Abstract

Computational ghost imaging is a structured-illumination active imager coupled with a single-pixel detector that has potential applications in remote sensing. Here we report on an architecture that acquires the two-dimensional spatial Fourier transform of the target object (which can be inverted to obtain a conventional image). We determine its image signature, resolution, and signal-to-noise ratio in the presence of practical constraints such as atmospheric turbulence, background radiation, and photodetector noise. We consider a bistatic imaging geometry and quantify the resolution impact of nonuniform Kolmogorov-spectrum turbulence along the propagation paths. We show that, in some cases, short-exposure intensity averaging can mitigate atmospheric-turbulence-induced resolution loss. Our analysis reveals some key performance differences between computational ghost imaging and conventional active imaging, and identifies scenarios in which theory predicts that the former will perform better than the latter.

© 2012 Optical Society of America

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  1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
    [CrossRef]
  2. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  3. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
    [CrossRef]
  4. Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [CrossRef]
  5. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
    [CrossRef]
  6. B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
    [CrossRef]
  7. We define a classical source as one whose photodetection statistics can be accurately described using the semiclassical (shot-noise) theory. This is equivalent to having a source state with a proper P-representation. [17] A quantum source is one whose photodetection statistics cannot be described by the semiclassical theory; i.e., the source state does not have a proper P-representation.
  8. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
  9. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
    [CrossRef]
  10. P. K. Baheti and M. A. Neifeld, “Feature-specific structured imaging,” Appl. Opt. 45, 7382–7391 (2006).
    [CrossRef]
  11. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
    [CrossRef]
  12. R. Meyers and K. S. Deacon, “Quantum ghost imaging experiments at ARL,” Proc. SPIE 7815, 78150I (2010).
    [CrossRef]
  13. N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
    [CrossRef]
  14. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
    [CrossRef]
  15. N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” S.M. thesis (Massachusetts Institute of Technology, 2011).
  16. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  18. M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
    [CrossRef]
  19. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef]
  20. J. H. Shapiro, Imaging and Optical Communication Through Atmospheric Turbulence (Springer-Verlag, 1978), Chap. 6.
  21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  22. Although the Eq. (7) structure function was initially derived using the weak-fluctuation (Rytov) approximation, it remains valid well beyond the regime in which the Rytov approximation can be used [20].
  23. M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  24. A more general and better filter is gn(t)≡ ∑kfk*(t)fk+n(t)-〈|fk(t)|2〉δ0,n, but because it complicates the forthcoming analysis, here we use the suboptimal filter in Eq. (14).
  25. D. Kryskowski and G. H. Suits, Sources of Radiation (SPIE, 1993), Vol. 1, Chap. 3.
  26. N. S. Kopeika and J. Bordogna, “Background noise in optical communication systems,” Proc. IEEE 58, 1571–1577 (1970).
    [CrossRef]
  27. B. I. Erkmen, “Computational ghost imaging for remote sensing applications,” Interplanet. Netw. Prog. Rep. 42–185, 1–23 (2011).
  28. Given that the atmospheric coherence time is typically longer than a millisecond, and SLMs have modulation bandwidths that are several MHz, correlations can be taken nominally over thousands of modulation symbols before the state of turbulence has significantly changed.
  29. J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts, 2007).
  30. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
    [CrossRef]
  31. P. S. Idell, J. R. Fienup, and R. S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987).
    [CrossRef]
  32. A raster-scanning laser radar would share the same limitations with computational ghost imaging regarding forward-path turbulence, as the focused-spot size on the target surface would become turbulence-limited when the transmitter-aperture diameter exceeds the transmitter-plane coherence length.
  33. B. M. Welsh and C. S. Gardner, “Bistatic imaging lidar technique for upper atmospheric studies,” Appl. Opt. 28, 82–88 (1989).
    [CrossRef]
  34. S. Komiyama, “Single-photon detectors in the terahertz range,” IEEE J. Sel. Top. Quantum Electron. 17, 54–66 (2011).
    [CrossRef]
  35. M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 (2004).

2011 (3)

B. I. Erkmen, “Computational ghost imaging for remote sensing applications,” Interplanet. Netw. Prog. Rep. 42–185, 1–23 (2011).

S. Komiyama, “Single-photon detectors in the terahertz range,” IEEE J. Sel. Top. Quantum Electron. 17, 54–66 (2011).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[CrossRef]

2010 (3)

R. Meyers and K. S. Deacon, “Quantum ghost imaging experiments at ARL,” Proc. SPIE 7815, 78150I (2010).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
[CrossRef]

2009 (3)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
[CrossRef]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

2008 (3)

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

2006 (2)

2005 (1)

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

2004 (3)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 (2004).

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

1989 (1)

1987 (1)

1976 (1)

1971 (1)

1970 (1)

N. S. Kopeika and J. Bordogna, “Background noise in optical communication systems,” Proc. IEEE 58, 1571–1577 (1970).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Baheti, P. K.

Baraniuk, R.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Bordogna, J.

N. S. Kopeika and J. Bordogna, “Background noise in optical communication systems,” Proc. IEEE 58, 1571–1577 (1970).
[CrossRef]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Cai, Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Cheng, J.

Davenport, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Deacon, K. S.

R. Meyers and K. S. Deacon, “Quantum ghost imaging experiments at ARL,” Proc. SPIE 7815, 78150I (2010).
[CrossRef]

Duarte, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Erkmen, B. I.

B. I. Erkmen, “Computational ghost imaging for remote sensing applications,” Interplanet. Netw. Prog. Rep. 42–185, 1–23 (2011).

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

Fienup, J. R.

Fitch, M. J.

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 (2004).

Gardner, C. S.

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts, 2007).

Goodman, R. S.

Hardy, N. D.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” S.M. thesis (Massachusetts Institute of Technology, 2011).

Holmes, J. F.

Idell, P. S.

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Kelly, K.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Kerr, J. R.

Komiyama, S.

S. Komiyama, “Single-photon detectors in the terahertz range,” IEEE J. Sel. Top. Quantum Electron. 17, 54–66 (2011).
[CrossRef]

Kopeika, N. S.

N. S. Kopeika and J. Bordogna, “Background noise in optical communication systems,” Proc. IEEE 58, 1571–1577 (1970).
[CrossRef]

Kryskowski, D.

D. Kryskowski and G. H. Suits, Sources of Radiation (SPIE, 1993), Vol. 1, Chap. 3.

Laska, J.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Lee, M. H.

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

Lutomirski, R. F.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Meyers, R.

R. Meyers and K. S. Deacon, “Quantum ghost imaging experiments at ARL,” Proc. SPIE 7815, 78150I (2010).
[CrossRef]

Neifeld, M. A.

Osiander, R.

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 (2004).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Sergienko, A. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Shapiro, J. H.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

J. H. Shapiro, Imaging and Optical Communication Through Atmospheric Turbulence (Springer-Verlag, 1978), Chap. 6.

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Suits, G. H.

D. Kryskowski and G. H. Suits, Sources of Radiation (SPIE, 1993), Vol. 1, Chap. 3.

Sun, T.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Takhar, D.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Welsh, B. M.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yura, H. T.

Zhu, S.-Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Appl. Phys. Lett. (1)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Komiyama, “Single-photon detectors in the terahertz range,” IEEE J. Sel. Top. Quantum Electron. 17, 54–66 (2011).
[CrossRef]

IEEE Signal Process. Mag. (1)

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Interplanet. Netw. Prog. Rep. (1)

B. I. Erkmen, “Computational ghost imaging for remote sensing applications,” Interplanet. Netw. Prog. Rep. 42–185, 1–23 (2011).

J. Opt. Soc. Am. (1)

Johns Hopkins APL Tech. Dig. (1)

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 (2004).

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (6)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

Phys. Rev. E (1)

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Phys. Rev. Lett (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

Proc. IEEE (1)

N. S. Kopeika and J. Bordogna, “Background noise in optical communication systems,” Proc. IEEE 58, 1571–1577 (1970).
[CrossRef]

Proc. SPIE (2)

R. Meyers and K. S. Deacon, “Quantum ghost imaging experiments at ARL,” Proc. SPIE 7815, 78150I (2010).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

Other (11)

N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” S.M. thesis (Massachusetts Institute of Technology, 2011).

We define a classical source as one whose photodetection statistics can be accurately described using the semiclassical (shot-noise) theory. This is equivalent to having a source state with a proper P-representation. [17] A quantum source is one whose photodetection statistics cannot be described by the semiclassical theory; i.e., the source state does not have a proper P-representation.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. H. Shapiro, Imaging and Optical Communication Through Atmospheric Turbulence (Springer-Verlag, 1978), Chap. 6.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Although the Eq. (7) structure function was initially derived using the weak-fluctuation (Rytov) approximation, it remains valid well beyond the regime in which the Rytov approximation can be used [20].

A more general and better filter is gn(t)≡ ∑kfk*(t)fk+n(t)-〈|fk(t)|2〉δ0,n, but because it complicates the forthcoming analysis, here we use the suboptimal filter in Eq. (14).

D. Kryskowski and G. H. Suits, Sources of Radiation (SPIE, 1993), Vol. 1, Chap. 3.

Given that the atmospheric coherence time is typically longer than a millisecond, and SLMs have modulation bandwidths that are several MHz, correlations can be taken nominally over thousands of modulation symbols before the state of turbulence has significantly changed.

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts, 2007).

A raster-scanning laser radar would share the same limitations with computational ghost imaging regarding forward-path turbulence, as the focused-spot size on the target surface would become turbulence-limited when the transmitter-aperture diameter exceeds the transmitter-plane coherence length.

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Figures (3)

Fig. 1.
Fig. 1.

Ghost-imaging architectures showing imaging in transmission. (a) In the dual-arm version of ghost imaging, the reference arm is measured with a high-resolution camera (or a scanning pinhole detector) to determine the spatial beam profile (speckle pattern). (b) In computational ghost imaging, the speckle pattern is computed using the transmitter-plane beam profile and paraxial free-space beam propagation theory.

Fig. 2.
Fig. 2.

A ghost-imaging configuration for remote sensing. The transmitter projects a spatiotemporally modulated laser beam onto a target located L1 meters away along a path with atmospheric turbulence. The receiver, not necessarily colocated with the transmitter, is L2 meters away from the target. The target is assumed to be rough on the order of a wavelength, giving rise to diffuse surface scattering and speckle.

Fig. 3.
Fig. 3.

A generic laser-radar imager that uses flood-light active illumination.

Equations (37)

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

An{r:r(nd-d/2,nd+d/2]×(md-d/2,md+d/2]}
R(τ)fn*(t)fn(t+τ)=e-τ2/2T02
ES(r,t)=PD2nfn(t)ξ(r-nd),
ξ(r){1,r(-d/2,d/2]×(-d/2,d/2]0,otherwise
E1(r,t)=ES(r,t-L1/c)hFS(r-r;L1)eψT(r,r)dr,
hFS(r;L)k0i2πLeik0L+ik0|r|2/2L,
Dψ,ψ(r,r)|ψT(r1,r1)-ψT(r1+r,r1+r)|2=2.91k02L101Cn,T2(sL1)|r(1-s)+rs|5/3ds,
E1(r,t)=PD2k0d2i2πL1Ξ(k0r/L1)×nfn(t-L1/c)eψT(r,nd)e-ik0L1r·nd,
Ξ(k)sin(kxd/2)kxd/2sin(kyd/2)kyd/2
ρ0(2.91k02L101Cn,T2(sL1)(1-s)5/3ds)-3/5.
T*(r1)T(r2)=λ02πT(r1)δ(r1-r2)
E2(r,t)=T(r)E1(r,t-L2/c)hFS(r-r;L2)eψR(r,r)dr.
cn=1T-T/2T/2gn(t)i(t)dt,
gn(t)f0*(t)fn(t)-|f0(t)|2δ0,n,
i(t)=hR(t-τ)i(τ)dτ,
HR(ω)=F[hR(t)]=e-2ω2/ωR2-e-2ω2/ωN2,
i(t)|P(t)=GηP(t)+λd
Δi(t1)Δi(t2)|P(·)=[FG2(ηP(t1)+λd)+NT]δ(t2-t1),
P(t)=AR|E2(r,t)+EB(r,t)|2dr.
EB*(r1,t1)EB(r2,t2)=IBKB(r2-r1)RB(t2-t1),
cn=1T-T/2T/2-hR(t-τ)gn(t)i(τ)dτdt.
cn=ΩR2π×2ηGPd2D2d2λ02L12×T¯ne-D(n)/2,
T¯nT(r)ei2πdn·r/λ0L1dr
D(n)D(0,nd)=|n|5/3(ρ0/d)5/3
rdλ0L1Md=2M+12M2λ0L1D2λ0L1D,
rmax(λ0L1Md,λ0L1Mtd)2λ0L1min(D,28/5ρ0).
cns=1πL22(ARe2χR(r,r)dr)(ηGPD2×d4λ02L12)eψT*(0,0)+ψT(0,nd)T¯n,
|cns|2=(ΩR2π×2ηGPd2D2d2λ02L12)2e4Kχ,χ(0,nd)|T¯n|2,
SNR=|cn|2Δcn2,
|Δcn|2=1T2-T/2T/2-T/2T/2hB(t1-τ1)hB(t2-τ2)×cov(gn*(t1)i(τ1),gn(t2)i(τ2))dτ1dτ2dt1dt2,
cov(gn*(t1)i(τ1),gn(t2)i(τ2))=[FG2ηgn*(t1)gn(t2)P2(τ1)+gn*(t1)gn(t2){FG2(ηIBAR+λd)+NT}]δ(τ1-τ2)+G2η2gn*(t1)gn(t2)P2(τ1)P2(τ2)-G2η2gn*(t1)P2(τ1)gn(t2)P2(τ2),
SNR=T|T¯n|2e-D(n)TD1+T0D2+D3,
D1ΩSΩRϵ0+|T¯n|2(1-e-D(n)),
D2π2(2M+1)4ϵ0(1+ΩSΩR),
D31κ2[2FGκT¯0+FG2ηIBAR+FG2λd+NT].
SNRmax=ΩRΩS×|T¯n|2n|T¯n|2/(2M+1)2.
rlr=2λ0LDR1+DR24ρR2,

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