Abstract

Heterodyne detection in the limit of weak (a few photons) local oscillator and signal power levels has been largely neglected in the past, as authors almost always assumed that the noise was dominated by the shot noise from a strong local oscillator. We present the theory for heterodyne detection of diffuse and specular targets at arbitrary power levels, including the case where the local oscillator power is only a few photons per coherent integration period. The theory was tested with experimental results, and was found to show good agreement. We show how to interpret the power spectral density of the heterodyne signal and how to determine the optimal number of signal and local oscillator photons per coherent integration.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
    [PubMed]
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    [CrossRef]
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2007 (1)

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number resolving detector with 10-bits of resolution,” Phys. Rev. A 75, 062325 (2007).
[CrossRef]

2006 (1)

2005 (1)

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

2002 (1)

2000 (1)

1998 (1)

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549-1555 (1998).
[CrossRef]

1997 (1)

1995 (2)

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

R. Dandliker, M. Geiser, C. Giunti, S. Zatti, and G. Margheri, “Improvement of speckle statistics in double-wavelength superheterodyne interferometry,” Appl. Opt. 34, 7197-7201 (1995).
[CrossRef] [PubMed]

1991 (1)

1984 (1)

1983 (1)

1981 (1)

1975 (1)

E. Jakeman, C. J. Oliver, and E. R. Pike, “Optical homodyne detection,” Adv. Phys. , 24, 349-405 (1975).
[CrossRef]

Ashcom, J. B.

J. B. Ashcom, S. Kaushik, and R. M. Heinrichs, “Coherent Single-Photon Counting with InGaAsP/InP Avalanche Photodiode Arrays,” presented at the 12th Coherent Laser Radar Conference, Bar Harbor, Maine, 15-20 June 2003.

Aull, B.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Aull, B. F.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Aversa, J. C.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Bartholomew, B. J.

Bennett, C. A.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Burns, J.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Capron, B. A.

Chan, K. P.

Chang, J. T.

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number resolving detector with 10-bits of resolution,” Phys. Rev. A 75, 062325 (2007).
[CrossRef]

Charbon, E.

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
[PubMed]

Chen, C.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Crutcher, R. I.

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Dabas, A.

Dandliker, R.

Dauler, E. A.

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number resolving detector with 10-bits of resolution,” Phys. Rev. A 75, 062325 (2007).
[CrossRef]

Delaval, A.

Delville, P.

Donnelly, J. P.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Duerr, E. K.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Emery, M. S.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Favi, C.

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
[PubMed]

Favreau, X.

Felton, B.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Flamant, P. H.

Geiser, M.

Gersbach, M.

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
[PubMed]

Giunti, C.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 46, 49, 52, 473, 476, 533, 534, and 536.

Groves, S. H.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Hanson, H.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Harney, R. C.

Heinrichs, R. M.

J. B. Ashcom, S. Kaushik, and R. M. Heinrichs, “Coherent Single-Photon Counting with InGaAsP/InP Avalanche Photodiode Arrays,” presented at the 12th Coherent Laser Radar Conference, Bar Harbor, Maine, 15-20 June 2003.

Huston, M. A.

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Hutchinson, D. P.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Jakeman, E.

E. Jakeman, C. J. Oliver, and E. R. Pike, “Optical homodyne detection,” Adv. Phys. , 24, 349-405 (1975).
[CrossRef]

Jelalian, A. V.

A. V. Jelalian, Laser Radar Systems, (Artech House, 1992), p. 20.

Jiang, L. A.

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number resolving detector with 10-bits of resolution,” Phys. Rev. A 75, 062325 (2007).
[CrossRef]

J. X. Luu and L. A. Jiang, “Saturation effects in heterodyne detection with Geiger-mode InGaAs avalanche photodiode detector arrays,” Appl. Opt. 45, 3798-3804 (2006).
[CrossRef] [PubMed]

Kaushik, S.

J. B. Ashcom, S. Kaushik, and R. M. Heinrichs, “Coherent Single-Photon Counting with InGaAsP/InP Avalanche Photodiode Arrays,” presented at the 12th Coherent Laser Radar Conference, Bar Harbor, Maine, 15-20 June 2003.

Keast, C.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Killinger, D. K.

Kluter, T.

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
[PubMed]

Knecht, J.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Leeb, W. R.

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549-1555 (1998).
[CrossRef]

Liau, Z. L.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Loomis, A.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Lucke, R. L.

Luu, J. X.

Mahan, J.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Mahoney, L. J.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Margheri, G.

McIntosh, K. A.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Miller, G. H.

Molvar, K.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Napoleone, A.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Niclass, C.

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
[PubMed]

Oakley, D. C.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Oliver, C. J.

E. Jakeman, C. J. Oliver, and E. R. Pike, “Optical homodyne detection,” Adv. Phys. , 24, 349-405 (1975).
[CrossRef]

Pike, E. R.

E. Jakeman, C. J. Oliver, and E. R. Pike, “Optical homodyne detection,” Adv. Phys. , 24, 349-405 (1975).
[CrossRef]

Renzi, M.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Richards, R. K.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Rickard, L. J.

Shapiro, J. H.

Shaver, D. C.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Simpson, M. L.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Sitter, D. N.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, and D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and applications,” Appl. Opt. 36, 6913-6920 (1997).
[CrossRef]

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Soares, A.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Starr, E. F.

Streiff, M. L.

Suntharalingam, V.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

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Vernon, S.

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Wachter, E. A.

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Wang, J. Y.

Warner, K.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

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P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549-1555 (1998).
[CrossRef]

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B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Yost, D.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

Young, D.

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
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[CrossRef]

Appl. Opt. (8)

J. Mod. Opt. (1)

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549-1555 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number resolving detector with 10-bits of resolution,” Phys. Rev. A 75, 062325 (2007).
[CrossRef]

Proc. SPIE (1)

J. P. Donnelly, K. A. McIntosh, D. C. Oakley, A. Napoleone, S. H. Groves, S. Vernon, L. J. Mahoney, K. Molvar, J. Mahan, J. C. Aversa, E. K. Duerr, Z. L. Liau, B. F. Aull, and D. C. Shaver, “1−μm Geiger-mode detector development,” Proc. SPIE 5791, 281-287 (2005).
[CrossRef]

Proc. SPIE-Int. Soc. Opt. Eng. (1)

D. P. Hutchinson, M. L. Simpson, C. A. Bennett, R. K. Richards, M. S. Emery, R. I. Crutcher, D. N. Sitter Jr., E. A. Wachter, and M. A. Huston, “Coherent infrared imaging camera (CIRIC),” Proc. SPIE-Int. Soc. Opt. Eng. , 2540, 204-209 (1995).

Other (6)

B. Aull, J. Burns, C. Chen, B. Felton, H. Hanson, C. Keast, J. Knecht, A. Loomis, M. Renzi, A. Soares, V. Suntharalingam, K. Warner, D. Wolfson, D. Yost, and D. Young, “Laser radar imager based on 3D integration of Geiger-mode avalanche photodiodes with two SOI timing circuit layers,” Solid-State Circuits, Conference Digest of Technical Papers (IEEE, 2006).
[CrossRef]

C. Niclass, C. Favi, T. Kluter, M. Gersbach, and E. Charbon, “A 128×128 single-photon imager with on-chip column-level 10b time-to-digital converter array capable of 97 ps resolution,” presented at the 2008 IEEE International Solid-State Circuits Conference San Francisco, Calif., 3-7 February 2008.
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SensL, DigitalAPD, Photon Counting and Photon Timing Camera.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 46, 49, 52, 473, 476, 533, 534, and 536.

J. B. Ashcom, S. Kaushik, and R. M. Heinrichs, “Coherent Single-Photon Counting with InGaAsP/InP Avalanche Photodiode Arrays,” presented at the 12th Coherent Laser Radar Conference, Bar Harbor, Maine, 15-20 June 2003.

A. V. Jelalian, Laser Radar Systems, (Artech House, 1992), p. 20.

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Figures (13)

Fig. 1
Fig. 1

Heterodyne detection laser radar that uses photon-number-resolving detectors. The photon-number-resolving detector is constructed out of an array of photon-counting detectors. By collecting and time stamping the photon arrivals across the entire array or macropixel, the beat signal can be recovered.

Fig. 2
Fig. 2

Power spectral density (PSD) of the detected current for a heterodyne laser radar receiver. The PSD is shown for two different amounts of incoherent averaging, (a) 1 pulse and (b) 100 pulses. Only the portion of the PSD in the vicinity of the beat frequency is shown.

Fig. 3
Fig. 3

Example that shows the photoelectron arrivals for three pulses. The sum of delta functions in time results in a phasor sum in the Fourier domain. The sum of the phasors have a 2D Gaussian PDF that is zero mean for the noise and nonzero mean for the signal.

Fig. 4
Fig. 4

Signal-to-noise ratio is plotted as a function of N ¯ S for the cases N ¯ LO (solid curves) and N ¯ LO = N ¯ S (dashed curves). Each plot has curves that correspond to mixing efficiencies of m 2 = 0.10 , 0.32, and 1.00. The single-pulse SNR is plotted in (upper left) for a diffuse target and in (lower left) for a specular target. The total SNR is plotted in (upper right) for a diffuse target and in (lower right) for a specular target.

Fig. 5
Fig. 5

Optimal value of N ¯ S for a given heterodyne mixing efficiency, m 2 , and diffuseness parameter M.

Fig. 6
Fig. 6

Probability density function of the PSD at f = f i f and f n for a single received pulse. For this plot, N ¯ D = 0 , K = 1 , M = 2 , m 2 = 0.32 , N ¯ S = 10.6 , and N ¯ LO = 5.7 .

Fig. 7
Fig. 7

Probability density function of the PSD at f = f i f and f n for K = 10 received pulses. For this plot, K = 10 , M = 2 , m 2 = 0.32 , N ¯ S = 10.6 , and N ¯ LO = 5.7 .

Fig. 8
Fig. 8

Experimental setup for validating the presented theory.

Fig. 9
Fig. 9

Probability density function of the number of local oscillator photoelectrons per pulse. The black bars denote the experimental data and the white bars denote the theoretically expected values given by Eq. (5). The average local oscillator photoelectron level per pulse, N ¯ LO , is equal to (top left) 6.7, (top right) 14.2, (middle left) 27, (middle right) 57.1, and (bottom) 80.5.

Fig. 10
Fig. 10

Probability density function of the number of signal photoelectrons per pulse. The black bars denote the data and the white bars denote the theoretically expected values given by Eq. (7). The average local oscillator photoelectron level per pulse, N ¯ S , is equal to (top left) 10.6, (top right) 21.2, (middle left) 42.7, (middle right) 71.7, and (bottom) 99.6.

Fig. 11
Fig. 11

Carrier-to-noise ratio as a function of the number of signal photoelectrons. The circles denote the measured CNRs, and the triangles denote the theoretical values as predicted by Eq. (32) with m 2 = 0.32 and M = 2 .

Fig. 12
Fig. 12

Single-pulse SNR as a function of the number of signal photons. The circles denote the measured SNRs, the upside-down triangles denote the theoretical values as predicted by Eq. (37) with m 2 = 0.32 , and the upright triangles denote the theoretical values as predicted by Eq. (37) with m 2 = 0.32 and M = 2 .

Fig. 13
Fig. 13

Probability density function of (top) the PSD background at f = f n and (bottom) the PSD peak at f = f i f . The theoretical curves are given by Eq. (23) with m 2 = 0.32 and M = 2 .

Equations (43)

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I ( k ) ( t ) = i = 1 N S ( k ) + N LO ( k ) + N D ( k ) δ ( t t i ) ,
I ˜ ( k ) ( f ) = i = 1 N S ( k ) + N LO ( k ) + N D ( k ) exp ( j 2 π f t i ) = i = 1 N S ( k ) + N LO ( k ) + N D ( k ) exp ( j φ i ) ,
I classical ( k ) ( t ) = N S ( k ) + N LO ( k ) + N D ( k ) + 2 m N S ( k ) N LO ( k ) cos ( 2 π f i f t + θ ( k ) ) ,
P N LO ( k ) ( q ) = ( N ¯ LO ) q exp ( N ¯ LO ) q ! ,
E [ N LO ( k ) ] = N ¯ LO ,
E ( N LO ( k ) ) 2 = N ¯ LO + ( N ¯ LO ) 2 .
P N S ( k ) ( q ) = Γ ( q + M ) Γ ( q + 1 ) Γ ( M ) ( 1 + M N ¯ S ) q ( 1 + N ¯ S M ) M ,
E [ N S ( k ) ] = N ¯ S , E ( N S ( k ) ) 2 = N ¯ S + ( 1 + 1 M ) ( N ¯ S ) 2 .
P N S ( k ) ( q ) = 1 1 + N ¯ S ( N ¯ S 1 + N ¯ S ) q .
E N S ( k ) = N ¯ S , E ( N S ( k ) ) 2 = N ¯ S + 2 ( N ¯ S ) 2 .
P φ i ( φ ) = { 1 2 π [ 1 + 2 m N S ( k ) N LO ( k ) N S ( k ) + N LO ( k ) + N D ( k ) cos ( φ ) ] f = f i f , 1 2 π f = f n ,
M φ i ( λ ) = 0 2 π P φ i ( φ ) exp ( j λ φ ) d φ = { δ ( λ ) + m N S ( k ) N LO ( k ) N S ( k ) + N LO ( k ) + N D ( k ) δ ( λ 1 ) + m N S ( k ) N LO ( k ) N S ( k ) + N LO ( k ) + N D ( k ) δ ( λ + 1 ) for f = f i f δ ( λ ) for f = f n .
P RI ( r , i ) = 1 2 π σ r σ i exp { ( r r ¯ ) 2 2 σ r 2 ( i i ¯ ) 2 2 σ i 2 } ,
r ¯ = i = 1 N S ( k ) + N LO ( k ) + N D ( k ) cos φ i = ( N S ( k ) + N LO ( k ) + N D ( k ) ) M φ i ( 1 ) = { m N S ( k ) N LO ( k ) r ¯ i f , f = f i f 0 r ¯ n , f = f n , i ¯ = i = 1 N S ( k ) + N LO ( k ) + N D ( k ) sin φ i = N S ( k ) + N LO ( k ) + N D ( k ) 2 j [ M φ i ( 1 ) M φ i ( 1 ) ] = 0 ,
σ r 2 = N S ( k ) + N LO ( k ) + N D ( k ) 2 ( 1 + M φ i ( 2 ) 2 M φ i 2 ( 1 ) ) = { N S ( k ) + N LO ( k ) + N D ( k ) 2 ( 1 2 m 2 N S ( k ) N LO ( k ) ( N S ( k ) + N LO ( k ) + N D ( k ) ) 2 ) , f = f if N S ( k ) + N LO ( k ) + N D ( k ) 2 , f = f n , σ i 2 = N S ( k ) + N LO ( k ) + N D ( k ) 2 ( 1 M φ i ( 2 ) ) = N S ( k ) + N LO ( k ) + N D ( k ) 2 , for f = f i f , f = f n .
P A ( a ) = a σ i 2 exp ( a 2 2 σ i 2 ) , f = f n .
P A ( a ) = a σ i 2 exp ( a 2 + r ¯ i f 2 2 σ i 2 ) I 0 ( a r ¯ i f σ i 2 ) , f = f i f ,
P B ( b ) = d a d b P A ( a ) = d a d b P A ( b ) = P A ( b ) 2 b = { 1 2 σ i 2 exp ( b + r ¯ i f 2 2 σ i 2 ) I 0 ( r ¯ i f b σ i 2 ) P B i f ( b ) , f = f if 1 2 σ i 2 exp ( b 2 σ i 2 ) P B n ( b ) , f = f n ,
E [ B if ] = 0 b P B if ( b ) d b = 0 b d a d b P A ( a ) d b = 0 a 2 P A ( a ) d a = E [ A 2 ] = r ¯ if 2 + 2 σ i 2 ,
E [ B i f 2 ] = 0 a 4 P A ( a ) d a = r ¯ i f 4 + 8 r ¯ i f 2 σ i 2 + 8 σ i 4 ,
E [ B n ] = 2 σ i 2 ,
E [ B n 2 ] = 8 σ i 4 ,
P G ( g ) = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 P B ( g | N S ( k ) , N LO ( k ) , N D ( k ) ) P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) .
E [ G ] = g = 0 g P G ( g ) = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { g = 0 g P B ( g | N S ( k ) , N LO ( k ) , N D ( k ) ) } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) , E [ G 2 ] = g = 0 g 2 P G ( g ) = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { g = 0 g 2 P B ( g | N S ( k ) , N LO ( k ) , N D ( k ) ) } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) ,
σ G 2 = E [ G 2 ] ( E [ G ] ) 2 .
g = 0 g P B ( g | N S ( k ) , N LO ( k ) , N D ( k ) ) = E [ B | N S ( k ) , N LO ( k ) , N D ( k ) ] , g = 0 g 2 P B ( g | N S ( k ) , N LO ( k ) , N D ( k ) ) = E [ B 2 | N S ( k ) , N LO ( k ) , N D ( k ) ] .
E B i f | N S ( k ) , N LO ( k ) , N D ( k ) = r ¯ i f 2 + 2 σ i 2 = m 2 N S ( k ) N LO ( k ) + ( N S ( k ) + N LO ( k ) + N D ( k ) ) , E [ B i f 2 | N S ( k ) , N LO ( k ) , N D ( k ) ] = r ¯ i f 4 + 8 r ¯ i f 2 σ i 2 + 8 σ i 4 , = m 4 ( N S ( k ) N LO ( k ) ) 2 + 4 m 2 N S ( k ) N LO ( k ) ( N S ( k ) + N LO ( k ) + N D ( k ) ) + 2 ( N S ( k ) + N LO ( k ) + N D ( k ) ) 2 , E [ B n | N S ( k ) , N LO ( k ) , N D ( k ) ] = 2 σ i 2 = N S ( k ) + N LO ( k ) + N D ( k ) , E [ B n 2 | N S ( k ) , N LO ( k ) , N D ( k ) ] = 8 σ i 4 = 2 ( N S ( k ) + N LO ( k ) + N D ( k ) ) 2 .
E [ G i f ] = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { m 2 N S ( k ) N LO ( k ) + ( N S ( k ) + N LO ( k ) + N D ( k ) ) } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) = m 2 E [ N S ( k ) ] E [ N LO ( k ) ] + E [ N S ( k ) ] + E [ N LO ( k ) ] + E [ N D ( k ) ] , E [ G i f 2 ] = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { m 4 ( N S ( k ) N LO ( k ) ) 2 + 4 m 2 N S ( k ) N LO ( k ) ( N S ( k ) + N LO ( k ) + N D ( k ) ) + 2 ( N S ( k ) + N LO ( k ) + N D ( k ) ) 2 } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) = m 4 E [ ( N S ( k ) ) 2 ] E [ ( N LO ( k ) ) 2 ] + 4 m 2 { E [ N LO ( k ) ] E [ ( N S ( k ) ) 2 ] + E [ N S ( k ) ] E [ ( N LO ( k ) ) 2 ] + E [ N S ( k ) ] E [ N LO ( k ) ] E [ N D ( k ) ] } + 2 E [ ( N S ( k ) ) 2 ] + 2 E [ ( N LO ( k ) ) 2 ] + 2 E [ ( N D ( k ) ) 2 ] + 4 E [ N S ( k ) ] E [ N LO ( k ) ] + 4 E [ N S ( k ) ] E [ N D ( k ) ] + 4 E [ N LO ( k ) ] E [ N D ( k ) ] , E [ G n ] = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { N S ( k ) + N LO ( k ) + N D ( k ) } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) = E [ N S ( k ) ] + E [ N LO ( k ) ] + E [ N D ( k ) ] , E [ G n 2 ] = N D ( k ) = 0 N LO ( k ) = 0 N S ( k ) = 0 { 2 ( N S ( k ) + N LO ( k ) + N D ( k ) ) 2 } P N S ( k ) ( N S ( k ) ) P N LO ( k ) ( N LO ( k ) ) P N D ( k ) ( N D ( k ) ) = 2 E [ ( N S ( k ) ) 2 ] + 2 E [ ( N L O ( k ) ) 2 ] + 2 E [ ( N D ( k ) ) 2 ] + 4 E [ N S ( k ) ] E [ N LO ( k ) ] + 4 E [ N S ( k ) ] E [ N D ( k ) ] + 4 E [ N LO ( k ) ] E [ N D ( k ) ] .
E [ G i f ] = m 2 N ¯ S N ¯ LO + ( N ¯ S + N ¯ LO + N ¯ D ) , E [ G i f 2 ] = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + 2 ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M + 4 m 2 N ¯ S N ¯ LO N ¯ D + m 4 ( 1 + 1 M ) N ¯ S 2 N ¯ LO 2 + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 4 ) N ¯ S N ¯ LO 2 + m 2 ( m 2 + 4 ) ( 1 + 1 M ) N ¯ S 2 N ¯ LO , E [ G n ] = N ¯ S + N ¯ LO + N ¯ D , E [ G n 2 ] = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + 2 ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M .
σ G i f 2 = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M + 2 m 2 N ¯ S N ¯ LO N ¯ D + m 4 N ¯ S 2 N ¯ LO 2 M + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 2 ) N ¯ S N ¯ LO 2 + m 2 ( m 2 + 2 + m 2 + 4 M ) N ¯ S 2 N ¯ LO , σ G n 2 = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M .
E [ G i f ] = m 2 N ¯ S N ¯ LO + ( N ¯ S + N ¯ LO + N ¯ D ) , E [ G i f 2 ] = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + 2 ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 4 m 2 N ¯ S N ¯ LO N ¯ D + m 4 N ¯ S 2 N ¯ LO 2 + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 4 ) ( N ¯ S N ¯ LO 2 + N ¯ S 2 N ¯ LO ) , E [ G n ] = N ¯ S + N ¯ LO + N ¯ D , E [ G n 2 ] = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + 2 ( N ¯ S + N ¯ LO + N ¯ D ) 2 .
σ G i f 2 = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 m 2 N ¯ S N ¯ LO N ¯ D + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 2 ) ( N ¯ S N ¯ LO 2 + N ¯ S 2 N ¯ LO ) , σ G n 2 = 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 .
CNR = G i f ¯ G n ¯ G n ¯ ,
CNR = m 2 N ¯ S N ¯ LO N ¯ S + N ¯ LO + N ¯ D ,
SNR = Q 2 = ( G i f ¯ G n ¯ σ G i f + σ G n ) 2 .
BER = 1 2 erfc ( Q 2 ) .
SNR = ( m 2 N ¯ S N ¯ LO ) 2 ( 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M + 2 m 2 N ¯ S N ¯ LO N ¯ D + m 4 N ¯ S 2 N ¯ LO 2 M + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 2 ) N ¯ S N ¯ LO 2 + m 2 ( m 2 + 2 + m 2 + 4 M ) N ¯ S 2 N ¯ LO + 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M ) 2 .
SNR = ( m 2 N ¯ S 1 + 1 + m 2 ( m 2 + 2 ) N ¯ S ) 2 .
SNR = m 4 N ¯ S 3 ( 4 + ( m 4 + 8 m 2 + 4 ) N ¯ S + 2 m 2 ( m 2 + 2 ) N ¯ S 2 + 4 + 4 N ¯ S ) 2 .
SNR = ( m 2 N ¯ S 1 + 1 + m 2 ( m 2 + 2 ) N ¯ S + m 4 M N ¯ S 2 ) 2 .
SNR = ( m 2 N ¯ S 2 4 N ¯ S + 4 N ¯ S 2 + 2 N ¯ S 2 M + m 2 ( m 2 + 8 ) N ¯ S 2 + m 2 ( 2 m 2 + 4 + m 2 + 4 M ) N ¯ S 3 + m 4 N ¯ S 4 M + 4 N ¯ S + 4 N ¯ S 2 + 2 N ¯ S 2 M ) 2 .
SNR total = ( μ 1 μ 0 σ 1 + σ 0 ) 2 = ( K G ¯ i f K G ¯ n K σ G i f + K σ G i f ) 2 = K ( G ¯ i f G ¯ n σ G i f + σ G i f ) 2 = K SNR = N total SNR N ¯ S ,
SNR total = N total m 4 N ¯ S N ¯ LO 2 ( 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M + 2 m 2 N ¯ S N ¯ LO N ¯ D + m 4 N ¯ S 2 N ¯ LO 2 M + m 2 ( m 2 + 8 ) N ¯ S N ¯ LO + m 2 ( m 2 + 2 ) N ¯ S N ¯ LO 2 + m 2 ( m 2 + 2 + m 2 + 4 M ) N ¯ S 2 N ¯ LO + 2 ( N ¯ S + N ¯ LO + N ¯ D ) + ( N ¯ S + N ¯ LO + N ¯ D ) 2 + 2 N ¯ S 2 M ) 2 .

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