Abstract

The informational aspects of optical design have attracted increasing attention in the last few years [see for example A. Blanc-Lapierre (1953), D. Gabor (1952)]. The closely related topic of the Fourier treatment of optical imaging has been discussed by P.-M. Dufflieux (1946) and by Elias, Grey, and Robinson (1952) among others. The present paper gives an account of analytical techniques which provide a basis for a discussion of the problem of maximizing the information content in images formed by high-quality optical systems, by means of aberration balancing under prescribed constraints on the design. A derivation is given of the principal results needed for this purpose.

© 1955 Optical Society of America

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  1. C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948). Reference should also be made to D. Gabor, (1952), to Elias, Grey, and Robinson, J. Optic. Soc. Am. 42, 127 (1952), and to A. Blana-Lapierre, McGill Symposium on Microwave Optics II, No.  46 (1953).
    [CrossRef]
  2. The imaginary part of f(x, y) is needed because, in the isoplanatism patch A, the function a(ξ, η; x, y) takes (with harmlessly small error) the more special form aA(ξ, η) times a positive function of (x, y).
  3. Many of the results in Secs. X–XII were first formulated by P.-M. Duffieux, on whose ideas these sections are largely based. An account of these ideas is given in his book L’Intégrale de Fourier et ses applications à l’Optique (Privately printed, Rennes, 1946).
  4. Sinct stands for (sinπt)/πt; |A| for the area of A.
  5. A fuller discussion of these two cases will be found in Sec. 3.2 of the writer’s joint paper with P. B. Fellgett, Trans. Roy. Soc. (London) A247, 369 (1955).
    [CrossRef]
  6. See Sec. 3.22 of Fellgett and Linfoot.5
  7. Shannon,1 p. 53. The statistical independence of the νpq in the entropy-maximizing distribution is a consequence of the isoplanatism patch A being an area which maps on to a rectangle x0−a<x<x0+a, y0−b<y<y0+b in the (x, y)-plane.
  8. Compare Shannon,1 Secs. 12, 24.

1955 (1)

A fuller discussion of these two cases will be found in Sec. 3.2 of the writer’s joint paper with P. B. Fellgett, Trans. Roy. Soc. (London) A247, 369 (1955).
[CrossRef]

1948 (1)

C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948). Reference should also be made to D. Gabor, (1952), to Elias, Grey, and Robinson, J. Optic. Soc. Am. 42, 127 (1952), and to A. Blana-Lapierre, McGill Symposium on Microwave Optics II, No.  46 (1953).
[CrossRef]

Duffieux, P.-M.

Many of the results in Secs. X–XII were first formulated by P.-M. Duffieux, on whose ideas these sections are largely based. An account of these ideas is given in his book L’Intégrale de Fourier et ses applications à l’Optique (Privately printed, Rennes, 1946).

Fellgett, P. B.

A fuller discussion of these two cases will be found in Sec. 3.2 of the writer’s joint paper with P. B. Fellgett, Trans. Roy. Soc. (London) A247, 369 (1955).
[CrossRef]

Shannon, C. E.

C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948). Reference should also be made to D. Gabor, (1952), to Elias, Grey, and Robinson, J. Optic. Soc. Am. 42, 127 (1952), and to A. Blana-Lapierre, McGill Symposium on Microwave Optics II, No.  46 (1953).
[CrossRef]

Bell System Tech. J. (1)

C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948). Reference should also be made to D. Gabor, (1952), to Elias, Grey, and Robinson, J. Optic. Soc. Am. 42, 127 (1952), and to A. Blana-Lapierre, McGill Symposium on Microwave Optics II, No.  46 (1953).
[CrossRef]

Trans. Roy. Soc. (London) (1)

A fuller discussion of these two cases will be found in Sec. 3.2 of the writer’s joint paper with P. B. Fellgett, Trans. Roy. Soc. (London) A247, 369 (1955).
[CrossRef]

Other (6)

See Sec. 3.22 of Fellgett and Linfoot.5

Shannon,1 p. 53. The statistical independence of the νpq in the entropy-maximizing distribution is a consequence of the isoplanatism patch A being an area which maps on to a rectangle x0−a<x<x0+a, y0−b<y<y0+b in the (x, y)-plane.

Compare Shannon,1 Secs. 12, 24.

The imaginary part of f(x, y) is needed because, in the isoplanatism patch A, the function a(ξ, η; x, y) takes (with harmlessly small error) the more special form aA(ξ, η) times a positive function of (x, y).

Many of the results in Secs. X–XII were first formulated by P.-M. Duffieux, on whose ideas these sections are largely based. An account of these ideas is given in his book L’Intégrale de Fourier et ses applications à l’Optique (Privately printed, Rennes, 1946).

Sinct stands for (sinπt)/πt; |A| for the area of A.

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Figures (2)

Fig. 1
Fig. 1

(a) Optical system. (b) Reference sphere MP in exit-pupil aperture.

Fig. 2
Fig. 2

The set A × [ A - ( u , v ) ] in the frequency plane.

Equations (94)

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i = 1 n p i log 1 / p i .
log N = i p i log 1 / p i .
A = [ i = - exp { - ( i - i 0 ) 2 / 2 σ 2 } ] - 1
log N = i = - p i log ( 1 / p i ) = i = - A exp { - ( i - i 0 ) 2 / 2 σ 2 } × log [ ( 1 / A ) exp { - ( i - i 0 ) 2 / 2 σ 2 } ] = log ( 1 / A ) + A i = - ( i - i 0 ) 2 2 σ 2 × exp { - ( i - i 0 ) 2 / 2 σ 2 } log { ( 2 π e ) 1 2 σ }
σ ( x , y ) = p , q α p q exp { 2 π i [ p ( x - x 0 ) 2 a + q ( y - y 0 ) 2 b ] }
S ( ξ , η ; x , y ) = e ( ξ , η ; x , y ) + i λ 2 π log { ( 1 - ξ 2 - η 2 ) - 1 2 a ( ξ , η ; x , y ) } ,
d ( x , y ; x , y ) = ξ 2 + η 2 < sin 2 α exp { - i k S ( ξ , η ; x , y ) } × e i k O O [ ξ ( x - x ) + η ( y - y ) ] · d ξ d η
λ u = O O ξ ,             λ v = O O η ,
E ( u , v ; x , y ) = exp { - i k S ( ξ , η ; x , y ) }             ( u , v ) in A = 0             ( u , v ) not in A .
d ( x , y ; x , y ) = b - E ( u , v ; x , y ) × e 2 π i [ u ( x - x ) + v ( y - y ) ] d u d v = b F * [ E ] ,
- w ( x , y ; x , y ) d x d y = 1 w ( x , y ; x , y ) = d ( x , y ; x , y ) 2 × const .
w ( x , y ; x , y ) = d ( x , y ; x , y ) 2 .
E ( u , v ; x , y ) = e - i k f ( x , y ) E A ( u , v ) ,
E A ( u , v ) = exp { - i k S A ( ξ , η ) }             ( u , v ) in A = 0             ( u , v ) not in A ,
d ( x , y ; x , y ) = e - i k f ( x , y ) g A ( x - x , y - y ) ,
g A ( x , y ) = b - E A ( u , v ) e 2 π i ( u x + v y ) d u d v = b F * [ E A ] .
w ( x , y ; x , y ) = w A ( x - x , y - y ) ,
w A ( x , y ) = g A ( x , y ) 2 .
I ( x , y ) = A σ ( x , y ) w ( x - x , y - y ) d x d y = - σ ( x , y ) w ( x - x , y - y ) d x d y .
( u , v ) = - σ ( x , y ) e - 2 π i ( u x + v y ) d x d y
τ A ( u , v ) = - w A ( x , y ) e - 2 π i ( u x + v y ) d x d y ,
- I ( x , y ) e - 2 π i ( u x + v y ) d x d y = ( u , v ) τ A ( u , v )
I ( x , y ) = - ( u , v ) τ A ( u , v ) e 2 π i ( u x + v y ) d u d v .
σ ( x , y ) = - ( u , v ) e 2 π i ( u x + v y ) d u d v ,
τ A ( u , v ) τ A ( 0 , 0 ) = - w A ( x , y ) d x d y = - w A ( x - x , y - y ) d x d y
= - w ( x , y ; x , y ) d x d y = 1 ,
( u , v ) = F [ σ ] ,             σ ( x , y ) = F * [ ] ,
τ A ( u , v ) = F [ w A ] ,             w A ( x , y ) = F * ( τ A ] ,
( u , v ) τ A ( u , v ) = F [ I ] ,             I ( x , y ) = F * ( τ A ] ,
FF * = 1.
- h ( u , v ) h * ( u + u , v + v ) d u d v ,
F [ C [ h ] ] = F [ h ] 2
τ A ( u , v ) = C [ E A ] u , v C [ E A ] 0 , 0
g A ( x , y ) = F * [ E A ] x , y ( C [ E A ] 0 , 0 ) - 1 2 .
E A ( u , v ) = F [ g A ] u , v ( C [ E A ] 0 , 0 ) 1 2 ,
S ( ξ , η ; x , y ) = S A ( ξ , η ) + f ( x , y ) = e A ( ξ , η ) - i λ 2 π log { ( 1 - ξ 2 - η 2 ) 1 2 a ( ξ , η ) } + f ( x , y ) .
A × [ A - ( u , v ) ] 0
I ( x , y ) = F * [ τ A F [ σ ] ] ,
A ( u , v ) = - e - 2 π i ( u x + v y ) σ ( x , y ) d x d y = A e - 2 π i ( u v + x y ) σ ( x , y ) d x d y = e - 2 π i ( u x 0 + v y 0 ) p q α p q A · exp { 2 π i [ ( p 2 a - u ) ( x - x 0 ) + ( q 2 b - v ) ( y - y 0 ) ] } · d x d y = A e - 2 π i ( u x 0 + v y 0 ) p q α p q sinc ( 2 a u - p ) × sinc ( 2 b v - q ) ,
A - 1 exp { 2 π i ( p x 0 2 a + q y 0 2 b ) }
α p q = A - 1 exp { 2 π i ( u p x 0 + v q y 0 ) } p q .
σ ( x , y ) = Q p q in F α p q exp { 2 π i ( p x 2 a + q y 2 b ) }             ( x , y ) in A = 0             ( x , y ) not in A ,
ɛ ( u , v ) = F [ σ ] Q p q in F A α p q sinc ( 2 a u - p ) sinc ( 2 b v - q )
I 2 ( x , y ) = I 1 ( x , y ) + n 2 ( x , y ) = F * [ τ A τ 1 F [ σ 0 + n 0 ] ] + n 2 ( x , y ) .
( p 0 ) i = t i - 1 t i p 0 ( t ) d t ,             ( p 1 ) i = t i - 1 t i p 1 ( t ) d t ,
- i ( p 0 ) i log ( p 0 ) i
- i ( p 1 ) i log ( p 1 ) i
- i ( p 0 ) i log ( p 0 ) i + i ( p 1 ) i log ( p 1 ) i = - i δ p 0 ( ζ i ) log [ δ p 0 ( ζ i ) + i δ p 1 ( ζ i ) log ( δ p 1 ( ζ i ) ] ,
= - δ i p 0 ( ζ i ) log p 0 ( ζ i ) + δ i p 1 ( ζ i ) log p 1 ( ζ i ) .
- p 0 ( t ) log p 0 ( t ) d t + p 1 ( t ) log p 1 ( t ) d t
H = p ( t ) log 1 p ( t ) d t = - p ( t ) log p ( t ) d t
H 0 - H 1 = - p 0 ( t ) log p 0 ( t ) d t + p 1 ( t ) log p 1 ( t ) d t ,
B = ( 1 A A ( σ ( x , y ) ) 2 d x d y ) 1 2
κ A ( x , y ) = 1 ( x , y ) in A = 0 ( x , y ) not in A .
A ( n 2 ( x , y ) ) 2 d x d y
τ 1 ( u , v ) = F [ w 1 ]
ν p q 2 = φ p q             for Q p q in F ,
ν 2 ( u , v ) = F [ n 2 ] ,             ν p q = ν 2 ( u p , v q ) ,             φ p q = φ ( u p , v q )
κ F ( u , v ) = 1 ( u , v ) in F = 0 ( u , v ) not in F .
F * [ τ A τ 1 F [ σ ] ] + n 2 0
p ( ν ) = p ( ν 00 , ν 01 , ν - 1 , 0 , ν 10 , ν p q , )             { = q 0 , Q p q i n F 1 2 π φ p q 1 2 exp { - ν p q 2 2 φ p q }             if ν p q = 0 for all Q p q F = 0 if ν p q 0 for some Q p q F .
ν p q = A exp { - 2 π i ( u p x 0 + v q y 0 ) } γ p q ,
¨ p q = ¨ ( u p , v q ) = F [ σ - B κ A ] u p , v q
ξ = P 1 2 F - 1 2 = ( A ( σ - B ) 2 d x d y ) 1 2 F - 1 2 .
p ( ¨ ) = p ( ¨ 00 , ¨ 01 , ¨ - 10 , ¨ 10 , ¨ p q , )             { = q 0 , Q p q in F 1 2 π ξ p q exp { - ¨ p q 2 2 ξ p q 2 }             if ¨ p q = 0 for all Q p q F = 0 if ¨ p q 0 for some Q p q F
¨ p q = A exp { - 2 π i ( u p x 0 + v q y 0 ) } α ¨ p q ,
p ( η ¨ ) = p ( η ¨ 00 , η ¨ 01 , η ¨ 10 , η ¨ p q , ) = q 0 , Q p q in F ( τ A τ 1 ) p q 2 π ξ p q exp { - ( τ A τ 1 ) p q ¨ p q 2 2 ξ p q 2 }             if ¨ p q = 0 for all Q p q F = 0 if ¨ p q 0 for some Q p q F
η ¨ p q = A exp { - 2 π i ( u p x 0 + v q y 0 ) } β p q , x p q = R ( β p q ) Q p q in F , q > 0 or q = 0 , p > 0 = J ( β p q ) Q p q in F , q < 0 or q = 0 , p < 0 = β 00 ( p , q ) = ( 0 , 0 )
p ( I 1 , I 2 ) = p ( I 1 ) p I 1 ( I 2 ) .
p ( s , s ) = p ( s ) p s ( s ) .
p ( s ) = E p ( s , s ) d V ,
p ( s , s ) = p ( s ) p s ( s ) .
n = s - s = - n .
log ρ ( s ) = H = - E p ( s ) log p ( s ) d V ;
log ρ ( n ) = H s = - E p s ( s ) log p s ( s ) d V
ρ ( s ) ρ ( n ) = ( spread of { s } spread of s -cell corresponding to s ) r .
( entropy of { s } relative to E ) - ( entropy of { s } s relative to E ) .
log N = ( entropy of { s } - ( entropy of an s -cell ) ,
log N = - p ( s ) log p ( s ) d V + p ( s ) d V p s ( s ) log p s ( s ) d V = - p ( s ) log p ( s ) d V p s ( s ) d V + p ( s ) d V p s ( s ) log p s ( s ) d V = - d V d V p ( s , s ) log p ( s ) + d V d V p ( s , s ) log p s ( s ) ,
= d V d V p ( s , s ) log p ( s , s ) p ( s ) p ( s ) .
ρ ( s ) ρ ( n ) = ( spread of { s } in E spread of { s } s in E ) r
log N = ( entropy of { s } ) - ( entropy of s - noise ) .
log { ( spread of expected image-set { s } spread of s - cell corresponding to s ) r } .
log N = ( entropy of { I 2 } ) - ( entropy of { n 2 } ) .
Q p q in F , ( τ A τ 1 ) p q 0 log [ ( π e ) 1 2 φ p q 1 2 A - 1 ] ,
( , A - 1 R J ( exp { 2 π i ( u p x 0 + v q y 0 ) } × [ ( τ A τ 1 ) p q ¨ p q + ν p q ] ) , )             ( Q p q in F ) .
1 2 A - 1 ( τ A τ 1 p q 2 ξ p q 2 + φ p q ) 1 2
Q p q in L , ( τ A τ 1 ) p q 0 log [ ( π e ) 1 2 × ( τ A τ 1 p q 2 ξ p q 2 + φ p q ) 1 2 A - 1 ] .
log N = Q p q in F , ( τ A τ 1 ) p q 0 log ( 1 + τ A τ 1 p q 2 ξ p q 2 φ p q ) .
log N = A F log ( 1 + τ A τ 1 2 A 2 ν 2 2 ) d u d v .
log N = i = 1 m log N i .
log N = F d x d y F log ( 1 + τ τ 1 2 2 ν 2 2 ) d u d v ,
= ( u , v ) = F σ ( x , y ) e - 2 π i ( u x + v y ) d x d y , ν 2 = ν 2 ( u , v ) = F n 2 ( x , y ) e - 2 π i ( u x + v y ) d x d y
τ = τ ( u , v ; x , y ) = e 2 π i ( u x + v y ) C [ E ] u , v ; x , y C [ E ] 0 , 0 ; x , y ,