Abstract

Spherical aberration was measured and, by carrying out the computer simulation for ray tracing with the approximate solution of ray equation, the refractive-index distribution parameters of a gradient-index rod lens were determined to most fit the observed data.

© 1983 Optical Society of America

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References

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  1. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
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1971 (1)

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Figures (6)

Fig. 1
Fig. 1

Path of rays in gradient-index rod lens.

Fig. 2
Fig. 2

Principle of spherical (transverse) aberration measurement in a mode-matching condition.

Fig. 3
Fig. 3

Optical setup for measuring the spherical (transverse) aberration of a gradient-index rod lens.

Fig. 4
Fig. 4

Rays between planes A and B. Planes A and B are placed at L1 = 200 μm and L2 = 600 μm.

Fig. 5
Fig. 5

Measured longitudinal aberration plotted against the input height (marked by x) and the computer-simulated curve (indicated by a solid line).

Fig. 6
Fig. 6

Measured transverse aberration vs input height of +ri/r0 (solid line) and −ri/r0 (dotted line).

Tables (1)

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Table I Optical Parameters of Refractive-Index Distribution for Selfoc Lenses (Measured at Wavelength λ = 6328 Å).

Equations (25)

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n 2 ( r ) = n 0 2 { 1 ( g r ) 2 + h 4 ( g r ) 4 + h 6 ( g r ) 6 + h 8 ( g r ) 8 + } ,
2 r z 2 = 1 2 n i 2 cos 2 θ i · ( n 2 ) r .
r ¨ + w 0 2 r = w 0 2 ( α g 2 r 3 + β g 4 r 5 + γ g 6 r 7 )
w 0 = ( n 0 / n i ) g , α = 2 h 4 , β = 3 h 6 , γ = 4 h 8 .
ξ ¨ + w 0 2 ξ = w 0 2 ( δ · α ξ 3 + δ 2 · β ξ 5 + δ 3 · γ ξ 7 ) ,
ψ = w z , ϕ = w / w 0 , ξ ¨ = d 2 ξ d ψ 2 ,
ϕ 2 ξ ¨ + ξ = δ · α ξ 3 + δ 2 · β ξ 5 + δ 3 · γ ξ 7 .
ξ = ξ 0 + δ ξ 1 + δ 2 ξ 2 + δ 3 ξ 3 ,
ϕ = 1 + δ ϕ 1 + δ 2 ϕ 2 + δ 3 ϕ 3 ,
ξ ¨ 0 + ξ 0 = 0 ,
ξ ¨ 1 + ξ 1 = 2 ϕ 1 ξ ˙ 0 + α ξ 0 3 ,
ξ ¨ 2 + ξ 2 = ( ϕ 1 2 + 2 ϕ 2 ) ξ ¨ 0 2 ϕ 1 ξ ¨ 1 + 3 α ξ 0 2 ξ 1 + β ξ 0 5 ,
ξ ¨ 3 + ξ 3 = ( 2 ϕ 1 ϕ 2 + 2 ϕ 3 ) ξ ¨ 0 ( ϕ 1 2 + 2 ϕ 2 ) ξ ¨ 1 2 ϕ 1 ξ ¨ 2 + 3 α ( ξ 0 ξ 1 2 + ξ 0 2 ξ 2 ) + 5 β ξ 0 4 ξ 1 + γ ξ 0 7 .
ξ 0 = cos ( ψ )
ϕ 1 = 2 2 ( 3 h 4 ) .
ξ 1 = 2 4 [ h 4 cos ( ψ ) h 4 cos ( 3 ψ ) ] .
ϕ 2 = 2 6 ( 21 h 4 2 + 60 h 6 ) ,
ξ 2 = 2 8 [ ( 23 h 4 2 + 32 h 6 ) cos ( ψ ) ( 24 h 4 2 + 30 h 6 ) cos ( 3 ψ ) + ( h 4 2 2 h 6 ) cos ( 5 ψ ) ] ,
ϕ 3 = 2 10 ( 324 h 4 3 + 912 h 4 h 6 + 1120 h 8 ) ,
ξ 3 = 2 12 [ ( 547 h 4 3 + 1210 h 4 h 6 + 752 h 8 ) cos ( ψ ) ( 594 h 4 3 + 1230 h 4 h 6 + 672 h 8 ) cos ( 3 ψ ) + ( 48 h 4 3 + 14 h 4 h 6 224 h 8 / 3 ) cos ( 5 ψ ) ( h 4 3 6 h 4 h 6 + 16 h 8 / 3 ) cos ( 7 ψ ) ] .
ψ = 4 [ 1 4 G 2 ( 3 h 4 ) G 4 ( 21 h 4 2 + 60 h 6 ) G 6 ( 324 h 4 3 + 912 h 4 h 6 + 1120 h 8 ) ] n 0 n i g z ,
G = g r i 4 .
ψ = 4 [ 1 4 G 2 ( 3 h 4 2 ) G 4 ( 21 h 4 2 + 56 h 4 + 60 h 6 24 ) G 6 ( 324 h 4 3 216 h 4 2 + 1056 h 4 + 912 h 4 h 6 + 992 h 6 + 1120 h 8 320 ) ] ( g z ) .
ξ = [ 1 + G 2 h 4 + G 4 ( 23 h 4 2 + 32 h 6 ) + G 6 ( 547 h 4 3 + 1210 h 4 h 6 + 752 h 8 ) ] cos ( ψ ) [ G 2 h 4 + G 4 ( 24 h 4 2 + 30 h 6 ) + G 6 ( 594 h 4 3 + 1230 h 4 h 6 + 672 h 8 ) ] cos ( 3 ψ ) + { G 4 ( h 4 2 2 h 6 ) + G 6 ( 48 h 4 3 + 14 h 4 h 6 224 h 8 / 3 ) } cos ( 5 ψ ) G 6 ( h 4 3 6 h 4 h 6 + 16 h 8 / 3 ) } cos ( 7 ψ ) .
h 4 = 2 / 3 , h 6 = 17 / 45 , h 6 = 62 / 315

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