Abstract

The YӮ diagram of Delano is applied to the design of gradient-index rods and is used to explain GRIN rod vignetting. Several detailed design examples are given which show how to use the diagram for GRIN rods. The YӮ diagram is a particularly useful tool to explore potential solutions involving diameter constraints and equips one with a rapid nonalgebraic investigation method.

© 1988 Optical Society of America

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References

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  1. J. Rogers, M. Harrigan, R. Loce, “The Y−Ӯ Diagram of Radial Gradient Systems,” Appl. Opt. 27, 452 (1988).
    [CrossRef] [PubMed]
  2. M. E. Harrigan, “Some First-Order Properties of Radial Gradient Lenses Compared to Homogeneous Lenses,” Appl. Opt. 23, 2702 (1984).
    [CrossRef] [PubMed]
  3. E. Delano, “First-Order Design and the Y,Ӯ Diagram,” Appl. Opt. 2, 1251 (1963).
    [CrossRef]
  4. K. Matsushita, M. Toyama, “Unevenness of Illuminance Caused by Gradient-Index Fiber Arrays,” Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]

1988 (1)

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1980 (1)

1963 (1)

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Figures (14)

Fig. 1
Fig. 1

(a) GRIN rod limited marginal and chief ray; (b) YӮ circular diagram for (a).

Fig. 2
Fig. 2

Finite object and circular YӮ diagram.

Fig. 3
Fig. 3

+1× GRIN rod YӮ diagram.

Fig. 4
Fig. 4

±0.5× design.

Fig. 5
Fig. 5

Fractional object Yӯ diagram.

Fig. 6
Fig. 6

Fractional pupil Yӯ diagram.

Fig. 7
Fig. 7

GRIN rod vignetting and the YӮ diagram.

Fig. 8
Fig. 8

Erecting afocal design.

Fig. 9
Fig. 9

2× beam expander YӮ solution.

Fig. 10
Fig. 10

Ellipse sector area.

Fig. 11
Fig. 11

2× GRIN beam expander.

Fig. 12
Fig. 12

2× beam expander solution with the second rod at half aperture.

Fig. 13
Fig. 13

2× beam expander solution with different materials.

Fig. 14
Fig. 14

1× telecentric solution.

Equations (15)

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n ( r ) = n 0 ( 1 A r 2 / 2 + b 4 r 4 + b 6 r 6 + ) .
y ( z ) = R · cos [ A ( z t s ) ] ,
y ¯ ( z ) = y ¯ y m · R · sin [ A ( z t s ) ] .
z = 2 · AREA · n 0 / H .
A · L = π + 2 A · t s .
T C = 2 t 0 + L ,
tan ( A · t s ) = ( n 0 · t 0 A ) 1 .
y ¯ m = R 1 + n 0 2 t 0 2 A .
A s = ( a · y ¯ p 1 y ¯ p 2 / b 2 + a · b · sin 1 ( y ¯ p / b ) y p · y ¯ p ) / 2 .
y ¯ 1 = ± b 2 r P 2 / P 1 , y ¯ 2 = ± r P 2 / P 1 , y 1 = ± a 1 b 2 P 2 / r 2 P 1 , y 2 = ± r 1 P 2 / P 1 ,
y ( z ) = B 0 sin ( A z + φ ) ; y ¯ ( z ) = B ¯ 0 sin ( A z + φ ¯ ) .
tan ( φ ) = y 0 n 0 A / u 0 = n 0 t 0 A .
u ¯ = y ¯ t 0 n 0 2 A / ( 1 + n 0 2 t 0 2 A ) .
y ¯ 0 = y ¯ / ( 1 + n 0 2 t 0 2 A ) .
y ( z ) = R cos ( A z + φ ¯ ) , y ¯ ( z ) = y ¯ R sin ( A z + φ ¯ ) / y ¯ m .

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