Abstract

Intensity correlation imaging (ICI) is a concept which has been considered for the task of providing images of satellites in geosynchronous orbit using ground-based equipment. This concept is based on the intensity interferometer principle first developed by Hanbury Brown and Twiss. It is the objective of this paper to establish that a sun-lit geosynchronous satellite is too faint a target object to allow intensity interferometry to be used in developing image information about it—at least not in a reasonable time and with a reasonable amount of equipment. An analytic treatment of the basic phenomena is presented. This is an analysis of one aspect of the statistics of the very high frequency random variations of a very narrow portion of the optical spectra of the incoherent (black-body like—actually reflected sunlight) radiation from the satellite, an analysis showing that the covariance of this radiation as measured by a pair of ground-based telescopes is directly proportional to the square of the magnitude of one component of the Fourier transform of the image of the satellite—the component being the one for a spatial frequency whose value is determined by the separation of the two telescopes. This analysis establishes the magnitude of the covariance. A second portion of the analysis considers shot-noise effects. It is shown that even with much less than one photodetection event (pde) per signal integration time an unbiased estimate of the covariance of the optical field’s random variations can be developed. Also, a result is developed for the standard deviation to be associated with the estimated value of the covariance. From these results an expression is developed for what may be called the signal-to-noise ratio to be associated with an estimate of the covariance. This signal-to-noise ratio, it turns out, does not depend on the measurement’s integration time, Δt (in seconds), or on the optical spectral bandwidth, Δν (in Hertz), utilized—so long as ΔtΔν1, which condition it would be hard to violate. It is estimated that for a D=3.16m diameter satellite, with a pair of D=1.0m diameter telescopes (which value of D probably represents an upper limit on allowable aperture diameter since the telescope aperture must be much too small to even resolve the size of the satellite) at least N=2.55×1016 separate pairs of (one integration time, pde count) measurement values must be collected to achieve just a 10 dB signal-to-noise ratio. Working with 10 pairs of telescopes (all with the same separation), and with 10 nearly adjacent and each very narrow spectral bands extracted from the light collected by each of the telescope—so that for each measurement integration time there would be 100 pairs of measurement values available—and with an integration time as short as Δt=1ns, it would take T=2.55×105s or about 71 h to collect the data for just a single spatial frequency component of the image of the satellite. It is on this basis that it is concluded that the ICI concept does not seem likely to be able to provide a timely responsive capability for the imaging of geosynchronous satellites.

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References

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  1. D. C. Hyland, “Calculation of signal-to-noise ratio for image formation using intensity correlation interferometry,” , August2006.
  2. D. C. Hyland, “Constellations using entry pupil processing for high resolution imaging of geosynchronous objects,” in Proceedings of AAS/AIAA Space Flight Mechanics Conference, Tampa, Florida, January22–26, 2006.
  3. A. Ofir and E. Ribak, “Micro-arcsec imaging from the ground with intensity interferometers,” Proc. SPIE 6268, 1181–1191 (2006).
  4. R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, 1974).
  5. A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
    [CrossRef]
  6. J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery Theory and Applications, H. Stark, ed. (Academic, 1987), pp. 231–275.
  7. P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
    [CrossRef]
  8. D. L. Snyder, Random Point Processes (Wiley-Interscience, 1975).
  9. B. Saleh, Photoelectron Statistics (Springer, 1978).

2012 (1)

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

2006 (1)

A. Ofir and E. Ribak, “Micro-arcsec imaging from the ground with intensity interferometers,” Proc. SPIE 6268, 1181–1191 (2006).

1955 (1)

A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Dainty, J. C.

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery Theory and Applications, H. Stark, ed. (Academic, 1987), pp. 231–275.

Fienup, J. R.

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery Theory and Applications, H. Stark, ed. (Academic, 1987), pp. 231–275.

Forrester, A. I.

A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Gudmundsen, R. A.

A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Hanbury Brown, R.

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

Holmes, R.

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

Hyland, D. C.

D. C. Hyland, “Constellations using entry pupil processing for high resolution imaging of geosynchronous objects,” in Proceedings of AAS/AIAA Space Flight Mechanics Conference, Tampa, Florida, January22–26, 2006.

D. C. Hyland, “Calculation of signal-to-noise ratio for image formation using intensity correlation interferometry,” , August2006.

Johnson, P. O.

A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Kieda, D.

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

LeBohec, S.

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

Nunez, P. D.

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

Ofir, A.

A. Ofir and E. Ribak, “Micro-arcsec imaging from the ground with intensity interferometers,” Proc. SPIE 6268, 1181–1191 (2006).

Ribak, E.

A. Ofir and E. Ribak, “Micro-arcsec imaging from the ground with intensity interferometers,” Proc. SPIE 6268, 1181–1191 (2006).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, 1978).

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley-Interscience, 1975).

Mon. Not. R. Astron. Soc. (1)

P. D. Nunez, R. Holmes, D. Kieda, and S. LeBohec, “Imaging sub-milliarcsecond features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
[CrossRef]

Phys. Rev. (1)

A. I. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[CrossRef]

Proc. SPIE (1)

A. Ofir and E. Ribak, “Micro-arcsec imaging from the ground with intensity interferometers,” Proc. SPIE 6268, 1181–1191 (2006).

Other (6)

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

D. C. Hyland, “Calculation of signal-to-noise ratio for image formation using intensity correlation interferometry,” , August2006.

D. C. Hyland, “Constellations using entry pupil processing for high resolution imaging of geosynchronous objects,” in Proceedings of AAS/AIAA Space Flight Mechanics Conference, Tampa, Florida, January22–26, 2006.

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery Theory and Applications, H. Stark, ed. (Academic, 1987), pp. 231–275.

D. L. Snyder, Random Point Processes (Wiley-Interscience, 1975).

B. Saleh, Photoelectron Statistics (Springer, 1978).

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Figures (1)

Fig. 1.
Fig. 1.

Wavelength dependence of K(ν¯). The function K(ν¯) serves to establish the value of the signal-to-noise (voltage) ratio as indicated by Eq. (61).

Equations (67)

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I(r)=12|u(r,t)|2inc,
u(r,t)=νLνUdνα(ν,r)exp(2πiνt),
νLνUdνα(ν,r)α*(ν,r)f(ν,r)inc=f(ν,r)βν|α(ν,r)|2inc,
drνLνUdνα(ν,r)α*(ν,r)f(ν,r)inc=f(ν,r)βrβν|α(ν,r)|2inc,
drdrνLνUdνdνα(ν,r)α(ν,r)α*(νr)α*(νr)g(ν,ν,r,r)inc=[g(ν,ν,r,r)+g(ν,ν,r,r)]βr2βν2|α(ν,r)|2inc|α(ν,r)|2inc,
ΔνI(r)=12|u(r,t)|2inc.
ΔνI(r)=12νLνUdννLνUdνα(ν,r)α*(ν,r)exp(2πi(νν)t)inc.
ΔνI(r)=12βννLνUdν|α(ν,r)|2inc.
2I(r)=βν|α(ν¯,r)|2inc,
P¯=12|v(ρ,t)|2inc.
S(r,ρ)=[Z2+|rρ|2]1/2Z+|rρ|22ZZ+r2+ρ22Zr·ρZ.
v(ρ,t+τ)=νLνUdνν/ciZdrα(ν,r)exp(2πiνt)exp(2πiνc[Z+r2+ρ22Zr·ρZ]).
P¯=12νLνUdνdrνLνUdνdrννc2Z2α(ν,r)α*(ν,r)exp(2πi(νν)t)×exp(2πi[νc(Z+r2+ρ22Zr·ρZ)νc(Z+r2+ρ22Zr·ρZ)])inc.
P¯=12νLνUdνdrν2c2Z2βrβν|α(ν,r)|2inc,
P¯=ν¯2Δνc2Z2βrdrI(r),
βr=c2Z2ν¯2P¯/ΔνdrI(r).
βrβν|α(ν¯,r)|2inc=2c2Z2ν¯2P¯ΔνI(r)drI(r)=2c2Z2ν¯2P¯ΔνI^(r),
I^(r)=I(r)/drI(r).
CA·B=[EAE¯][EBE¯]inc=EAEBincE¯2,whereE¯=EAinc=EBinc.
EA=12(14πD2)Δt/2+Δt/2dtνLνUdνdνdrdrννc2Z2α(ν,r)α*(ν,r)exp(2πi(νν)t)×exp(2πi[νc(Z+r2+ρA22Zr·ρAZ)νc(Z+r2+ρA22Zr·ρAZ)]),
EB=12(14πD2)Δt/2+Δt/2dtνLνUdνdνdrdrννc2Z2α(ν,r)α*(ν,r)exp(2πi(νν)t)×exp(2πi[νc(Z+r2+ρB22Zr·ρBZ)νc(Z+r2+ρB22Zr·ρBZ)]).
E¯=12(14πD2)ΔtνLνUdνdrν2c2Z2βrβν|α(ν,r)|2inc.
E¯=(14πD2)Δtc2Z2ν¯2P¯,
Δt/2+Δt/2dtexp(2π(νν)t)=Δtsinc(π(νν)Δt),
EA=12(14πD2)ΔtνLνUdνdνdrdrννc2Z2α(ν,r)α*(ν,r)sinc(π(νν)Δt)×exp(2πi[νc(Z+r2+ρA22Zr·ρAZ)νc(Z+r2+ρA22Zr·ρAZ)]),
EB=12(14πD2)ΔtνLνUdνdνdrdrννc2Z2α(ν,r)α*(ν,r)sinc(π(νν)Δt)×exp(2πi[νc(Z+r2+ρB22Zr·ρBZ)νc(Z+r2+ρB22Zr·ρBZ)]).
EAEBinc=14(14πD2)2(Δt)2νLνUdνdνdrdrννc2Z2νLνUdνdνdrdrννc2Z2×α(ν,r)α(ν,r)α*(ν,r)α*(ν,r)sinc(π(νν)Δt)sinc(π(νν)Δt)×exp(2πi[νc(Z+r2+ρA22Zr·ρAZ)νc(Z+r2+ρA22Zr·ρAZ)])×exp(2πi[νc(Z+r2+ρB22Zr·ρBZ)νc(Z+r2+ρB22Zr·ρBZ)])inc.
EAEBinc=Q1+Q2.
Q1=14(14πD2)2(Δt)2νLνUdνdνdrdrν2ν2c4Z4βr2βν2|α(ν,r)|2inc|α(ν,r)|2inc.
Q1=[12(14πD2)ΔtνLνUdνdrν2c2Z2βrβν|α(ν,r)|2inc]×[12(14πD2)ΔtνLνUdνdrν2c2Z2βrβν|α(ν,r)|2inc].
Q1=E¯2.
CA·B=Q2.
Q2=14(14πD2)2(Δt)2νLνUdνdνdrdrν2ν2c4Z4βr2βν2|α(ν,r)|2inc|α(ν,r)|2incsinc2(π(νν)Δt)×exp(2πi[νc(ρA2ρB22Zr·(ρAρB)Z)νc(ρA2ρB22Zr·(ρAρB)Z)]).
CA·B=14(14πD2)2(Δt)2νLνUdνdνdrdrν2ν2c4Z4βr2βν2|α(ν,r)|2inc|α(ν,r)|2incsinc2(π(νν)Δt)×exp(2πi[ννcρA2ρB22Z(νrνr)·(ρAρB)cZ]).
r+=12(r+r),r=rr,ν+=12(ν+ν),andν=νν,
CA·B=14(14πD2)2(Δt)2νLνUdν+LlLudνdr+dr(ν+214ν2)2c4Z4βr2βν2|α(ν++12ν,r++12r)|2inc×|α(ν+12ν,r+12r)|2incsinc2(πνΔt)×exp(2πi[νcρA2ρB22Z(νr++ν+r)·(ρAρB)cZ]),
+dξsinc2(πξK)=K1,
CA·B=14(14πD2)2ΔtνLνUdνdr+drν4c4Z4βr2βν2|α(ν,r++12r)|2inc|α(ν,r+12r)|2inc×exp(2πiνr·(ρAρB)cZ).
CA·B=14(14πD2)2ΔtΔνdr+drν¯4c4Z4βr2βν2|α(ν¯,r++12r)|2inc|α(ν¯,r+12r)|2inc×exp(2πiν¯r·(ρAρB)cZ).
CA·B=(14πD2)2P¯2ΔtΔν{drexp(2πiν¯r·(ρAρB)cZ)I^(r)}×{drexp(+2πiν¯,r·(ρAρB)cZ)I^(r)}.
I˜(κ)=drexp(2πiκ·r)I^(r),
CA·B=(14πD2P¯Δt)2ΔtΔν|I˜(ρAρBλ¯Z)|2,
λ¯=c/ν¯.
CA·B=(14πD2P¯Δthν¯)2ΔtΔν|I˜(ρAρBλ¯Z)|2,
xAnpde=ξAn,xBnpde=ξBn,xn2Apde=ξAn2+ξAn,xn2Bpde=ξBn2+Aξn.
CA·B=[ξAnμ][ξBnμ]inc=ξAnξBnincμ2,whereξAninc=ξBninc=μ,
μ=14πD2P¯Δthν¯,
C^A·B=1N1n=1NxnAxnB1N(N1)(n=1NxnA)(n=1NxnB),
C^A·Bpdeinc=1N1n=1NxnAxnBpdeinc1N(N1)n,n=1nnNxnAxnBpdeinc1N(N1)n=1NxnAxnBpdeinc.
C^A·Bpdeinc=1Nn=1NξAnξBninc1N(N1)n,n=1nnNξAnξBninc.
C^A·Bpdeinc=ξAnξBnincμ2.
Var{C^A·B}=[1N1n=1NxnAxnB1N(N1)(n=1NxnA)(n=1NxnB)]×[1N1n=1NxnAxnB1N(N1)(n=1NxnA)(n=1NxnB)]pde.
Var{C^A·B}=S2(N1)22S3N(N1)2+S4N2(N1)2,
S2=n,n=1NxnAxnAxnBxnBpde,
S3=n,n,n=1NxnAxnAxnBxnBpde,
S4=n,n,n,n=1NxnAxnAxnBxnBpde.
S2=n=1Nxn2Apdexn2Bpde+n,n=1nnNxnApdexnApdexnBpdexnBpde=n=1N(ξn2A+ξnA)(ξn2B+ξnB)+n,n=1nnNξnAξnAξnBξnB=N(μ2+μ)2+N(N1)μ4=N2μ4+2Nμ3+Nv2.
S3=N3μ4+2N2μ3+Nμ2.
S4=n,n=1NxnAxnApden,n=1NxnBxnBpde.
S4=N4μ4+2N3μ3+N2μ2.
Var{C^A·B}=μ2N1μ2/N,
SNRV=CA·BVar{C^A·B}=14πD2P¯ΔtΔtΔνhν¯N1/2|I˜(ρAρBλ¯Z)|.
SNRV<14πD2(P¯/Δν)hν¯N1/2.
P¯=1.4×103(14πD2)π(3.6×107)2S(ν)Δν=2.70×1013D2S(ν)Δν,
SNRV=K(ν¯)(DD)2N1/2,
K(ν¯)=3.19×1020S(ν¯)ν¯.
Z(λ¯/D)D,or equivalentlyDDλ¯Z.

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