Abstract

The main subject of this paper is the aberrations of optical systems in which the various media may have axial distributions of index. The general formulas for the transfer contributions to the third-order aberrations take a particularly simple form for an axial distribution. A numerical illustration of the use of the various formulas is given. In particular, the depth of a given index distribution has little effect on the transfer contributions, whereas the inhomogeneous surface contributions may be quite sensitive to depth.

© 1971 Optical Society of America

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References

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  1. E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]
  2. L. Montagnino, J. Opt. Soc. Am. 58, 1667 (1968).
    [CrossRef]
  3. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  4. P. J. Sands, J. Opt. Soc. Am. 61, 777 (1971).
    [CrossRef] [PubMed]
  5. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [CrossRef]
  6. References 3, 4, and 5 will respectively be referred to by the abbreviations I, II, and III. Equations in these papers will be referenced by preceding the relevant equation number by the abbreviation of the paper concerned.
  7. D. T. Moore, thesis, Institute of Optics, University of Rochester, Rochester, N. Y.14627, 1970.
  8. E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 753 (1970).
    [CrossRef] [PubMed]

1971 (2)

1970 (3)

1968 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

Other (2)

References 3, 4, and 5 will respectively be referred to by the abbreviations I, II, and III. Equations in these papers will be referenced by preceding the relevant equation number by the abbreviation of the paper concerned.

D. T. Moore, thesis, Institute of Optics, University of Rochester, Rochester, N. Y.14627, 1970.

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Tables (2)

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Table I Expressions for I1 and I3 for various axial index distributions.

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Table II Computation of third-order aberration coefficients of systems with axial index distributions.

Equations (33)

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a 1 * = 1 2 N 0 3 v a 3 a             a 2 * = 1 2 N 0 3 v a 2 v b a a 3 * = 1 2 N 0 3 v a v b 2 a             a 4 * = 0 a 5 * = 1 2 N 0 3 v b 3 a ,
a = [ y a / N 0 ( x ) 2 ] - N 0 v a d x / N 0 ( x ) 3 .
a = [ y b / N 0 ( x ) 2 ] - N 0 v b d x / N 0 ( x ) 3 .
a 1 * = 1 2 ( N 0 y a v a 3 ) + ( 4 N 2 y a 4 + 2 N 1 y a 2 v a 2 - 1 2 N 0 v a 4 ) d x
a 1 * = 1 2 ( N 0 y a v a 3 ) - 1 2 N 0 v a 4 d x .
a 1 * = 1 2 N 0 3 v a 3 [ ( y a / N 0 2 ) - N 0 v a d x / N 0 ( x ) 3 ] ,
Λ a = y a ( x ) F d x ,
F = ( L * / Y ) - [ ( d / d x ) ( L * / V ) ] .
Λ a = - y a ( x ) d d x ( L * V ) d x .
Λ a = 1 2 y a ( x ) d d x ( N 0 V ζ ) d x + O ( 5 ) .
Δ Λ a = 1 2 N 0 3 V ζ a + O ( 5 ) ,
a = y a d d x 1 N 0 ( x ) 2 d x = - 2 y a N ˙ 0 ( x ) d x N 0 ( x ) 3 .
V ( x ) = v a ( x ) S + v b ( x ) T + O ( 3 ) ζ ( x ) = V ( x ) 2 + W ( x ) 2 .
y a ( x ) = A * ( x ) y a + N 0 B * ( x ) v a .
A * = 1 ,             B * = d x / N 0 ( x ) ;
I n = d x / [ N 0 ( x ) ] n ,
y a ( x ) = y a + N 0 v a I 1 a = ( y a / N 0 2 ) - N 0 v a I 3 .
N 0 ( x ) = 1.5 - 0.158 197 6 ( e - x - 1 ) .
a 1 = κ y a 4 , a 2 = ( y b / y a ) a 1 , a 3 = ( y a / y a ) 2 a 1 , a 5 = ( y b / y a ) 3 a 1 ,
κ = - 1 2 c 2 Δ N ˙ 0 .
a 1 * = - N ˙ 0 ( x ¯ ) y a ( x ¯ ) v a ( x ¯ ) 3 t ,
a 1 * - ( N 0 ) y a ( x ¯ ) v a ( x ¯ ) 3 ,
case a :             N 0 ( x ) = N 0 [ 1 + α ( e - a x - 1 ) ]
case b :             N 0 ( x ) = N 0 [ ( 1 + α 2 x 2 ) / ( 1 + a 2 x 2 ) ] ,
case a :             N 0 ( x ) = N 0 ( 1 + α x )
case b :             N 0 ( x ) = N 0 ( 1 + α 2 x 2 ) .
case a :             N 0 α = [ N 0 ( t ) - N 0 ] / ( e - a t - 1 )
case b :             α 2 = ( N 0 ( t ) N 0 - 1 ) 1 t 2 + N 0 ( t ) N 0 a 2 .
a = y a ( 1 / N 0 2 ) + N 0 v a [ I 1 / N 0 ( t ) 2 - I 3 ] .
I 1 = t ( ln N 0 ) / N 0 = 0.645 385 I 3 = - 1 2 t ( 1 / N 0 2 ) / N 0 = 0.269 097 ,
a = - 0.053 819 y a - 0.025 490 v a .
a = - 0.053 819 y a - 0.034 735 v a .
N ˙ 0 ( x ) = 2 N 0 ( α 2 - a 2 ) x / ( 1 + a 2 x 2 ) 2 .