Abstract

Starting from the ray equation, analytic solutions for the ray paths are found and used to determine the equivalent focal length and the cardinal points of tapered gradient-index (GRIN) rods.

© 1980 Optical Society of America

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References

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  1. F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970).
    [CrossRef]

1970

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Figures (4)

Fig. 1
Fig. 1

Specification of a ray.

Fig. 2
Fig. 2

Illustrating the scaled coordinates.

Fig. 3
Fig. 3

Construction for calculating the Gaussian properties δ′ and F′.

Fig. 4
Fig. 4

Construction for calculating the Gaussian P properties δ and F.

Equations (66)

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d dl ( n dR dl ) = grad n ,
d dl = ( dz dl ) d dz = [ 1 + ( dx dz ) 2 + ( dy dz ) 2 ] 1 / 2 d dz = d dz ,
d dz ( n dR dz ) = grad n .
d dz ( n dx dz ) = n x ,
d dz ( n dy dz ) = n y ,
x = x ρ ( z ) , = y ρ ( z ) , z = z ρ ( z ) ,
d d z [ n ( x , ) dx dz ] d z dz = n x d x dx ,
n ( x , ) d d z ( dx dz ) d z dz = n x d x dx ,
d z dz = 1 ρ ( z ) z [ ρ ( z ) ] 2 d ρ dz = 1 ρ ( z ) ( 1 z d ρ dz ) , d x dx = 1 ρ ( z ) .
ρ ( z ) = ρ 0 mz ,
d z dz = 1 + m z ρ ( z ) ; d x dx = 1 ρ ( z ) .
n ( 1 + m z ) d d z ( dx dz ) = n x ,
dx dz = d dz [ x ρ ( z ) ] = ρ ( z ) d x dz + x d ρ dz ,
dx dz = [ ρ ( z ) d x d z ] d z dz m x ,
dx dz = ( 1 + m z ) d x d z m x .
n ( 1 + m z ) d d z [ ( 1 + m z ) d x d z m x ] = n x ,
n ( 1 + m z ) 2 d 2 x d z 2 = n x ,
n = n 0 [ 1 ½ A 2 ( x 2 + y 2 ) ] ,
n = n 0 [ 1 ½ Ã 2 ( x 2 + 2 ) ] ,
1 n n x = 1 n 0 [ 1 ½ Ã 2 ( x 2 + 2 ) ] ( n 0 Ã 2 x ) = Ã 2 x ,
( 1 + m z ) 2 d 2 x d z 2 = Ã 2 x ,
τ = ( 1 + m z ) ,
τ 2 d 2 x d τ 2 = ( Ã m ) 2 x ,
p = ½ ± i [ ( Ã m ) 2 ¼ ] 1 / 2
x ( τ ) = K 1 τ ( 1 / 2 + ib ) + K 2 τ ( 1 / 2 ib ) ,
b 2 = ( Ã m ) 2 ¼ ,
x ( τ ) = K 3 τ 1 / 2 cos ( b ln τ ) + K 4 τ 1 / 2 sin ( b ln τ ) ,
x ( z ) = K 3 ( 1 + m z ) 1 / 2 cos [ b ln ( 1 + m z ) ] + K 4 ( 1 + m z ) 1 / 2 sin [ b ln ( 1 + m z ) ] ,
x 0 = ( x h 0 ) = K 3
L 0 = ( dx dz ) z = 0 = [ ( 1 + m z ) d x d z mx ] z = 0 = ( d x d z ) z = 0 m x 0 ,
d x d z = K 3 m 2 ( 1 + m z ) 1 / 2 cos [ b ln ( 1 + m z ) ] K 3 ( 1 + m z ) 1 / 2 sin [ b ln ( 1 + m z ) ] bm ( 1 + m z ) + K 4 m 2 ( 1 + m z ) sin [ b ln ( 1 + m z ) ] + K 4 ( 1 + m z ) 1 / 2 cos [ b ln ( 1 + m z ) ] bm ( 1 + m z ) ,
( d x d z ) z = 0 = ½ x 0 m + K 4 bm ,
L 0 = K 4 bm ½ m x 0
x ( z ) = x 0 ( 1 + m z ) 1 / 2 cos [ b ln ( 1 + m z ) ] + ( L 0 + ½ m x 0 mb ) ( 1 + m z ) 1 / 2 sin [ b ln ( 1 + m z ) ] ,
( z ) = 0 ( 1 + m z ) 1 / 2 cos [ b ln ( 1 + m z ) ] + ( M 0 + ½ m 0 mb ) ( 1 + m z ) 1 / 2 sin [ b ln ( 1 + mz ) ] ,
b ln ( 1 + m z ) = { b [ ( m z ) ½ ( m z ) 2 + . . . ] } m = 0 ,
b = 1 m ( Ã 2 m 2 / 4 ) 1 / 2 ,
x ( z ) = x 0 cos ( Ã z ) + L 0 Ã sin ( Ã z ) ,
x ( z ) = x 0 cos ( A z ) + L 0 A sin ( Az ) ,
β 2 = ¼ ( Ã m ) 2 .
x ( τ ) = K 1 τ ( 1 / 2 + β ) + K 2 τ ( 1 / 2 β ) ,
x ( z ) = K 1 ( 1 + m z ) 1 / 2 exp [ β ln ( 1 + m z ) ] + K 2 ( 1 + m z ) 1 / 2 exp [ β ln ( 1 + m z ) ] .
x ( z ) = K 3 ( 1 + m z ) 1 / 2 cosh [ β ln ( 1 + m z ) ] + K 4 ( 1 + m z ) 1 / 2 sinh [ β ln ( 1 + m z ) ] ,
| Ã | > | m | 2 = | tan α | 2 ,
x ( z ) = K 1 τ + K 2 = K 1 ( 1 + m z ) + K 2 ,
τ = ρ 0 / [ ρ ( z ) ]
x ( z ) = x 0 ( ρ ρ 0 ) 1 / 2 cos ( b ln ρ 0 ρ ) + L 0 ρ 0 + m x 0 2 mb ( ρ ρ 0 ) 1 / 2 sin ( b ln ρ 0 ρ ) ,
L = L 0 ( ρ 0 ρ ) 1 / 2 cos ( b ln ρ 0 ρ ) 1 b ρ 0 ( ρ 0 ρ ) 1 / 2 ( 1 2 L 0 ρ 0 + Ã 2 x 0 m ) sin ( b ln ρ 0 ρ ) ,
X = g X 0 = g 0 ; L = ( υ ) L 0 = ( υ 0 ) Y = h Y 0 = h 0 ; M = ( u ) M 0 = ( u 0 ) } ,
g ( z ) = g 0 ( ρ ρ 0 ) 1 / 2 cos ( b ln ρ 0 ρ ) ( υ 0 ρ 0 m g 0 2 m b ) ( ρ ρ 0 ) 1 / 2 sin [ b ln ( ρ 0 ρ ) ] ,
υ ( z ) = υ 0 ( ρ 0 ρ ) 1 / 2 cos ( b ln ρ 0 ρ ) 1 b ρ 0 ( ρ 0 ρ ) 1 / 2 ( 1 2 υ 0 ρ 0 Ã 2 g 0 m ) sin ( b ln ρ 0 ρ ) ,
Y ( D ) = h k = h 0 ( ρ 2 ρ 1 ) 1 / 2 cos ( b ln ρ 1 ρ 2 ) + h 0 2 b ( ρ 1 ρ 2 ) 1 / 2 sin ( b ln ρ 1 ρ 2 ) ,
M ( D ) = u k = Ã 2 h 0 mb ρ 1 ( ρ 1 ρ 2 ) 1 / 2 sin ( b ln ρ 1 ρ 2 ) ,
u = n u k = n à 2 h 0 mb ρ 1 ( ρ 1 ρ 2 ) 1 / 2 sin b ln ( ρ 1 ρ 2 ) ,
u = n 0 Ã 2 mb ρ 1 h 0 ( ρ 1 ρ 2 ) 1 / 2 sin ( b ln ρ 1 ρ 2 )
F = mb ρ 1 n 0 A ( ρ 2 ρ 1 ) 1 / 2 1 sin ( b ln ρ 1 ρ 2 )
δ = ( h 0 h k ) / u ,
δ = { 1 ( ρ 2 ρ 1 ) 1 / 2 [ cos ( b ln ρ 1 ρ 2 ) + 1 2 b sin ( b ln ρ 1 ρ 2 ) ] } n 0 Ã 2 mb ρ 1 ( ρ 1 ρ 2 ) 1 / 2 sin ( b ln ρ 1 ρ 2 ) ,
δ = mb ρ 2 n 0 Ã 2 { 1 2 b + [ cos ( b ln ρ 1 ρ 2 ) ( ρ 1 ρ 2 ) 1 / 2 ] sin ( b ln ρ 1 ρ 2 ) } ,
h k = h 0 ( ρ 2 ρ 1 ) 1 / 2 [ cos ( b ln ρ 1 ρ 2 ) + 1 2 b sin ( b ln ρ 1 ρ 2 ) ] u 0 ρ 1 mb ( ρ 2 ρ 1 ) 1 / 2 sin ( b ln ρ 1 ρ 2 ) ,
u k = u 0 ( ρ 1 ρ 2 ) 1 / 2 cos ( b ln ρ 1 ρ 2 ) 1 b ρ 1 ( ρ 1 ρ 2 ) 1 / 2 ( 1 2 u 0 ρ 1 Ã 2 h 0 m ) sin ( b ln ρ 1 ρ 2 ) .
u 0 = Ã 2 n 0 mb ρ 1 sin ( b ln ρ 1 ρ 2 ) [ cos ( b ln ρ 1 ρ 2 ) 1 2 b sin ( b ln ρ 1 ρ 2 ) ] .
F = h k u = h k n 0 u 0 = 1 n 0 u 0 h 0 ( ρ 2 ρ 1 ) 1 / 2 × 1 [ cos ( b ln ρ 1 ρ 2 ) 1 2 b sin ( b ln ρ 1 ρ 2 ) ] .
F = mb ρ 1 n 0 Ã 2 ( ρ 2 ρ 1 ) 1 / 2 1 sin ( b ln ρ 1 ρ 2 ) ,
δ = mb ρ 1 n 0 Ã 2 { 1 2 b [ cos ( b ln ρ 1 ρ 2 ) ( ρ 2 ρ 1 ) ] 1 / 2 sin ( b ln ρ 1 ρ 2 ) } .
cos ( b ln ρ 1 ρ 2 ) = cos ( b ln ρ 2 ρ 1 ) ; sin ( b ln ρ 1 ρ 2 ) = sin ( b ln ρ 2 ρ 1 ) .

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