Abstract

A single-element photographic lens was designed to be constructed with a radial gradient-index function. The lens in an f/6.3 version is corrected for spherical aberration, coma, tangential, and sagittal curvature with a 25° half-field, the aberrations being defined as total, rather than third order, fifth order, etc. The performance of the lens roughly matches that of optimal homogeneous lenses, of the same general type, operating at f/11.

© 1974 Optical Society of America

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References

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  1. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [Crossref]
  2. D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [Crossref]
  3. E. W. Marchand, Appl. Opt. 11, 1104 (1972).
    [Crossref] [PubMed]

1972 (1)

1971 (2)

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Figures (5)

Fig. 1
Fig. 1

Gradient-index singlet.

Fig. 2
Fig. 2

Meridional coma.

Fig. 3
Fig. 3

Meridional field curvature.

Fig. 4
Fig. 4

Front meniscus lens with stop.

Fig. 5
Fig. 5

(a) Spherical aberration, (b) astigmatism.

Tables (2)

Tables Icon

Table I Lens data and spherical aberration.

Tables Icon

Table II Meridional rays, field curvature, distortion, and coma.

Equations (20)

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n = N 0 + N 1 r 2 + N 2 r 4 + .
r ¨ = m m / l 0 2 ,             θ ˙ = c / ( l 0 r 2 ) ,
m 2 = n 2 - c 2 / r 2 ;
n 0 = n ( r 0 ) = N 0 + N 1 r 0 2 + N 2 r 0 4 + , l 0 = ( n 0 2 - p 0 2 - q 0 2 ) 1 2 , c = x 0 q 0 - y 0 p 0 ;
x = r cos θ ,             y = r sin θ , p = n ( r ˙ cos θ - r sin θ θ ˙ ) , q = n ( r ˙ sin θ + r cos θ θ ˙ ) , n = N 0 + N 1 r 2 + N 2 r 4 .
n 2 = N 0 2 ± b 2 r 2 ,             b > 0
x = x 0 { cosh z ¯ cos z ¯ } + ( p 0 / b ) { sinh z ¯ sin z ¯ } , y = y 0 { cosh z ¯ cos z ¯ } + ( q 0 / b ) { sinh z ¯ sin z ¯ } , p = p 0 { cosh z ¯ cos z ¯ } + b x 0 { sinh z ¯ - sin z ¯ } , q = q 0 { cosh z ¯ cos z ¯ } + b y 0 { sinh z ¯ - sin z ¯ } ,
z ¯ = b ( z - z ¯ 0 ) / l 0 ,
n 0 2 = N 0 2 ± b 2 r 0 2 = N 0 2 ± b 2 ( x 0 2 + y 0 2 ) .
n 2 = N 0 2 ± b 2 r 2 = N 0 2 ± b 2 ( x 2 + y 2 ) ,
y 0 = z 0 σ 0 ,             q 0 = - y 0 [ 1 / z 0 + c 1 ( N 0 - 1 ) ] , q 1 = q 0 { cosh ( b d / N 0 ) cos ( b d / N 0 ) } + b y 0 { sinh ( b d / N 0 ) - sin ( b d / N 0 ) } , y 1 = y 0 { cosh ( b d / N 0 ) cos ( b d / N 0 ) } + ( q 0 / b ) { sinh ( b d / N 0 ) sin ( b d / N 0 ) } , 1 / - z = q 1 / y 1 + c 2 ( N 0 - 1 ) ,             σ 1 = y 1 / z .
1 / f * = lim h 0 ( σ 1 / h ) .
1 / f * = ( c 1 - c 2 ) ( N 0 - 1 ) { cosh ( b d / N 0 ) cos ( b d / N 0 ) } + c 1 c 2 d ( N 0 - 1 ) 2 N 0 { [ sinh ( b d / N 0 ) ] / ( b d / N 0 ) [ sin ( b d / N 0 ) ] / ( b d / N 0 ) } + b { - sinh ( b d / N 0 ) sin ( b d / N 0 ) } .
L A = z - f .
C m = 1 2 ( y A + y B ) - y p .
D = 100 ( y p - y 0 ) / y 0 ,
u = - tan σ ,             u p = - tan σ p
Z m = lim h 0 y p - y u P - u .
Z m ~ y p - y u P - u .
A = Z m - Z s .