Abstract

Expressions are derived for the maximum precision with which an optical instrument can determine the direction of a light source and the orientation of a source about an axis parallel to the line of sight.

© 1964 Optical Society of America

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References

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  1. Derived from ΔpΔq= h/4π, where Δp and Δq are the rms errors. See R. W. Ditchburn, Light (Interscience Publishers Inc., New York, 1959), p. 612.
  2. The answers are, respectively, Δθσ=312/2k(n)12 and Δθσ=k/n12.
  3. W. Gröbner and N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), 3rd ed., Vol. II, p. 202, Eq. (5).

Ditchburn, R. W.

Derived from ΔpΔq= h/4π, where Δp and Δq are the rms errors. See R. W. Ditchburn, Light (Interscience Publishers Inc., New York, 1959), p. 612.

Gröbner, W.

W. Gröbner and N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), 3rd ed., Vol. II, p. 202, Eq. (5).

Hofreiter, N.

W. Gröbner and N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), 3rd ed., Vol. II, p. 202, Eq. (5).

Other (3)

Derived from ΔpΔq= h/4π, where Δp and Δq are the rms errors. See R. W. Ditchburn, Light (Interscience Publishers Inc., New York, 1959), p. 612.

The answers are, respectively, Δθσ=312/2k(n)12 and Δθσ=k/n12.

W. Gröbner and N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), 3rd ed., Vol. II, p. 202, Eq. (5).

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Figures (7)

F. 1
F. 1

Illustration for derivation of Eq. (8).

F. 2
F. 2

Illustration for derivation of Eq. (11).

F. 3
F. 3

Illustration for derivation of Eqs. (16)(18)

F. 4
F. 4

A simple detection system.

F. 5
F. 5

Twist measurement by use of polarized light.

F. 6
F. 6

Twist measurement without use of polarized light.

F. 7
F. 7

Illustration for derivation of Eq. (39).

Tables (1)

Tables Icon

Table I The value of (A/M) and (1/M) for various aperture configurations.

Equations (65)

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Δ θ ( m + n ) = Δ θ 1 / ( m + n ) 1 2 = [ ( Δ θ m ) 2 + ( Δ θ n ) 2 ] 1 2 .
( Δ θ σ ) 2 = 1 r ( Δ θ i ) 2 ,
I = B f ( θ ) .
d I / d θ = B f ( θ ) ,
d S = Δ I d θ = ( Δ I / Δ θ ) Δ θ d θ = ( d I / d θ ) Δ θ d θ .
( d N ) 2 av = I d θ .
( d S / d N ) 2 = ( d S ) 2 av / ( d N ) 2 av = [ ( d I / d θ ) Δ θ d θ ] 2 / I d θ .
( Δ θ min ) 2 = I / [ ( d I / d θ ) 2 d θ ] .
( Δ θ σ ) 2 = ( Δ θ min ) 2 .
( Δ θ σ ) 2 = B [ f ( θ ) ] 2 d θ f ( θ ) .
B = n / f ( θ ) d θ .
( Δ θ σ ) 2 = f ( θ ) d θ / n [ f ( θ ) ] 2 d θ f ( θ ) .
I / B = f ( θ ) = 1 + cos ( 2 π D θ / λ ) ,
f ( θ ) = ( 2 π D / λ ) sin ( 2 π D θ / λ ) .
Δ θ σ = λ / 2 π D ( n ) 1 2 .
Δ E Δ t = h / 4 π ,
Δ E = h ν Δ n .
ν = Δ ϕ / 2 π Δ t .
Δ ϕ = 1 2 n 1 2 .
Δ θ σ = ( λ / 4 π D ) [ ( 1 / n a ) + ( 1 / n b ) ] 1 2 .
Δ θ σ = ( λ / 4 π ) ( n a D a 2 + n b D b 2 ) 1 2 .
Δ θ σ = λ / 4 π ( M p ) 1 2 .
Δ θ σ = ( λ / 4 π ) ( A / n M ) 1 2 .
( Δ θ σ ) 2 = 1 k ( Δ θ σ i ) 2 ,
Δ θ σ = ( λ / 4 π ) ( 1 k M p i ) 1 2 .
Δ θ σ = ( λ / 4 π ) ( n A k M i ) 1 2 .
Δ θ σ = λ / π D ( 5 n ) 1 2 .
Δ θ σ = λ / π D n 1 2 .
Δ θ rms = ( λ / π D ) ( Δ f / 2 n ) 1 2 .
( Signal ) o = Δ n o = 2 n sin α cos α Δ α i , ( Signal ) e = Δ n e = 2 n sin α cos α Δ α i .
( Noise ) o = n o 1 2 = n 1 2 cos α , ( Noise ) e = n e 1 2 = n 1 2 sin α .
( S / N ) o = 2 ( n ) 1 2 sin α Δ α i , ( S / N ) e = 2 ( n ) 1 2 cos α Δ α i .
( Δ α min ) o = 1 / [ 2 ( n ) 1 2 sin α ] , ( Δ α min ) e = 1 / [ 2 ( n ) 1 2 cos α ] .
( Δ α ) 2 = o , e ( Δ α min ) j 2 ,
Δ α = 1 2 n 1 2 ,
Δ ϕ = Δ α ,
Δ β = 2 δ / 1.22 π γ ( n ) 1 2 .
Δ β / Δ α δ / γ .
Δ θ σ γ Δ B δ Δ α .
( A 1 + A 2 ) 2 A 1 A 2 D 2
A 1 + A 2 A 1 A 2 D 2
12 D 2
12 H D 3
16 D 2
64 / π H D 3
16 D 2 2 + D 1 2
64 / π D 2 4 D 1 4
n 1 a D 1 = n s D s , n 1 b D 1 = n ( s 1 ) a D ( s 1 ) , n 2 a D 2 = n ( s 1 ) b D ( s 1 ) etc . ,
Δ θ i i = [ λ / 4 π ( D i + D i ) ] [ ( 1 / n i ) + ( 1 / n i ) ] 1 2 ,
( λ 4 π ) 2 ( Δ θ σ ) 2 = 1 r [ ( D i + D i ) 2 / ( 1 n i + 1 n i ) ] ,
( λ 4 π ) 2 ( Δ θ σ ) 2 = 1 r ( n i D i 2 + n i D i 2 ) ,
( λ / 4 π ) 2 ( Δ θ σ ) 2 = M p ,
I = B J 1 2 ( x ) / x 2 ,
x = ( π D / λ F ) r ,
n = 2 π B ( λ F π D ) 2 0 J 1 2 ( x ) d x x ,
B = ( n / π ) ( π D / λ F ) 2 .
I = ( n / π ) ( π D / λ F ) 2 [ J 1 2 ( x ) / x 2 ] .
d S = Δ I d A = Δ I Δ r Δ r Δ p Δ p d A = d I d r d r d p Δ p d A .
Δ p min = [ ( d I / d r ) ( d r / d p ) ( d A / I ) 1 2 ] 1 .
( Δ p σ ) 2 = i ( Δ p min ) i 2 = 0 2 π 0 ( d I d r ) 2 r d r cos 2 ϕ d ϕ I ,
d I d r = 2 n π ( π D λ F ) 3 J 1 ( x ) J 2 ( x ) x 2 .
0 2 π cos 2 ϕ d ϕ = π ,
( Δ p σ ) 2 = 4 n ( π D λ F ) 2 0 J 2 2 ( x ) d x x .
Δ p σ / F = Δ θ σ = λ / π D ( n ) 1 2 .
Δ θ σ = λ / 4 π ( M p ) 1 2 ,