Abstract

We derive the time-variant second-order statistics of the depth-scan photocurrent in time-domain optical coherence tomography (TD-OCT) systems using polarized thermal light sources and superluminescent diodes (SLDs). Since the asymptotic-joint-probability-distribution function (JPDF) of the photocurrent due to polarized thermal light is Gaussian and the signal-noise-ratio in TD-OCT is typically high (>80dB), the JPDF of the depth-scan photocurrent could be approximated as a Gaussian random process that is completely determined by its second-order statistics. We analyze both direct and differential light detection schemes and include the effect of electronic thermal fluctuations. Our results are a necessary prerequisite for future development of statistical image processing techniques for TD-OCT.

© 2008 Optical Society of America

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
    [CrossRef] [PubMed]
  2. B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).
  3. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
    [CrossRef]
  4. M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
    [CrossRef]
  5. J. M. Schmitt, "Restoration of optical coherence images of living tissue using the CLEAN algorithm," J. Biomed. Opt. 3, 66-75 (1998).
    [CrossRef]
  6. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, "Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media," Appl. Opt. 42, 204-217 (2003).
    [CrossRef] [PubMed]
  7. D. L. Marks, T. S. Ralston, and S. A. Boppart, "Speckle reduction by I-divergence regularization in optical coherence tomography," J. Opt. Soc. Am. A 22, 2366-2371 (2005).
    [CrossRef]
  8. A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
    [CrossRef] [PubMed]
  9. D. C. Adler, T. H. Ko, and J. G. Fujimoto, "Speckle reduction in optical coherence tomography images by use of a spatially adaptive wavelet filter," Opt. Lett. 29, 2878-2880 (2004).
    [CrossRef]
  10. D. Levitz, L. Thrane, M. Frosz, P. Andersen, C. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P. Hansen, "Determination of optical scattering properties of highly scattering media in optical coherence tomography images," Opt. Express 12, 249-259 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. A. G. Podoleanu and D. A. Jackson, "Noise analysis of a combined optical coherence tomograph and a confocal scanning ophthalmoscope," Appl. Opt. 38, 2116-2127 (1999).
    [CrossRef]
  14. A. M. Rollins and J. A. Izatt, "Optical interferometer designs for optical coherence tomography," Opt. Lett. 24, 1484-1486 (1999).
    [CrossRef]
  15. A. G. Podoleanu, "Unbalanced versus balanced operation in an optical coherence tomography system," Appl. Opt. 39, 173-182 (2000).
    [CrossRef]
  16. C. C. Rosa and A. G. Podoleanu, "Limitation to the achievable signal-to-noise ratio in optical coherence tomography due to mismatch of the balanced receiver," Appl. Opt. 43, 4802-4815 (2004).
    [CrossRef] [PubMed]
  17. M. Bashkansky and J. Reintjes, "Statistics and reduction of speckle in optical coherence tomography," Opt. Lett. 25, 545-547 (2000).
    [CrossRef]
  18. Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
    [CrossRef]
  19. B. Saleh, Photoelectron Statistics (Springer-Verlag, 1978).
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    [CrossRef]
  21. J. Rice, "On generalized shot noise," Adv. Appl. Probab. 9, 553-565 (1977).
    [CrossRef]
  22. C. L. Mehta, "Theory of photoelectron counting," in Progress in Optics, Vol. VIII, E.Wolf, ed. (Elsevier, 1970), pp. 373-440.
    [CrossRef]
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    [CrossRef]
  25. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).

2006 (1)

2005 (1)

2004 (4)

2003 (2)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, "Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media," Appl. Opt. 42, 204-217 (2003).
[CrossRef] [PubMed]

2001 (1)

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

2000 (2)

1999 (3)

1998 (2)

K. Takada, "Noise in optical low coherence refiectometry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
[CrossRef]

J. M. Schmitt, "Restoration of optical coherence images of living tissue using the CLEAN algorithm," J. Biomed. Opt. 3, 66-75 (1998).
[CrossRef]

1997 (2)

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).

M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
[CrossRef]

1995 (1)

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

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1978 (1)

B. Saleh, Photoelectron Statistics (Springer-Verlag, 1978).

1977 (1)

J. Rice, "On generalized shot noise," Adv. Appl. Probab. 9, 553-565 (1977).
[CrossRef]

1970 (2)

C. L. Mehta, "Theory of photoelectron counting," in Progress in Optics, Vol. VIII, E.Wolf, ed. (Elsevier, 1970), pp. 373-440.
[CrossRef]

B. Picinbono, C. Bendjaballah, and J. Pouget, "Photoelectron shot noise," J. Math. Phys. 11, 2166-2176 (1970).
[CrossRef]

1965 (1)

B. M. Oliver, "Thermal and Quantum Noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Adler, D. C.

Andersen, C.

Andersen, P.

Andersson-Engels, S.

Bashkansky, M.

Bendjaballah, C.

B. Picinbono, C. Bendjaballah, and J. Pouget, "Photoelectron shot noise," J. Math. Phys. 11, 2166-2176 (1970).
[CrossRef]

Boppart, S. A.

Bouma, B. E.

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

Cao, H.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Carney, P. S.

Chang, R.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Charalambous, I.

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Dogariu, A.

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Du, G.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Frosz, M.

Fujimoto, J. G.

D. C. Adler, T. H. Ko, and J. G. Fujimoto, "Speckle reduction in optical coherence tomography images by use of a spatially adaptive wavelet filter," Opt. Lett. 29, 2878-2880 (2004).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Gao, D.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Han, W.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Hansen, P.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Izatt, J. A.

A. M. Rollins and J. A. Izatt, "Optical interferometer designs for optical coherence tomography," Opt. Lett. 24, 1484-1486 (1999).
[CrossRef]

M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
[CrossRef]

Jackson, D. A.

Ko, T. H.

Kulkarni, M. D.

M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
[CrossRef]

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Levitz, D.

Li, X.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Liu, Y.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Mandel, L.

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

Marks, D. L.

Mehta, C. L.

C. L. Mehta, "Theory of photoelectron counting," in Progress in Optics, Vol. VIII, E.Wolf, ed. (Elsevier, 1970), pp. 373-440.
[CrossRef]

Oldenburg, A. L.

Oliver, B. M.

B. M. Oliver, "Thermal and Quantum Noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Picinbono, B.

B. Picinbono, C. Bendjaballah, and J. Pouget, "Photoelectron shot noise," J. Math. Phys. 11, 2166-2176 (1970).
[CrossRef]

Pleasea, L.

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Podoleanu, A.

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Podoleanu, A. G.

Pouget, J.

B. Picinbono, C. Bendjaballah, and J. Pouget, "Photoelectron shot noise," J. Math. Phys. 11, 2166-2176 (1970).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Ralston, T. S.

Reintjes, J.

Reynolds, J. J.

Rice, J.

J. Rice, "On generalized shot noise," Adv. Appl. Probab. 9, 553-565 (1977).
[CrossRef]

Rollins, A. M.

Rosa, C. C.

Rosen, R.

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, 1978).

Schmitt, J. M.

J. M. Schmitt, "Restoration of optical coherence images of living tissue using the CLEAN algorithm," J. Biomed. Opt. 3, 66-75 (1998).
[CrossRef]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Song, J.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Swartling, J.

Takada, K.

K. Takada, "Noise in optical low coherence refiectometry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
[CrossRef]

Tearney, G. J.

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

Thomas, C. W.

M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
[CrossRef]

Thrane, L.

Valanciunaite, J.

Wolf, E.

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

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).

Zhao, Y.

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

Adv. Appl. Probab. (1)

J. Rice, "On generalized shot noise," Adv. Appl. Probab. 9, 553-565 (1977).
[CrossRef]

Appl. Opt. (4)

Electron. Lett. (1)

M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 33, 1365-1376 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Takada, "Noise in optical low coherence refiectometry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
[CrossRef]

J. Appl. Phys. (1)

Y. Zhao, W. Han, J. Song, X. Li, Y. Liu, D. Gao, G. Du, H. Cao, and R. Chang, "Spontaneous emission factor for semiconductor super luminescent diodes," J. Appl. Phys. 85, 3945-3948 (1999).
[CrossRef]

J. Biomed. Opt. (1)

J. M. Schmitt, "Restoration of optical coherence images of living tissue using the CLEAN algorithm," J. Biomed. Opt. 3, 66-75 (1998).
[CrossRef]

J. Math. Phys. (1)

B. Picinbono, C. Bendjaballah, and J. Pouget, "Photoelectron shot noise," J. Math. Phys. 11, 2166-2176 (1970).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (3)

Phys. Med. Biol. (1)

A. Podoleanu, I. Charalambous, L. Pleasea, A. Dogariu, and R. Rosen, "Correction of distortions in optical coherence tomography imaging of the eye," Phys. Med. Biol. 49, 1277-1294 (2004).
[CrossRef] [PubMed]

Proc. IEEE (1)

B. M. Oliver, "Thermal and Quantum Noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other (5)

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

B. Saleh, Photoelectron Statistics (Springer-Verlag, 1978).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).

C. L. Mehta, "Theory of photoelectron counting," in Progress in Optics, Vol. VIII, E.Wolf, ed. (Elsevier, 1970), pp. 373-440.
[CrossRef]

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

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Figures (2)

Fig. 1
Fig. 1

Typical TD-OCT setup with direct optical detection.

Fig. 2
Fig. 2

Typical TD-OCT setup with differential optical detection.

Equations (24)

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

i ( t ) = h det ( t t ) d N ( t ) ,
E [ i ( t ) ] = h det ( t t ) E [ d N ( t ) ] ,
E [ i ( t 1 ) i ( t 2 ) ] = h det ( t 1 t 1 ) h det ( t 2 t 2 ) E [ d N ( t 1 ) d N ( t 2 ) ] ,
E [ d N ( t ) ] = ρ E [ I ( t ) ] d t ,
E [ d N ( t 1 ) d N ( t 2 ) ] = { ρ 2 Γ 1 ( t 1 , t 2 ) + ρ E [ I ( t 1 ) ] δ ( t 2 t 1 ) } d t 1 d t 2 ,
E [ i dif ( t ) ] = E [ i 2 ( t ) i 1 ( t ) ] = E [ i 2 ( t ) ] E [ i 1 ( t ) ] ,
E [ i dif ( t 1 ) i dif ( t 2 ) ] = E [ i 1 ( t 1 ) i 1 ( t 2 ) ] 2 E [ i 1 ( t 1 ) i 2 ( t 2 ) ] + E [ i 2 ( t 1 ) i 2 ( t 2 ) ] ,
E [ i 1 ( t 1 ) i 2 ( t 2 ) ] = h det ( t 1 t 1 ) h det ( t 2 t 2 ) E [ d N 1 ( t 1 ) d N 2 ( t 2 ) ] ,
E [ d N 1 ( t 1 ) d N 2 ( t 2 ) ] = ρ 2 Γ I 1 , I 2 d t 1 d t 2 ,
I OCT ( t ) = V r [ t f ( t ) ] 2 + 0 V s ( t τ ) R ( τ ) d τ 2 + 2 Re { V r [ t f ( t ) ] 0 V s * ( t τ ) R ( τ ) d τ } ,
E [ I OCT ( t ) ] I OCT ( t ) a V src [ t f ( t ) ] 2 + 2 Re { a b * V src [ t f ( t ) ] 0 V src * ( t τ ) R ( τ ) d τ } ,
E [ I OCT ( t ) ] = a 2 I src + I src 2 Re { a b * 0 γ src [ τ f ( t ) ] R ( τ ) d τ } ,
E [ I OCT ( t ) ] = a 2 I src + I src 2 Re { a b * 0 γ src * ( α t τ ) R ( τ ) d τ } .
E [ i OCT ( t ) ] = ρ a 2 I src h det ( t ) d t + ρ I src h det ( t ) 2 Re { a b * [ γ src R ] ( α t ) } .
Γ I OCT ( t 1 , t 2 ) = E I OCT dc ( t 1 ) I OCT dc ( t 2 ) + 2 E I OCT dc ( t 1 ) I OCT int ( t 2 ) + E I OCT int ( t 1 ) I OCT int ( t 2 ) ,
E [ V 1 * V 2 * V 3 V 4 ] = E [ V 1 * V 3 ] E [ V 2 * V 4 ] + E [ V 1 * V 4 ] E [ V 2 * V 3 ] .
E [ I OCT dc ( t 1 ) I OCT dc ( t 2 ) ] = a 4 I src 2 ( 1 + γ src ( t 2 t 1 ) 2 ) ,
2 E [ I OCT dc ( t 1 ) I OCT int ( t 2 ) ] = 4 a 2 I src 2 Re { a b * [ 0 γ src * ( α t 2 τ ) R ( τ ) d τ + γ src ( t 2 t 1 ) 0 γ src * ( t 2 t 1 τ ) R ( τ ) d τ ] } ,
E [ I OCT int ( t 1 ) I OCT int ( t 2 ) ] = I src 2 2 Re { ( a b * ) 2 [ 0 γ src * ( α t 1 τ ) R ( τ ) d τ 0 γ src * ( α t 2 τ ) R ( τ ) d τ + 0 γ src ( t 2 t 1 + τ ) R ( τ ) d τ 0 γ src ( t 1 t 2 + τ ) R ( τ ) d τ ] + a 2 b 2 [ 0 γ src * ( α t 1 τ ) R ( τ ) d τ 0 γ src ( α t 2 τ ) R * ( τ ) d τ + γ src ( t 1 t 2 ) 0 0 γ src ( t 2 t 1 + τ 1 τ 2 ) R ( τ 1 ) R * ( τ 2 ) d τ 1 d τ 2 ] } .
E [ i OCT ( t 1 ) i OCT ( t 2 ) ] = ρ 2 h det ( t 1 t 1 ) h det ( t 2 t 2 ) Γ I OCT ( t 1 , t 2 ) d t 1 d t 2 + ρ h det ( t 1 t 1 ) h det ( t 2 t 1 ) E [ I OCT ( t 1 ) ] d t 1 ,
E [ i OCT dif ( t ) ] = ρ I src h det ( t ) 2 Re { a b * [ γ src R ] ( α t ) } .
Γ I OCT 1 , I OCT 2 ( t 1 , t 2 ) = E { [ I OCT 1 dc ( t 1 ) + I OCT 1 int ( t 1 ) ] [ I OCT 2 dc ( t 2 ) I OCT 2 int ( t 2 ) ] } = E [ I OCT 1 dc ( t 1 ) I OCT 2 dc ( t 2 ) ] E [ I OCT 1 int ( t 1 ) I OCT 2 int ( t 2 ) ] .
E [ i OCT dif ( t 1 ) i OCT dif ( t 2 ) ] = ρ 2 h det ( t 1 t 1 ) h det ( t 2 t 2 ) Γ OCT int ( t 1 , t 2 ) d t 1 d 2 + ρ a 2 I src h det ( t 1 t 1 ) h det ( t 2 t 1 ) d t 1 ,
E [ n elect ( t 1 ) n elect ( t 2 ) ] = 4 K B T B R L δ ( t 2 t 1 ) ,

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