Abstract

The properties of the YӮ diagram as applied to radial gradient lens systems are developed. It is shown that the diagram consists of elliptical arcs, which are circular arcs for certain conditions. The effects of pupil and object shifts are discussed as well as the effects of scaling the pupil size. It is shown how the constructional parameters of the system are read from the diagram. Finally, for the purpose of design, it is shown how one may obtain the YӮ diagram of a system which satisfies a certain set of design constraints.

© 1988 Optical Society of America

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References

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  1. E. Delano, “First-Order Design and the Y−Ӯ Diagram,” Appl. Opt. 2, 1251 (1963).
    [CrossRef]
  2. S. R. Lange et al., “APART, A First-Order Deterministic Radiation Analysis Program,” Proc. Soc. Photo-Opt. Instrum. Eng. 107, 89 (1977).
  3. W. Besenmater, “Designing Zoom Lenses Aided by the Delano Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 242 (1980).
  4. D. Kessler, R. V. Shack, “First-Order Design of Laser Systems with the YӮ Diagram,” J. Opt. Soc. Am. A 1, 1219 (1984).
  5. R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, “Semiautomatic Generation of Optical Prototypes,” Appl. Opt. 6, 969 (1967).
    [CrossRef] [PubMed]
  6. R. V. Shack, “Analytical System Design with Pencil and Ruler—the Advantages of the Y−Ӯ Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 384, 127 (1973).
  7. F. J. López-López, “Inhomogeneous Media in the Y−Ӯ Diagram,” Annual Optical Society Meeting (1972).
  8. P. J. Sands, “Inhomogeneous Lenses, III. Paraxial Optics,” J. Opt. Soc. Am. 61, 879 (1971).
    [CrossRef]
  9. R. W. Wood, Physical Optics (Macmillan, New York, 1905), p. 71.
  10. M. E. Harrigan, R. P. Loce, J. R. Rogers, “Application of the Y−Ӯ Diagram to GRIN Rod Design,” Appl. Opt. 27, 459 (1988).
    [CrossRef] [PubMed]

1988 (1)

1984 (1)

D. Kessler, R. V. Shack, “First-Order Design of Laser Systems with the YӮ Diagram,” J. Opt. Soc. Am. A 1, 1219 (1984).

1980 (1)

W. Besenmater, “Designing Zoom Lenses Aided by the Delano Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 242 (1980).

1977 (1)

S. R. Lange et al., “APART, A First-Order Deterministic Radiation Analysis Program,” Proc. Soc. Photo-Opt. Instrum. Eng. 107, 89 (1977).

1973 (1)

R. V. Shack, “Analytical System Design with Pencil and Ruler—the Advantages of the Y−Ӯ Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 384, 127 (1973).

1971 (1)

1967 (1)

1963 (1)

Besenmater, W.

W. Besenmater, “Designing Zoom Lenses Aided by the Delano Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 242 (1980).

Delano, E.

Harrigan, M. E.

Kessler, D.

D. Kessler, R. V. Shack, “First-Order Design of Laser Systems with the YӮ Diagram,” J. Opt. Soc. Am. A 1, 1219 (1984).

Lange, S. R.

S. R. Lange et al., “APART, A First-Order Deterministic Radiation Analysis Program,” Proc. Soc. Photo-Opt. Instrum. Eng. 107, 89 (1977).

Loce, R. P.

López-López, F. J.

F. J. López-López, “Inhomogeneous Media in the Y−Ӯ Diagram,” Annual Optical Society Meeting (1972).

Pegis, R. J.

Rigler, A. K.

Rogers, J. R.

Sands, P. J.

Shack, R. V.

D. Kessler, R. V. Shack, “First-Order Design of Laser Systems with the YӮ Diagram,” J. Opt. Soc. Am. A 1, 1219 (1984).

R. V. Shack, “Analytical System Design with Pencil and Ruler—the Advantages of the Y−Ӯ Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 384, 127 (1973).

Vogl, T. P.

Walters, R.

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), p. 71.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

D. Kessler, R. V. Shack, “First-Order Design of Laser Systems with the YӮ Diagram,” J. Opt. Soc. Am. A 1, 1219 (1984).

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

R. V. Shack, “Analytical System Design with Pencil and Ruler—the Advantages of the Y−Ӯ Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 384, 127 (1973).

S. R. Lange et al., “APART, A First-Order Deterministic Radiation Analysis Program,” Proc. Soc. Photo-Opt. Instrum. Eng. 107, 89 (1977).

W. Besenmater, “Designing Zoom Lenses Aided by the Delano Diagram,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 242 (1980).

Other (2)

F. J. López-López, “Inhomogeneous Media in the Y−Ӯ Diagram,” Annual Optical Society Meeting (1972).

R. W. Wood, Physical Optics (Macmillan, New York, 1905), p. 71.

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Figures (7)

Fig. 1
Fig. 1

Shape and orientation of the yӯ ellipse.

Fig. 2
Fig. 2

Gradient and equivalent thin lens systems.

Fig. 3
Fig. 3

Deriving the area–distance relationship.

Fig. 4
Fig. 4

Effect of stop shift.

Fig. 5
Fig. 5

Equivalent focal lengths.

Fig. 6
Fig. 6

Obtaining a circular diagram.

Fig. 7
Fig. 7

YӮ diagram for negative A.

Equations (33)

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n ( r ) = n 0 ( 1 A r 2 2 + b 4 r 4 + b 6 r 6 + ) .
y = ( u 1 A ) sin ( A z ) + y 1 cos ( A z ) , u = u 1 cos ( A z ) y 1 A sin ( A z ) .
y ¯ = ( u ¯ 1 A ) sin ( A z ) + y ¯ 1 cos ( A z ) , u ¯ = u ¯ 1 cos ( A z ) y ¯ 1 A sin ( A z ) ,
y = B sin ( A z + ϕ ) ,
tan ( ϕ ) = y 1 A u 1 , B 2 = ( u 1 A ) 2 + ( y 1 ) 2 .
y ¯ = B ¯ sin ( A z + ϕ ¯ ) ,
tan ( ϕ ¯ ) = y ¯ 1 A u ¯ 1 , B ¯ 2 = ( u ¯ 1 A ) 2 + ( y ¯ 1 ) 2 ,
y ¯ = [ B ¯ cos ( Δ ϕ ) B ] y + B ¯ sin ( Δ ϕ ) cos ( A z + ϕ ) .
y ¯ = [ B ¯ cos ( Δ ϕ ) B ] y
tan θ = B ¯ cos ( Δ ϕ ) B .
a 2 = 2 B 2 B ¯ 2 sin ( Δ ϕ ) B 2 + B ¯ 2 + C , b 2 = 2 B 2 B ¯ 2 sin ( Δ ϕ ) B 2 + B ¯ 2 C ,
C = 4 B 2 B ¯ 2 cos 2 ( Δ ϕ ) + ( B 2 B ¯ 2 ) 2 .
tan ( 2 α ) = 2 B B ¯ cos ( Δ ϕ ) B 2 B ¯ 2 .
H τ A B = 2 ( area swept out ) ,
y = r ( θ ) cos θ ,
y ¯ = r ( θ ) cos θ ,
tan θ = y ¯ ( θ ) y ( θ ) .
d ( area ) = 1 2 r 2 ( θ ) d θ .
d θ = cos 2 θ d ( tan θ ) .
d θ d z = cos 2 θ y 2 ( y d y ¯ d z y ¯ d y d z ) .
2 d ( area ) = H n 0 d z .
y ¯ * = P y + Q y ¯ , u ¯ * = P y + Q u ¯ ,
P = B ¯ cos ( Δ ϕ ) B ,
pupil shift = P y 1 P u 1 + u ¯ 1 .
z 2 π = 2 π A .
H = n 0 A r 2 ,
y = ( u 1 | A | ) sinh ( | A | z ) + y 1 cosh ( | A | z ) , u = u 1 cosh ( | A | z ) + y 1 | A | sinh ( | A | z ) ,
y ¯ = ( u ¯ 1 | A | ) sinh ( | A | | ) + y ¯ 1 cosh ( | A | z ) , u ¯ = u ¯ 1 cosh ( | A | z ) + y ¯ 1 | A | sinh ( | A | z ) ,
y 1 2 ( u 1 | A | ) 2
B 2 = ( y 1 ) 2 ( u 1 | A | ) , tanh ϕ = ( u 1 y 1 | A | ) .
y ¯ 1 2 ( u 1 | A | ) 2
B ¯ 2 2 = ( u 1 | A | ) 2 ( y ¯ 1 ) 2 , tanh ϕ ¯ = y ¯ 1 | A | u 1 .
y ¯ = [ B ¯ sin h ( Δ ϕ ) B ] y + B ¯ cos h ( Δ ϕ ) sin h ( | A | z + ϕ ) .

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