Abstract

Marechal’s treatment of tolerance theory shows that in designing high quality systems one should aim at minimizing the variance E of the wave aberration. Since the value of E is essentially positive, a useful criterion for the whole field is the sum of the suitably weighted values of E for a typical set of image points. It is shown here that the variance E (for both the axial and extraaxial images) may be calculated very simply by means of a set of universal coefficients [P(i,j), Q(i,j), and R(i,j)] once the wave aberrations of selected rays are known. The values of these coefficients are uniquely determined by the form of polynomial assumed for the wave aberration and by the pattern of rays traced. Tables of P(i,j), Q(i,j), and R(i,j) are presented for the different cases that can arise in practice.

© 1968 Optical Society of America

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References

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  1. A. Maréchal, Thesis, University of Paris (1948).
  2. H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).
  3. H. H. Hopkins, Opt. Acta, 13, 343 (1966).
    [CrossRef]
  4. W. B. King, Appl. Opt. 7, 197 (1968).
    [CrossRef] [PubMed]

1968 (1)

1966 (1)

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

1965 (1)

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

Hopkins, H. H.

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

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

King, W. B.

Maréchal, A.

A. Maréchal, Thesis, University of Paris (1948).

Appl. Opt. (1)

Japan. J. Appl. Phys. (1)

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

Opt. Acta (1)

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

Other (1)

A. Maréchal, Thesis, University of Paris (1948).

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Figures (2)

Fig. 1
Fig. 1

The scaled pupil coordinates (xs/as, ys/bs) of the rim rays plotted against a unit circle for a 15.24-cm (6-in.), f/2 system at full field (50°).

Fig. 2
Fig. 2

The specification of rays is determined by the values of A and S. An example of the ray pattern for A = 3, S = 3 is shown.

Tables (1)

Tables Icon

Table I The Permissible Values of (n, q) are Tabulated for Five Different Classes of Optical Systems

Equations (41)

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I = | A exp [ i k W ( x , y ) ] d A | 2 ,
d A = d x d y / A d x d y
I = [ 1 - ( 2 π 2 / λ 2 ) E ] 2 ,
E = A [ W ( x , y ) ] 2 d A - [ A W ( x , y ) d A ] 2 .
( 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 ) .
J = 2 A + S ,
ϕ i = i [ π / ( S + 1 ) ] , i = 1 , 2 , , S .
( x s ) j = a s x j ; ( y s ) j = b s y j ,
x j = 0 , y i = + [ ( A + 1 - j ) / A ] 1 2 .
S = 1 : i = 1 , r 1 = 1 S = 3 { i = 1 , r 1 = y 2 = + [ ( A - 1 ) / A ] 1 2 i = 2 , r 2 = 1 ,
x i = r i sin ϕ i , y i = r i cos ϕ i .
x j = x j - N y j = - y j - N } ,
W ( x , y ) = W 11 y + W 20 ( x 2 + y 2 ) + W 22 y 2 + W 31 ( x 2 + y 2 ) y + W 33 y 3 + W 40 ( x 2 + y 2 ) 2 + W 42 ( x 2 + y 2 ) y 2 + W 51 ( x 2 + y 2 ) 2 y + W 60 ( x 2 + y 2 ) 3 .
( W j ) even = U j = ( W j + W j + N ) / 2 ; j = 1 , 2 , , N - 1 U j = W j , j = N .
U j = W 20 ( x 2 + y 2 ) j + W 40 ( x 2 + y 2 ) j 2 + W 60 ( x 2 + y 2 ) j 3 + W 22 y j 2 + W 42 ( x 2 + y 2 ) j y j 2 .
( W j ) o d d = V j = ( W j - W j + N ) / 2 ; j = 1 , 2 , , N - 1.
V j = W 11 y j + W 31 ( x 2 + y 2 ) j y j + W 51 ( x 2 + y 2 ) j 2 y j + W 33 y j 3 .
W j = W 20 y j 2 + W 40 y j 4 + W 60 y j 6 ,
U j = n q W n , q ( x 2 + y 2 ) j ( n - q ) / 2 y j q ,
W n , q = j = 1 N K ( n , q ; j ) U j .
V j = q n W n , q ( x 2 + y 2 ) j ( n - q ) / 2 y j q ,
W n , q = j = 1 N - 1 K ( n , q ; j ) V j .
W j = n W n , 0 y j n ,
W n , 0 = j = 1 A K ( n , 0 ; j ) W j ,
W = n q W n , q r n cos q ϕ .
E = E even + E odd ,
E even = 1 π 0 1 0 2 π U 2 r d r d ϕ - [ 1 π 0 1 0 2 π U r d r d ϕ ] 2 ,
E odd = 1 π 0 1 0 2 π V 2 r d r d ϕ ,
E = 1 π 0 1 0 2 π [ n q W n , q ( r n cos q ϕ ) ] 2 r d r d ϕ - [ 1 π 0 1 0 2 π n q W n , q ( r n cos q ϕ ) r d r d ϕ ] 2 .
E even = m n p q W m , p W n , q { ( 2 m + n + 2 ) × [ 1 × 3 ( p + q - 1 ) 2 × 3 ( p + q ) ] - 4 ( m + 2 ) ( n + 2 ) × [ 1 × 3 ( p - 1 ) 2 × 4 p ] [ 1 × 3 ( q - 1 ) 2 × 4 q ] } .
E odd = m n p q W m , p W n , q × { ( 2 m + n + 2 ) [ 1 × 3 ( p + q - 1 ) 2 × 4 ( p + q ) ] } .
E axial = m n W m , 0 W n , 0 [ 2 m + n + 2 - 4 ( m + 2 ) ( n + 2 ) ] .
E even = i = 1 N j = 1 N Q ( i , j ) U i U j
E odd = i = 1 N - 1 j = 1 N - 1 R ( i , j ) V i V j ,
E axial = i = 1 A j = 1 A P ( i , j ) W i W j ,
Q ( i , j ) = n n p q { ( 2 m + n + 2 ) [ 1 × 3 ( p + q - 1 ) 2 × 4 ( p + q ) ] - 4 ( m + 2 ) ( n + 2 ) [ 1 × 3 ( p - 1 ) 2 × 4 p ] × [ 1 × 3 ( q - 1 ) 2 × 4 q ] } K ( m , p ; i ) K ( n , q , j ) ,
R ( i , j ) = m n p q { ( 2 m + n + 2 ) × [ 1 × 3 ( p + q - 1 ) 2 × 4 ( p + q ) ] } K ( m , p ; i ) K ( n , q ; j ) ,
P ( i , j ) = m n { 2 m n ( m + 2 ) ( n + 2 ) ( m + n + 2 ) } × K ( m , 0 ; i ) K ( n , 0 ; j ) .
Φ = λ ( α E axial ) λ + λ τ ( β E even + γ E odd ) λ , τ ,

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