Abstract

Amplitude interferometry and intensity interferometry are methods for processing the complex amplitude and the intensity, respectively, of an electromagnetic field to estimate the mutual intensity of the field at two spatial locations. Whereas complex amplitude measurements allow for the direct estimation of the mutual-intensity phase, intensity measurements do not. We consider applications for which the estimation of the magnitude or the squared magnitude of the mutual intensity is adequate, and we provide fundamental limits on the estimation accuracy of any unbiased estimator of the squared magnitude of the mutual intensity from coherent (complex amplitude) or incoherent (intensity) measurements. Our analysis is performed for the high-light-level (classical-noise-limit) case and quantifies the advantages of making the more difficult coherent measurements.

© 1998 Optical Society of America

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References

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  1. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1980).
  3. R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).
  4. R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio-astronomy,” Philos. Mag. 45, 663–682 (1954).
  5. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  6. M. Elbaum, M. King, M. Greenebaum, “Laser correlography: transmission of high-resolution object signatures through the turbulent atmosphere,” (Riverside Research Institute, New York, 1974).
  7. B. E. A. Saleh, “Statistical accuracy in estimating parameters of the spatial coherence function by photon counting techniques,” J. Phys. A 6, 980–986 (1973).
    [CrossRef]
  8. C. W. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. 59, 164–175 (1969).
    [CrossRef]
  9. B. E. A. Saleh, “Minimum variance of estimators of the optical area of coherence,” Proc. IEEE 61, 250 (1973).
    [CrossRef]
  10. B. E. A. Saleh, “Lower bound on the variances of estimators of the spatial coherence function of an optical field,” J. Opt. Soc. Am. 63, 390–391 (1973).
    [CrossRef]
  11. J. Nowakowski, M. Elbaum, “Fundamental limits in estimating light pattern position,” J. Opt. Soc. Am. 73, 1744–1758 (1983).
    [CrossRef]
  12. S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991).
    [CrossRef]
  13. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).
  14. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).
  15. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  16. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1984), pp. 9–75.
  17. T. J. Schulz, “Estimation of the squared modulus of the mutual intensity from high-light-level intensity measurements,” J. Opt. Soc. Am. A 12, 1331–1337 (1995).
    [CrossRef]
  18. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theor. IT-8, 194–195 (1962).
    [CrossRef]

1995 (1)

1991 (1)

1983 (1)

1973 (3)

B. E. A. Saleh, “Lower bound on the variances of estimators of the spatial coherence function of an optical field,” J. Opt. Soc. Am. 63, 390–391 (1973).
[CrossRef]

B. E. A. Saleh, “Statistical accuracy in estimating parameters of the spatial coherence function by photon counting techniques,” J. Phys. A 6, 980–986 (1973).
[CrossRef]

B. E. A. Saleh, “Minimum variance of estimators of the optical area of coherence,” Proc. IEEE 61, 250 (1973).
[CrossRef]

1969 (1)

1965 (1)

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theor. IT-8, 194–195 (1962).
[CrossRef]

1954 (1)

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio-astronomy,” Philos. Mag. 45, 663–682 (1954).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1980).

Ebstein, S. M.

Elbaum, M.

J. Nowakowski, M. Elbaum, “Fundamental limits in estimating light pattern position,” J. Opt. Soc. Am. 73, 1744–1758 (1983).
[CrossRef]

M. Elbaum, M. King, M. Greenebaum, “Laser correlography: transmission of high-resolution object signatures through the turbulent atmosphere,” (Riverside Research Institute, New York, 1974).

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1984), pp. 9–75.

Greenebaum, M.

M. Elbaum, M. King, M. Greenebaum, “Laser correlography: transmission of high-resolution object signatures through the turbulent atmosphere,” (Riverside Research Institute, New York, 1974).

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio-astronomy,” Philos. Mag. 45, 663–682 (1954).

R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

Helstrom, C. W.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).

King, M.

M. Elbaum, M. King, M. Greenebaum, “Laser correlography: transmission of high-resolution object signatures through the turbulent atmosphere,” (Riverside Research Institute, New York, 1974).

Nowakowski, J.

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theor. IT-8, 194–195 (1962).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, “Statistical accuracy in estimating parameters of the spatial coherence function by photon counting techniques,” J. Phys. A 6, 980–986 (1973).
[CrossRef]

B. E. A. Saleh, “Minimum variance of estimators of the optical area of coherence,” Proc. IEEE 61, 250 (1973).
[CrossRef]

B. E. A. Saleh, “Lower bound on the variances of estimators of the spatial coherence function of an optical field,” J. Opt. Soc. Am. 63, 390–391 (1973).
[CrossRef]

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

Schulz, T. J.

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio-astronomy,” Philos. Mag. 45, 663–682 (1954).

Van Trees, H. L.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1980).

IRE Trans. Inf. Theor. (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theor. IT-8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

B. E. A. Saleh, “Statistical accuracy in estimating parameters of the spatial coherence function by photon counting techniques,” J. Phys. A 6, 980–986 (1973).
[CrossRef]

Philos. Mag. (1)

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio-astronomy,” Philos. Mag. 45, 663–682 (1954).

Proc. IEEE (1)

B. E. A. Saleh, “Minimum variance of estimators of the optical area of coherence,” Proc. IEEE 61, 250 (1973).
[CrossRef]

Other (8)

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

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1984), pp. 9–75.

M. Elbaum, M. King, M. Greenebaum, “Laser correlography: transmission of high-resolution object signatures through the turbulent atmosphere,” (Riverside Research Institute, New York, 1974).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1980).

R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

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Figures (4)

Fig. 1
Fig. 1

Normalized error variance versus coherence factor |μ(x, y)| for estimation of the squared magnitude of the mutual intensity |J(x, y)|2. The estimator for amplitude measurements is given by Eq. (3), whereas the estimators for intensity measurements are given by Eq. (9) for F3, Eq. (14) for F6, and Eq. (15) for F7.

Fig. 2
Fig. 2

SNR gain (in dB) versus coherence factor |μ(x, y)| for estimation of the squared magnitude of the mutual intensity |J(x, y)|2. This gain is the ratio of the SNR for estimation from amplitude measurements according to Eq. (3) to the SNR for estimation from intensity measurements according to Eq. (9) (solid curve), Eq. (14) (dashed curve), and Eq. (15) (dotted–dashed curve).

Fig. 3
Fig. 3

Cramér–Rao lower bounds on the normalized error variance versus coherence factor |μ(x, y)| for estimation of the squared magnitude of the mutual intensity |J(x, y)|2.

Fig. 4
Fig. 4

SNR gain (in dB) versus coherence factor |μ(x, y)| for estimation of the squared magnitude of the mutual intensity |J(x, y)|2. The solid curve shows the gain for the efficient processing of amplitude measurements as opposed to the efficient processing of intensity measurements. The SNR gains for the estimators discussed in Section 2 are also shown for comparison.

Tables (1)

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Table 1 Terms Used to Compute E[|Jˆ(x, y)|4]

Equations (59)

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Γ(x, y, τ)=E[u(x, t+τ)u*(y, t)],
{u(x, tk),u(y, tk)}k=1K,
Jˆ(x, y)=1Kk=1Ku(x, tk)u*(y, tk).
var[|Jˆ(x, y)|2]=1K2|J(x, y)|2[I¯2+|J(x, y)|2],
J(x, x)=J(y, y)=I¯,
μ(x, y)=defJ(x, y)/I¯,
var[|Jˆ(x, y)|2]=1K2I¯4|μ(x, y)|2[1+|μ(x, y)|2],
varnorm[|Jˆ(x, y)|2]=KI¯4var[|Jˆ(x, y)|2]=2|μ(x, y)|2[1+|μ(x, y)|2].
{I(x, tk), I(y, tk)}k=1K,
F3(x, y)=1Kk=1K[I(x, tk)-Iˆ(x)][I(y, tk)-Iˆ(y)],
Iˆ(x)=1Kk=1KI(x, tk),
Iˆ(y)=1Kk=1KI(y, tk).
var[F3(x, y)]=1K[3|J(x, y)|4+4I¯2|J(x, y)|2+I¯4],
varnorm[F3(x, y)]=KI¯4var[F3(x, y)]=3|μ(x, y)|4+4|μ(x, y)|2+1.
F6(x, y)=1Kk=1KI(x, tk)I(y, tk)-I2(x, tk)+I2(y, tk)4,
F7(x, y)=1Kk=1KI(x, tk)I(y, tk)-I2(x, tk)+I2(y, tk)2+k=1K Ik(x, tk)+Ik(y, tk)2K2,
varnorm[F6(x, y)]=(7/2)|μ(x, y)|4+3/2
varnorm[F7(x, y)]=5|μ(x, y)|4-4|μ(x, y)|2+3,
var[αˆp][I-1]pp,
[I]pq=E ln p(z; α)αp ln p(z; α)αq=-E2 ln p(z; α)αpαq,
pA(ux, uy; I¯, J)
=π-2(I¯2-|J|2)-1×exp-I¯|ux|2-I¯|uy|2+Jux*uy+J*uxuy*I¯2-|J|2,
E[ux]=E[uy]=0,
E[|ux|2]=J(x, x)=defI¯,
E[|uy|2]=J(y, y)=defI¯,
E[uxuy*]=J(x, y)=defJ.
J=|J|exp(jβ).
AMP_CRLB[|J(x, y)|2]
=2|J(x, y)|2[I¯2+|J(x, y)|2]/K,
AMP_CRLBnorm[|J(x, y)|2]
=(K/I¯4)AMP_CRLB[|J(x, y)|2]=2|μ(x, y)|2[1+|μ(x, y)|2].
pI(Ix, Iy; I¯, J)=(I¯2-|J|2)-1 exp-I¯Ix+I¯IyI¯2-|J|2I02|J|IxIyI¯2-|J|2,Ix,Iy00otherwise,
|Jˆ(x, y)|2=1Kk=1Ku(x, tk)u*(y, tk)2,=1K2k=1Kk=1Ku(x, tk)u*(y, tk)×u(y, tk)u*(x, tk),
E[|Jˆ(x, y)|2]=1K2k=1Kk=1KE[u(x, tk)u*(y, tk)×u(y, tk)u*(x, tk)],
E[|Jˆ(x, y)|2]=1K2k=1Kk=1K|J(x, y)|2+I¯2δk-k=|J(x, y)|2+I¯2K,
E[|Jˆ(x, y)|4]=1K4k=1Kk=1Km=1Km=1K×E[u(x, tk)u*(y, tk)u(y, tk)×u*(x, tk)u(x, tm)u*(y, tm)×u(y, tm)u*(x, tm)].
E[|Jˆ(x, y)|4]=|J(x, y)|4+(1/K)[2|J(x, y)|4+4I¯2|J(x, y)|2],
var[|J(x, y)|4]=E[|J(x, y)|4]-{E[|J(x, y)|2]}2=|J(x, y)|4+1K[2|J(x, y)|4+4I¯2|J(x, y)|2]-|J(x, y)|4-2|J(x, y)|2I¯2K-I¯4K21K2|J(x, y)|2[I¯2+|J(x, y)|2].
p(ux, uy; I¯, J)
=π-2(I¯2-|J|2)-1 ×exp-I¯|ux|2-I¯|uy|2+Jux*uy+J*uxuy*I¯2-|J|2,
J=JR+jJI=|J|exp(jβ),
θr=[I¯JRJI]T,
θp=[I¯ |J|β]T,
θ=[I¯|J|2β]T.
[I(θ)]pq=trC-1 CθpC-1 Cθq,
C=I¯JJ*I¯
I(θr)=2(I¯2+|J|2)-4I¯JR-4I¯JI-4I¯JR2(I¯2+JR2-JI2)4JIJR-4I¯JI4JIJR2(I¯2-JR2+JI2)(I¯2-|J|2)-2,
I-1(θr)=(I¯2+|J|2)2I¯JRI¯JII¯JR(I¯2+JR2-JI2)2JRJII¯JIJRJI(I¯2-JR2+JI2)2.
var(I¯ˆ)(I¯2+|J|2)/2,
var(JRˆ)(I¯2+JR2-JI2)/2,
var(JIˆ)(I¯2-JR2+JI2)/2.
θ=g(θr)=[I¯(JR2+JI2)tan-1(JI/JR)]T
I-1(θ)=g(θr)θI-1(θr) g(θr)Tθ,
g(θr)θ=I¯I¯(JR2+JI2)I¯ tan-1(JI/JR)I¯I¯JR(JR2+JI2)JR tan-1(JI/JR)JRI¯JI(JR2+JI2)JI tan-1(JI/JR)JI
=10002JR2JI0-JI|J|2JR|J|2.
I-1(θ)=(I¯2+|J|2)22I¯|J|202I¯|J|22|J|2(I¯2+|J|2)000(I¯2-|J|2)2|J|2
var(I¯ˆ)I¯2+|J|22,
var(|J|2ˆ)2|J|2(I¯2+|J|2),
var(βˆ)I¯2-|J|22|J|2.

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