Abstract

A new fifth-order algebraic ray-tracing formula valid in any radial gradient is proposed. This leads to algebraic fifth-order aberration formulas for gradient-index (GRIN) rods or Wood lenses. Such formulas can be used for profiling GRIN rod lens samples.

© 1985 Optical Society of America

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References

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  1. W. Streifer, K. Paxton, “Analytical Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media: 1: Meridional Rays,” Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  2. E. Marchand, Gradient Index Optics (Academic, New York, 1978).
  3. E. Marchand, “Distortion in a Gradient-Index Rod,” Appl. Opt. 22, 404 (1983).
    [CrossRef] [PubMed]
  4. N. Yamamoto, K. Iga, “Evaluation of Gradient-Index Rod Lenses by Imaging,” Appl. Opt. 19, 1101 (1980).
    [CrossRef] [PubMed]
  5. T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
    [CrossRef] [PubMed]

1983 (2)

1980 (1)

1971 (1)

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Equations (42)

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Y + Y = 2 h 4 Y 3 + 3 h 6 Y 5 .
Y = g y , Z = g z , t = N 0 Z / l ,
n 2 = N 0 2 ( 1 Y 2 + h 4 Y 4 + h 6 Y 6 ) .
Y 0 = g y 0 , V 0 = Y 0 .
V = Y = q / N 0 = ( n sin γ ) / N 0 ,
Y + Y = 0
Y 1 = Y 0 c + V 0 s , V 1 = V 0 c Y 0 s ,
c = cos t , s = sin t .
Y + Y = 2 h 4 Y 1 3 = 2 h 4 ( Y 0 c + V 0 s ) 3 .
Y + Y = f ( t ) .
Y = Y 0 c + V 0 s + s 0 t c f d f c 0 t s f d f , V = V 0 c Y 0 s + c 0 t c f d f + s 0 t s f d f .
Y = Y 1 + 2 h 4 Y 3 , V = Y 1 + 2 h 4 Y 3 ,
Y 3 = Y 0 3 M 1 + 3 Y 0 2 V 0 M 2 + 3 Y 0 V 0 2 M 3 + V 0 3 M 4 ,
M 1 = s ( 3 A 0 c s ) / 4 , A 0 = ( t + c s ) / 2 , M 2 = ( s c A 0 ) / 4 , B 0 = ( t c s ) / 2 . . M 3 = s B 0 / 4 , M 4 = ( s 3 3 c B 0 ) / 4 ,
Y 3 Y 1 3 + 6 h 4 Y 1 2 Y 3 , Y 5 Y 1 5 ,
f ( t ) = 2 h 4 Y 1 3 + 12 h 4 2 Y 1 2 Y 3 + 3 h 6 Y 1 5 .
Y = Y 1 + 2 h 4 Y 3 + 12 h 4 2 F 1 + 3 h 6 F 2 , V = Y 1 + 2 h 4 Y 3 + 12 h 4 2 F 1 + 3 h 6 F 2 .
F 2 = Y 0 5 N 1 + 5 Y 0 4 V 0 N 2 + 10 Y 0 3 V 0 2 N 3 + 10 Y 0 2 V 0 3 N 4 + 5 Y 0 V 0 4 N 5 + V 0 5 N 6 ,
N 1 = ( c 5 c + 5 s A 1 ) / 6 , N 4 = M 2 N 2 , N 2 = ( s c A 1 ) / 6 , N 5 = M 3 N 3 , N 3 = M 1 N 1 , N 6 = M 4 N 4 , A 1 = 0 t c 4 d t = ( 3 A 0 + c 3 s ) / 4 .
F 1 = Y 0 5 L 1 + Y 0 4 V 0 L 2 + Y 0 3 V 0 2 L 3 + Y 0 2 V 0 3 L 4 + Y 0 V 0 4 L 5 + V 0 5 L 6 ,
L 1 = s M 11 c M 12 , L 2 = 2 ( s M 12 c M 13 ) + 3 ( s M 21 c M 22 ) , L 3 = ( s M 13 c M 14 ) + 6 ( s M 22 c M 23 ) + 3 ( s M 31 c M 32 ) , L 4 = 3 ( s M 23 c M 24 ) + 6 ( s M 32 c M 33 ) + ( s M 41 c M 42 ) , L 5 = 3 ( s M 33 c M 34 ) + 2 ( s M 42 c M 43 ) , L 6 = s M 43 c M 44 , }
M 11 = 0 t c 3 M 1 d t , M 12 = 0 t c 2 s M 1 d t , M 13 = 0 t c s 2 M 1 d t ,
l = l 0 = ( n 0 2 q 0 2 ) 1 / 2 = N 0 [ 1 ( Y 0 2 + V 0 2 ) + h 4 Y 0 4 + h 6 Y 0 6 ] 1 / 2 .
t = Z + Z 2 + Z 4 , c = c 0 ( 1 Z 2 2 / 2 ) s 0 ( Z 2 + Z 4 ) , s = s 0 ( 1 Z 2 2 / 2 ) + c 0 ( Z 2 + Z 4 ) ,
c 0 = cos Z , s 0 = sin Z , Z 2 = a Z , Z 4 = b Z , a = ( Y 0 2 + V 0 2 ) / 2 , b = ( 3 a 2 h 4 Y 0 4 ) / 2 .
Y = Y ¯ 1 ( 1 Z 2 2 / 2 ) + V ¯ 1 ( Z 2 + Z 4 ) + 2 h 4 [ Y 0 3 ( M ¯ 1 + Z 2 P ¯ 1 ) + 3 Y 0 2 V 0 ( M ¯ 2 + Z 2 P ¯ 2 ) + 3 Y 0 V 0 2 ( M ¯ 3 + Z 2 P ¯ 3 ) + V 0 3 ( M ¯ 4 + Z 2 P ¯ 4 ) ] + 12 h 4 2 F ¯ 1 + 3 h 6 F ¯ 2 ,
V = V ¯ 1 ( 1 Z 2 2 / 2 ) Y ¯ 1 ( Z 2 + Z 4 ) + 2 h 4 [ Y 0 3 ( P ¯ 1 + Z 2 R ¯ 1 ) + 3 Y 0 2 V 0 ( P ¯ 2 + Z 2 R ¯ 2 ) + 3 Y 0 V 0 2 ( P ¯ 3 + Z 2 R ¯ 3 ) + V 0 3 ( P ¯ 4 + Z 2 R ¯ 4 ) ] + 12 h 4 2 G ¯ 1 + 3 h 6 G ¯ 2 .
Y ˆ = Y + g L tan γ , tan γ = q ( 1 q ) 1 / 2 = q + 1 2 q 3 + ( 3 / 8 ) q 5 , q = N 0 V .
V = V 1 + 2 h 4 V 3 + 12 h 4 2 G 1 + 3 h 6 G 2 ,
V 1 = V 0 c Y 0 s , V 3 = Y 0 3 P 1 + 3 Y 0 2 V 0 P 2 + 3 Y 0 V 0 2 P 3 + V 0 3 P 4 , G 1 = Y 0 5 J 1 + Y 0 4 V 0 J 2 + Y 0 3 V 0 2 J 3 , + Y 0 2 V 0 3 J 4 + Y 0 V 0 4 J 5 + V 0 5 J 6 G 2 = Y 0 5 Q 1 + 5 Y 0 4 V 0 Q 2 + 10 Y 0 3 V 0 2 Q 3 + 10 Y 0 2 V 0 3 Q 4 + 5 Y 0 V 0 4 Q 5 + V 0 5 Q 6 . }
A 1 = 0 t c 4 d t , A 2 = 0 t c 3 s d t , , C 1 = 0 t c 6 d t , C 2 = 0 t c 5 s d t , , E 1 = 0 t A 1 d t , E 2 = 0 t A 2 d t , . }
A 1 = ( 3 A 0 + c 3 s ) / 4 , A 0 = ( t + c s ) / 2 , A 2 = ( 1 c 4 ) / 4 , B 0 = ( t c s ) / 2 . A 3 = A 0 A 1 , A 4 = s 4 / 4 , A 5 = ( 3 B 0 c s 3 ) / 4 , }
C 1 = ( c 5 s + 5 A 1 ) / 6 , C 5 = A 3 C 3 , C 2 = ( 1 c 6 ) / 6 , C 6 = A 4 C 4 , C 3 = A 1 C 1 , C 7 = A 5 C 5 . C 4 = A 2 C 2 , }
E 1 = 3 ( E 0 A 2 ) / 4 , E 0 = ( t 2 + s 2 ) / 4 , E 2 = ( t A 1 ) / 4 , F 0 = ( t 2 s 2 ) / 4 . E 3 = E 0 E 1 , E 4 = A 5 / 4 , E 5 = ( 3 F 0 A 4 ) / 4 , }
D i = A i t E i .
M i = s A i c A i + 1 , N i = s C i c C i + 1 , P i = c A i + s A i + 1 , Q i = c C i + s C i + 1 ,
M 1 i = ( 3 D i + 1 + C i + 2 ) / 8 , M 2 i = ( A i + 1 + C i + 3 D i ) / 8 , M 3 i = ( D i + 1 C i + 2 ) / 8 , M 4 i = 3 ( A i + 1 D i ) / 8 , }
R 1 = ( 9 c 3 1 3 s t ) / 8 , R 3 = ( 9 c s 2 s t ) / 8 , R 2 = ( 7 s 9 s 3 + c t ) / 8 , R 4 = 3 ( 3 s 3 s + c t ) / 8 ,
length = 100 mm , N 0 = 1.5 , g = 2 π / 67 , pitch ( period ) = 67 mm , h 4 = 2 / 3 , h 6 = 17 / 45 .
n = N 0 sech ( g r ) ,
E = Y 2 Y 0 2 + V 2 V 0 2 h 4 ( Y 4 Y 0 4 ) h 6 ( Y 6 Y 0 6 ) = 0 ,
Ray 1 Ray 2 y 0 0.01 0.01 q 0 0.02 0.03 y 0.0032419 9.33 × 10 6 q 0.0200439542 0.0300329612 E 10 13 10 13 Δ y 1.18 × 10 6 1.29 × 10 6 Δ q 2.18 × 10 8 1.42 × 10 8

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