Abstract

A new procedure for the tracing of rays in inhomogéneous media is described. The method is based on a polynomial solution of the differential equation for ray paths. The index of refraction is expressed in a polynomial in the optical-axis-direction coordinate and in the coordinate in the direction orthogonal to the optical axis. The technique is incorporated into a lens-design program to provide optimization of gradient indices. The results of the methods have been verified in two other ways.

© 1975 Optical Society of America

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References

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  1. H. A. Buchdahl, J. Opt. Soc. Am. 63, 46 (1973).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  8. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [Crossref]
  9. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [Crossref]
  10. P. J. Sands, J. Opt. Soc. Am. 61, 777 (1971).
    [Crossref] [PubMed]
  11. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [Crossref]
  12. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1971).
    [Crossref]
  13. P. J. Sands, J. Opt. Soc. Am. 61, 1495 (1971).
    [Crossref]
  14. P. J. Sands, J. Opt. Soc. Am. 63, 1210 (1973).
    [Crossref]
  15. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  16. D. T. Moore, Ph.D. thesis (University of Rochester, 1974), pp. 173–187.
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969, pp. 147–149.

1973 (2)

1972 (1)

1971 (7)

1970 (3)

1968 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969, pp. 147–149.

Buchdahl, H. A.

H. A. Buchdahl, J. Opt. Soc. Am. 63, 46 (1973).
[Crossref]

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Herriott, D. R.

Marchand, E. W.

McKenna, J.

Montagnino, L.

Moore, D. T.

D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
[Crossref]

D. T. Moore, Ph.D. thesis (University of Rochester, 1974), pp. 173–187.

Paxton, K. B.

Rawson, E. G.

Sands, P. J.

Steiffer, W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969, pp. 147–149.

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Tables (3)

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TABLE I Lens-system configuration.

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TABLE II Polynomial expansion of ray coefficient.

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TABLE III Coordinates of ray.

Equations (42)

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δ N d s = 0 ,
δ N ( x , Y , Z ) ( 1 + Y ˙ 2 + Z ˙ 2 ) 1 / 2 d x = 0 ,
L = N ( x , Y , Z ) ( 1 + Y ˙ 2 + Z ˙ 2 ) 1 / 2 .
d d x ( L Y ˙ ) L Y = 0 , d d x ( L Z ˙ ) L Z = 0 .
( N x Y ˙ N Y ) ( 1 + Y ˙ 2 + Z ˙ 2 ) + N Y ¨ = 0 , ( N x Z ˙ N Z ) ( 1 + Y ˙ 2 + Z ˙ 2 ) + N Z ¨ = 0 .
N ( x , ξ ) = n = 0 N m = 0 M N n m x m ξ n ,
Y ( x ) = j = 1 A j x j 1 Z ( x ) = j = 1 B j x j 1 .
2 N 00 A 3 + ( 1 + A 2 2 + B 2 2 ) ( A 2 j = 0 N N j 1 ( A 1 2 + B 1 2 ) j A 1 j = 1 N j N j 0 ( A 1 2 + B 1 2 ) j 1 ) = 0 .
A 3 = ( 1 + A 2 2 + B 2 2 ) 2 N 00 ( A 1 j = 1 N j N j 0 ( A 1 2 + B 1 2 ) j 1 A 2 j = 0 N N j 1 ( A 1 2 + B 1 2 ) j ) .
1 + Y ˙ 2 + Z ˙ 2 = j = 1 H j x j 1 ,
N x = j = 1 G j x j 1 ,
N Y = j = 1 E j x j 1 ,
N Z = j = 1 F j x j 1 ,
N = j = 1 D j x j 1 .
j = 1 G j x j 1 j = 1 j A j + 1 x j 1 j = 1 E j x j 1 j = 1 H j x j 1 + j = 1 D j x j 1 j = 1 j ( j + 1 ) A j + 2 x j 1 = 0 .
A k + 2 = 1 k ( k + 1 ) D 1 ( m = 1 k m A m + 1 n = 1 k + 1 m G k + 2 m n H n m = 1 k E m H k + 1 m + m = 1 k 1 m ( m + 1 ) A m + 2 D k + 1 m )
B k + 2 = 1 k ( k + 1 ) D 1 ( m = 1 k m B m + 1 n = 1 k + 1 m G k + 2 m n H n m = 1 k E m H k + 1 m + m = 1 k 1 m ( m + 1 ) B m + 2 D k + 1 m ) .
j = 1 H j x j 1 = 1 + Y ˙ 2 + Z ˙ 2 ,
H j = m = 1 j m ( j + 1 m ) ( A m + 1 A j + 1 m + B m + 1 B j + 1 m ) , j > 1 .
ξ = j = 1 c 1 , j x j 1 ,
c 1 , j = n = 1 j A n A j + 1 n + B n B j + 1 n ,
ξ 2 = j = 1 c 2 , j x j 1 = ( j = 1 c 1 , j x j 1 ) 2 .
c 2 , j = m = 1 j c 1 , m c 1 , j + 1 m .
c n . j = m = 1 j c 1 , m c n 1 , j + 1 m .
N = j = 1 j D j x j 1 ,
D j = N 0 , j 1 + n = 1 N m = 1 j N n , m 1 c n , j + 1 m ,
N x = j = 1 G j x j 1 , G j = j N 0 , j + n = 1 N m = 1 j m N n m c n , j + 1 m .
j = 1 c a n , j x j 1 = 2 j = 1 A j x j 1 j = 1 c n , j x j 1
j = 1 c b n , j x j 1 = 2 j = 1 B j x j 1 j = 1 c n , j x j 1 ,
c a n , j = 2 m = 1 j c n , m A j + 1 m , c b n , j = 2 m = 1 j c n , m B j + 1 m , for n > 0 .
c a 0 , j = 2 A j and c b 0 , j = 2 B j .
N Y = j = 1 E j x j 1 ,
E j = n = 1 N m = 1 j n N n , m 1 c a n 1 , j + 1 m ,
N Z = j = 1 F j x j 1 , F j = n = 1 N m = 1 j n N n , m 1 c b n 1 , j + 1 m .
y = σ 1 ρ 3 cos θ + σ 2 ρ 2 h ( 2 + cos 2 θ ) + ( 3 σ 3 + σ 4 ) ρ h cos θ + σ 5 h 3 , κ = σ 1 ρ 3 sin θ + σ 2 ρ 2 h sin 2 θ + ( σ 3 + σ 4 ) ρ h 2 sin θ ,
N = 1.55 0.005 Y 2 + 0.02 x 0.005 x 2 .
N e N e = ( N cos I N cos I ) p .
e = 1.0 i + 0.0 j + 0.0 k ,
p = 0.9991997 i 0.04 j .
e = 0.9998993 i 0.0141902 j .
0.9996893 i + 0.024925 j
0.9992019 i + 0.03994406 j .