Abstract

We demonstrate a flexible cross-correlated (C2) imaging method in the time domain by application of a tunable and highly flexible light source. An advantage of the flexible C2 method is shown by characterization of the step-index fiber (SMF28) over a broad range of wavelengths from 870nm to 1090nm and by the modal analysis of the distributed modal filtering (DMF) rod fiber within a wavelength range from 1050nm to 1090nm. Also, the influence of the spectral shape and bandwidth on the imaging trace is investigated by deliberately adjusting the input spectrum of the light source. The modal intensity as well as the phase distribution are extracted by the alternative method of 2D FT filtering. Being exceptionally tunable the flexible C2 method gives an ability to adapt the system’s parameters in a desired manner satisfying even measurements of very specific fiber designs opening up new possibilities for advanced modal characterization of fibers over broad range of wavelengths.

© 2017 Optical Society of America

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References

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    [Crossref]
  4. T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2014 (3)

2013 (2)

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

M. Laurila, R. Barankov, M. M. Jørgensen, T. T. Alkeskjold, J. Broeng, J. Lægsgaard, and S. Ramachandran, “Cross-correlated imaging of single-mode photonic crystal rod fiber with distributed mode filtering,” Opt. Express 21(8), 9215–9229 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

2008 (2)

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
[Crossref] [PubMed]

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

2007 (2)

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl.Opt. 46(23), 5719–5727 (2007).

S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15(23), 15402–15409 (2007).
[Crossref] [PubMed]

2004 (1)

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

2003 (1)

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

2001 (1)

1985 (1)

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. B 72(1), 156–160 (1982).
[Crossref]

Alexeenko, I.

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

Alkeskjold, T. T.

Amezcua Correa, R.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Antonio Lopez, J. E.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Barankov, R.

Barankov, R. A.

R. A. Barankov, “Cross-correlation imaging for waveguide characterization,” M. S. Thesis, http://arxiv.org/abs/1206.0666v1 [physics.optics] 24.

Ben Yoo, S. J.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Benitez, A. V.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Born, M.

M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]

Broeng, J.

Burke, J.

Chen, H.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Demas, J.

Dunn, C.

Ercan, B.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Essiambre, R.-J.

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100(5), 1035–1055 (2012).
[Crossref]

Fontaine, N. K.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Fricke-Begemann, T.

Fu, Y.

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl.Opt. 46(23), 5719–5727 (2007).

Ghalmi, S.

Gruner-Nielsen, L.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Gu, G.

Guan, B.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Hawkins, T. W.

Hipp, M.

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

Hypszer, R.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. B 72(1), 156–160 (1982).
[Crossref]

Jansen, F.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Jauregui, C.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Jedrzejewska-Szczerska, M.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Jones, M.

Jørgensen, M. M.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. B 72(1), 156–160 (1982).
[Crossref]

Kong, F.

Kosmowski, B. B.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Lægsgaard, J.

Laurila, M.

Limpert, J.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Lingle, R. J.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Maciejewski, M.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Mori, T.

Nakadate, S.

Neger, T.

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

Nicholson, J. W.

Osten, W.

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl.Opt. 46(23), 5719–5727 (2007).

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

Otto, H. - J.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Parsons, J.

Pedrini, G.

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl.Opt. 46(23), 5719–5727 (2007).

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

Petersen, S. R.

Plucinski, J.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Ramachandran, S.

Reiterer, P.

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

Ryf, R.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Saito, H.

Sakamoto, T.

Saleh, B. E. A.

B. E. A. Saleh and C. M. Teich, Fundamentals of Photonics (Wiley-Interscience, 2013)

Schimpf, D. N.

Scolari, L.

Scott, R. P.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Strakowski, M.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Stutzki, F.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Sun, Y.

N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. Antonio Lopez, R. Amezcua Correa, B. Guan, B. Ercan, R. P. Scott, S. J. Ben Yoo, L. Gruner-Nielsen, Y. Sun, and R. J. Lingle, “30×30 MIMO transmission over 15 spatial modes,” in Optical Fiber Communication Conference, OFC 2015, Th5C.1.

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. B 72(1), 156–160 (1982).
[Crossref]

Teich, C. M.

B. E. A. Saleh and C. M. Teich, Fundamentals of Photonics (Wiley-Interscience, 2013)

Tiziani, H. J.

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

Tkach, R. W.

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100(5), 1035–1055 (2012).
[Crossref]

Tünnermann, A.

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Wada, M.

Wielandy, S.

Wierzba, P.

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

Woisetschlager, J.

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]

Yablon, A. D.

Yamamoto, F.

Yamamoto, T.

Appl. Opt. (2)

Appl.Opt. (2)

G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl.Opt. 42(29), 5846–5854 (2003).

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl.Opt. 46(23), 5719–5727 (2007).

Bull. Pol. Ac.:Tech. (1)

J. Plucinski, R. Hypszer, P. Wierzba, M. Strakowski, M. Jedrzejewska-Szczerska, M. Maciejewski, and B. B. Kosmowski, “Optical low-coherence interferometry for selected technical applications,” Bull. Pol. Ac.:Tech. 56(2), 155–172 (2008).

J. Lightwave Technol. (2)

H. - J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

T. Mori, T. Sakamoto, M. Wada, T. Yamamoto, and F. Yamamoto, “Few-mode fibers supporting more than two LP modes for mode-division-multiplexed transmission with MIMO DSP,” J. Lightwave Technol. 32 (14), 2468–2479 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. B 72(1), 156–160 (1982).
[Crossref]

Meas. (1)

M. Hipp, J. Woisetschlager, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Meas. 36(1), 53–66 (2004).
[Crossref]

Opt. Express (8)

S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15(23), 15402–15409 (2007).
[Crossref] [PubMed]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
[Crossref] [PubMed]

T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011).
[Crossref] [PubMed]

D. N. Schimpf, R. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13008–13019 (2011).
[Crossref] [PubMed]

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express 20(7), 7263–7373 (2012).
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Figures (12)

Fig. 1
Fig. 1 (a) The scheme of the setup is presented as an unbalanced Mach-Zehnder interferometer. BS and BC are a beam splitter and a beam combiner, FUT is a fiber under test and DL is a delay line. (b) An example of the cross-correlated trace, where the intensity1 recorded by a camera is represented as function of the time-delay τ.
Fig. 2
Fig. 2 The scheme of 2D FT analysis of the fiber modal content.
Fig. 3
Fig. 3 The scheme of the measurement setup for the flexible C2 imaging. L - lens, HWP - half-wave plate, P - polarizer, BS - beams splitter, CCD - the camera.
Fig. 4
Fig. 4 The output spectrum of SuperK filtered by SuperSELECT at 1050nm, when the only one channel is turned on.
Fig. 5
Fig. 5 Calculated effective refractive indices for the FM and the first HOM as a function of wavelength for SMF28. LP11 experience the cutoff around 1260nm and is not guided at longer wavelengths.
Fig. 6
Fig. 6 The output spectrum of SuperK filtered by SuperSELECT with different FWHM centered at 1050nm (a) and 890nm (b).
Fig. 7
Fig. 7 Calculated and measured temporal resolution depending on the different spectral widths of the input spectrum at the central wavelength of 1050nm.
Fig. 8
Fig. 8 Comparison of the C2 traces measured with the spectral bandwidths of 7.23nm (blue line) and 28.25nm (red line) at the wavelength of 1050nm.
Fig. 9
Fig. 9 The measured C2 trace for the spectrum combined with 2 beams at the central wavelengths of 1040nm and 1050nm. At the inset is depicted the input spectrum.
Fig. 10
Fig. 10 Calculated and measured differential time delay between LP01 and LP11 propagating in the FUT.
Fig. 11
Fig. 11 The C2 trace recorded at the central wavelength of 890nm (a) with spectral widths of 7.23nm (blue line) and 28.25nm (red line) and 870nm (b) with spectral width of 28.25nm (red line). As wavelength was varied the relative modal power was maintained to be kept constant. Insets on the right side represent reconstructed modal intensity profiles of LP01 and LP11 at the maxima of the interference peaks and corresponding phase distributions zoomed in the area of interest.
Fig. 12
Fig. 12 (a) A spectrogram of the intermodal time delay vs wavelength. Every horizontal line on the spectrogram corresponds to a C2 measurement with the spectrum centered on a specific wavelength. (b) The cross-correlated trace measured with offset coupling at 1054nm. (c) Reconstructed modes presented at the measured C2 trace.

Tables (1)

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Table 1 Combination of beams from SuperSELECT used in the measurements.

Equations (12)

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I = I 0 ( x , y ) + I Int ( x , y , τ )
I 0 = I r ( x , y ) + I s ( x , y )
I Int ( x , y , τ ) = m 2 I r ( x , y ) I m ( x , y ) α m | c rm ( τ τ m ) | cos ( Ψ )
c rm ( t ) = 1 2 π S ( Ω ) exp ( i Δ φ mr ( Ω ) ) exp ( i Ω t ) d Ω
Ω = ω ω 0
𝒫 ( r , τ ) = m α m 2 | c rm ( τ τ m ) | I m ( r )
I ( x , y ) = I 0 ( x , y ) + m ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ]
f 0 = k 1 k 2 k 0 ( θ 1 θ 2 ) = k 0 θ
I ( x , y ) = I 0 ( x , y ) + c ( x , y ) e 2 π i f 0 x + c * ( x , y ) e 2 π i f 0 x
c ( x , y ) = 1 2 m ( x , y ) e i ϕ ( x , y )
I ( f x , f y ) = I 0 ( f x , f y ) + C ( f x f 0 , f y ) + C * ( f x + f 0 , f y )
L r = β m ( 2 ) β r ( 2 ) L

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