Abstract

A general theoretical model is developed to improve the novel Spectral Domain Interferometry method denoted as Master/Slave (MS) Interferometry. In this model, two functions, g and h are introduced to describe the modulation chirp of the channeled spectrum signal due to nonlinearities in the decoding process from wavenumber to time and due to dispersion in the interferometer. The utilization of these two functions brings two major improvements to previous implementations of the MS method. A first improvement consists in reducing the number of channeled spectra necessary to be collected at Master stage. In previous MSI implementation, the number of channeled spectra at the Master stage equated the number of depths where information was selected from at the Slave stage. The paper demonstrates that two experimental channeled spectra only acquired at Master stage suffice to produce A-scans from any number of resolved depths at the Slave stage. A second improvement is the utilization of complex signal processing. Previous MSI implementations discarded the phase. Complex processing of the electrical signal determined by the channeled spectrum allows phase processing that opens several novel avenues. A first consequence of such signal processing is reduction in the random component of the phase without affecting the axial resolution. In previous MSI implementations, phase instabilities were reduced by an average over the wavenumber that led to reduction in the axial resolution.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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2015 (4)

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “On the possibility of producing true real-time retinal cross-sectional images using a graphics processing unit enhanced master-slave optical coherence tomography system,” J. Biomed. Opt. 20(7), 076008 (2015).
[Crossref] [PubMed]

J. Wang, A. Bradu, G. Dobre, and A. Podoleanu, “Full-field swept source master slave optical coherence tomography,” IEEE Photonics J. 7(4), 1943 (2015).
[Crossref]

A. Bradu, M. Maria, and A. G. Podoleanu, “Demonstration of tolerance to dispersion of master/slave interferometry,” Opt. Express 23(11), 14148–14161 (2015).
[Crossref] [PubMed]

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “Master slave en-face OCT/SLO,” Biomed. Opt. Express 6(9), 3655–3669 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (1)

2012 (2)

2011 (1)

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[Crossref]

2010 (2)

Y. Watanabe and T. Itagaki, “Real-time display SD-OCT using a linear-in-wavenumber spectrometer and a graphics processing unit,” Proc. SPIE 7554, 75542S (2010).
[Crossref]

K. K. H. Chan and S. Tang, “High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform,” Biomed. Opt. Express 1(5), 1309–1319 (2010).
[Crossref] [PubMed]

2009 (1)

V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin, “Linear-wavenumber spectrometer for high-speed spectral-domain optical coherence tomography,” Opt. Spectrosc. 106(3), 459–465 (2009).
[Crossref]

2007 (2)

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

2005 (2)

2004 (1)

2003 (2)

C. Yang, S. Yazdanfar, and J. A. Izatt, “Fast scanning, dispersion-adjustable reference delay for OCT using fiber Bragg gratings,” Proc. SPIE 5140, 53–59 (2003).
[Crossref]

J. G. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. 21(11), 1361–1367 (2003).
[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]

Akiba, M.

Barnes, F.

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “On the possibility of producing true real-time retinal cross-sectional images using a graphics processing unit enhanced master-slave optical coherence tomography system,” J. Biomed. Opt. 20(7), 076008 (2015).
[Crossref] [PubMed]

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “Master slave en-face OCT/SLO,” Biomed. Opt. Express 6(9), 3655–3669 (2015).
[Crossref] [PubMed]

Bradu, A.

Chan, K. K. H.

Chan, K.-P.

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]

Chong, C.

Coen, S.

N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
[Crossref] [PubMed]

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[Crossref]

Dobre, G.

J. Wang, A. Bradu, G. Dobre, and A. Podoleanu, “Full-field swept source master slave optical coherence tomography,” IEEE Photonics J. 7(4), 1943 (2015).
[Crossref]

Drexler, W.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Duker, J.

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]

Fujimoto, J.

Fujimoto, J. G.

J. G. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. 21(11), 1361–1367 (2003).
[Crossref] [PubMed]

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]

Gelikonov, G. V.

V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin, “Linear-wavenumber spectrometer for high-speed spectral-domain optical coherence tomography,” Opt. Spectrosc. 106(3), 459–465 (2009).
[Crossref]

Gelikonov, V. M.

V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin, “Linear-wavenumber spectrometer for high-speed spectral-domain optical coherence tomography,” Opt. Spectrosc. 106(3), 459–465 (2009).
[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(5035), 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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hermann, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Hlawatsch, F.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Hofer, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Hu, Z.

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]

Itagaki, T.

Y. Watanabe and T. Itagaki, “Real-time display SD-OCT using a linear-in-wavenumber spectrometer and a graphics processing unit,” Proc. SPIE 7554, 75542S (2010).
[Crossref]

Itoh, M.

Izatt, J. A.

C. Yang, S. Yazdanfar, and J. A. Izatt, “Fast scanning, dispersion-adjustable reference delay for OCT using fiber Bragg gratings,” Proc. SPIE 5140, 53–59 (2003).
[Crossref]

Kapinchev, K.

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “On the possibility of producing true real-time retinal cross-sectional images using a graphics processing unit enhanced master-slave optical coherence tomography system,” J. Biomed. Opt. 20(7), 076008 (2015).
[Crossref] [PubMed]

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “Master slave en-face OCT/SLO,” Biomed. Opt. Express 6(9), 3655–3669 (2015).
[Crossref] [PubMed]

Ko, T.

Kowalczyk, A.

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]

Lippok, N.

Madjarova, V. D.

Makita, S.

Maria, M.

Matz, G.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Morosawa, A.

Nielsen, P.

Payne, A.

Podoleanu, A.

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “On the possibility of producing true real-time retinal cross-sectional images using a graphics processing unit enhanced master-slave optical coherence tomography system,” J. Biomed. Opt. 20(7), 076008 (2015).
[Crossref] [PubMed]

J. Wang, A. Bradu, G. Dobre, and A. Podoleanu, “Full-field swept source master slave optical coherence tomography,” IEEE Photonics J. 7(4), 1943 (2015).
[Crossref]

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “Master slave en-face OCT/SLO,” Biomed. Opt. Express 6(9), 3655–3669 (2015).
[Crossref] [PubMed]

Podoleanu, A. G.

Povazay, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Rogers, J.

Rollins, A. M.

Rosa, C. C.

Sakai, T.

Schroeder, J.

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[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]

Shilyagin, P. A.

V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin, “Linear-wavenumber spectrometer for high-speed spectral-domain optical coherence tomography,” Opt. Spectrosc. 106(3), 459–465 (2009).
[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]

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]

Tang, S.

Unterhuber, A.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

Vanholsbeeck, F.

N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
[Crossref] [PubMed]

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[Crossref]

Wang, J.

J. Wang, A. Bradu, G. Dobre, and A. Podoleanu, “Full-field swept source master slave optical coherence tomography,” IEEE Photonics J. 7(4), 1943 (2015).
[Crossref]

Watanabe, Y.

Y. Watanabe and T. Itagaki, “Real-time display SD-OCT using a linear-in-wavenumber spectrometer and a graphics processing unit,” Proc. SPIE 7554, 75542S (2010).
[Crossref]

Wojtkowski, M.

Yang, A.

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[Crossref]

Yang, C.

C. Yang, S. Yazdanfar, and J. A. Izatt, “Fast scanning, dispersion-adjustable reference delay for OCT using fiber Bragg gratings,” Proc. SPIE 5140, 53–59 (2003).
[Crossref]

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

C. Yang, S. Yazdanfar, and J. A. Izatt, “Fast scanning, dispersion-adjustable reference delay for OCT using fiber Bragg gratings,” Proc. SPIE 5140, 53–59 (2003).
[Crossref]

Biomed. Opt. Express (3)

IEEE Photonics J. (1)

J. Wang, A. Bradu, G. Dobre, and A. Podoleanu, “Full-field swept source master slave optical coherence tomography,” IEEE Photonics J. 7(4), 1943 (2015).
[Crossref]

J. Biomed. Opt. (1)

A. Bradu, K. Kapinchev, F. Barnes, and A. Podoleanu, “On the possibility of producing true real-time retinal cross-sectional images using a graphics processing unit enhanced master-slave optical coherence tomography system,” J. Biomed. Opt. 20(7), 076008 (2015).
[Crossref] [PubMed]

Nat. Biotechnol. (1)

J. G. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. 21(11), 1361–1367 (2003).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (4)

Opt. Spectrosc. (1)

V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin, “Linear-wavenumber spectrometer for high-speed spectral-domain optical coherence tomography,” Opt. Spectrosc. 106(3), 459–465 (2009).
[Crossref]

Proc. SPIE (4)

C. Yang, S. Yazdanfar, and J. A. Izatt, “Fast scanning, dispersion-adjustable reference delay for OCT using fiber Bragg gratings,” Proc. SPIE 5140, 53–59 (2003).
[Crossref]

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach,” Proc. SPIE 6443, 64430O (2007).
[Crossref]

A. Yang, F. Vanholsbeeck, S. Coen, and J. Schroeder, “Chromatic dispersion compensation of an OCT system with a programmable spectral filter,” Proc. SPIE 8091, 809125 (2011).
[Crossref]

Y. Watanabe and T. Itagaki, “Real-time display SD-OCT using a linear-in-wavenumber spectrometer and a graphics processing unit,” Proc. SPIE 7554, 75542S (2010).
[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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1 Block diagram for a spectral domain OCT system. C1 and C2, collimators; M, reference mirror; O, object. Two channeled spectra are shown underneath, for a mirror as a sample. IDC represents the power spectrum of the optical source, shown by the red Gaussian shape solid line. The A shape is shown by the dashed blue line, determined by the interference contrast of the modulating signal proportional to the channeled spectrum. On the left, the usual case in practice is shown when IDC≠A. Here, the interference contrast A is deliberately shown smaller on the left side of the spectrum. In this case, A varies below IDC on the left hand side and regains the IDC value on the right. The channeled spectrum on the right shows the ideal case, when IDC = A, i.e. when the contrast profile A and the IDC profile are superposed on each other.
Fig. 2
Fig. 2 Step by step procedure to infer the Mask function from a reduced set of experimentally measured channeled spectra.
Fig. 3
Fig. 3 Derivative of the experimental phase with respect to ν ˜ for different positions of the reference mirror M in Fig. 1 (black dots) adjusting the OPD = 2z. The derivative phase is evaluated at the center of the spectrum ν ˜ c . Continuous line, linear fit of experimental measurements.
Fig. 4
Fig. 4 The functions g (a) and h (b) versus the pixels of the spectrometer according to the number P of CSexp acquired in the Master stage. Blue line, P = 2. Green line, P = 11. Red line, P = 71. Black line, normalized channeled spectrum at the Master stage for OPD = 0.
Fig. 5
Fig. 5 A-scans for 3 OPD = 2z values (z = 220 μm, 720 μm and 1320 μm measured from OPD = 0). Black line, A-scan peaks obtained using FT. Blue line, A-scans obtained using CMSI with P = 2 CSexp in the Master stage. Green line, A-scans obtained using CMSI with P = 11. Red line, A-scans obtained using CMSI with P = 71. All peaks are normalized with respect to the first peaks at z = 220 μm. The inset shows a zoom in the peaks around 1320 μm.
Fig. 6
Fig. 6 PSF corresponding to the channeled spectrum I for the OPD at position 3 (z = 1340 mm) in Fig. 5. The PSF is obtained by calculating FT−1[|I( ν ˜ )|] that is equal to FT−1[|A(g( ν ˜ ))|]. The complex form I has been calculated in Appendix C. FT−1[|I( ν ˜ )|] corresponds to the Fourier transformation of a channeled spectrum with no chirp.
Fig. 7
Fig. 7 A-scan for z = 1001 μm. Black line, A scan obtained with FT. The other three graphs are A-scan peaks obtained using the CMSI method with Q = 776, evaluated from different numbers of P-CSexp used at the Master stage. Blue line, P = 2. Green line, P = 11. Red line, P = 491. Inset, details of the A-scans from 980 μm to 1030 μm.
Fig. 8
Fig. 8 PSF corresponding to the channeled spectrum I for an OPD = 2z, where z = 1001 mm. The complex form I has been calculated in Appendix C. FT−1[|I( ν ˜ )|] corresponds to the Fourier transformation of a channeled spectrum with no chirp.
Fig. 9
Fig. 9 (a) A-scans (vertical axis) for a mirror as object, represented in time (horizontal axis) calculated with MSI and P = 100 CSexp utilized as masks. (b) A-scans (vertical axis) for a mirror as object represented in time (horizontal axis) calculated with CMSI using Q = 100 masks obtained from P = 2 CSexp. (c) Reflectance profiles calculated by MSI (red) and CMSI (blue) for t = 100 s in each respective image.
Fig. 10
Fig. 10 (a) B-scan of the lens and the iris of a human eye with CMSI. (b) B-scan of the same raw data as in (a) but with MSI. Both images are normalized according to the maximum of each of them. To demonstrate the slight improvement in contrast at large depths of the CMSI image, we display their bottom only, showing the lens and the iris. The 2 mm-axial range of the B-scans is considered in air.
Fig. 11
Fig. 11 Schematic representation of the peaks obtained by calculating the inverse FT of I (top) and CS (bottom) for a single layer object. The OPD is chosen so that Î does not overlap ÎDC.
Fig. 12
Fig. 12 Diagram explaining the process of changing a real sinusoidal function into a complex form. The parameter t0 is chosen to eliminate the DC component of the real sinusoidal function.

Tables (1)

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Table 1 Axial resolution according to the position of the reference mirror M in Fig. 1 (determining the optical path difference value) and the numerical tool used. ΔLDC is the width of the peak FT−1[IDC(g( ν ˜ ))]. ΔLinterf is the width of the peak FT−1[|I( ν ˜ )|]. All widths are evaluated via a Gaussian fit.

Equations (45)

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d( ν )= 2π c ν[ ( n 2 ( ν ) n 2 ( ν 0 ) ) e 2 ( n 1 ( ν ) n 1 ( ν 0 ) ) e 1 ],
I( ν ˜ )= I DC (g( ν ˜ ))+ 1 2 ( I ¯ ( ν ˜ )+ I ¯ ( ν ˜ ) ),
I ¯ ( ν ˜ )= r( ρ )A( g( ν ˜ ) )Exp[ i( 2π c g( ν ˜ )2ρ+h( ν ˜ ) ) ]dρ ,
I ^ ( 2z c )= I ^ DC ( 2z c )+ 1 2 r( z ) P 0 ( 2z c )+ 1 2 r ( z ) P 0 ( 2z c ) ,
P 0 ( t )=F T 1 [ A( ν ) ].
MSI( z )= [ C( N ˜ ,z) ] N ˜ =0 = [ C S exp ( ν ˜ + N ˜ ,z )I( ν ˜ )d ν ˜ ] N ˜ =0 ,
CS ¯ ( ν ˜ ,z)=A( g( ν ˜ ) )Exp[ i( 2π c g( ν ˜ )2z+h( ν ˜ ) ) ].
C S exp ( ν ˜ ,z )= 1 2 α( z ) CS ¯ ( ν ˜ ,z ) e i φ rand( z ) + 1 2 α( z ) CS ¯ ( ν ˜ ,z ) * e i φ rand( z ) .
MSI( z )= 1 2 α( z ) e i φ rand( z ) CS ¯ ( ν ˜ ,z ) I( ν ˜ )d ν ˜ +CC,
MSI( z )= 1 2 e{ α( z ) e i φ rand( z ) CS ¯ ( ν ˜ ,z ) * I ¯ ( ν ˜ )d ν ˜ },
γ( z )= CS ¯ ( ν ˜ ,z ) * I ¯ ( ν ˜ )d ν ˜ .
γ( z )= r( ρ ) | A( g( ν ˜ ) ) | 2 Exp[ i( 2π c g( ν ˜ )×2( zρ ) ) ]dρd ν ˜ .
γ( z )= r( zδ ) | A( g( ν ˜ ) ) | 2 exp[ i( 2π c g( ν ˜ )2δ ) ]dδd ν ˜ .
P 1 ( t )= | A( g( ν ˜ ) ) | 2 Exp[ i2πg( ν ˜ )t ]d ν ˜ .
γ( z )= r( zδ ) P 1 ( 2δ/c )dδ =r( z ) P 1 ( 2z/c ),
P 1 ( t )= | A(ν) | 2 Exp[ iνt ] G ( ν )dν ,
P 1 ( t )=F T 1 [ | A(ν) | 2 G ( ν ) ].
MSI( z )= α( z ) 2 e{ e i φ rand( z ) ( r( z ) P 1 ( 2z/c ) ) }.
ϕ exp ( ν ˜ ,z )= 2π c g( ν ˜ )2z+h( ν ˜ )+ φ rand ( z ).
ν ˜ ϕ exp ( ν ˜ ,z )= 2π c g'( ν ˜ )2z+h'( ν ˜ ),
M built ( ν ˜ ,z)= g ( ν ˜ )Exp[ i( 2π c g( ν ˜ )2z+h( ν ˜ ) ) ],
CMSI( z )= M built ( ν ˜ ,z )I( ν ˜ )d ν ˜ .
CMSI( z )= 1 2 M built ( ν ˜ ,z ) I ¯ ( ν ˜ )d ν ˜ ,
CMSI( z )= 1 2 r( ρ )A( g( ν ˜ ) )Exp[ i( 2π c g( ν ˜ )×2( zρ ) ) ] g ( ν ˜ )d ν ˜ dρ ,
CMSI( z )= 1 2 r( zδ )A( ν )Exp[ i 2π c ν2δ ]dνdδ .
CMSI( z )= 1 2 r( z ) P 0 ( 2z/c ),
M built (n,q)= g ( n )Exp[ i( 2π c g( n )qΔOPD+h( n ) ) ],
CMSI( q )= n=1 N M built ( n,q )I( n ) ,
FT[ f ^ ( t ) ]= f ^ ( t ) Exp[ i2πtν ]dt,
F T 1 [ f( ν ) ]= f( ν ) Exp[ i2πtν ]dν.
I ^ ( t )= I ^ DC ( t )+ 1 2 I ^ ¯ ( t )+ 1 2 I ^ ¯ ( t ) ,
F T 1 [ f ( ν ) * ]= f ^ ( t ) * .
I ^ ( 2z /c )= I ^ DC ( 2z /c )+ 1 2 I ^ ¯ ( 2z /c )+ 1 2 I ^ ¯ ( 2z /c ) .
I ¯ (ν)= r( ρ )A( ν )Exp[ i 2π c ν2ρ ]dρ ,
I ¯ ^ (t=2z/c)= r( ρ )A( ν )Exp[ i 2π c ν2( zρ ) ]dρdν ,
I ¯ ^ (2z/c)= A ^ ( 2 c ( zρ ) )r( ρ )dρ = A ^ ( 2z/c )r( z ),
MSI( z )= 1 2 α( z ) e i φ rand( z ) CS ¯ ^ ( t ˜ ,z ) I ^ ( t ˜ )d t ˜ +CC,
f 1 ( ν ˜ ) f 2 ( ν ˜ )d ν ˜ = f ^ 1 ( t ˜ ) f ^ 2 ( t ˜ )d t ˜ ,
I ^ ( t ˜ )= I ^ DC ( t ˜ )+ 1 2 I ^ ¯ ( t ˜ )+ 1 2 I ^ ¯ ( t ˜ ) .
CS ¯ ^ ( t ˜ ,z ) * I ^ ( t ˜ )d t ˜ = 1 2 CS ¯ ^ ( t ˜ ,z ) * I ^ ¯ ( t ˜ )d t ˜ .
CS ¯ ^ ( t ˜ ,z ) * I ^ ( t ˜ )d t ˜ = 1 2 CS ¯ ( ν ˜ ,z ) * I ¯ ( ν ˜ )d ν ˜ ,
MSI( z )= 1 2 α( z )e{ e i φ rand( z ) CS ¯ ( ν ˜ ,z ) I ¯ ( ν ˜ )d ν ˜ },
f( ν )= I DC ( ν )+ 1 2 I ¯ ( ν ) e ia2πν + 1 2 I ¯ ( ν ) * e ia2πν .
CMSI( z )= M ^ built ( t ˜ ,z ) I ^ ( t ˜ )d t ˜ .
CMSI( z )= 1 2 M built ( ν ˜ ,z ) I ¯ ( ν ˜ )d ν ˜ .

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