Abstract

We have developed a technique for imaging dark, i.e. non-radiating, objects by intensity interferometry measurements using a thermal light source in the background. This technique is based on encoding the dark object’s profile into the spatial coherence of such light. We demonstrate the image recovery using an adaptive error-minimizing Gerchberg-Saxton algorithm in case of a completely opaque object, and outline the steps for imaging purely refractive objects.

© 2014 Optical Society of America

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  1. H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons In coherent beams of radiation,” Proc. R. Soc. London A 242300–324 (1957).
    [CrossRef]
  2. H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London A 243291–319 (1958).
    [CrossRef]
  3. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  4. D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
    [CrossRef]
  5. D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
    [CrossRef]
  6. P. D. Nunez, R. Holmes, D. Kieda, J. Rou, S. LeBohec, “Imaging submilliarcsecond stellar features with intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012).
    [CrossRef]
  7. I. Klein, M. Guelman, S. G. Lipson, “Space-based intensity interferometer,” Appl. Opt. 46, 4237 (2007).
    [CrossRef] [PubMed]
  8. D. Dravins, S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
    [CrossRef]
  9. S. LeBohec et al., “Stellar intensity interferometry: Experimental steps toward long-baseline observations,” Proc. SPIE 7734, 77341D (2010).
    [CrossRef]
  10. R. Holmes, B. Calef, D. Gerwe, P. Crabtree, “Cramer-Rao bounds for intensity interferometry measurements,” Appl. Opt. 52, 5235–5246 (2013).
    [CrossRef] [PubMed]
  11. M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
    [CrossRef]
  12. R. B. Holmes, M. S. Belenkii, “Investigation of the CauchyRiemann equations for one-dimensional image recovery in intensity interferometry,” J. Opt. Soc. Am. A 21, 697–706 (2004).
    [CrossRef]
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl.Opt. 21, 2758–2769 (1982).
  14. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A. 7, 450–458 (1990).
    [CrossRef]
  15. D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
    [CrossRef]
  16. J. Wambsganss, “Gravitational Lensing in Astronomy,” Living Rev. Relativity 1, 12 (1998).
    [CrossRef]
  17. M. Moniez, Gen. Relativ. Gravit., “Microlensing as a probe of the Galactic structure: 20 years of microlensing optical depth studies,” Gen. Realtiv. Gravit. 42, 2047–2074 (2010).
  18. M. Moniez, “Does transparent hidden matter generate optical scintillation?” Astron. Astrophys. 412, 105–120 (2003).
    [CrossRef]
  19. F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
    [CrossRef]

2013 (3)

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

R. Holmes, B. Calef, D. Gerwe, P. Crabtree, “Cramer-Rao bounds for intensity interferometry measurements,” Appl. Opt. 52, 5235–5246 (2013).
[CrossRef] [PubMed]

D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
[CrossRef]

2012 (2)

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

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

2011 (1)

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

2010 (2)

S. LeBohec et al., “Stellar intensity interferometry: Experimental steps toward long-baseline observations,” Proc. SPIE 7734, 77341D (2010).
[CrossRef]

M. Moniez, Gen. Relativ. Gravit., “Microlensing as a probe of the Galactic structure: 20 years of microlensing optical depth studies,” Gen. Realtiv. Gravit. 42, 2047–2074 (2010).

2008 (1)

D. Dravins, S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

2007 (1)

2004 (1)

2003 (1)

M. Moniez, “Does transparent hidden matter generate optical scintillation?” Astron. Astrophys. 412, 105–120 (2003).
[CrossRef]

1998 (1)

J. Wambsganss, “Gravitational Lensing in Astronomy,” Living Rev. Relativity 1, 12 (1998).
[CrossRef]

1995 (1)

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
[CrossRef]

1990 (1)

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A. 7, 450–458 (1990).
[CrossRef]

1982 (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl.Opt. 21, 2758–2769 (1982).

1978 (1)

1958 (1)

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London A 243291–319 (1958).
[CrossRef]

1957 (1)

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons In coherent beams of radiation,” Proc. R. Soc. London A 242300–324 (1957).
[CrossRef]

Ansari, R.

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

Belenkii, M. S.

Brown, H. R.

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London A 243291–319 (1958).
[CrossRef]

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons In coherent beams of radiation,” Proc. R. Soc. London A 242300–324 (1957).
[CrossRef]

Calef, B.

Crabtree, P.

Dravins, D.

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

D. Dravins, S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

Erkmen, B. I.

D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
[CrossRef]

Fienup, J. R.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A. 7, 450–458 (1990).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl.Opt. 21, 2758–2769 (1982).

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

Gerwe, D.

Guelman, M.

Habibi, F.

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

Holmes, R.

R. Holmes, B. Calef, D. Gerwe, P. Crabtree, “Cramer-Rao bounds for intensity interferometry measurements,” Appl. Opt. 52, 5235–5246 (2013).
[CrossRef] [PubMed]

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

Holmes, R. B.

Jensen, H.

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

Kieda, D.

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

Klein, I.

Klibanov, M. V.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
[CrossRef]

Kowalczyk, A. M.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A. 7, 450–458 (1990).
[CrossRef]

LeBohec, S.

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

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

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

S. LeBohec et al., “Stellar intensity interferometry: Experimental steps toward long-baseline observations,” Proc. SPIE 7734, 77341D (2010).
[CrossRef]

D. Dravins, S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

Lipson, S. G.

Moniez, M.

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

M. Moniez, Gen. Relativ. Gravit., “Microlensing as a probe of the Galactic structure: 20 years of microlensing optical depth studies,” Gen. Realtiv. Gravit. 42, 2047–2074 (2010).

M. Moniez, “Does transparent hidden matter generate optical scintillation?” Astron. Astrophys. 412, 105–120 (2003).
[CrossRef]

Nunez, P. D.

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

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

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

Rahvar, S.

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

Rou, J.

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

Sacks, P. E.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
[CrossRef]

Strekalov, D. V.

D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
[CrossRef]

Tikhonravov, A. V.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
[CrossRef]

Twiss, R. Q.

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London A 243291–319 (1958).
[CrossRef]

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons In coherent beams of radiation,” Proc. R. Soc. London A 242300–324 (1957).
[CrossRef]

Wambsganss, J.

J. Wambsganss, “Gravitational Lensing in Astronomy,” Living Rev. Relativity 1, 12 (1998).
[CrossRef]

Yu, N.

D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
[CrossRef]

Appl. Opt. (2)

Appl.Opt. (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl.Opt. 21, 2758–2769 (1982).

Astron. Astrophys. (2)

M. Moniez, “Does transparent hidden matter generate optical scintillation?” Astron. Astrophys. 412, 105–120 (2003).
[CrossRef]

F. Habibi, M. Moniez, R. Ansari, S. Rahvar, “Searching for Galactic hidden gas through interstellar scintillation: results from a test with the NTT-SOFI detector,” Astron. Astrophys. 525, A108 (2011).
[CrossRef]

Astroparticle Phys. (1)

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Optical intensity interferometry with the Cherenkov Telescope Array,” Astroparticle Phys. 43, 331–347 (2013).
[CrossRef]

Gen. Realtiv. Gravit. (1)

M. Moniez, Gen. Relativ. Gravit., “Microlensing as a probe of the Galactic structure: 20 years of microlensing optical depth studies,” Gen. Realtiv. Gravit. 42, 2047–2074 (2010).

Inverse Problems (1)

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Problems 11, 1–28 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A. 7, 450–458 (1990).
[CrossRef]

Living Rev. Relativity (1)

J. Wambsganss, “Gravitational Lensing in Astronomy,” Living Rev. Relativity 1, 12 (1998).
[CrossRef]

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

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

New Astronomy Rev. (1)

D. Dravins, S. LeBohec, H. Jensen, P. D. Nunez, “Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging,” New Astronomy Rev. 56, 143–167 (2012).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

D. V. Strekalov, B. I. Erkmen, N. Yu, “Intensity interferometry for observation of dark objects,” Phys. Rev. A 88, 053837 (2013).
[CrossRef]

Proc. R. Soc. London A (2)

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons In coherent beams of radiation,” Proc. R. Soc. London A 242300–324 (1957).
[CrossRef]

H. R. Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London A 243291–319 (1958).
[CrossRef]

Proc. SPIE (2)

D. Dravins, S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

S. LeBohec et al., “Stellar intensity interferometry: Experimental steps toward long-baseline observations,” Proc. SPIE 7734, 77341D (2010).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

A transfer function (solid line) applied to the object’s absorption profile between n − 1th and nth Gerchberg-Saxton iterations gradually drives it towards black-and-white solution. A dashed line represents an identity transfer function An = An−1.

Fig. 2
Fig. 2

A logarithmic plot of the correlation function C(ρi,j) reveals the structure which encodes the shape of the object of interest. The solid contour lines correspond to C(ρi,j)/C0(0) = 2−12 (red) and C(ρi,j)/C0(0) = 2−13 (green). The object is shown on the inset.

Fig. 3
Fig. 3

The results of a simple image recovery algorithm. The step numbers are given in the low left corner of each frame.

Fig. 4
Fig. 4

Left: fractional change (given by square root of variances (6)) of the test object’s image and its Fourier transform in a simple image recovery corresponding to Fig. 3. Right: the same in the alternating algorithm; shown are images 40, 130 and 200.

Fig. 5
Fig. 5

Left: fractional change of the test object’s image and its Fourier transform in the alternating adaptive algorithm; shown are images 10, 75, 125 and 200. Right: the transfer function slope.

Fig. 6
Fig. 6

The image reconstruction results with the threshold set at 2−13 (left) and 2−12 (right).

Fig. 7
Fig. 7

Left: numerically modeled intensity distribution of a Gaussian source modified by a rectangular phase object. Right: the same reconstructed by Gerchberg-Saxton algorithm after 5000 iterations. Side of each image is 2 cm, array size 680 × 680 pixels.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

C ( ρ 1 , ρ 2 ) T 1 T / 2 T / 2 d t I 1 ( t ) I 2 ( t ) ,
C ( ρ d ) K | 𝒯 ( k ρ d L + L s ) I s ( 0 ) β 2 𝒜 ( k ρ d L ) | 2 .
𝒜 ( q ) d 2 ρ A ( ρ ) e i q ρ , 𝒯 ( q ) d 2 ρ I s ( ρ ) e i q ρ .
a ˜ = k b / L and c ˜ = k b / ( L + L s ) .
A ( ρ ) = exp { π ρ 2 C 0 ( 0 ) C ( 0 ) } .
σ n i , j | A n ( ρ i , j ) A n 1 ( ρ i , j ) | 2 i , j | A n 1 ( ρ i , j ) | 2 , σ ˜ n i , j | 𝒜 n ( ρ i , j ) 𝒜 n 1 ( ρ i , j ) | 2 i , j | 𝒜 n 1 ( ρ i , j ) | 2 ,
I ( ρ ) = 1 β 2 π R s 2 d 2 ξ I s ( β ξ ( β 1 ) ρ ) A ( ξ ) .
b λ L R o b N d ,
C ( ρ d ) K | 𝒜 ( k ρ d L ) | 2
T * ( ρ ) T ( ρ ) | T ( ρ s ) | 2 ,
T * ( ρ ) T ( ρ ) = e i [ ϕ ( ρ ) ϕ ( ρ ) ] e i ρ d ϕ ( ρ s ) .
C ( ρ d ) K | d 2 ρ s I s [ β ρ s + L s k ϕ ( ρ s ) ] e i k L ρ d ρ s | 2 .

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