American Institute of Mathematical Sciences

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May  2016, 10(2): 409-432. doi: 10.3934/ipi.2016006

Ghost imaging in the random paraxial regime

Josselin Garnier 1,

1. 

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris Diderot, 75205 Paris Cedex 13

Received  October 2014 Published  May 2016

In this paper we analyze a wave-based imaging modality called ghost imaging that can produce an image of an object illuminated by a partially coherent source. The image of the object is obtained by correlating the intensities measured by two detectors, one that does not view the object and another one that does view the object. More exactly, a high-resolution detector measures the intensity of a wave field emitted by a partially coherent source which has not interacted with the object to be imaged. A bucket (or single-pixel) detector collects the total (spatially-integrated) intensity of the wave field emitted by the same source that has interacted with the object. The correlation of the intensity measured at the high-resolution detector with the intensity measured by the bucket detector gives an image of the object. In this paper we analyze this imaging modality when the medium through which the waves propagate is random. We discuss the relation with time reversal focusing and with correlation-based imaging using ambient noise sources. We clarify the role of the partial coherence of the source and we study how scattering affects the resolution properties of the ghost imaging function in the paraxial regime: the image resolution is all the better as the source is less coherent, and all the worse as the medium is more scattering.
Keywords: Passive sensor imaging, ambient noise sources, scattering media, ghost imaging..
Mathematics Subject Classification: Primary: 35R30, 35R60; Secondary: 78A46, 78A4.
Citation: Josselin Garnier. Ghost imaging in the random paraxial regime. Inverse Problems & Imaging, 2016, 10 (2) : 409-432. doi: 10.3934/ipi.2016006
References:
[1]

D. G. Alfaro Vigo, J.-P. Fouque, J. Garnier and A. Nachbin, Robustness of time reversal for waves in time-dependent random media, Stochastic Process. Appl., 113 (2004), 289-313. doi: 10.1016/j.spa.2004.04.002.  Google Scholar

[2]

G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Disc. Cont. Dyn. Syst. A, 12 (2005), 793-815. doi: 10.3934/dcds.2005.12.793.  Google Scholar

[3]

P. Blomgren, G. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Amer., 111 (2002), 230-248. doi: 10.1121/1.1421342.  Google Scholar

[4]

M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.  Google Scholar

[5]

J. Cheng, Ghost imaging through turbulent atmosphere, Opt. Express, 17 (2009), 7916-7921. doi: 10.1364/OE.17.007916.  Google Scholar

[6]

M. Fink, Time reversed acoustics, Scientific American, 281 (1999), 91-97. Google Scholar

[7]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[8]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739. doi: 10.1137/060659235.  Google Scholar

[9]

J. Garnier and G. Papanicolaou, Passive sensor imaging using cross correlations of noisy signals in a scattering medium, SIAM J. Imaging Sciences, 2 (2009), 396-437. doi: 10.1137/080723454.  Google Scholar

[10]

J. Garnier and G. Papanicolaou, Resolution analysis for imaging with noise, Inverse Problems, 26 (2010), 074001, 22pp. doi: 10.1088/0266-5611/26/7/074001.  Google Scholar

[11]

J. Garnier and G. Papanicolaou, Fluctuation theory of ambient noise imaging, CRAS Geoscience, 343 (2011), 502-511. doi: 10.1016/j.crte.2011.01.004.  Google Scholar

[12]

J. Garnier and K. Sølna, Coupled paraxial wave equations in random media in the white-noise regime, Ann. Appl. Probab., 19 (2009), 318-346. doi: 10.1214/08-AAP543.  Google Scholar

[13]

J. Garnier and K. Sølna, Fourth-moment analysis for wave propagation in the white-noise paraxial regime, Arch. Rational Mech. Anal., 220 (2016), 37-81. doi: 10.1007/s00205-015-0926-2.  Google Scholar

[14]

N. D. Hardy and J. H. Shapiro, Reflective Ghost Imaging through turbulence, Phys. Rev. A, 84 (2011), 063824. doi: 10.1103/PhysRevA.84.063824.  Google Scholar

[15]

P. Hariharan, Optical Holography, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781139174039.  Google Scholar

[16]

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, Piscataway, 1997.  Google Scholar

[17]

O. Katz, Y. Bromberg and Y. Silberberg, Compressive ghost imaging, Appl. Phys. Lett., 95 (2009), 131110. doi: 10.1063/1.3238296.  Google Scholar

[18]

C. Li, T. Wang, J. Pu, W. Zhu and R. Rao, Ghost imaging with partially coherent light radiation through turbulent atmosphere, Appl. Phys. B, 99 (2010), 599-604. doi: 10.1007/s00340-010-3969-y.  Google Scholar

[19]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. Google Scholar

[20]

J. H. Shapiro, Computational ghost imaging, Phys. Rev. A, 78 (2008), 061802(R). doi: 10.1364/IQEC.2009.IThK7.  Google Scholar

[21]

J. H. Shapiro and R. W. Boyd, The physics of ghost imaging, Quantum Inf. Process., 11 (2012), 949-993. doi: 10.1007/s11128-011-0356-5.  Google Scholar

[22]

F. D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, Springer Lecture Notes in Physics, 70 (1977), 224-287.  Google Scholar

[23]

V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover, New York, 1961.  Google Scholar

[24]

B. J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw Hill, New York, 1977. Google Scholar

[25]

A. Valencia, G. Scarcelli, M. D'Angelo and Y. Shih, Two-photon imaging with thermal light, Phys. Rev. Lett., 94 (2005), 063601. Google Scholar

[26]

P. Zhang, W. Gong, X. Shen and S. Han, Correlated imaging through atmospheric turbulence, Phys. Rev. A, 82 (2010), 033817. doi: 10.1103/PhysRevA.82.033817.  Google Scholar

show all references

References:
[1]

D. G. Alfaro Vigo, J.-P. Fouque, J. Garnier and A. Nachbin, Robustness of time reversal for waves in time-dependent random media, Stochastic Process. Appl., 113 (2004), 289-313. doi: 10.1016/j.spa.2004.04.002.  Google Scholar

[2]

G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Disc. Cont. Dyn. Syst. A, 12 (2005), 793-815. doi: 10.3934/dcds.2005.12.793.  Google Scholar

[3]

P. Blomgren, G. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Amer., 111 (2002), 230-248. doi: 10.1121/1.1421342.  Google Scholar

[4]

M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.  Google Scholar

[5]

J. Cheng, Ghost imaging through turbulent atmosphere, Opt. Express, 17 (2009), 7916-7921. doi: 10.1364/OE.17.007916.  Google Scholar

[6]

M. Fink, Time reversed acoustics, Scientific American, 281 (1999), 91-97. Google Scholar

[7]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[8]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739. doi: 10.1137/060659235.  Google Scholar

[9]

J. Garnier and G. Papanicolaou, Passive sensor imaging using cross correlations of noisy signals in a scattering medium, SIAM J. Imaging Sciences, 2 (2009), 396-437. doi: 10.1137/080723454.  Google Scholar

[10]

J. Garnier and G. Papanicolaou, Resolution analysis for imaging with noise, Inverse Problems, 26 (2010), 074001, 22pp. doi: 10.1088/0266-5611/26/7/074001.  Google Scholar

[11]

J. Garnier and G. Papanicolaou, Fluctuation theory of ambient noise imaging, CRAS Geoscience, 343 (2011), 502-511. doi: 10.1016/j.crte.2011.01.004.  Google Scholar

[12]

J. Garnier and K. Sølna, Coupled paraxial wave equations in random media in the white-noise regime, Ann. Appl. Probab., 19 (2009), 318-346. doi: 10.1214/08-AAP543.  Google Scholar

[13]

J. Garnier and K. Sølna, Fourth-moment analysis for wave propagation in the white-noise paraxial regime, Arch. Rational Mech. Anal., 220 (2016), 37-81. doi: 10.1007/s00205-015-0926-2.  Google Scholar

[14]

N. D. Hardy and J. H. Shapiro, Reflective Ghost Imaging through turbulence, Phys. Rev. A, 84 (2011), 063824. doi: 10.1103/PhysRevA.84.063824.  Google Scholar

[15]

P. Hariharan, Optical Holography, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781139174039.  Google Scholar

[16]

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, Piscataway, 1997.  Google Scholar

[17]

O. Katz, Y. Bromberg and Y. Silberberg, Compressive ghost imaging, Appl. Phys. Lett., 95 (2009), 131110. doi: 10.1063/1.3238296.  Google Scholar

[18]

C. Li, T. Wang, J. Pu, W. Zhu and R. Rao, Ghost imaging with partially coherent light radiation through turbulent atmosphere, Appl. Phys. B, 99 (2010), 599-604. doi: 10.1007/s00340-010-3969-y.  Google Scholar

[19]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. Google Scholar

[20]

J. H. Shapiro, Computational ghost imaging, Phys. Rev. A, 78 (2008), 061802(R). doi: 10.1364/IQEC.2009.IThK7.  Google Scholar

[21]

J. H. Shapiro and R. W. Boyd, The physics of ghost imaging, Quantum Inf. Process., 11 (2012), 949-993. doi: 10.1007/s11128-011-0356-5.  Google Scholar

[22]

F. D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, Springer Lecture Notes in Physics, 70 (1977), 224-287.  Google Scholar

[23]

V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover, New York, 1961.  Google Scholar

[24]

B. J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw Hill, New York, 1977. Google Scholar

[25]

A. Valencia, G. Scarcelli, M. D'Angelo and Y. Shih, Two-photon imaging with thermal light, Phys. Rev. Lett., 94 (2005), 063601. Google Scholar

[26]

P. Zhang, W. Gong, X. Shen and S. Han, Correlated imaging through atmospheric turbulence, Phys. Rev. A, 82 (2010), 033817. doi: 10.1103/PhysRevA.82.033817.  Google Scholar

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