Abstract

Fundamental limitations of estimating the amplitudes and phases of interference fringes at low light levels are determined by the finite number of photoevents registered in the measurement. By modeling the receiver as a spatial array of photon-counting detectors, results are obtained that permit specification of the minimum number of photoevents required for estimation of fringe parameters to a given accuracy. Both a discrete Fourier-transform estimator and an optimum joint maximum-likelihood estimator are considered. In addition, the Cramér–Rao statistical error bounds are derived, specifying the limiting performance of all unbiased estimators in terms of the collected light flux. The performance of the spatial sampling receiver is compared with that of an alternate technique for fringe-parameter estimation that uses a barred grid and temporal sampling of a moving fringe.

© 1973 Optical Society of America

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References

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  1. W. I. Beavers, Astron. J. 68, 273 (1963).
    [Crossref]
  2. J. L. Elliot, M. S. thesis, Department of Physics, Massachusetts Institute of Technology (1965).
  3. W. I. Beavers and W. D. Swift, Appl. Opt. 7, 1975 (1968).
    [Crossref] [PubMed]
  4. E. S. Kulagin, Opt. Spektrosk. 23, 839 (1967) [Opt. Spectrosc. 23, 459 (1967)].
  5. J. S. Wilczynski, J. Opt. Soc. Am. 57, 579 (1967).
  6. W. T. Rhodes, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-16 780).
  7. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 501.
  8. Reference 7, p. 515.
  9. L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), pp. 181–248.
    [Crossref]
  10. L. Mandel, Proc. Phys. Soc. Lond. 74, 233 (1959).
    [Crossref]
  11. G. D. Bergland, IEEE Spectrum 6 (7), 41 (1969).
    [Crossref]
  12. J. Walkup, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-11 685).
  13. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, and World, New York, 1967), p. 106.
  14. Reference 12, p. 157.
  15. Reference 13, p. 118.
  16. J. B. Thomas, Introduction to Statistical Communication Theory (Wiley, New York, 1969), p. 160.
  17. Reference 16, p. 167.
  18. Reference 16, p. 161.
  19. Reference 12, p. 39.
  20. D. C. Rife and G. A. Vincent, Bell Syst. Tech. J. 49, 197 (1970).
    [Crossref]
  21. Reference 12, p. 44.
  22. R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).
  23. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).
  24. Reference 12, p. 49.
  25. It is possible to maximize Γ0 with respect to the expanded parameter set (ϕ,xt, V,as). However, of the four equations generated, only three are linearly independent. The estimate V̂ obtained is identical with Eq. (37).
  26. J. W. Goodman, J. Opt. Soc. Am. 60, 506 (1970).
    [Crossref]

1970 (2)

D. C. Rife and G. A. Vincent, Bell Syst. Tech. J. 49, 197 (1970).
[Crossref]

J. W. Goodman, J. Opt. Soc. Am. 60, 506 (1970).
[Crossref]

1969 (1)

G. D. Bergland, IEEE Spectrum 6 (7), 41 (1969).
[Crossref]

1968 (1)

1967 (2)

E. S. Kulagin, Opt. Spektrosk. 23, 839 (1967) [Opt. Spectrosc. 23, 459 (1967)].

J. S. Wilczynski, J. Opt. Soc. Am. 57, 579 (1967).

1963 (1)

W. I. Beavers, Astron. J. 68, 273 (1963).
[Crossref]

1959 (1)

L. Mandel, Proc. Phys. Soc. Lond. 74, 233 (1959).
[Crossref]

Beavers, W. I.

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, and World, New York, 1967), p. 106.

Bergland, G. D.

G. D. Bergland, IEEE Spectrum 6 (7), 41 (1969).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 501.

Deutsch, R.

R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).

Elliot, J. L.

J. L. Elliot, M. S. thesis, Department of Physics, Massachusetts Institute of Technology (1965).

Goodman, J. W.

Kulagin, E. S.

E. S. Kulagin, Opt. Spektrosk. 23, 839 (1967) [Opt. Spectrosc. 23, 459 (1967)].

Mandel, L.

L. Mandel, Proc. Phys. Soc. Lond. 74, 233 (1959).
[Crossref]

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), pp. 181–248.
[Crossref]

Rhodes, W. T.

W. T. Rhodes, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-16 780).

Rife, D. C.

D. C. Rife and G. A. Vincent, Bell Syst. Tech. J. 49, 197 (1970).
[Crossref]

Swift, W. D.

Thomas, J. B.

J. B. Thomas, Introduction to Statistical Communication Theory (Wiley, New York, 1969), p. 160.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

Vincent, G. A.

D. C. Rife and G. A. Vincent, Bell Syst. Tech. J. 49, 197 (1970).
[Crossref]

Walkup, J.

J. Walkup, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-11 685).

Wilczynski, J. S.

J. S. Wilczynski, J. Opt. Soc. Am. 57, 579 (1967).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 501.

Appl. Opt. (1)

Astron. J. (1)

W. I. Beavers, Astron. J. 68, 273 (1963).
[Crossref]

Bell Syst. Tech. J. (1)

D. C. Rife and G. A. Vincent, Bell Syst. Tech. J. 49, 197 (1970).
[Crossref]

IEEE Spectrum (1)

G. D. Bergland, IEEE Spectrum 6 (7), 41 (1969).
[Crossref]

J. Opt. Soc. Am. (2)

J. S. Wilczynski, J. Opt. Soc. Am. 57, 579 (1967).

J. W. Goodman, J. Opt. Soc. Am. 60, 506 (1970).
[Crossref]

Opt. Spektrosk. (1)

E. S. Kulagin, Opt. Spektrosk. 23, 839 (1967) [Opt. Spectrosc. 23, 459 (1967)].

Proc. Phys. Soc. Lond. (1)

L. Mandel, Proc. Phys. Soc. Lond. 74, 233 (1959).
[Crossref]

Other (18)

Reference 12, p. 44.

R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

Reference 12, p. 49.

It is possible to maximize Γ0 with respect to the expanded parameter set (ϕ,xt, V,as). However, of the four equations generated, only three are linearly independent. The estimate V̂ obtained is identical with Eq. (37).

J. L. Elliot, M. S. thesis, Department of Physics, Massachusetts Institute of Technology (1965).

W. T. Rhodes, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-16 780).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 501.

Reference 7, p. 515.

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), pp. 181–248.
[Crossref]

J. Walkup, Ph.D. thesis, Department of Electrical Engineering, Stanford University (1971) (University Microfilms, Ann Arbor, Mich., order No. 72-11 685).

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, and World, New York, 1967), p. 106.

Reference 12, p. 157.

Reference 13, p. 118.

J. B. Thomas, Introduction to Statistical Communication Theory (Wiley, New York, 1969), p. 160.

Reference 16, p. 167.

Reference 16, p. 161.

Reference 12, p. 39.

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Figures (4)

F. 1
F. 1

Receiver model.

F. 2
F. 2

Phasor diagram for the m0th DFT component.

F. 3
F. 3

Mean-square error in the DFT phase estimate (gaussian approximation) vs 1 / 2 γ ( m 0 ). Also shown (dotted) is a comparison with the asymptotic approximation[γ2(m0)]−1.

F. 4
F. 4

Michelson stellar interferometer as modified for temporal-sampling fringe-parameter estimation.

Equations (53)

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I ( ξ ) = I s [ 1 + V s cos ( 2 π f 0 ξ + ϕ ) ] + I b , 0 ξ L
n ¯ ( j ) T Δ ν , j = 0 , 1 , , N 1
P n ( j ) ( k ) = { [ n ¯ ( j ) ] k k ! exp [ n ¯ ( j ) ] , k = 0 , 1 , 0 , otherwise .
n ¯ ( j ) = η ATI ( j ) h ν ¯ ,
n ¯ ( j ) = x s [ 1 + V s cos ( 2 π j m 0 N + ϕ ) ] + x b = x t [ 1 + V cos ( 2 π j m 0 N + ϕ ) ] , j = 0 , 1 , , N 1 .
N s = N x s = [ j = 0 N 1 n ¯ ( j ) ] x b = 0 .
X ( m ) = 1 N j = 0 N 1 n ( j ) exp [ i 2 π j m / N ] , m = 0 , 1 , , N 1 .
R ( m ) = 1 N j = 0 N 1 n ( j ) cos ( 2 π j m N ) ,
I ( m ) = 1 N j = 0 N 1 n ( j ) sin ( 2 π j m N ) .
X ( m ) = r ( m ) exp [ i θ ( m ) ] ,
r ( m ) | X ( m ) | = [ R 2 ( m ) + I 2 ( m ) ] 1 2
θ ( m ) arg X ( m ) = tan 1 [ I ( m ) R ( m ) ] .
R ¯ ( m 0 ) = 1 N j = 0 N 1 n ¯ ( j ) cos ( 2 π j m 0 N ) = x s V s 2 cos ϕ ,
( m 0 ) = 1 N j = 0 N 1 n ¯ ( j ) sin ( 2 π j m 0 N ) = x s V s 2 sin ϕ ,
σ 2 [ R ( m 0 ) ] = σ 2 [ I ( m 0 ) ] = 1 N 2 j = 0 N 1 n ¯ ( j ) cos 2 ( 2 π j m 0 N ) = x s + x b 2 N σ 2 ( m 0 ) ,
cov [ R ( m 0 ) , I ( m 0 ) ] = 1 N 2 j = 0 N 1 n ¯ ( j ) cos ( 2 π j m 0 N ) sin ( 2 π j m 0 N ) = 0 .
p Δ ϕ ( Δ ϕ ) = 1 2 π exp [ 1 2 γ 2 ( m 0 ) ] + ( 2 π ) 1 2 γ ( m 0 ) cos Δ ϕ × Θ [ 1 2 γ ( m 0 ) cos Δ ϕ ] exp [ 1 2 γ 2 ( m 0 ) sin 2 Δ ϕ ] .
Θ ( b ) = ( 2 π ) 1 2 b exp ( y 2 2 ) d y ,
γ ( m 0 ) [ R ¯ 2 ( m 0 ) + 2 ( m 0 ) σ 2 ( m 0 ) ] 1 2 = V s [ N s / 2 ] 1 2 [ 1 + x b x s ] 1 2 .
Δ ϕ 2 1 γ 2 ( m 0 ) = 2 ( 1 + x b / x s ) V s 2 N s .
N s = 2 ( 1 + x b / x s ) V s 2 Δ ϕ 2
â s ( n ; DFT ) 2 r ( m 0 ) = 2 [ R 2 ( m 0 ) + I 2 ( m 0 ) ] 1 2 .
p ( a s ) = N â s x s + x b exp { N [ ( â s ) 2 + ( x s V s ) 2 ] 4 ( x s + x b ) } × I 0 [ V s N s â s 2 ( x s + x b ) ]
[ â s ( n ; DFT ) a s ] 2 2 ( x s + x b ) N .
R = { â s ( n ; DFT ) 2 | â s ( n ; DFT ) a s | 2 } 1 2 { N ( x s V s ) 2 2 ( x s + x b ) } 1 2 = γ ( m 0 ) .
N s = 2 R 2 ( 1 + x b / x s ) V s 2
x ̂ t ( n ; DFT ) = 1 N j = 0 N 1 n ( j ) = X ( 0 )
V ̂ = â s x ̂ t .
P ( n α ) = j = 0 N 1 P ( n j α ) = j = 0 N 1 [ n ¯ j ( α ) ] n j n j ! exp [ n ¯ j ( α ) ] .
Λ ( n α ) ln P ( n α ) = j = 0 N 1 [ ln n j ! n ¯ j ( α ) + n j ln n ¯ j ( α ) ] .
Λ 0 ( n α ) = j = 0 N 1 [ n ¯ j ( α ) + n j ln n ¯ j ( α ) ] ,
n ¯ j ( α ) = x t [ 1 + V cos ( 2 π j m 0 N + ϕ ) ] , j = 0 , 1 , , N 1
Λ 0 ( n α ) α i = 0 , i = 1 , 2 , 3 ,
ϕ ̂ ( n ; JML ) = tan 1 { j = 0 N 1 n j sin ( 2 π j m 0 N ) / j = 0 N 1 n j cos ( 2 π j m 0 N ) } ,
x ̂ t ( n ; JML ) = 1 N j = 0 N 1 n j ,
â s ( n ; JML ) = x ̂ t ( n ; JML ) × { j = 0 N 1 n j cos [ 2 π j m 0 N + ϕ ̂ ( n ; JML ) ] / j = 0 N 1 n j cos 2 [ 2 π j m 0 N + ϕ ̂ ( n ; JML ) ] } .
V ̂ ( n ; JML ) = â s x ̂ t = j = 0 N 1 n j cos [ 2 π j m 0 N + ϕ ̂ ( n ; JML ) ] / j = 0 N 1 n j cos 2 [ 2 π j m 0 N + ϕ ̂ ( n ; JML ) ] .
â s ( n ; JML ) = â s ( n ; DFT ) 1 + [ C ( n , ϕ ̂ ) / x ̂ t ( n ; JML ) ] ,
C ( n , ϕ ̂ ) 1 N j = 0 N 1 n j cos [ 4 π j m 0 N + 2 ϕ ̂ ( n ; JML ) ] ,
C ( n , ϕ ̂ ) = 0 , σ 2 [ C ( n , ϕ ̂ ) ] x t 2 N , x ̂ t = x t , σ 2 [ x ̂ t ] = x t N .
E [ α i ( n ) α i ] 2 J i i = CRB ( α i ) , i = 1 , 2 , , k
J i j = E [ 2 ln P ( n α ) α i α j ] , i , j = 1 , 2 , , k .
E [ V ̂ ( n ) V ] 2 i , j = 1 3 V α i J i j V α j .
CRB ( ϕ ) = 2 N x t V 2 , γ ( m 0 ) 1
CRB ( a s ) 2 x t N ,
CRB ( x t ) = x t N ,
CRB ( V ) = 2 N x t .
τ ( ξ ) = 1 2 [ 1 + cos ( 2 π f 0 ξ ) ] , L 2 ξ L 2 .
I ( ξ , t ) = I [ 1 + V cos ( 2 π f 0 ξ + 2 π Δ ν s t + ϕ ) ] ,
L / 2 L / 2 I d ( ξ , t ) d ξ = L I 2 [ 1 + V 2 cos ( 2 π Δ ν s t + ϕ ) ] .
n ¯ ( j ) = x t [ 1 + V 2 cos ( 2 π j m 0 N + ϕ ) ] , j = 0 , 1 , , N 1
x t = η I G Δ t 2 h ν ¯ .
γ T ( m 0 ) = V 2 ( N x t 2 ) 1 2 = 1 2 γ ( m 0 ) .