Abstract

Hopkins’ treatment of tolerance theory shows that, in designing quality optical systems, we should aim at minimizing the variance K of the wave-aberration difference function. Since the value of K is essentially positive, a useful criterion for the whole field is the sum of the suitably weighted values of K for a typical set of image points, corresponding to a range of selected values of spatial frequencies s and azimuths ψ. In this paper we show that the variance K (for both the axial and the extraaxial images) may be calculated by means of a set of universal coefficients P(i,j;s),Q(i,j;s,ψ) and R(i,j;s,ψ) once the wave-aberrations of certain rays are known. The values of these coefficients are uniquely determined by the function of the wave-aberration polynomial assumed and by the pattern of rays traced. Examples of the coefficients are presented.

© 1968 Optical Society of America

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References

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  1. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [CrossRef]
  2. K. Strehl, Z. Instrumk. 22, 213 (1902).
  3. M. Maréchal, Thesis, U. of Paris (1948). See also M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), p. 62.
  4. W. B. King, Appl. Opt. 7, 489 (1968).
    [CrossRef] [PubMed]
  5. H. H. Hopkins, Proc. Phys. Soc. 70, 5B, 449 (1957).
  6. W. B. King, Appl. Opt. 7, 197 (1968).
    [CrossRef] [PubMed]
  7. See, for example, H. H. Hopkins, Wave Theory of Aberration, (Clarendon Press, Oxford, 1950), Chap. 4.
  8. See, for example, K. Levenberg, Q. Appl. Math. 2, 164 (1944); C. G. Wynne, Proc. Phys. Soc. 73, 777 (1959); J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
    [CrossRef]

1968 (2)

1966 (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. 70, 5B, 449 (1957).

1944 (1)

See, for example, K. Levenberg, Q. Appl. Math. 2, 164 (1944); C. G. Wynne, Proc. Phys. Soc. 73, 777 (1959); J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

1902 (1)

K. Strehl, Z. Instrumk. 22, 213 (1902).

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. 70, 5B, 449 (1957).

See, for example, H. H. Hopkins, Wave Theory of Aberration, (Clarendon Press, Oxford, 1950), Chap. 4.

King, W. B.

Levenberg, K.

See, for example, K. Levenberg, Q. Appl. Math. 2, 164 (1944); C. G. Wynne, Proc. Phys. Soc. 73, 777 (1959); J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

Maréchal, M.

M. Maréchal, Thesis, U. of Paris (1948). See also M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), p. 62.

Strehl, K.

K. Strehl, Z. Instrumk. 22, 213 (1902).

Appl. Opt. (2)

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

Proc. Phys. Soc. (1)

H. H. Hopkins, Proc. Phys. Soc. 70, 5B, 449 (1957).

Q. Appl. Math. (1)

See, for example, K. Levenberg, Q. Appl. Math. 2, 164 (1944); C. G. Wynne, Proc. Phys. Soc. 73, 777 (1959); J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

Z. Instrumk. (1)

K. Strehl, Z. Instrumk. 22, 213 (1902).

Other (2)

M. Maréchal, Thesis, U. of Paris (1948). See also M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), p. 62.

See, for example, H. H. Hopkins, Wave Theory of Aberration, (Clarendon Press, Oxford, 1950), Chap. 4.

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Figures (1)

Fig. 1
Fig. 1

Ray positions corresponding to j = 1, 2, … 14. The effective pupil ray passes through the center (0,0) of the equivalent circular pupil.

Tables (3)

Tables Icon

Table I P(i,j;s) Coefficients. s = 0.15 [(i,j) Refers to the ith Row, jth Column]

Tables Icon

Table II Q(i,j;s,ψ) Coefficients. s = 0.15, ψ = 45°

Tables Icon

Table III R(i,j;s,ψ) Coefficients. s = 0.15, ψ = 45°

Equations (37)

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M ( s o , ψ o ) exp [ i θ ( s o , ψ 0 ) ] = D ( s o , ψ o ) / D o ( s o , ψ o ) ,
M ( s o , ψ o ) = 1 - ( 2 π 2 s o 2 / λ 2 ) K ( s o , ψ o ) ,
K ( s o , ψ o ) = ( V ¯ 2 ) - [ V ¯ ] 2 ,
V ¯ = 1 S S V ( x o , y o ; s o , ψ o ) d x o d y o ( V ¯ 2 ) = 1 S S V 2 ( x o , y o ; s o , ψ o ) d x o d y o } ,
V ( x o , y o ; s o , ψ o ) = 1 s o { W ( x o + s o 2 , y o ) - W ( x o - s o 2 , y o ) } .
( x s / a s ) 2 + ( y s / b s ) 2 = 1 ,
x 2 + y 2 = 1
x = ( x s / a s ) , y = ( y s / b s ) .
s = s o [ ( cos ψ o / a s ) 2 + ( sin ψ o / b s ) 2 ] 1 2 ψ = arctan [ ( a s / b s ) tan ψ o ] } .
W ( x , y ) = U ( x , y ) + V ( x , y ) ,
U ( x , y ) = W 20 ( x 2 + y 2 ) + W 40 ( x 2 + y 2 ) 2 + W 60 ( x 2 + y 2 ) 3 + W 80 ( x 2 + y 2 ) 4 + W 100 ( x 2 + y 2 ) 5 + W 22 y 2 + W 42 ( x 2 + y 2 ) y 2 + W 44 y 4
V ( x , y ) = W 11 y + W 31 ( x 2 + y 2 ) y + W 51 ( x 2 + y 2 ) 2 y + W 71 ( x 2 + y 2 ) 3 y + W 91 ( x 2 + y 2 ) 4 y + W 22 y 3 .
x = x ψ cos ψ - y ψ sin ψ , y = x ψ sin ψ + y ψ cos ψ .
V ( x ψ , y ψ ; s , ψ ) = 1 s { W ( x ψ + s 2 , y ψ ) - W ( x ψ - s 2 , y ψ ) } ,
H k , l ( s ) = 1 S S x ψ k y ψ l d x ψ d y ψ .
K ( s , ψ ) = K E ( s , ψ ) + K 0 ( s , ψ ) ,
K E ( s , ψ ) = m p n q E ( m , p , n , q ; s , ψ ) W m , p W n , q
m , n = 2 , 4 , 6 , 8 , 10 ,             p , q = 0 , 2 , 4 ,
K 0 ( s , ψ ) = m p n q 0 ( m , p , n , q ; s , ψ ) W m , p W n , q
m , n = 1 , 3 , 5 , 7 , 9             p , q , = 1 , 3.
( x s , ) j = a s x j , ( y s ) j = b s y j ,
x i = 0 y j = + [ ( 6 - j ) / 5 ] 1 2 } ,
j = 6 , ( x 6 = y 2 / 2 1 2 , y 6 = y 2 / 2 1 2 ) j = 7 , ( x 7 = y 2 , y 7 = 0 ) j = 8 , ( x 8 = 1.0 , y 8 = 0 ) } .
x j = x ( j - 8 ) y j = - y ( j - 8 ) } ,
U j = 1 2 { W j + W ( j + 8 ) } , j = 1 , 2 , 6 , U j = W j , j = 7 , 8 ,
V j = 1 2 { W j - W ( j + 8 ) } ,     j = 1 , 2 , 6.
U j = n q W n , q ( x 2 + y 2 ) j ( n - q ) / 2 y j q
W n , q = j = 1 8 K ( n , q ; j ) U j .
V j = n q W n , q ( x 2 + y 2 ) j ( n - q ) / 2 y j q
W n , q = j = 1 6 K ( n , q ; j ) V j .
W j = n W n , o y j n ,
W n , o = j = 1 5 K ( n , o ; j ) W j .
K E ( s , ψ ) = i = 1 8 j = i 8 Q ( i , j ; s , ψ ) U i U j ,
K 0 ( s , ψ ) = i = 1 6 j = i 8 R ( i , j ; s , ψ ) V i V j ,
K axial ( s ) = m n A ( m , o , n , o ; s ) W m , o W n , o
K axial ( s ) = i = 1 5 j = i 5 P ( i , j ; s ) W i W j ,
Φ = λ s { α s K axial ( s ) } λ + λ τ { s ψ [ β s , ψ K E ( s , ψ ) + γ s , ψ K 0 ( s , ψ ) ] } λ , τ ,

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