 ) .
C m = 4 h ( y ) x 1 + ( h ( y ) ) 2 , H m ( 0 , y ) = 2 h ( y ) 1 + ( h ( y ) ) 2 .
x ( ( 1 + ( h ( y ) ) 2 ) C m ) = 2 y ( ( 1 + ( h ( y ) ) 2 ) H m ) .
H r = 1 2 ( 1 + β | u | 2 ) 1 / 2 μ 1 + | u | 2 ( u η η 1 + β | u | 2 + u ξ ξ ) .
K r = ( ( 1 + β | u | 2 ) 1 / 2 μ ) 2 1 + β | u | 2 u ξ ξ u η η u ξ η 2 ( 1 + | u | 2 ) 2 .
u ( ξ , η ) = h ( η ) + 1 2 h 1 + β h 2 ξ 2 .
H r = ( 1 + β h 2 ) 1 / 2 μ 1 + h 2 h 1 + β h 2 .
C r = 2 ( 1 + β h 2 ) 1 / 2 μ 1 + h 2 1 ( 1 + β h 2 ) 1 / 2 ( h 1 + β h 2 ) ξ .
x ( 1 + h 2 ( 1 + β h 2 ) 1 / 2 μ ( 1 + β h 2 ) 1 / 2 C r ) = 2 y ( 1 + h 2 ( 1 + β h 2 ) 1 / 2 μ H r ) .

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