Abstract

The conditions under which a two-element variable power lens can be created are examined. Such a lens is defined as one in which the functional form of the optical effect created does not change as the elements translate with respect to one another—only the magnitude of the effect changes. It is found that only variable power optical effects that can be described by quadratic functions can be formed by laterally translating two-element variable power lenses. In the case of rotationally translating two-element variable power lenses, possible designs are found by mapping possible laterally translating designs from a Cartesian space to the polar coordinate space of the rotationally translating lens.

© 2011 Optical Society of America

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References

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  1. C. E. Campbell, “Conditions under which two-element variable power lenses can be created. Part 2. Application to specific designs,” J. Opt. Soc. Am. A 28, 2153–2159 (2011)
    [CrossRef]
  2. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (February 21, 1967).
  3. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
    [CrossRef] [PubMed]
  4. W. E. Humphrey, “Variable astigmatic lens and method for constructing lens,” U.S. patent 3,751,138 (August 7, 1973).
  5. J. G. Baker, “Variable power, analytic function, optical component in the form of a pair of laterally adjustable plates having shaped surfaces, and optical systems including such components,” U.S. patent 3,583,790 (November 7, 1968).
  6. I. A. Palusinski, J. M. Sasián, and J. E. Greivenkamp, “Lateral-shift variable aberration generators,” Appl. Opt. 38, 86–90(1999).
    [CrossRef]
  7. E. Acosta and J. Sasián, “Phase plates for the generation of variable amount of primary spherical aberration,” Opt. Express 19, 13171–13178 (2011).
    [CrossRef] [PubMed]
  8. J. G. Baker and W. I. Plummer, “Analytic function optical component,” U.S. patent 4,650,292 (December 28, 1983).
  9. G. G. Stokes, “On a mode of measuring the astigmatism of a defective eye,” Report of the British Association for 1849, part II, 10 in Mathematical and Physical Papers, Vol.  II (Cambridge University Press, 1883), pp. 172–175.
  10. J. P. Foley and C. E. Campbell, “An optical device with variable astigmatic power,” Optom. Vis. Sci. 76, 664–667 (1999).
    [CrossRef] [PubMed]
  11. C. E. Campbell, “Variable power phase lenses,” Appl. Opt. 44, 3438–3441 (2005).
    [CrossRef] [PubMed]
  12. E. Acosta and S. Bara, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A 22, 1993–1996 (2005).
    [CrossRef]

2011 (2)

2005 (2)

1999 (2)

I. A. Palusinski, J. M. Sasián, and J. E. Greivenkamp, “Lateral-shift variable aberration generators,” Appl. Opt. 38, 86–90(1999).
[CrossRef]

J. P. Foley and C. E. Campbell, “An optical device with variable astigmatic power,” Optom. Vis. Sci. 76, 664–667 (1999).
[CrossRef] [PubMed]

1970 (1)

Acosta, E.

Alvarez, L. W.

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

Baker, J. G.

J. G. Baker, “Variable power, analytic function, optical component in the form of a pair of laterally adjustable plates having shaped surfaces, and optical systems including such components,” U.S. patent 3,583,790 (November 7, 1968).

J. G. Baker and W. I. Plummer, “Analytic function optical component,” U.S. patent 4,650,292 (December 28, 1983).

Bara, S.

Campbell, C. E.

Foley, J. P.

J. P. Foley and C. E. Campbell, “An optical device with variable astigmatic power,” Optom. Vis. Sci. 76, 664–667 (1999).
[CrossRef] [PubMed]

Greivenkamp, J. E.

Humphrey, W. E.

W. E. Humphrey, “Variable astigmatic lens and method for constructing lens,” U.S. patent 3,751,138 (August 7, 1973).

Lohmann, A. W.

Palusinski, I. A.

Plummer, W. I.

J. G. Baker and W. I. Plummer, “Analytic function optical component,” U.S. patent 4,650,292 (December 28, 1983).

Sasián, J.

Sasián, J. M.

Stokes, G. G.

G. G. Stokes, “On a mode of measuring the astigmatism of a defective eye,” Report of the British Association for 1849, part II, 10 in Mathematical and Physical Papers, Vol.  II (Cambridge University Press, 1883), pp. 172–175.

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Optom. Vis. Sci. (1)

J. P. Foley and C. E. Campbell, “An optical device with variable astigmatic power,” Optom. Vis. Sci. 76, 664–667 (1999).
[CrossRef] [PubMed]

Other (5)

J. G. Baker and W. I. Plummer, “Analytic function optical component,” U.S. patent 4,650,292 (December 28, 1983).

G. G. Stokes, “On a mode of measuring the astigmatism of a defective eye,” Report of the British Association for 1849, part II, 10 in Mathematical and Physical Papers, Vol.  II (Cambridge University Press, 1883), pp. 172–175.

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

W. E. Humphrey, “Variable astigmatic lens and method for constructing lens,” U.S. patent 3,751,138 (August 7, 1973).

J. G. Baker, “Variable power, analytic function, optical component in the form of a pair of laterally adjustable plates having shaped surfaces, and optical systems including such components,” U.S. patent 3,583,790 (November 7, 1968).

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Figures (2)

Fig. 1
Fig. 1

The two elements of a rotationally translating variable power lens are shown rotated about a common rotation point by angle Δ α in opposite directions from a line drawn from the rotation point through the center of area in which the variable power is generated. Element 1 is shown with thin-line borders. Element 2 is shown with thick, dashed-line borders. The area where the variable power is generated is indicated by the dashed circle whose center is at distance a from the common rotation point.

Fig. 2
Fig. 2

Point ( x , y ) in the Cartesian coordinate system is mapped from a location ( x , y ) in the deformed coordinate system via an intermediate mapping from ( x , y ) to coordinates θ ( x ) and r ( y ) and then by a standard polar to Cartesian transformation from ( θ , r ) to ( x , y ) with a final transformation of the y coordinate by distance a .

Equations (29)

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F var ( x , y , Δ m ) = V ( x , y ) · s ( Δ m ) ,
Δ x = a · Δ r and Δ y = b · Δ r .
F var ( x , y , Δ r ) = V ( x , y ) · s ( Δ r ) .
f ( x + a · Δ r , y + b · Δ r ) = f ( x , y ) + n = 1 ( a x + b y ) n f ( x , y ) n ! Δ r n .
f ( x a · Δ r , y b · Δ r ) = f ( x , y ) n = 1 ( a x + b y ) n f ( x , y ) n ! ( Δ r ) n .
f ( x + a · Δ r , y + b · Δ r ) f ( x a · Δ r , y b · Δ r ) = f ( x , y ) f ( x , y ) + n = 1 ( a x + b y ) n f ( x , y ) · Δ r n ( a x + b y ) n f ( x , y ) · ( Δ r ) n n ! ,
f ( x + a · Δ r , y + b · Δ r ) f ( x a · Δ r , y b · Δ r ) = n = 1 ( a x + b y ) n f ( x , y ) n ! ( Δ r n ( Δ r ) n ) .
f ( x + a · Δ r , y + b · Δ r ) f ( x a · Δ r , y b · Δ r ) = 2 n = 0 ( a x + b y ) ( 2 n + 1 ) f ( x , y ) ( 2 n + 1 ) ! Δ r ( 2 n + 1 ) .
F var ( x , y , Δ r ) = V ( x , y ) · s ( Δ r ) = 2 n = 0 ( a x + b y ) ( 2 n + 1 ) f ( x , y ) ( 2 n + 1 ) ! Δ r ( 2 n + 1 ) .
n = 0 V ( x , y ) · Δ r ( 2 n + 1 ) = 2 n = 0 ( a x + b y ) ( 2 n + 1 ) f ( x , y ) ( 2 n + 1 ) ! Δ r ( 2 n + 1 ) ,
V ( x , y ) { Δ r + Δ r 3 + } = { 2 ( a f ( x , y ) x + b f ( x , y ) y ) Δ r + 1 3 ( a 3 3 f ( x , y ) x 3 + 3 a 2 b 3 f ( x , y ) x 2 y + 3 a b 2 3 f ( x , y ) x y 2 + b 3 3 f ( x , y ) y 3 ) Δ r 3 + 1 60 ( a 5 5 f ( x , y ) x 5 + 5 a 4 b 5 f ( x , y ) x 4 y + 10 a 3 b 2 5 f ( x , y ) x 3 y 2 + 10 a 2 b 3 5 f ( x , y ) x 2 y 3 + 5 a b 4 5 f ( x , y ) x y 4 + 5 f ( x , y ) y 5 ) Δ r 5 + } .
n + 1 f ( x , y ) x n + 1 = x ( n f ( x , y ) x n ) ,
n + 1 f ( x , y ) x n + 1 = x ( B ) = 0.
n + m f ( x , y ) x n y m = m y m ( n f ( x , y ) x n ) , if     f n ( x , y ) x n = B then     n + m f ( x , y ) x n y m = m y m ( B ) = 0 .
n + m f ( x , y ) x n y m = B then     n + m + 1 f ( x , y ) x n y m + 1 = 0 and     n + m + 1 f ( x , y ) x n + 1 y m = 0 .
( a f ( x , y ) x + b f ( x , y ) y ) .
V ( x , y ) = 2 ( a f ( x , y ) x + b f ( x , y ) y ) ,
( a 3 3 f ( x , y ) x 3 + 3 a 2 b 3 f ( x , y ) x 2 y + 3 a b 2 3 f ( x , y ) x y 2 + b 3 3 f ( x , y ) y 3 ) = k ,
f ( x , y ) = c · x 3 + d · x 2 y + e · x y 2 + g · y 3 + h · x 2 + i · x y + j · y 2 + l · x + m · y ,
V ( x , y ) = 2 ( ( 3 a c + b d ) x 2 + ( 2 a d + 2 b e ) x y + ( a e + 3 b g ) y 2 ( 2 a h + b i ) x + ( a i + 2 b j ) y + ( a l + b m ) ) .
V ( x , y ) = k x 2 + p ( y ) x + q ( y ) ,
θ = π / 2 S · x ,
r = y + a ,
x = r cos ( θ ) y = r sin ( θ ) a ,
x = ( y + a ) cos ( π / 2 S · x ) y = ( y + a ) sin ( π / 2 S · x ) a ,
S = arctan ( D / 2 a ) ( D / 2 ) ,
θ = arctan ( ( y + a ) / x ) ,
x = ( π / 2 arctan ( ( y + a ) / x ) ) S ,
y = x 2 + ( y + a ) 2 a .

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