Abstract

The effects of many phase errors that are distributed in any manner over the aperture of the objective are evaluated in terms of a coefficient B that governs contrast in the image of an unresolvably small object. The object and its surround are assumed self-radiant. The corresponding formulation for the contrast coefficient is solved for n zones that have different, stepped phase errors and for n zones within which the phase error has parabolic dependence on radial distance from the center of the zone. The resulting formulas for B are so simple and the significance of B sufficiently broad that the method is advantageous for estimating the effects of phase errors which are too complex to be treated in a practical manner by other methods.

© 1964 Optical Society of America

Full Article  |  PDF Article

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Illustration of the coordinate systems (ζ,η) and (ρ,ϕ).

Fig. 2
Fig. 2

The contrast coefficient B for a single zone of constant phase error Δ1 and relative area r ¯ 1.

Fig. 3
Fig. 3

The contrast coefficient B for a single zone of maximum parabolic phase error Δ1 and relative area r ¯ 1.

Fig. 4
Fig. 4

Illustration of the aperiodic nature of the contrast coefficient B as a function of the maximum parabolic phase error Δ1.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

C = H ( 0 ) / H s ,
K = [ H ( 0 ) - H s ] / H s = C - 1.
H ( x , y ) = - - f ( x 0 , y 0 ) × U ( x - M x 0 , y - M y 0 ) 2 d x 0 d y 0 ,
U ( ζ , η ) = 0 ρ m 0 2 π P ( ρ , ϕ ) × exp [ 2 π i ρ ( ζ cos ϕ + η sin ϕ ) ] d ϕ ρ d ρ ,
M 2 H ( 0 ) = σ 0 + A i ( f 0 - 1 ) U ( 0 ) 2 ,
M 2 H s = σ 0 ,
σ 0 = π ( ρ m 2 - ρ 0 2 ) ,
U ( 0 ) = 0 ρ m 0 2 π P ( ρ , ϕ ) d ϕ ρ d ρ .
P ( ρ , ϕ ) = 0             when             0 ρ ρ 0 .
U ( 0 ) = F - 2 P ( ζ , η ) ( 1 + ζ 2 + η 2 F 2 ) - 2 d ζ d η ,
[ 1 + ( ζ 2 + η 2 ) / F 2 ] 2 = 1.
U ( 0 ) = F - 2 P ( ζ , η ) d ζ d η area of the pupil ,
ρ m = R m / F ,             ρ 0 = R 0 / F ,
P ( ζ , η ) = P a ( ζ , η ) P f ( ζ , η ) P e ( ζ , η ) ,
P a ( ζ , η ) P f ( ζ , η ) = 1 ,
P ( ζ , η ) = P e ( ζ , η ) .
C = K + 1 ,
K = A i ( f 0 - 1 ) U ( 0 ) 2 / σ 0 ,
P e ( ζ , η ) = P ( ζ , η ) = exp ( i Δ ν )
U ( 0 ) = 1 F 2 R 0 R m 0 2 π u d ϕ d u + 1 F 2 ν = 1 n [ exp ( i Δ ν ) - 1 ] d ζ d η ,
d ζ d η = π R ¯ ν 2 ,
U ( 0 ) = σ 0 { 1 + ν = 1 n [ exp ( i Δ ν ) - 1 ] R ¯ ν 2 R m 2 - R 0 2 } ,
σ 0 = π ( 1 - R 0 2 / R m 2 ) R m 2 / F 2 .
r ¯ ν = R ¯ ν 2 / ( R m 2 - R 0 2 ) .
U ( 0 ) = σ 0 { 1 + ν = 1 n r ¯ ν [ exp ( i Δ ν ) - 1 ] } .
K = σ 0 A i ( f 0 - 1 ) 1 + ν = 1 n r ¯ ν [ exp ( i Δ ν ) - 1 ] 2 .
r a = 0.61 / ρ m ,
A i = π r a 2 r i 2 / r a 2 = π N 2 ( 0.61 / ρ m ) 2 ,
K = π 2 ( 0.61 ) 2 N 2 ( f 0 - 1 ) B ,
B = ( 1 - R 0 2 R m 2 ) 1 + ν = 1 n r ¯ ν [ exp ( i Δ ν ) - 1 ] 2 .
B = ( 1 - R 0 2 R m 2 ) 1 + [ exp ( i Δ j ) - 1 ] × ν = 1 j r ¯ ν + ν = j + 1 n r ¯ ν [ exp ( i Δ ν ) - 1 ] 2 .
Δ = Δ ν + a ν u 2 ,
a ν u ν , m 2 = - Δ ν .
P ( ζ , η ) P ( u ) = exp [ i ( Δ ν + a ν u 2 ) ] .
U ( 0 ) = π R m 2 - R 0 2 F 2 + 2 π F 2 ν = 1 n 0 u ν , m { exp [ i ( Δ ν + a ν u 2 ) ] - 1 } u d u .
U ( 0 ) = π F 2 { R m 2 - R 0 2 - ν = 1 n u ν , m 2 + ν = 1 n e i Δ ν e i a ν u ν , m 2 - 1 i a ν } .
r ¯ ν = u ν , m 2 / ( R m 2 - R 0 2 ) .
U ( 0 ) = σ 0 [ 1 - ν = 1 n r ¯ ν + i ν = 1 n r ¯ ν 1 - exp ( i Δ ν ) Δ ν ] .
B = ( 1 - R 0 2 R m 2 ) | 1 - ν = 1 n r ¯ ν + i ν = 1 n r ¯ ν 1 - exp ( i Δ ν ) Δ ν | 2 .