Abstract

Traditional analysis of spectroscopic optical coherence tomography (SOCT) signals is limited by an uncertainty relationship between time (depth) and frequency (wavelength). The use of a bilinear time–frequency distribution for analysis, such as those that compose Cohen’s class of functions, may provide a way to avoid this limitation. Here we present the relationship between traditional SOCT analysis and the relevant Cohen class functions: the Wigner and Choi–Williams distributions. While cross terms that arise in these bilinear time–frequency distributions have been viewed as an artifact, here we identify these terms with temporal coherence, which contains significant information about the signal through phase relationships. The utility of time–frequency distributions is illustrated through analysis of calculated signals.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
    [CrossRef]
  3. A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
    [CrossRef] [PubMed]
  4. A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
    [PubMed]
  5. M. D. Kulkarni and J. A. Izatt, in Conference on Lasers and Electro-Optics, Vol. 9 of OSA Technical Digest Series (Optical Society of America, 1996), pp. 59-60.
  6. U. Morgner, W. Drexler, F. X. Kartner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, "Spectroscopic optical coherence tomography," Opt. Lett. 25, 111-113 (2000).
    [CrossRef]
  7. C. Y. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, "Spectroscopic spectral-domain optical coherence microscopy," Opt. Lett. 31, 1079-1081 (2006).
    [CrossRef] [PubMed]
  8. A. Wax, C. H. Yang, and J. A. Izatt, "Fourier-domain low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 28, 1230-1232 (2003).
    [CrossRef] [PubMed]
  9. R. N. Graf and A. Wax, "Nuclear morphology measurements using Fourier domain low coherence interferometry," Opt. Express 13, 4693-4698 (2005).
    [CrossRef]
  10. C. Y. Xu, F. Kamalabadi, and S. A. Boppart, "Comparative performance analysis of time-frequency distributions for spectroscopic optical coherence tomography," Appl. Opt. 44, 1813-1822 (2005).
    [CrossRef]
  11. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).
    [CrossRef]
  12. J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, "Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28, 2067-2069 (2003).
    [CrossRef]
  13. M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
    [CrossRef]
  14. L. Cohen, "Time frequency-distributions-a review," Proc. IEEE 77, 941-981 (1989).
    [CrossRef]
  15. U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).
  16. C. Y. Xu, P. S. Carney, and S. A. Boppart, "Wavelength-dependent scattering in spectroscopic optical coherence tomography," Opt. Express 13, 5450-5462 (2005).
    [CrossRef] [PubMed]
  17. D. C. Adler, T. H. Ko, P. R. Herz, and J. G. Fujimoto, "Optical coherence tomography contrast enhancement using spectroscopic analysis with spectral autocorrelation," Opt. Express 12, 5487-5501 (2004).
    [CrossRef] [PubMed]

2006 (1)

2005 (4)

2004 (1)

2003 (5)

2000 (1)

1996 (1)

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[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, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1989 (1)

L. Cohen, "Time frequency-distributions-a review," Proc. IEEE 77, 941-981 (1989).
[CrossRef]

Adler, D. C.

Boone, C. W.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

Boppart, S. A.

Bouma, B. E.

Carney, P. S.

Cense, B.

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]

Choma, M. A.

Cohen, L.

L. Cohen, "Time frequency-distributions-a review," Proc. IEEE 77, 941-981 (1989).
[CrossRef]

Dasari, R. R.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

de Boer, J. F.

Drexler, W.

Feld, M. S.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

Fercher, A. F.

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]

Fujimoto, J. G.

Graf, R. N.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

R. N. Graf and A. Wax, "Nuclear morphology measurements using Fourier domain low coherence interferometry," Opt. Express 13, 4693-4698 (2005).
[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]

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]

Herz, P. R.

Hitzenberger, C. K.

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]

Ippen, E. P.

Izatt, J. A.

M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
[CrossRef]

A. Wax, C. H. Yang, and J. A. Izatt, "Fourier-domain low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 28, 1230-1232 (2003).
[CrossRef] [PubMed]

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

M. D. Kulkarni and J. A. Izatt, in Conference on Lasers and Electro-Optics, Vol. 9 of OSA Technical Digest Series (Optical Society of America, 1996), pp. 59-60.

Kamalabadi, F.

Kartner, F. X.

Ko, T. H.

Kobayashi, K.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

Kulkarni, M. D.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

M. D. Kulkarni and J. A. Izatt, in Conference on Lasers and Electro-Optics, Vol. 9 of OSA Technical Digest Series (Optical Society of America, 1996), pp. 59-60.

Leitgeb, R.

Leonhardt, U.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).

Li, X. D.

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]

Luo, W.

Morgner, U.

Muller, M. G.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

Nines, R.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

Park, B. H.

Pierce, M. C.

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Pyhtila, J. W.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

Ralston, T. S.

Sarunic, M. V.

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]

Sivak, M. V.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

Steele, V. E.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

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]

Stoner, G. D.

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[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]

Tan, W.

Tearney, G. J.

Vinegoni, C.

Wang, H. W.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

Wax, A.

R. N. Graf and A. Wax, "Nuclear morphology measurements using Fourier domain low coherence interferometry," Opt. Express 13, 4693-4698 (2005).
[CrossRef]

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

A. Wax, C. H. Yang, and J. A. Izatt, "Fourier-domain low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 28, 1230-1232 (2003).
[CrossRef] [PubMed]

Xu, C. Y.

Yang, C. H.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

A. Wax, C. H. Yang, and J. A. Izatt, "Fourier-domain low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 28, 1230-1232 (2003).
[CrossRef] [PubMed]

M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
[CrossRef]

Appl. Opt. (1)

Cancer Res. (1)

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, and M. S. Feld, "In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry," Cancer Res. 63, 3556-3559 (2003).
[PubMed]

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

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, and M. V. Sivak, "Optical coherence tomography and microscopy in gastrointestinal tissues," IEEE J. Sel. Top. Quantum Electron. 2, 1017-1028 (1996).
[CrossRef]

J. Biomed. Opt. (1)

A. Wax, J. W. Pyhtila, R. N. Graf, R. Nines, C. W. Boone, R. R. Dasari, M. S. Feld, V. E. Steele, and G. D. Stoner, "Prospective grading of neoplastic change in rat esophagus epithelium using angle-resolved low-coherence interferometry," J. Biomed. Opt. 10, 051604 (2005).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (4)

Proc. IEEE (1)

L. Cohen, "Time frequency-distributions-a review," Proc. IEEE 77, 941-981 (1989).
[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 (2)

U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).

M. D. Kulkarni and J. A. Izatt, in Conference on Lasers and Electro-Optics, Vol. 9 of OSA Technical Digest Series (Optical Society of America, 1996), pp. 59-60.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

(a) Michelson interferometry scheme, in which modulation of the reference arm path length generates (b) an interferogram as in a time-domain OCT system using a low-coherence source.

Fig. 2
Fig. 2

Wigner distributions for a two-component signal with delay between pulses of (a) T = 0 , (b) T = 3 ; (c), (d) T = 5 .

Fig. 3
Fig. 3

Wigner distributions for time delay T and a center frequency of ω 0 = 12.5 .

Fig. 4
Fig. 4

Time marginals for the Wigner distributions of Fig. 3.

Fig. 5
Fig. 5

Interferogram from the distributions in Figs. 3, 4.

Fig. 6
Fig. 6

(a) Wigner distribution with T = 5 , (b) s ( t ) 2 marginal, (c) S ( ω ) 2 marginal.

Fig. 7
Fig. 7

Wigner distribution of a two-component sample field with windows of narrow frequency width (green/dashed) and narrow time width (red/solid).

Fig. 8
Fig. 8

Comparison of processing of SOCT signals using (a) broad and (b) narrow spectral windows. Using a broad spectral window maintains high temporal resolution but cannot resolve two closely spaced peaks (c). In comparison, using a narrow spectral window does not permit the peaks to be resolved but achieves spectral resolution (d) that shows a modulation of the spectral profile (blue/solid) compared with the original spectrum (green/dashed).

Fig. 9
Fig. 9

(a) Spectral modulation due to temporal coherence induced by sample. (b) Fourier-transforming spectral modulation yields a correlation function with the peak indicating the spacing of the two peaks in the signal field.

Equations (26)

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

I T = E R + E S 2 = E R 2 + E S 2 + 2 Re ( E R E S * cos ϕ ) ,
I T ( k , t 0 ) = I T ( t ) W ( t , T , t 0 ) e i k t d t ,
W ( k , t ) = E ( k + q 2 ) E * ( k q 2 ) e i q t d q
W T ( k , t ) = ( E T ( k + q 2 ) E T * ( k q 2 ) ) e i q t d q = W R + W S + W Cross ,
I ( k ) = E T 2 = E R 2 + E S 2 + 2 Re ( E R E S cos ( φ ( k ) ) ) .
I int ( k ) = E R E R * + c.c. = 2 Re ( E R E S cos ( φ ( k ) ) ) .
Γ ( z ) = I int ( k ) exp ( i k z ) d k ,
I ̃ int ( k ) 2 = Γ ( z ) 2 = E R ( k ) E S * ( k ) exp ( i k z ) d k E R * ( k ) E S ( k ) exp ( i k z ) d k .
k ¯ = k + k 2 , q = k k , k = k ¯ + q 2 , k = k ¯ q 2 ,
Γ ( z ) 2 = d k ¯ d q E R ( k ¯ + q 2 ) E R * ( k ¯ q 2 ) E S * ( k ¯ + q 2 ) E S ( k ¯ q 2 ) exp ( i q z ) .
W S ( k , z ) = 1 2 π E S ( k + q 2 ) E S * ( k q 2 ) exp ( i q z ) d q ,
2 π W S ( k , z ) exp ( i q z ) d z = E S ( k + q 2 ) E S * ( k q 2 ) .
Γ ( z ) 2 = d k ¯ d q E R ( k ¯ + q 2 ) E R * ( k ¯ q 2 ) [ 2 π W S ( k ¯ , z ) exp ( i q z ) d z ] exp ( i q z ) = 2 π d k ¯ d z W S ( k ¯ , z ) d q E R ( k ¯ + q 2 ) E R * ( k q 2 ) exp ( i q ( z + z ) ) = ( 2 π ) 2 W S ( k ¯ , z ) W R ( k ¯ , z + z ) d k ¯ d z .
Γ ( z ) = E R ( k ) W ( k , k w ) E S * ( k ) exp ( i k z ) d k .
W ( k , k w ) = A exp ( ( k k w ) 2 2 Δ k w 2 ) .
Γ ( z ) E R ( k w ) W ( k , k w ) E S * ( k ) exp ( i k z ) d k .
S ( k w , z ) = Γ ( z , k w ) 2 = E R ( k w ) 2 W ( k , k w ) E S * ( k ) exp ( i k z ) d k W * ( k , k w ) E S ( k ) exp ( i k z ) d k
S ( k w , z ) = 2 π E R ( k w ) 2 W S ( k ¯ , z ) W W ( k ¯ k w , z + z ) d k ¯ d z .
E T = E S + E R ,
E S = E 0 exp ( t 2 a 2 ) exp ( i ω 0 t ) ,
E R = E 0 exp ( ( t T ) 2 a 2 ) exp ( i ω 0 t ) .
W ( t , ω ) = 1 2 π E T * ( t τ 2 ) E T ( t + τ 2 ) exp ( i ω τ ) d τ ,
W ( t , ω ) = E 0 2 a 2 π exp ( 2 t 2 a 2 a 2 ( ω 0 + ω ) 2 2 ) + E 0 2 a 2 π exp ( 2 ( t T ) 2 a 2 a 2 ( ω 0 + ω ) 2 2 ) + E 0 2 a 2 π cos ( T ( ω 0 + ω ) ) exp ( 2 ( t T 2 ) 2 a 2 a 2 ( ω 0 + ω ) 2 2 ) .
W ( t , ω ) d ω = s ( t ) 2 .
W ( t , ω ) d t = S ( ω ) 2 .
E S = E 1 + E 2 .

Metrics