## 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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### Figures (14)

Fig. 1

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

Fig. 2

Finite object and circular YӮ diagram.

Fig. 3

+1× GRIN rod YӮ diagram.

Fig. 4

±0.5× design.

Fig. 5

Fractional object Yӯ diagram.

Fig. 6

Fractional pupil Yӯ diagram.

Fig. 7

GRIN rod vignetting and the YӮ diagram.

Fig. 8

Erecting afocal design.

Fig. 9

2× beam expander YӮ solution.

Fig. 10

Ellipse sector area.

Fig. 11

2× GRIN beam expander.

Fig. 12

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

Fig. 13

2× beam expander solution with different materials.

Fig. 14

1× telecentric solution.

### Equations (15)

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