Abstract

An exact expression is obtained that evaluates the error associated with angular-position measurement of an incoherently illuminated object when a quadrant detector is used in the image plane of an optical system. The accuracy of such a measurement is inversely proportional to the signal-to-noise voltage ratio associated with the four quadrants summed to act as a single detector. In the particular case of a circular object, the rms angular measurement error is given by the expression σθ = π[(3/16)2 + (n/8)2]1/2(λ/D)/SNRv, where n is the angular subtense of the object divided by the diffraction angle λ/D, SNRv is the signal-to-noise voltage ratio, λ is the wavelength of the light used, and D is the diameter of the limiting aperture in the optical system under consideration.

© 1982 Optical Society of America

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References

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  1. G. B. Parent and B. J. Thompson, in Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), Article 4.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968, p. 114, Eq. (6–24).
  3. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 486, Eq. (11.4.16).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 486, Eq. (11.4.16).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968, p. 114, Eq. (6–24).

Parent, G. B.

G. B. Parent and B. J. Thompson, in Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), Article 4.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 486, Eq. (11.4.16).

Thompson, B. J.

G. B. Parent and B. J. Thompson, in Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), Article 4.

Other (3)

G. B. Parent and B. J. Thompson, in Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), Article 4.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968, p. 114, Eq. (6–24).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 486, Eq. (11.4.16).

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Figures (2)

Fig. 1
Fig. 1

Quadrant detector in the object plane of an optical system. This array consists of four single detectors labeled I through IV with their edges parallel to the x and y axes as shown. A determination of the x displacement is made by making the following calculation: Form the sum of the signals from detectors I and IV and the sum from detectors II and III and then take the difference of these two sums. The x displacement is proportional to this difference divided by the sum of the signals from all four detectors.

Fig. 2
Fig. 2

Position-error constant as a function of the normalized object diameter. The normalized object diameter is the ratio of the angular subtense of the target object disk to the diffraction angle, λ/D, associated with the aperture of the measurement system. The position-error constant relates the rms one-axis position-measurement error, measured in units of λ/D, to the signal-to-noise voltage ratio. The position-error constant divided by the signal-to-noise ratio is equal to the rms one-axis position-measurement error divided by λ/D. The signal-to-noise voltage ratio corresponds to the target-measurement signal obtained by summing the signals from all four of the detectors making up the quadrant shown in Fig. 1. The bandwidth associated with this signal-to-noise ratio is the same as that associated with the position-measurement process. The dashed curve corresponds to the approximate analytic expression given in Eq. (40), and it obviously compares favorably with the exact numerical evaluation results shown by the solid curve, obtained from Eq. (36).

Equations (50)

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A + ( x 0 ) + A - ( x 0 ) = A + ( x 0 ) + A - ( x 0 ) + N + + N - ,
σ N 2 = [ A + ( x 0 ) + A - ( x 0 ) - A + ( x 0 ) + A - ( x 0 ) ] 2 .
N + = 0 ,
N - = 0 ,
N + 2 = σ 2 ,
N - 2 = σ 2 ,
N + N - = 0.
σ N 2 = 2 σ 2 .
SNR v = A + ( x 0 ) + A - ( x 0 ) σ N - 1 .
A + ( x 0 ) - A - ( x 0 ) = A + ( x 0 ) - A - ( x 0 ) + N + - N - ,
[ A + ( x 0 ) - A - ( x 0 ) - A + ( x 0 ) - A - ( x 0 ) ] 2 = 2 σ 2 .
{ [ A + ( x 0 ) - A - ( x 0 ) - A + ( x 0 ) - A - ( x 0 ) ] 2 } 1 / 2 = A + ( x 0 ) + A - ( x 0 ) SNR v - 1 .
A + ( x 0 ) + A - ( x 0 ) = d x d y I ( x , y ) ,
A + ( x 0 ) - A - ( x 0 ) = d y 0 d x I ( x , y ) - d y - 0 d x I ( x , y ) .
I ( x , y ) = 1 λ 4 z 1 2 z 2 2 d x 1 d y 1 O ( x 1 , y 1 ) | P ˜ ( x 1 + x 0 λ z 1 + x λ z 2 , y 1 + y 0 λ z 1 + y λ z 2 ) | 2 .
P ˜ ( α , β ) = d x d y P ( x , y ) exp [ 2 π i ( x α + y β ) ] ,
P ( x , y ) = { 1 , inside entrance pupil 0 , outside entrance pupil .
d α d β P ˜ ( α , β ) 2 = d x d y P ( x , y ) 2 ,
A + ( x 0 ) + A - ( x 0 ) = a λ 2 z 1 2 d x 1 d y 1 O ( x 1 , y 1 ) ,
α = d x d y P ( x , y ) .
α 1 = x 1 λ z 1 ,             β 1 = y 1 λ z 1 ,
α 0 = x 0 λ z 1 ,             β 1 = y 0 λ z 1 ,
α = x λ z 2 ,             β = y λ z 2 ,
O 0 ( α 1 , β 1 ) = O ( λ z 1 α 1 , λ z 1 β 1 ) .
A + ( x 0 ) - A - ( x 0 ) = d β 0 d α d α 1 d β 1 O 0 ( α 1 , β 1 ) × P ˜ ( α 1 + α 0 + α , β 1 + β 0 + β ) 2 - d β 0 d α d α 1 d β 1 O 0 ( α 1 , β 1 ) × P ˜ ( α 1 + α 0 + α , β 1 + β 0 + β ) 2 .
O 0 ( α 1 , β 1 ) P ˜ ( α 1 + α 0 + α , β 1 + β 0 + β ) 2 = d f α d f β exp [ 2 π i ( α 1 f α + β 1 f β ) ] × d f α d f β I ˜ ( f α - f α , f β - f β ) T ( f α , f β ) × exp { 2 π i [ ( α 0 + α ) f α + ( β 0 + β ) f β ] } ,
I ˜ ( f α , f β ) = d α d β O 0 ( α , β ) exp [ - 2 π i ( α f α + β f β ) ] ,
T ( f α , f β ) = d α d β P ˜ ( α , β ) 2 exp [ - 2 π i ( α f α + β f β ) ] .
A + ( x 0 ) - A - ( x 0 ) = d β 0 d α d α 1 d β 1 × d f α d f β exp [ 2 π i ( α 1 f α + β 1 f β ) ] × d f α d f β I ˜ ( f α - f α , f β - f β ) T ( f α , f β ) × exp { 2 π i [ ( α 0 + α ) f α + ( β 0 + β ) f β ] } - d β - 0 d α d α 1 d β 1 × d f α d f β exp [ 2 π i ( α 1 f α + β 1 f β ) ] × d f α d f β I ˜ ( f α - f α , f β - f β ) T ( f α , f β ) × exp { 2 π i [ ( α 0 + α ) f α + ( β 0 + β ) f α ] } .
A + ( x 0 ) - A - ( x 0 ) = 0 d α F ( α + α 0 ) - - d α F ( α + α 0 ) ,
F ( α ) = d f α I ˜ ( f α , 0 ) T ( f α , 0 ) exp ( 2 π i α f α ) .
A + ( x 0 ) - A - ( x 0 ) = - 2 x 0 λ z 1 F ( 0 ) .
θ 0 = x 0 z 1 .
θ = θ 0 + n θ ,
n θ = 0 ,
( n θ ) 2 = σ θ 2 ,
θ = θ 0 .
A + ( x 0 ) - A - ( x 0 ) = - 2 ( θ 0 + n θ ) λ F ( 0 ) .
[ A + ( x 0 ) - A - ( x 0 ) - A + ( x 0 ) - A - ( x 0 ) ] 2 = 4 σ θ 2 λ 2 [ F ( 0 ) ] 2 .
σ θ = a 2 SNR v F ( 0 ) λ z 1 2 d x 1 d y 1 O ( x 1 , y 1 ) .
O ( x , y ) = { I 0 , if ( x 2 + y 2 ) 1 / 2 ½ b 0 , otherwise ,
I ˜ ( f α , f β ) = π b 2 I 0 4 λ 2 z 1 2 2 J 1 [ k b ( f α 2 + f β 2 ) 1 / 2 2 z 1 ] [ k b ( f α 2 + f β 2 ) 1 / 2 2 z 1 ] ,
T ( f α , f β ) = { { cos - 1 [ ( f α 2 + f β 2 ) 1 / 2 D ] - ( f α 2 + f β 2 ) 1 / 2 D ( 1 - f α 2 + f β 2 D 2 ) 1 / 2 } D 2 2 , if f α 2 + f β 2 D 2 0 , otherwise .
σ θ = H ( n ) λ / D SNR v ,
H ( n ) = π 8 { 0 1 d x 2 J 1 ( π n x ) ( π n x ) [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] - 1 } .
H ( n ) = 3 π 16             when n 1.
H ( n ) = π n 8 [ 0 π n d α J 1 ( α ) α ] - 1 .
H ( n ) = n π 8             when n 1.
H ( n ) = π [ ( 3 16 ) 2 + ( n 8 ) 2 ] 1 / 2 ,
σ θ = π [ ( 3 / 16 ) 2 + ( n / 8 ) 2 ] 1 / 2 ( λ / D ) SNR v .