Abstract

The formulas presented in an earlier paper for the first-degree chromatic paraxial-aberration coefficients of a symmetric system with inhomogeneous media are specialized to two cases of practical importance, those of axial and cylindrical index distributions. For a given medium, the distribution of dispersion is not independent of the distribution of index for the base wavelength. From some simple assumptions pertaining to the origin of the inhomogeneities, one possible form of this dependence is deduced. On the basis of this, it is shown by numerical examples that it is possible to utilize a cylindrical distribution to achromatize a singlet, whereas this is not the case for an axial distribution.

© 1971 Optical Society of America

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References

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  1. P. J. Sands, J. Opt. Soc. Am. 61, 777 (1971). This paper will hereafter be referred to as II and equations in II will be referenced by prefixing the equation number by II.
    [Crossref] [PubMed]
  2. An optical system is said to be symmetric if it has (i) an axis of rotational symmetry with points in common with both the object and image space, and (ii) a plane of symmetry that contains the axis.
  3. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1971) and D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971). These two papers will be referred to, respectively, as IV and MS.
    [Crossref]
  4. The general formulas for the third-order aberration coefficients are derived in the first paper of this series, P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [Crossref]
  5. From Ref. 2, and from the third paper of this series, P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [Crossref]
  6. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  7. For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
    [Crossref] [PubMed]

1971 (3)

1970 (1)

1969 (1)

For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
[Crossref] [PubMed]

Buchdahl, H. A.

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

French, W. G.

For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
[Crossref] [PubMed]

Pearson, A. D.

For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
[Crossref] [PubMed]

Rawson, E. G.

For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
[Crossref] [PubMed]

Sands, P. J.

Appl. Phys. Letters (1)

For example, by ion diffusion, see A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969), D. P. Hamblen (U. S. patent3 486 808), Navias (U. S. patent3 212 401), and Deutsche Patentschrift1 191 980; or by neutron irradiation, see Ph. Sinai, Appl. Opt. 10, 99 (1971).
[Crossref] [PubMed]

J. Opt. Soc. Am. (4)

Other (2)

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

An optical system is said to be symmetric if it has (i) an axis of rotational symmetry with points in common with both the object and image space, and (ii) a plane of symmetry that contains the axis.

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Tables (2)

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Table I Chromatic paraxial-aberration coefficients for an axial distribution.

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Table II Chromatic paraxial-aberration coefficients of cylindrical distribution.

Equations (41)

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N N ( x , ξ , ω ) = N 0 ( x , ω ) + N 1 ( x , ω ) ξ + ,
ω = ( λ - λ 0 ) / [ 1 + α ( λ - λ 0 ) ] .
N 0 ( x , ω ) = N 0 ( x ) + m = 1 ν 0 m ( x ) ω m , N 1 ( x , ω ) = N 1 ( x ) + m = 1 ν 1 m ( x ) ω m ,
N ( x , ξ , ω ) = N 0 ( x , ξ ) + m = 1 ν m ( x , ξ ) ω m ,
N 0 ( x , ξ ) = N 0 ( x ) + N 1 ( x ) ξ + , ν m ( x , ξ ) = ν 0 m ( x ) + ν 1 m ( x ) ξ + .
f a 1 * = - N 0 ( x ) y a ( x ) v a ( x ) d ϕ 1 ( x ) + 2 y a ( x ) 2 ϕ ¯ 1 ( x ) d x , f a 1 * = - N 0 ( x ) y a ( x ) v b ( x ) d ϕ 1 ( x ) + 2 y a ( x ) y b ( x ) ϕ ¯ 1 ( x ) d x ,
ϕ 1 ( x ) = ν 01 ( x ) / N 0 ( x ) , ϕ ¯ 1 ( x ) = ν 11 ( x ) - N 1 ( x ) ϕ 1 ( x ) .
f a 1 = - N 0 y a i a Δ ϕ 1 ,             f ¯ a 1 = - N 0 y a i b Δ ϕ 1 ,
N 1 ( x ) = 0 ,             ν 11 ( x ) = 0 ,             and             ϕ ¯ 1 ( x ) = 0 1 ,
N 0 ( x ) V ( x ) = N 0 V 0 ,             Y ( x ) = Y 0 + N 0 V 0 I 1 ,
I 1 = 0 x N 0 ( x ) - 1 d x ;
f a 1 * = - N 0 v a ,             f ¯ a 1 * = - N 0 v b ,
= 0 t y a ( x ) d ϕ 1 ( x ) ,
= { y a ( x ¯ ) ϕ 1             for some x ¯ such that 0 x ¯ t , ( y a ϕ 1 ) - N 0 v a I * y a ( ϕ 1 ) + N 0 v a [ ϕ 1 ( t ) I 1 - I * ] ,
I * = 0 t [ ν 01 ( x ) / N 0 ( x ) 2 ] d x .
f a 1 * = 2 ϕ ¯ 1 0 t y a 2 ( x ) d x ,             f ¯ a 1 * = 2 ϕ ¯ 1 0 t y a ( x ) y b ( x ) d x ,
ϕ ¯ 1 = ν 11 - N 1 ϕ 1 , ϕ 1 = ν 01 / N 0 .
y a ( x ) = y a C ( x ) + v a S ( x ) ,             y b ( x ) = y b C ( x ) + v b S ( x ) ,
0 t y a ( x ) 2 d x = y a 2 0 + 2 y a v a 1 + v a 2 2 0 t y a ( x ) y b ( x ) d x = y a y b 0 + ( y a v b + y b v a ) 1 + v a v b 2 ,
n = 0 t C ( x ) 2 - n S ( x ) n d x .
0 = 1 2 ( t + C S ) 1 = ( C 2 - 1 ) / N ¯ 1 = S 2 2 = ( C S - t ) / 2 N ¯ 1 ,
N ¯ 1 = 2 N 1 / N 0 ,
f a 1 * = 2 ϕ ¯ 1 [ y a 2 0 + 2 y a v a 1 + v a 2 2 ] f ¯ a 1 * = 2 ϕ ¯ 1 [ y a y b 0 + ( y b v a + y a v b ) 1 + v a v b 2 ]
N ( x , ξ ) = N * + ρ θ ( x , ξ ) ,
N ( x , ξ , ω ) = N * ( ω ) + ρ ( ω ) θ ( x , ξ ) .
N * = N 0 * + ν 1 * ω + ν 2 * ω 2 + ρ = ρ 0 + ρ 1 ω + ρ 2 ω 2 + θ = θ 0 ( x ) + θ 1 ( x ) ξ + θ 2 ( x ) ξ 2 + .
N n ( x ) = N 0 * δ n 0 + ρ 0 θ n ( x ) ν n m ( x ) = ν m * δ n 0 + ρ m θ n ( x ) ,
θ 0 ( x ) = e - a x ,             θ n ( x ) = 0             for             n 1 ,
N 0 ( x ) = N 0 * + ρ 0 e - a x ,             ν 01 ( x ) = ν 1 * + ρ 1 e - a x
I 1 = ( 1 / N 0 * ) { x + ( 1 / a ) l n [ N 0 ( x ) / N 0 ] } I * = ( ν 1 * / N 0 * ) I 1 + ( 1 / a ) ( ρ 1 ρ 0 - ν 1 * N 0 * ) [ 1 / N 0 ( x ) ] .
ρ 0 = - 0.158 198 ρ 1 = 0.0625 , N 0 ( 1 ) = 1.6 ν 01 ( 1 ) = - 0.079 508 ,
I 1 = 0.641 985 ,             I * = - 0.025 798.
N 0 ( x , ξ ) = N 0 * + ρ 0 ( θ 0 + θ 1 ξ ) = 1.603 42 ν 1 ( x , ξ ) = ν 1 * + ρ 1 ( θ 0 + θ 1 ξ ) = - 0.080 57.
N 0 ( x , 0 ) = N 0 = N 0 * + ρ 0 θ 0 = 1.5225.
ρ 0 = 0.022 02 ,             ρ 1 = - 0.008 75 ,             and             θ 1 = 3.674 88 ,
N 1 = ρ 0 θ 1 = 0.080 92 ν 01 = ν 1 * + ρ 1 θ 0 = - 0.048 41 ν 11 = ρ 1 θ 1 = - 0.032 16
ϕ 1 = ν 01 / N 0 = - 0.031 79 ,             ϕ ¯ 1 = ν 11 - N 1 ϕ 1 = - 0.029 58.
f a T = - c ν 1 y a 2 ,             f ¯ a T = - c ν 1 y a y b ,
ϕ 1 2 ν 1 / ( N - 1 ) ,
f a 1 * 2 ϕ ¯ 1 y a 2 t ,             f ¯ a 1 * 2 ϕ ¯ 1 y a y b t .
ϕ ¯ 1 1 2 c ν 01 / t ,