Abstract

An inhomogeneous medium has a cylindrical index distribution if the refractive index depends only on the distance from some fixed line, the axis of the distribution. Formulas are given whereby the third-order aberration coefficients of a symmetric optical system may be computed exactly when the optical media have such cylindrical index distributions. The formulas are valid for both diverging and converging media. A comprehensive numerical illustration of the use of the various formulas is included.

© 1971 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. Reference 1, p. 72.
  3. A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).
  4. The first electron optical system was built by H. Busch [H. Busch, Arch. Elektrotechn. 18, 583 (1927)].
    [CrossRef]
  5. E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
    [CrossRef]
  6. D. T. Moore, thesis, Institute of Optics, University of Rochester, 1970.
  7. D. T. Moore, J. Opt. Soc. Am. 60, 1557A (1970); J. Opt. Soc. Am. 61, 886 (1971).
  8. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  9. P. J. Sands, J. Opt. Soc Am. 61, 777 (1971).
    [CrossRef] [PubMed]
  10. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [CrossRef]
  11. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1971).
    [CrossRef]
  12. This equation follows directly from Eq. (209.4) of Optical Aberration Coefficients, by H. A. Buchdahl (Dover, New York, 1969). See also Sec. II of Ref. 10.
  13. L. Montagnino, J. Opt. Soc. Am. 58, 1667 (1968).
    [CrossRef]
  14. E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]

1971 (3)

1970 (4)

E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
[CrossRef]

P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
[CrossRef]

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
[CrossRef]

D. T. Moore, J. Opt. Soc. Am. 60, 1557A (1970); J. Opt. Soc. Am. 61, 886 (1971).

1969 (1)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).

1968 (1)

1927 (1)

The first electron optical system was built by H. Busch [H. Busch, Arch. Elektrotechn. 18, 583 (1927)].
[CrossRef]

Buchdahl, H. A.

This equation follows directly from Eq. (209.4) of Optical Aberration Coefficients, by H. A. Buchdahl (Dover, New York, 1969). See also Sec. II of Ref. 10.

Busch, H.

The first electron optical system was built by H. Busch [H. Busch, Arch. Elektrotechn. 18, 583 (1927)].
[CrossRef]

French, W. G.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).

Herriott, D. R.

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
[CrossRef]

Marchand, E. W.

McKenna, J.

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
[CrossRef]

Montagnino, L.

Moore, D. T.

D. T. Moore, J. Opt. Soc. Am. 60, 1557A (1970); J. Opt. Soc. Am. 61, 886 (1971).

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

Pearson, A. D.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).

Rawson, E. G.

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
[CrossRef]

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).

Sands, P. J.

Wood, R. W.

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

Appl. Opt. (1)

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 3, 753 (1970).
[CrossRef]

Appl. Phys. Letters (1)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 2, 76 (1969).

Arch. Elektrotechn. (1)

The first electron optical system was built by H. Busch [H. Busch, Arch. Elektrotechn. 18, 583 (1927)].
[CrossRef]

J. Opt. Soc Am. (1)

P. J. Sands, J. Opt. Soc Am. 61, 777 (1971).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

Other (4)

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

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

Reference 1, p. 72.

This equation follows directly from Eq. (209.4) of Optical Aberration Coefficients, by H. A. Buchdahl (Dover, New York, 1969). See also Sec. II of Ref. 10.

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Tables (1)

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Table I Third-order aberrations of a system with a cylindrical index distribution.

Equations (37)

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N = N 0 + N 1 ξ + N 2 ξ 2 + ,
d 2 y / d x 2 - N ¯ 1 y = 0 ,             N ¯ 1 = 2 N 1 / N 0 .
N 1 0 ,             α 2 = N ¯ 1 C ( x ) = cosh ( α x ) ,             S ( x ) = ( 1 / α ) sinh ( α x )
N 1 < 0 ,             α 2 = - N ¯ 1 C ( x ) = cos ( α x ) ,             S ( x ) = ( 1 / α ) sin ( α x ) .
y ( x ) = y 0 C ( x ) + v 0 S ( x ) , v ( x ) = y ˙ ( x ) = N ¯ 1 y 0 S ( x ) + v 0 C ( x ) , y ¨ ( x ) = v ˙ ( x ) = N ¯ 1 y ( x ) ,
n = 0 t C ( x ) 4 - n S ( x ) n d x ,             n = 0 , 1 , 2 , 3 , 4 ,
0 = 1 8 [ 3 t + ( 5 + 2 N ¯ 1 S 2 ) C S ] 1 = ( 1 / 4 N ¯ 1 ) ( C 4 - 1 ) 2 = ( 1 / 8 N ¯ 1 ) [ - t + ( 1 + 2 N ¯ 1 S 2 ) C S ] 3 = 1 4 S 4 4 = ( 1 / 8 N ¯ 1 2 ) [ 3 t - ( 3 - 2 N ¯ 1 S 2 ) C S ] ,
n 0 t x n d x = t ( n + 1 ) / ( n + 1 ) .
a 1 * = 1 2 N 0 ( y a v a 3 ) + 4 N 2 y a 4 d x + 2 N 1 y a 2 v a 2 d x - 1 2 N 0 v a 4 d x a 2 * = 1 2 N 0 ( y a v a 2 v b ) + 4 N 2 y a 3 y b d x + N 1 y a v a ( y a v b + y b v a ) d x - 1 2 N 0 v a 3 v b d x a 3 * = 1 2 N 0 ( y a v a v b 2 ) + 4 N 2 y a 2 y b 2 d x + 2 N 1 y a y b v a v b d x - 1 2 N 0 v a 2 v b 2 d x a 4 * = ( N 1 g 2 t / N 0 2 ) a 5 * = 1 2 N 0 ( y a v b 3 ) + 4 N 2 y a y b 3 d x + N 1 y b v b ( y a v b + y b v a ) d x - 1 2 N 0 v a v b 3 d x ,
N 1 f ( y a , y b , v a , v b ) + N ¯ 2 I n ,
N ¯ 2 = 4 N 2 - 10 N 1 2 / 3 N 0 ,             I n = 0 t y a ( x ) 4 - n y b ( x ) n d x
1 2 N 0 v a 4 d x = 1 2 N 0 v a 3 d y a = 1 2 N 0 ( y a v a 3 ) - 3 2 N 0 y a v a 2 v ˙ a d x = 1 2 N 0 ( y a v a 3 ) - 3 N 1 y a 2 v a 2 d x .
a 1 * = 4 N 2 y a 4 d x + 5 N 1 y a 2 v a 2 d x a 2 * = 4 N 2 y a 3 y b d x + 3 N 1 y a 2 v a v b d x + 2 N 1 y a y b v a 2 d x a 3 * = 4 N 2 y a 2 y b 2 d x + N 1 y a 2 v b 2 d x + 4 N 1 y a y b v a v b d x a 5 * = 4 N 2 y a y b 3 d x + N 1 y b 2 v a v b d x + 4 N 1 y a y b v b 2 d x .
y a 2 v a v b d x = ( y a 3 v b ) - y a ( y a 2 v ˙ b + 2 y a y ˙ a v b ) d x = ( y a 3 v b ) - N ¯ 1 y a 3 y b d x - 2 y a 2 v a v b d x .
y a 2 v a v b d x = 1 3 ( y a 3 v b ) - 1 3 N ¯ 1 I 1 .
y a y b v a 2 d x = ( y a 2 y b v a ) - y a ( y a y b v ˙ a + y a v a v b + y b v a 2 ) d x = ( y a 2 y b v a ) - N ¯ 1 I 1 - y a 2 v a v b d x - y a y b v a 2 d x ,
y a y b v a 2 d x = 1 2 y a 2 ( y b v a - 1 3 y a v b ) - 1 3 N ¯ 1 I 1 .
y a y b v a v b d x = ( y a y b 2 v a ) - y b ( y a v a v b + y b v a 2 + y a y b v ˙ a ) d x ,
2 y a y b v a v b d x = ( y a y b 2 v a ) - N ¯ 1 I 2 - y b 2 v a 2 d x .
y a y b v a v b d x = ( y a 2 y b v b ) - y a ( y b v a v b + y a v b 2 + y a y b v ˙ b ) d x ,
2 y a y b v a v b d x = ( y a 2 y b v b ) - N ¯ 1 I 2 - y a 2 v b 2 d x .
6 y a y b v a v b d x = y a y b ( y a v b + y b v a ) - 2 N ¯ 1 I 2 - ( g 2 t / N 0 2 ) ,
3 y a 2 v a 2 d x = ( y a 3 v a ) - N ¯ 1 I 0 3 y a 2 v a v b d x = ( y a 3 v b ) - N ¯ 1 I 1 6 y a y b v a 2 d x = y a 2 ( 3 y b v a - y a v b ) - 2 N ¯ 1 I 1 6 y a y b v a v b d x = y a y b ( y a v b + y b v a ) - 2 N ¯ 1 I 2 - ( g 2 t / N 0 2 ) 3 y a 2 v b 2 d x = y a y b ( 2 y a v b - y b v a ) - N ¯ 1 I 2 + ( g 2 t / N 0 2 ) 3 y b 2 v a v b d x = ( y b 3 v a ) - N ¯ 1 I 3 6 y a y b v b 2 d x = y b 2 ( 3 y a v b - y b v a ) - 2 N ¯ 1 I 3 .
a 1 * = ( 5 N 1 / 3 ) ( y a 3 v a ) + N ¯ 2 I 0 a 2 * = ( N 1 / 3 ) y a 2 ( 2 y a v b + 3 y b v a ) + N ¯ 2 I 1 a 3 * = ( N 1 / 3 ) y a y b ( 4 y a v b + y b v a ) + N ¯ 2 I 2 - 1 3 a 4 * a 5 * = ( N 1 / 3 ) y b 2 ( 6 y a v b - y b v a ) + N ¯ 2 I 3 .
I 0 = y a ( x ) 4 d x = [ y a C ( x ) + v a S ( x ) ] 4 d x = y a 4 C 4 d x + 4 y a 3 v a C 3 S d x + 6 y a 2 v a 2 C 2 S 2 d x + 4 y a v a 3 C S 3 d x + v a 4 S 4 d x ,
I 0 = y a 4 0 + 4 y a 3 v a 1 + 6 y a 2 v a 2 2 + 4 y a v a 3 3 + v a 4 4 , I 1 = y a 3 y b 0 + y a 2 ( 3 y b v a + y a v b ) 1 + 3 y a v a ( y a v b + y b v a ) 2 + v a 2 ( y b v a + 3 y a v b ) 3 + v a 3 v b 4 , I 2 = y a 2 y b 2 0 + 2 y a y b ( y a v b + y b v a ) 1 + [ ( y a v b + y b v a ) 2 + 2 y a y b v a v b ] 2 + 2 v a v b ( y a v b + y b v a ) 3 + v a 2 v b 2 4 , I 3 = y a y b 3 0 + y b 2 ( y b v a + 3 y a v b ) 1 + 3 y b v b ( y a v b + y b v a ) 2 + v b 2 ( 3 y b v a + y a v b ) 3 + v a v b 3 4 .
l ˆ = v a / y a ,             p ˆ = v b / y b .
I 0 = y a 4 ( 0 + 4 l ˆ 1 + 6 l ˆ 2 2 + 4 l ˆ 3 3 + l ˆ 4 4 ) I 1 = y a 3 y b [ 0 + ( 3 l ˆ + p ˆ ) 1 + 3 l ˆ ( l ˆ + p ˆ ) 2 + l ˆ 2 ( l ˆ + 3 p ˆ ) 3 + l ˆ 3 p ˆ 4 ] I 2 = y a 2 y b 2 { 0 + 2 ( l ˆ + p ˆ ) 1 + [ ( l ˆ + p ˆ ) 2 + 2 l ˆ p ˆ ] 2 + 2 l ˆ p ˆ ( l ˆ + p ˆ ) 3 + l ˆ 2 p ˆ 2 4 } I 3 = y a y b 3 [ 0 + ( l ˆ + 3 p ˆ ) 1 + 3 p ˆ ( l ˆ + p ˆ ) 2 + p ˆ 2 ( 3 l ˆ + p ˆ ) 3 + l ˆ p ˆ 3 4 ] .
I 0 = v a 4 4 I 1 = y b v a 3 ( 3 + p ˆ 4 ) I 2 = y b 2 v a 2 ( 2 + 2 p ˆ 3 + p ˆ 2 4 ) I 3 = y b 3 v a ( 1 + 3 p ˆ 2 + 3 p ˆ 2 3 + p ˆ 3 4 ) .
I 1 = y a 3 v b ( 1 + 3 l ˆ 2 + 3 l ˆ 2 3 + l ˆ 3 4 ) I 2 = y a 2 v b 2 ( 2 + 2 l ˆ 3 + l ˆ 2 4 ) I 3 = y a v b 3 ( 3 + l ˆ 4 ) .
C ( x ) 2 - N ¯ 1 S ( x ) 2 = 1.
v = v + ( k - 1 ) i ,             i = c 0 y + v ,             k = N 0 / N 0 .
a 0 1 = 1 2 N 0 ( k - 1 ) y a i a 2 ( i a + v a ) ,             q = ( i b / i a ) , a 0 2 = q a 0 1 ,             a 0 3 = q a 0 2 , a 0 4 = 1 2 g 2 c 0 ( k - 1 ) / N 0 ,             a 0 5 = q ( a 0 3 + a 0 4 ) .
κ = - 2 c 0 Δ N 1 - c 1 Δ N 0 ,             a 1 = κ y a 4 a 2 = κ y a 3 y b ,             a 3 = κ y a 2 y b 2 ,             a 5 = κ y a y b 3 .
a 1 * = 4 N 2 I 0 ,             a 2 * = 4 N 2 I 1 ,             a 3 * = 4 N 2 I 2 , a 4 * = 0 ,             and             a 5 * = 4 N 2 I 3 ,
a 1 * = 4 y a 4 ( 0 + 4 l ˆ 1 + 6 l ˆ 2 2 + 4 l ˆ 3 3 + l ˆ 4 4 ) ,
n = 0 t N 2 ( x ) x n d x