Abstract

An analysis is made of refracting systems consisting of two spherical surfaces. Solutions are found for those systems having zero third-order spherical aberration. These are in the form of four one-parameter families of functions. Expressions for third-order coma and astigmatism are derived. Parameter domains for useful solutions are indicated. A method for applying these results to optical design is described.

© 1969 Optical Society of America

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References

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  1. O. N. Stavroudis, J. Opt. Soc. Am. 57, 741 (1967).
    [Crossref]
  2. L. E. Sutton, Appl. Opt. 2, 1275 (1963).
    [Crossref]
  3. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [Crossref]
  4. T. Smith, in A Dictionary of Applied Physics, R. Glazebrook, Ed. (Macmillan and Company, Ltd., London, 1923), Vol. IV, p. 290.
  5. M. Herzberger, Modern Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1958), Ch. 19.
  6. R. J. Pegis, T. P. Vogl, A. K. Rigler, and R. Walters, Appl. Opt. 6, 969 (1967).
    [Crossref] [PubMed]

1967 (2)

1963 (1)

1951 (1)

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Other (2)

T. Smith, in A Dictionary of Applied Physics, R. Glazebrook, Ed. (Macmillan and Company, Ltd., London, 1923), Vol. IV, p. 290.

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1958), Ch. 19.

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Figures (3)

Fig. 1
Fig. 1

The optical parameters. Spherical surfaces with curvatures c1 and c2 separate media with refractive indices N0, N1, and N2. t0 is the object distance in the medium with refractive index = N0. t1 is the axial thickness of the N1 medium. The marginal ray is parallel to the axis in the N2 medium.

Fig. 2
Fig. 2

Values of q and the optical parameters as functions of k, showing the critical values of k. N0 = 1. N1 = 1.5731. N2 = 1.9525.

Fig. 3
Fig. 3

Values of q and the optical parameters as functions of k, showing the critical values of k. N0 = 1. N1 = 1.9525. N2 = 1.5731.

Tables (2)

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Table I Values of q and the optical parameters corresponding to the critical values of k.

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Table II Critical values of θ and corresponding values of q and t0.

Equations (51)

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( y 1 u 1 ) = ( 1 - t 0 ( N 1 - N 0 ) c 1 / N 1 [ N 0 - ( N 1 - N 0 ) c 1 t 0 ] / N 1 ) ( y 0 u 0 ) ,
( y 2 u 2 ) = ( [ N 1 - ( N 1 - N 0 ) c 1 t 1 ] / N 1 - N 0 f [ N 1 ( N 2 - N 1 ) c 2 + N 1 ( N 1 - N 0 ) c 1 - ( N 2 - N 1 ) ( N 1 - N 0 ) c 2 t 1 c 1 ] / N 2 N 1 [ N 0 - ( N 1 - N 0 ) c 1 t 0 - N 0 f ( N 2 - N 1 ) c 2 ] / N 2 ) ( y 0 u 0 ) ,
f = [ N 1 t 0 + N 0 t 1 - ( N 1 - N 0 ) t 1 c 1 t 0 ] / N 1 N 0
N 0 - ( N 1 - N 0 ) c 1 t 0 - N 0 f ( N 2 - N 1 ) c 2 = 0.
q = N 0 - ( N 1 - N 0 ) c 1 t 0
c 2 = q / ( N 2 - N 1 ) N 0 f t 1 = N 1 ( N 0 f - t 0 ) / q c 1 = ( N 0 - q ) / ( N 1 - N 0 ) t 0 .
( y 1 u 1 ) = ( 1 - t 0 ( N 0 - q ) / N 1 t 0 q / N 1 ) ( y 0 u 0 )
( y 2 u 2 ) = ( N 0 ( t 0 - N 0 f + f q ) / t 0 q - N 0 f 1 / N 2 f 0 ) ( y 0 u 0 ) .
i 1 = c 1 y 1 - u 0 ,
i 1 = ( N 0 - q ( N 1 - N 0 ) t 0 ) y 0 - ( N 1 - q N 1 - N 0 ) u 0 .
i 2 = c 2 y 2 - u 1 ,
i 2 = ( N 1 t 0 - f N 2 ( N 0 - q ) ( N 2 - N 1 ) f t 0 N 1 ) y 0 - ( N 2 q ( N 2 - N 1 ) N 1 ) u 0 .
y 1 = - t 0 u 0 , u 1 = q u 0 / N 1 , i 1 = - ( N 1 - q ) u 0 / ( N 1 - N 0 ) ,
y 2 = - N 0 f u 0 , u 2 = 0 , i 2 = - N 2 q u 0 / ( N 2 - N 1 ) N 1 .
y ¯ 1 = y ¯ 0 - t c ū 0 , ū 1 = y ¯ 0 ( N 0 - q ) / N 1 t 0 + ū 0 q / N 1 , i ¯ 1 = ( N 0 - q ( N 1 - N 0 ) t 0 ) y ¯ 0 - ( N 1 - q N 1 - N 0 ) ū 0 ,
y ¯ 2 = y ¯ 0 N 0 ( t 0 - N 0 f + f q ) / t 0 q - ū 0 N 0 f , ū 2 = y ¯ 0 / N 2 f , i ¯ 2 = ( N 1 t 0 - f N 2 ( N 0 - q ) ( N 2 - N 1 ) f t 0 N 1 ) y ¯ 0 - ( N 2 q ( N 2 - N 1 ) N 1 ) ū 0 .
S j = N j - 1 y j ( u j - i j ) ( N j - N j - 1 ) / 2 I N j ,             ( j = 1 , 2 )
2 I S 1 = - u 0 2 N 0 t 0 ( N 1 2 - N 0 q ) / N 1 2 ,
2 I S 2 = - u 0 2 N 0 f q .
spherical             B = S 1 i 1 2 + S 2 i 2 2 ,
coma             F = S 1 i 1 i ¯ 1 + S 2 i 2 i ¯ 2 ,
astigmatism             C = S 1 i ¯ 1 2 + S 2 i ¯ 2 2 .
a = ( N 0 - q ) / ( N 1 - N 0 ) t 0 b = - ( N 1 - q ) / ( N 1 - N 0 ) c = [ N 1 t 0 - f N 2 ( N 0 - q ) ] / ( N 2 - N 1 ) f t 0 N 1 d = - N 2 q / ( N 2 - N 1 ) N 1 .
i 1 = b u 0 ,             i 2 = d u 0
i ¯ 1 = a y ¯ 0 + b ū 0 ,             i ¯ 2 = c y ¯ 0 + d ū 0 .
H = - N 0 t 0 ( N 1 2 - N 0 q ) / N 1 2
K = - N 0 f q .
2 I S 1 = H u 0 2 ,             2 I S 2 = K u 0 2 .
2 I B = u 0 4 [ H b 2 + K d 2 ] ,
2 I F = ( H u 0 2 ) ( b u 0 ) ( a y ¯ 0 + b ū 0 ) + ( K u 0 2 ) ( d u 0 ) ( c y ¯ 0 + d ū 0 ) = u 0 3 [ ( H a b + K c d ) y ¯ 0 + ( H b 2 + K d 2 ) ū 0 ]
2 I C = ( H u 0 2 ) ( a y ¯ 0 + b ū 0 ) 2 + ( K u 0 2 ) ( c y ¯ 0 + d ū 0 ) 2 = u 0 2 [ ( H a 2 + K c 2 ) y ¯ 0 2 + 2 ( H a b + K c d ) y ¯ 0 ū 0 + ( H b 2 + K d 2 ) ū 0 2 ] .
B = S 1 i 1 2 + S 2 i 2 2 ,
2 I B = ( - u 0 2 N 0 t 0 ( N 1 2 - N 0 q ) N 1 2 ) ( - ( N 1 - q ) u 0 N 1 - N 0 ) 2 + ( - u 0 2 N 0 f q ) ( - N 2 q u 0 ( N 2 - N 1 ) N 1 ) 2 .
t 0 ( N 2 - N 1 ) 2 ( N 1 2 - N 0 q ) ( N 1 - q ) 2 + N 2 2 f q 3 ( N 1 - N 0 ) 2 = 0.
t 0 = 27 N 2 2 N 1 2 f ( k 2 - 1 ) / 4 ( N 2 - N 1 ) 2 ( N 1 - N 0 )
q r = 3 N 1 2 / { ( 2 N 1 + N 0 ) - ( N 1 - N 0 ) × ( [ ( k + 1 ) / ( k - 1 ) ] 1 3 ω r + [ ( k - 1 ) / ( k + 1 ) ] 1 3 ω - r ) } ( r = 0 , 1 , 2 )
q = 3 N 1 2 / { ( 2 N 1 + N 0 ) - ( N 1 - N 0 ) × ( [ ( k + 1 ) / ( k - 1 ) ] 1 3 + [ ( k - 1 ) / ( k + 1 ) ] 1 3 ) } .
t 0 = - 27 N 2 2 N 1 2 f sec 2 θ / 4 ( N 2 - N 1 ) 2 ( N 1 - N 0 ) .
k + 1 k - 1 = i tan θ + 1 i tan θ - 1 = - e i θ e - i θ = - e 2 i θ ;
( k + 1 k - 1 ) 1 3 ω r + ( k - 1 k + 1 ) 1 3 ω - r = - [ e 2 i θ / 3 e 2 π i r / 3 + e - 2 i θ / 3 e - 2 π i r / 3 ] = - 2 cos 2 3 ( θ + π r ) .
q r = 3 N 1 2 / [ ( 2 N 1 + N 0 ) + 2 ( N 1 - N 0 ) cos 2 3 ( θ + π r ) ]             ( r = 0 , 1 , 2 ) .
H b 2 + K d 2 = 0 ,
K = - H b 2 / d 2 .
2 I F = u 0 3 y ¯ 0 H b ( a d - b c ) / d ,
2 I C = u 0 2 H ( a d - b c ) [ ( a d + b c ) y ¯ 0 + 2 b d ū 0 ] / d 2 .
a d - b c = [ t 0 ( N 1 - q ) - f N 2 ( N 0 - q ) ] / ( N 2 - N 1 ) ( N 1 - N 0 ) t 0 f .
k 0 = ( 3 N 1 + 2 N 0 ) ( 3 N 1 - N 0 ) 1 2 3 3 N 1 ( N 1 + N 0 ) 1 2 = ( 1 - 4 N 0 3 27 N 1 2 ( N 1 + N 0 ) ) 1 2 , k = ( N 1 + 2 N 0 ) ( 4 N 1 - N 0 ) 1 2 3 3 N 1 ( N 0 ) 1 2 = ( 1 + 4 ( N 1 - N 0 ) 3 27 N 1 2 N 0 ) 1 2 ,
k * = ( 1 + 4 N 0 ( N 1 - N 0 ) ( N 2 - N 1 ) 2 27 N 2 2 N 1 2 ) 1 2 .
N 0 < N 1 < 2 N 0
N 0 < N 2 < N 1 N 0 / ( 2 N 0 - N 1 ) ,
0 < k 0 < 1 < k * < k < .