Ghost imaging in the random paraxial regime

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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

Abstract
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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 and 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.
[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.
[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.
[4]	M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.
[5]	J. Cheng, Ghost imaging through turbulent atmosphere, Opt. Express, 17 (2009), 7916-7921. doi: 10.1364/OE.17.007916.
[6]	M. Fink, Time reversed acoustics, Scientific American, 281 (1999), 91-97.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	P. Hariharan, Optical Holography, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781139174039.
[16]	A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, Piscataway, 1997.
[17]	O. Katz, Y. Bromberg and Y. Silberberg, Compressive ghost imaging, Appl. Phys. Lett., 95 (2009), 131110. doi: 10.1063/1.3238296.
[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.
[19]	L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995.
[20]	J. H. Shapiro, Computational ghost imaging, Phys. Rev. A, 78 (2008), 061802(R). doi: 10.1364/IQEC.2009.IThK7.
[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.
[22]	F. D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, Springer Lecture Notes in Physics, 70 (1977), 224-287.
[23]	V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover, New York, 1961.
[24]	B. J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw Hill, New York, 1977.
[25]	A. Valencia, G. Scarcelli, M. D'Angelo and Y. Shih, Two-photon imaging with thermal light, Phys. Rev. Lett., 94 (2005), 063601.
[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.

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.
[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.
[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.
[4]	M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.
[5]	J. Cheng, Ghost imaging through turbulent atmosphere, Opt. Express, 17 (2009), 7916-7921. doi: 10.1364/OE.17.007916.
[6]	M. Fink, Time reversed acoustics, Scientific American, 281 (1999), 91-97.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	P. Hariharan, Optical Holography, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781139174039.
[16]	A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, Piscataway, 1997.
[17]	O. Katz, Y. Bromberg and Y. Silberberg, Compressive ghost imaging, Appl. Phys. Lett., 95 (2009), 131110. doi: 10.1063/1.3238296.
[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.
[19]	L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995.
[20]	J. H. Shapiro, Computational ghost imaging, Phys. Rev. A, 78 (2008), 061802(R). doi: 10.1364/IQEC.2009.IThK7.
[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.
[22]	F. D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, Springer Lecture Notes in Physics, 70 (1977), 224-287.
[23]	V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover, New York, 1961.
[24]	B. J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw Hill, New York, 1977.
[25]	A. Valencia, G. Scarcelli, M. D'Angelo and Y. Shih, Two-photon imaging with thermal light, Phys. Rev. Lett., 94 (2005), 063601.
[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.

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