Abstract

Simple algebraic formulas have been developed for the third-order aberrations of a single-element lens having a radial refractive-index gradient, plane faces, and object plane at infinity. Such formulas have some advantages over previous methods of analysis that require ray tracing and that are well suited to multielement lens systems. The derivation technique adopted in the present paper can be extended to deal with a finite object distance, but the formulas are somewhat more complicated.

© 1976 Optical Society of America

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References

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  1. R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.
  2. D. Hamblen, U. S. Patent No. 3,486,808, December, 1969.
  3. P. Sinai, Appl. Opt. 10, 99 (1971).
    [Crossref] [PubMed]
  4. R. S. Moore, U. S. Patents No. 3,718,383, February, 1971, and 3,816,160, June, 1974.
  5. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [Crossref]
  6. D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [Crossref]
  7. D. T. Moore and P. J. Sands, U. S. Patent No. 3, 729, 253, April, 1973.
  8. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London and New York, 1970), p. 122.
  9. E. W. Marchand and D. J. Janeczko, J. Opt. Soc. Am. 64, 846 (1974).
    [Crossref]

1974 (1)

1971 (3)

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London and New York, 1970), p. 122.

Hamblen, D.

D. Hamblen, U. S. Patent No. 3,486,808, December, 1969.

Janeczko, D. J.

Marchand, E. W.

Moore, D. T.

Moore, R. S.

R. S. Moore, U. S. Patents No. 3,718,383, February, 1971, and 3,816,160, June, 1974.

Sands, P. J.

D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
[Crossref]

D. T. Moore and P. J. Sands, U. S. Patent No. 3, 729, 253, April, 1973.

Sinai, P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London and New York, 1970), p. 122.

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Other (5)

D. T. Moore and P. J. Sands, U. S. Patent No. 3, 729, 253, April, 1973.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London and New York, 1970), p. 122.

R. S. Moore, U. S. Patents No. 3,718,383, February, 1971, and 3,816,160, June, 1974.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.

D. Hamblen, U. S. Patent No. 3,486,808, December, 1969.

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Figures (3)

FIG. 1
FIG. 1

Meridional coma.

FIG. 2
FIG. 2

Meridional curvature.

FIG. 3
FIG. 3

Sagittal curvature.

Tables (2)

Tables Icon

TABLE I Wood lens data.

Tables Icon

TABLE II Maximum third-order aberrations.

Equations (54)

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d d s ( n d r d s ) = n ,
n d z d s = l = l 0 = const . ,
r ¨ = n n / l 0 2 ,
n = N 0 = N 1 r 2 + N 2 r 4 ,             r 2 = x 2 + y 2 ,
n = N 0 ( 1 + N ¯ 1 r 2 ) , d n d r = 2 N 1 r ( 1 + 2 N ¯ 2 r 2 ) , N ¯ 1 = N 1 / N 0 , N ¯ 2 = N 2 / N 1 .
x ¨ = 2 N 0 N 1 x [ 1 + r 2 ( 2 N ¯ 2 + N ¯ 1 ) ] / l 0 2 , y ¨ = 2 N 0 N 1 y [ 1 + r 2 ( 2 N ¯ 2 + N ¯ 1 ) ] / l 0 2 .
b = ( 2 N 1 N 0 ) 1 / 2 ,             k = b / N 0 ,             N ˆ = - ( 2 N ¯ 2 + N ¯ 1 ) ,
x ¨ ( l 0 / b ) 2 + x = N ˆ x r 2 ,
z ¯ = b z / l 0 ,
x + x = N ˆ x r 2 ,             y + y = N ˆ y r 2 ,
x = x 0 cos z ¯ + p ¯ 0 sin z ¯ , y = y 0 cos z ¯ + q ¯ 0 sin z ¯ ,
p ¯ 0 = p 0 / b ,             q ¯ 0 = q 0 / b ,
p 0 = n 0 ( d x d s ) 0 ,             q 0 = n 0 ( d y d s ) 0
r 2 u cos 2 z ¯ + v cos z ¯ sin z ¯ + w sin 2 z ¯ ,
u = x 0 2 + y 0 2 ,             v = 2 ( x 0 p ¯ 0 + y 0 q ¯ 0 ) , w = p ¯ 0 2 + q ¯ 0 2 .
x 1 = c x 0 + s p ¯ 0 + k ¯ W ( s x 0 - c p ¯ 0 ) + ( 1 8 N ˆ ) [ a ¯ 1 c s 2 + a ¯ 2 s 3 + b ¯ 1 ( s - c k ¯ ) + b ¯ 2 s k ¯ ] , p ¯ 1 = c p ¯ 0 - s x 0 + k ¯ W ( s p ¯ 0 + c x 0 ) + ( 1 8 N ˆ ) [ a ¯ 1 s ( 3 c 2 - 1 ) + a ¯ 2 ( 3 c s 2 ) + b ¯ 1 s k ¯ + b ¯ 2 ( s + c k ¯ ) ] .
c = cos k ¯ ,             s = sin k ¯ ,             k ¯ = k d ,             W = N ¯ 1 ( u + w )
a ¯ 1 = x 0 ( u - w ) - p ¯ 0 v , b ¯ 1 = p ¯ 0 ( u + 3 w ) + x 0 v , a ¯ 2 = p ¯ 0 ( u - w ) + x 0 v , b ¯ 2 = x 0 ( 3 u + w ) + p ¯ 0 v .
h / f tan σ - q 1 ,             f = - lim h 0 ( h / q 1 ) ,
f = 1 / b s .
f - 1 / ( 2 N 1 d ) ,
y 1 / f tan σ - q 1 ,
f = c f = c / ( b s ) ,
( y - y 1 ) / f = - tan σ = q 1 ( 1 - q 1 2 ) - 1 / 2
y = y 1 + f ( q 1 + 1 2 q 1 3 ) ,
y 0 = q 0 f .
y = q 0 f + c N y 0 ( u + w ) + ( 1 2 c 3 q 0 f ) × [ q 0 2 + q 0 ( y 0 / f ) ( t 2 - 3 ) + 3 ( y 0 / f ) 2 ] + N ˆ c 3 { ( 1 4 t q ¯ 0 ) [ 3 u ( 2 + t 2 ) + w t 2 ] - y 0 w } ,
t = s / c , N = N ¯ 1 k ¯ / c s - 1 / 2 f 2 + ( 1 8 N ˆ ) [ 2 c 2 + 3 ( 1 + k ¯ / c s ) ] .
S ¯ = c h 3 N .
S = f h 2 N .
N ˆ = 8 ( 1 / 2 f 2 - N ¯ 1 k ¯ / c s ) / [ 2 c 2 + 3 ( 1 + k ¯ / c s ) ] .
C = 1 2 ( y a + y b ) - y p .
q 0 = - sin σ
y 0 p = z e tan σ - z e q 0 ( 1 + 1 2 q 0 2 ) , y 0 a = y 0 p + a ,             y 0 b = y 0 p - a
C = 3 q 0 a 2 c 2 2 f [ 1 - 2 N f 2 z ˆ + N ˆ 2 b 2 ( 1 + 1 c 2 ) ] ,
z ˆ = z e / f .
D = 100 ( y p - y 0 ) / y 0 ,
D = 50 q 0 2 c 4 { 1 + z ˆ ( 3 - t 2 ) + 3 z ˆ 2 - 2 N f 2 z ˆ ( t 2 + z ˆ 2 ) + ( N ˆ / 2 b 2 ) [ 3 z ˆ ( 2 + t 2 ) + 4 z ˆ + t 4 ] } .
Z m = lim h 0 y - y p u - u p ,
y 0 = y 0 p + h ,
u - u p q 1 - q 1 p - b s h = - h / f .
Z m = - f lim h 0 [ ( y - y p ) / h ] .
Z m = ( 1 2 f q 0 2 c 2 ) { 3 ( 1 + 2 z ˆ ) - t 2 - 2 N f 2 ( t 2 + 3 z ˆ 2 ) + ( N ˆ / b 2 ) [ 2 + 3 z ˆ ( 2 + t 2 ) ] }
x 0 = h ,             p 0 = 0 ,             y 0 = - z e q 0 ( 1 - q 0 2 ) - 1 / 2 ,             q 0 = - sin σ .
Z s = f + lim h 0 ( x 1 l / p 1 ) ,
l = ( 1 - p 1 2 - q 1 2 ) 1 / 2 .
x 1 = c x 0 { 1 + k ¯ W s / c + ( N ˆ / 8 x 0 ) × [ a ¯ 1 s 2 + a ¯ 2 s 3 / c + b ¯ 1 ( s / c - k ¯ ) + b ¯ 2 k ¯ s / c ] } , p 1 = - b s x 0 { 1 - k ¯ W c / s - ( N ˆ / 8 x 0 ) × [ a ¯ 1 ( 3 c 2 - 1 ) + a ¯ 2 ( 3 c s ) + b ¯ 1 k ¯ + b ¯ 2 ( 1 + k ¯ c / s ) ] } ,
x 1 / p 1 = - f { 1 + k ¯ W / c s + ( N ˆ / 8 x 0 ) [ 2 a ¯ 1 c 2 + a ¯ 2 ( 3 c s + s 3 / c ) + b ¯ 1 s / c + b ¯ 2 ( 1 + k ¯ / c s ) ] } ,
l 1 - ( 1 2 b 2 ) ( p ¯ 1 2 + q ¯ 1 2 ) = 1 - ( 1 2 b 2 ) × [ ( x 0 s ) 2 + ( c q ¯ 0 - y 0 s ) 2 ] 1 - 1 2 ( c q 0 - y 0 / f ) 2 .
Z s = - f { W k ¯ / c s - 1 2 ( c q 0 - y 0 / f ) 2 + ( N ˆ / 8 x 0 ) [ 2 c 2 a ¯ 1 + a ¯ 2 ( 3 c s + s 3 / c ) + b ¯ 1 s / c + b ¯ 2 ( 1 + k ¯ / c s ) ] } ,
a ¯ 1 = x 0 ( u - w ) ,             a ¯ 2 = b ¯ 1 = x 0 v , b ¯ 2 = x 0 ( 3 u + w ) ,             u = x 0 2 + y 0 2 y 0 2 , v = 2 y 0 q ¯ 0 ,             w = q ¯ 0 2 ,             W = N ¯ 1 ( y 0 2 + q ¯ 0 2 ) ,
Z s = ( 1 2 f q 0 2 c 2 ) { 1 + 2 z ˆ - t 2 - 2 N f 2 ( t 2 + z ˆ 2 ) + ( N ˆ / b 2 ) × [ 1 + ( 1 + k ¯ / c s ) / 2 c 2 + ( 2 + t 2 ) z ˆ ] } ,
A = ( 1 2 f q 0 2 c 2 ) { 1 + 2 z ˆ - 2 N f 2 z ˆ 2 + ( N ˆ / 2 b 2 ) × [ 1 + 2 z ˆ ( 2 + t 2 ) - ( 1 + k ¯ / c s ) / 2 c 2 ] }
P = f q 0 2 c 2 { t 2 ( 1 + 2 N f 2 ) - ( N ˆ / 2 b 2 ) [ 1 + ( 3 / 2 c 2 ) ( 1 + k ¯ / c s ) ] } ,