Abstract

Optimization of an optical coherence imaging (OCI) system on the basis of task performance is a challenging undertaking. We present a mathematical framework based on task performance that uses statistical decision theory for the optimization and assessment of such a system. Specifically, we apply the framework to a relatively simple OCI system combined with a specimen model for a detection task and a resolution task. We consider three theoretical Gaussian sources of coherence lengths of 2, 20, and 40μm. For each of these coherence lengths we establish a benchmark performance that specifies the smallest change in index of refraction that can be detected by the system. We also quantify the dependence of the resolution performance on the specimen model being imaged.

© 2005 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
    [CrossRef]
  3. J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
    [CrossRef]
  4. H. H. Barrett, K. J. Myers, “Image quality,” in Foundations of Image Science, Series in Pure and Applied Optics (Wiley, Hoboken, New Jersey, 2004), Chap. 14, pp. 913-1000.
  5. M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
    [CrossRef]
  6. E. Clarkson, H. H. Barrett, “Approximation to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
    [CrossRef]
  7. W. E. Smith, H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
    [CrossRef]
  8. P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
    [CrossRef]
  9. H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics (Wiley, New York, 2000).
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    [CrossRef] [PubMed]
  12. J. L. Harris, “Resolving power and decision theory,” J. Opt. Soc. Am. 54, 606–611 (1964).
    [CrossRef]
  13. B. E. Bouma, G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).
  14. J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

2003 (2)

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

M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

2002 (1)

2000 (2)

P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
[CrossRef]

E. Clarkson, H. H. Barrett, “Approximation to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
[CrossRef]

1999 (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

1986 (1)

1964 (1)

1931 (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[CrossRef]

Akcay, C.

Akcay, C. A.

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

Barrett, H. H.

Bonetto, P.

P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
[CrossRef]

Bouma, B. E.

B. E. Bouma, G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Cheong, K.

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

Clarkson, E.

M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

E. Clarkson, H. H. Barrett, “Approximation to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
[CrossRef]

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

DeLemos, T.

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

Drexler, W.

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

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Harris, J. L.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

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

Hoppin, J. W.

Hotelling, H.

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Kupinski, M. A.

Lasser, T.

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

Leahy, R. M.

P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
[CrossRef]

Lee, K. S.

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Myers, K. J.

H. H. Barrett, K. J. Myers, “Image quality,” in Foundations of Image Science, Series in Pure and Applied Optics (Wiley, Hoboken, New Jersey, 2004), Chap. 14, pp. 913-1000.

O’Daniel, J.

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

Parrein, P.

C. Akcay, P. Parrein, J. P. Rolland, “Estimation of longitudinal resolution in optical coherence imaging,” Appl. Opt. 41, 5256–5262 (2002).
[CrossRef] [PubMed]

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Qi, J.

P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
[CrossRef]

Rolland, J. P.

C. Akcay, P. Parrein, J. P. Rolland, “Estimation of longitudinal resolution in optical coherence imaging,” Appl. Opt. 41, 5256–5262 (2002).
[CrossRef] [PubMed]

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Smith, W. E.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

B. E. Bouma, G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).

Ann. Math. Stat. (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[CrossRef]

Appl. Opt. (2)

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

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

P. Bonetto, J. Qi, R. M. Leahy, “Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics,” IEEE Trans. Nucl. Sci. 47, 1567–1572 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Other (4)

H. H. Barrett, K. J. Myers, “Image quality,” in Foundations of Image Science, Series in Pure and Applied Optics (Wiley, Hoboken, New Jersey, 2004), Chap. 14, pp. 913-1000.

B. E. Bouma, G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).

J. P. Rolland, J. O’Daniel, E. Clarkson, K. Cheong, C. A. Akcay, P. Parrein, T. DeLemos, K. S. Lee, “AUC-based resolution quantification in optical coherence tomography,” in Medical Imaging: Image Perception, Observer Performance and Technology Assessment, D. P. Chakraborty and M. P. Eckstein, eds., Proc. SPIE5372, 334–353 (2004).

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

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Figures (5)

Fig. 1
Fig. 1

Basic free-space OCI system setup.

Fig. 2
Fig. 2

Illustration of the specimen model.

Fig. 3
Fig. 3

Detection task. (a) Detectability index, (b) AUC for Gaussian sources of 2, 20, and 40 μ m coherence length.

Fig. 4
Fig. 4

Resolution task. Detectability index versus separation distance for sources with coherence length of (a) 2 μ m , (b) 20 μ m , and (c) 40 μ m . AUC versus separation distance for sources with coherence length of (d) 2 μ m , (e) 20 μ m , and (f) 40 μ m .

Fig. 5
Fig. 5

Signal X for the detection task given Δ n 0 = 0.1 for coherence length sources of (a) 40 μ m and (b) 2 μ m .

Equations (48)

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d 2 = X K 1 X .
X n = I 1 ( t n ) I 0 ( t n ) .
K i ( t , t ) = I i ( t ) I i ( t ) I i ( t ) I i ( t ) .
K n m = p ( H 0 ) K 0 ( t n , t m ) + p ( H 1 ) K 1 ( t n , t m ) ,
K n m = 1 2 K 0 ( t n , t m ) + 1 2 K 1 ( t n , t m ) .
AUC = 1 2 + 1 2 erf ( d 2 2 ) .
I ( t ) = e Δ t t Δ t t N ( t ) d t ,
I ( t ) = e Δ t r ( t t ) N ( t ) d t ,
r ( t ) = { 1 0 t Δ t 0 otherwise } .
E s ( t ) = exp ( i ω t ) E ̂ s ( ω ) d ω ,
E ( t ) = { α ̂ ( ω ) exp [ i ϕ 1 ( ω , t ) + i ω t ] + β ̂ ( ω ) exp [ i ϕ 2 ( ω , t ) + i ω t ] } E ̂ s ( ω ) d ω .
m ( ω , t ) = α ̂ ( ω ) exp [ i ϕ 1 ( ω , t ) ] + β ̂ ( ω ) exp [ i ϕ 2 ( ω , t ) ] .
E ( t ) = m ( ω , t ) exp ( i ω t ) E ̂ s ( ω ) d ω .
I ( t ) = e Δ t r ( t t ) N ( t ) d t = e Δ t r ( t t ) N ( t ) d t .
N ¯ ( t ) = R A e η 0 E ( t ) E ( t ) = ρ E ( t ) E ( t ) ,
ρ = R A e η 0 .
N ( t ) = N ¯ ( t ) = ρ E ( t ) E ( t ) .
N ( t ) = ρ m * ( ω , t ) m ( ω , t ) exp [ i ( ω ω ) t ] × E ̂ s ( ω ) E ̂ s ( ω ) d ω d ω .
G ( τ ) = E s ( t τ ) E s ( t ) .
G * ( τ ) = G ( τ ) .
E ̂ s ( ω ) E ̂ s ( ω ) = 1 4 π 2 exp [ i ( ω t 2 ω t 1 ) ] × E s ( t 1 ) E s ( t 2 ) d t 1 d t 2 = 1 4 π 2 exp [ i ( ω t 2 ω t 1 ) ] G ( t 2 t 1 ) d t 1 d t 2 = 1 2 π δ ( ω ω ) exp [ i ( ω + ω 2 ) s ] G ( s ) d s = δ ( ω ω ) S ( ω ) .
N ( t ) = ρ m ( ω , t ) 2 S ( ω ) d ω .
I ( t ) = ρ e Δ t r ( t t ) [ m ( ω , t ) 2 S ( ω ) d ω ] d t .
K ( t , t ) = ( e Δ t ) 2 r ( t t 1 ) r ( t t 2 ) K N ( t 1 , t 2 ) d t 1 d t 2 ,
K N ( t 1 , t 2 ) = N ( t 1 ) N ( t 2 ) N ( t 1 ) N ( t 2 ) = N ( t 1 ) δ ( t 1 t 2 ) + K N ¯ ( t 1 , t 2 ) .
K N ¯ ( t 1 , t 2 ) = N ¯ ( t 1 ) N ¯ ( t 2 ) N ¯ ( t 1 ) N ¯ ( t 2 ) .
N ¯ ( t 1 ) N ¯ ( t 2 ) = ρ 2 E ( t 1 ) E ( t 1 ) E ( t 2 ) E ( t 2 ) = ρ 2 E ( t 1 ) E ( t 1 ) E ( t 2 ) E ( t 2 ) + ρ 2 tr [ E ( t 1 ) E ( t 2 ) E ( t 2 ) E ( t 1 ) ] = N ¯ ( t 1 ) N ¯ ( t 2 ) + ρ 2 tr [ E ( t 1 ) E ( t 2 ) E ( t 2 ) E ( t 1 ) ] ,
K N ¯ ( t 1 , t 2 ) = ρ 2 tr [ E ( t 1 ) E ( t 2 ) E ( t 2 ) E ( t 1 ) ] = ρ 2 tr [ J ( t 1 , t 2 ) J ( t 2 , t 1 ) ] = ρ 2 tr [ J ( t 1 , t 2 ) J ( t 1 , t 2 ) ] ,
J ( t 1 , t 2 ) = E ( t 1 ) E ( t 2 ) .
G ( τ ) = E s ( t ) E s ( t τ ) .
G ( τ ) = G ( τ ) ,
G ( τ ) = tr [ G ( τ ) ] .
J ( t 1 , t 2 ) = m ( ω , t 1 ) m * ( ω , t 2 ) exp [ i ( ω t 1 ω t 2 ) ] × E ̂ s ( ω ) E ̂ s ( ω ) d ω d ω .
E ̂ s ( ω ) E ̂ s ( ω ) = δ ( ω ω ) G ̂ ( ω ) .
K N ¯ ( t 1 , t 2 ) = ρ 2 m ( ω , t 1 ) m * ( ω , t 2 ) m * ( ω , t 1 ) × m ( ω , t 2 ) exp [ i ( ω ω ) ( t 1 t 2 ) ] × tr [ G ̂ ( ω ) G ̂ ( ω ) ] d ω d ω .
K ( t , t ) = ( e Δ t ) 2 r ( t t 1 ) r ( t t 1 ) N ( t 1 ) d t 1 + ( e Δ t ) 2 r ( t t 1 ) r ( t t 2 ) K N ¯ ( t 1 , t 2 ) d t 1 d t 2
K ( t , t ) ( e Δ t ) 2 r ( t t 1 ) r ( t t 1 ) N ( t 1 ) d t 1
β ̂ ( ω ) = r 1 + ( 1 r 1 2 ) r 2 δ 2 ( ω ) ,
r 1 = 1 n 1 + n , r 2 = Δ n 2 n + Δ n , δ ( ω ) = exp ( i ω n l c ) .
S ( ω ) = 2 ( ln 2 ) 1 2 π 1 2 Δ ω exp { [ 2 ( ln 2 ) 1 2 ω ω 0 Δ ω ] 2 } ,
Δ ω = 8 ln 2 l c c .
K G ( t , t ) = ( e Δ t ) 2 r ( t t 1 ) r ( t t 2 ) K N ¯ ( t 1 , t 2 ) d t 1 d t 2 ,
K N ¯ ( t 1 , t 2 ) = ρ 2 m ( ω , t 1 ) m * ( ω , t 2 ) m * ( ω , t 1 ) m ( ω , t 2 ) × exp [ i ( ω ω ) ( t 1 t 2 ) ] tr [ G ̂ ( ω ) G ̂ ( ω ) ] d ω d ω .
m ( ω , t ) = α ̂ ( ω ) exp [ i ϕ 1 ( ω , t ) ] + β ̂ ( ω ) exp [ i ϕ 2 ( ω , t ) ] = C 1 ( ω , t ) + C 2 ( ω , t ) .
tr G ̂ ( ω ) G ̂ ( ω ) = G ̂ x x ( ω ) G ̂ x x * ( ω ) + G ̂ x y ( ω ) G ̂ y x * ( ω ) + G ̂ y x ( ω ) G ̂ x y * ( ω ) + G ̂ y y ( ω ) G ̂ y y * ( ω ) = m = 1 2 n = 1 2 G ̂ m n ( ω ) G ̂ m n * ( ω ) , x = 1 , y = 2 .
K N ¯ ( t 1 , t 2 ) = ρ 2 h = 1 2 j = 1 2 k = 1 2 l = 1 2 m = 1 2 n = 1 2 C h ( ω , t 1 ) C j * ( ω , t 2 ) × exp [ i ω ( t 1 t 2 ) ] G ̂ m n ( ω ) d ω × C k * ( ω , t 1 ) C 1 ( ω , t 2 ) exp [ i ω ( t 1 t 2 ) ] × G ̂ m n * ( ω ) d ω .
K N ¯ ( t 1 , t 2 ) = O ̱ [ 6.4 × 10 5 δ ( t 1 t 2 ) ] ,
N ( t 1 ) δ ( t 1 t 2 ) = O ̱ [ 1 × 10 10 δ ( t 1 t 2 ) ] .

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