## 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

### Cited By

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

### Figures (4)

Fig. 1

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

Fig. 2

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

Fig. 3

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

Fig. 4

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

### Equations (39)

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