Abstract

Hand-held OCT systems that offer physicians greater freedom to access imaging sites of interest could be useful for many clinical applications. In this study, by incorporating the theoretical speckle model into the decorrelation function, we have explicitly correlated the cross-correlation coefficient to the lateral displacement between adjacent A-scans. We used this model to develop and study a freehand-scanning OCT system capable of real-time scanning speed correction and distortion-free imaging—for the first time to the best our knowledge. To validate our model and the system, we performed a series of calibration experiments. Experimental results show that our method can extract lateral scanning distance. In addition, using the manually scanned hand-held OCT system, we obtained OCT images from various samples by freehand manual scanning, including images obtained from human in vivo.

© 2012 OSA

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2011 (1)

2010 (6)

2009 (4)

2008 (2)

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[CrossRef] [PubMed]

S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. 13(2), 020505 (2008).
[CrossRef] [PubMed]

2007 (1)

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403–051421 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (2)

J. K. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express 13(14), 5234–5239 (2005).
[CrossRef] [PubMed]

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

2001 (1)

P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48(4), 1084–1091 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. 4(1), 95–105 (1999).
[CrossRef]

1997 (2)

J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8(1), 38–44 (1997).
[CrossRef]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. G. Fujimoto, and M. E. Brezinski, “Forward-imaging instruments for optical coherence tomography,” Opt. Lett. 22(21), 1618–1620 (1997).
[CrossRef] [PubMed]

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

1987 (1)

Adie, S. G.

Ahmad, A.

Barton, J. K.

Boas, D. A.

Boppart, S. A.

Bouma, B. E.

Brezinski, M. E.

Brown, D. G.

Carson, P. L.

J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8(1), 38–44 (1997).
[CrossRef]

Chaney, E. J.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, J.-F.

J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8(1), 38–44 (1997).
[CrossRef]

Chen, Z. P.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Cheng, C. J.

P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48(4), 1084–1091 (2001).
[CrossRef] [PubMed]

Chudoba, C.

Conry, M.

Curatolo, A.

Ferguson, R. A.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Fowlkes, J. B.

J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8(1), 38–44 (1997).
[CrossRef]

Fujimoto, J. G.

Gehlbach, P.

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

Gerstmann, D. K.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Grimwood, A.

Gu, C.

Han, J.

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[CrossRef] [PubMed]

Han, J. H.

Han, S.

S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. 13(2), 020505 (2008).
[CrossRef] [PubMed]

Hart, C.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Humayun, M.

S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. 13(2), 020505 (2008).
[CrossRef] [PubMed]

Huo, L.

Insana, M. F.

Jung, W. G.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Kang, J. U.

K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express 18(22), 23472–23487 (2010).
[CrossRef] [PubMed]

X. Liu and J. U. Kang, “Progress toward inexpensive endoscopic high-resolution common-path OCT,” Proc. SPIE 7559, 755902, 755902-11 (2010).
[CrossRef]

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[CrossRef] [PubMed]

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[CrossRef] [PubMed]

Kirk, R. W.

Ko, T.

Lau, B.

Lee, J.

Li, P. C.

P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48(4), 1084–1091 (2001).
[CrossRef] [PubMed]

Li, X.

Liang, Y.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Liu, X.

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

X. Liu and J. U. Kang, “Progress toward inexpensive endoscopic high-resolution common-path OCT,” Proc. SPIE 7559, 755902, 755902-11 (2010).
[CrossRef]

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[CrossRef] [PubMed]

Liu, Y.

Marks, D. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403–051421 (2007).
[CrossRef] [PubMed]

McCormick, D. T.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

McDowell, E. J.

McLaughlin, R. A.

Mu, G.

Nguyen, F. T.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403–051421 (2007).
[CrossRef] [PubMed]

Oldenburg, A. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403–051421 (2007).
[CrossRef] [PubMed]

Pitris, C.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Radhakrishnan, H.

Ren, J.

Rubin, J. M.

J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8(1), 38–44 (1997).
[CrossRef]

Sampson, D. D.

Sarunic, M. V.

S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. 13(2), 020505 (2008).
[CrossRef] [PubMed]

Schmitt, J. M.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. 4(1), 95–105 (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, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Sharma, U.

Song, C.

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

Srinivasan, V.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Stromski, S.

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(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

Tien, N. C.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Tomlins, P. H.

Wagner, R. F.

Wang, F.

Wang, L.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Wang, W.

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[CrossRef] [PubMed]

Wilder-Smith, P.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Woolliams, P. D.

Wu, J.

Wu, Y.

Xi, J.

Xiang, S. H.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. 4(1), 95–105 (1999).
[CrossRef]

Yang, C.

Yaqoob, Z.

Yeh, C. K.

P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48(4), 1084–1091 (2001).
[CrossRef] [PubMed]

Yung, K. M.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. 4(1), 95–105 (1999).
[CrossRef]

Zhang, J.

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

Zhang, K.

K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express 18(22), 23472–23487 (2010).
[CrossRef] [PubMed]

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[CrossRef] [PubMed]

Zhu, X.

Zysk, A. M.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403–051421 (2007).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (2)

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[CrossRef]

W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. 11(4), 806–810 (2005).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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Figures (11)

Fig. 1
Fig. 1

Schematic of manual-scanned OCT imaging with time-varying scanning speed.

Fig. 2
Fig. 2

Flow chart for the scanning speed correction using cross-correlation coefficient.

Fig. 3
Fig. 3

(a) OCT system diagram; (b) principle of common path interferometer based on single mode fiber; (c) single mode fiber probe integrated with needle.

Fig. 4
Fig. 4

Block diagram of signal processing for the OCT system with real-time scanning speed variance correction.

Fig. 5
Fig. 5

(a) Image obtained in calibration experiment (∆x = 0.96μm); (b) relationship between ρ and ∆x obtained by calculating XCC between adjacent A-scans from different B-scans (red circles: experimental; black, dashed line: theoretical); (c) relationship between ρ and ∆x obtained by calculating XCC using A-scans with different offsets from the same B-scan (red, solid line: experimental; black, dashed line: theoretical); (d) the ratio between standard deviation and mean of ρi, at different sampling intervals; (e) ratio between σΔxtotal and Δxtotal.

Fig. 6
Fig. 6

(a) Displacement of probing beam versus time (upper); ∆x as a function of time (lower); (b) pseudo B-scan obtained from sinusoidal scanning pattern; (c) upper inset: XCC calculated from adjacent A-scans; lower inset: ∆x calculated from XCC (red) and ground truth ∆x calculated from the driving voltage (black); (d) pseudo B-scan with artifact induced by non-constant scanning speed; (e) B-scan after non-constant scanning speed correction.

Fig. 7
Fig. 7

OCT images obtained from manual scan: (a) before scanning speed correction; (b) after scanning speed correction. Red arrows in Fig. 7(a) indicates areas with motion artifacts.

Fig. 8
Fig. 8

Images of IR viewing card with sampling interval ∆xs equal to 1μm (a), 2μm (b), and 4μm (c).

Fig. 9
Fig. 9

(a) photo of quality resolution chart; (b)OCT image obtained from manual scan with scanning speed correction; (c) Blue curve: Mean OCT signal of difference A-scans in Fig. 9(a); red circles: zero-crossing points of the blue curve; (d) OCT image obtained from manual scan without scanning speed correction

Fig. 10
Fig. 10

Manually scanned OCT image of human skin from finger tip (a) and palm (b).

Fig. 11
Fig. 11

Image of onion cells obtained from manual scanning.

Equations (21)

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Δx= v f A
v m = f A 2 F n
ρ I x,y (z), I x+Δx,y+Δy (z+Δz) = [ I x,y (z) I x,y (z) ][ I x+Δx,y+Δy (z+Δz) I x+Δx,y+Δy (z+Δz) ] σ I x,y (z) σ I x+Δx,y+Δy (z+Δz)
ρ I x,y (z), I x+Δx,y (z) = [ I x,y (z) I x,y (z) ][ I x+Δx,y (z) I x+Δx,y (z) ] σ I x,y (z) σ I x+Δx,y (z)
σ I x,y (z) 2 = [ I x,y (z) I x,y (z) ] 2 = I x,y (z) 2 I x,y (z) 2 = I RMS 2 I 0 2 σ I x+Δx,y (z) 2 = [ I x+Δx,y (z) I x+Δx,y (z) ] 2 = I x+Δx,y (z) 2 I x+Δx,y (z) 2 = I RMS 2 I 0 2 [ I x,y (z) I x,y (z) ][ I x+Δx,y (z) I x+Δx,y (z) ] = I x,y (z) I x+Δx,y (z) I 0 2
ρ= I x,y (z) I x+Δx,y (z) I 0 2 I RMS 2 I 0 2
I x,y (z) I x+Δx,y (z) = | S x,y (z) S x+Δx,y * (z) | 2 + I 0 2
S x,y ( z )= x',y',z' a( xx',yy',zz' )P( x',y',z' )dx'dy'dz'
S x+Δx,y ( z )= x',y',z' a( x+Δxx',yy',zz' )P( x',y',z' )dx'dy'dz'
I x,y (z) I x+Δx,y (z) = | x',y',z' x'',y'',z'' a( xx',yy',zz' ) a( x+Δxx'',yy'',zz'' ) P( x',y',z' ) P * ( x'',y'',z'' )dx'dy'dz'dx''dy''dz'' | 2 + I 0 2
a( xx',yy',zz' )a( x+Δxx'',yy'',zz'' ) = a 0 2 δ( x'+Δxx'' )δ( y'y'' )δ( z'z'' )
I x,y (z) I x+Δx,y (z) = | x',y',z' x'',y'',z'' a 0 2 δ( x'+Δxx'' )δ( y''y' )δ( z''z' ) P( x',y',z' ) P * ( x'',y'',z'' )dx'dy'dz'dx''dy''dz'' | 2 + I 0 2 = | x',y',z' a 0 2 P( x',y',z' ) P * ( x'+Δx,y',z' )dx'dy'dz' | 2 + I 0 2
I x,y (z) I x+Δx,y (z) I 0 2 = | a 0 2 P x ( x' ) P y ( y' ) P z ( z' ) P x * ( x'+Δx ) P y * ( y' ) P z * ( z' )dx'dy'dz' | 2 + I 0 2 I 0 2 = | a 0 2 [ + P x ( x' ) P x * ( x'+Δx ) dx' ][ + P y ( y' ) P y * ( y' ) dy' ][ + P z ( z' ) P z * ( z' ) dz' ] | 2 = | a 0 2 + P y ( y' ) P y * ( y' ) dy' | 2 | + P z ( z' ) P z * ( z' ) dz' | 2 | + P x ( x' ) P x * ( x'+Δx ) dx' | 2
P x ( x )= P 0 exp( x 2 w 0 2 )
ρ= | a 0 2 + P y ( y' ) P y * ( y' )dy' | 2 | + P z ( z' ) P z * ( z' ) dz' | 2 | + P x ( x' ) P x * ( x'+Δx ) dx' | 2 | a 0 2 + P y ( y' ) P y * ( y' ) dy' | 2 | + P z ( z' ) P z * ( z' ) dz' | 2 | + P x ( x' ) P x * ( x' )dx' | 2 = | P x ( x' ) P x * ( x'+Δx ) dx' | 2 | P x ( x' ) P x * ( x' ) dx' | 2
ρ= | [ P 0 2 exp( x ' 2 w 0 2 )exp( ( x'+Δx ) 2 w 0 2 ) ] dx' | 2 | [ P 0 2 exp( x ' 2 w 0 2 )exp( x ' 2 w 0 2 ) ] dx' | 2 = | exp( 2 ( x'+ Δx 2 ) 2 + Δ x 2 2 w 0 2 ) dx' | 2 | exp( 2 x ' 2 w 0 2 ) dx' | 2
+ exp( 2 x ' 2 w 0 2 ) dx'= + exp( 2 ( x'+ Δx 2 ) 2 w 0 2 ) dx= π/2 w 0
ρ= | exp( Δ x 2 2 w 0 2 ) exp( 2 ( x'+ Δx 2 ) 2 w 0 2 ) dx' | 2 | exp( 2 x ' 2 w 0 2 ) dx' | 2 = | exp( Δ x 2 2 w 0 2 ) π/2 w 0 | 2 | π/2 w 0 | 2 =exp[ ( Δx ) 2 w 0 2 ]
Δx= w 0 ln( 1 ρ )
ρ j,j+1 = i= i f i l ( I ij I j )( I i( j+1 ) I ( j+1 ) ) [ i= i f i l ( I ij I j ) 2 ][ i= i f i l ( I i( j+1 ) I ( j+1 ) ) 2 ]
ρ= ( i=1 N1 ρ i ) / ( N-1 )

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