Abstract

We report on the use of the acousto-optic frequency combs generated by frequency shifting loops as compact and versatile optical waveforms generators for pulse compression systems in the optical coherent domain. The high degree of tunability and mutual coherence of these sources permits an efficient use of the available detection bandwidth, and represent simple alternatives to broadband lasers that do not require fast electronics. The full, complex optical field is retrieved using heterodyne measurements in bandwidths as high as 20 GHz. Compression ratios up to 150 at 80-MHz repetition rate, with autocorrelation peak-to-sidelobe ratios in excess of 28 dB, are demonstrated. In a proof-of-concept ranging experiment, we obtain resolutions of 4 mm in free space at meter scales, limited by detection bandwidth. Systems based on frequency shifting loops thus enable compact implementations of the pulse compression concept in the optical coherent domain, for its use in general optical metrology systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (4)

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
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A. Zadok, E. Preter, and Y. London, “Phase-coded and noise-based Brillouin optical correlation-domain analysis,” Appl. Sci.-Basel 8, 1482 (2018).
[Crossref]

T. Hariyama, P. A. M. Sandborn, M. Watanabe, and M. C. Wu, “High-accuracy range-sensing system based on FMCW using low-cost VCSEL,” Opt. Express 26, 9285–9297 (2018).
[Crossref] [PubMed]

V. Durán, C. Schnébelin, and H. Guillet de Chatellus, “Coherent multi-heterodyne spectroscopy using acousto-optic frequency combs,” Opt. Express 26, 13800–13809 (2018).
[Crossref] [PubMed]

2017 (1)

2016 (4)

L. Romero-Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self imaging effects on periodic objects,” Opt. Lett. 41, 340–343 (2016). Erratum, Opt. Lett. 41, 5748 (2016).
[Crossref]

A. Bergman, T. Langer, and M. Tur, “Coding-enhanced ultrafast and distributed Brillouin dynamic gratings sensing using coherent detection,” J. Lightwave Technol. 34, 5593–5600 (2016).
[Crossref]

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. 8, 1–11 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (2)

2011 (1)

2009 (1)

2005 (1)

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 45, 557–571 (2005).
[Crossref]

2004 (1)

2001 (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron.,  7, 728–744 (2001).
[Crossref]

2000 (1)

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique-proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

1989 (1)

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

1983 (1)

1982 (1)

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. 18, 690–693 (1982).
[Crossref]

Agmon, A.

Ai, X.

Arbel, N.

N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. 8, 1–11 (2016).
[Crossref]

Azaña, J.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
[Crossref] [PubMed]

L. Romero-Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self imaging effects on periodic objects,” Opt. Lett. 41, 340–343 (2016). Erratum, Opt. Lett. 41, 5748 (2016).
[Crossref]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron.,  7, 728–744 (2001).
[Crossref]

Baba, H.

Barber, Z. W.

Baumann, E.

Berg, T.

Bergman, A.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

A. Bergman, T. Langer, and M. Tur, “Coding-enhanced ultrafast and distributed Brillouin dynamic gratings sensing using coherent detection,” J. Lightwave Technol. 34, 5593–5600 (2016).
[Crossref]

Biton, M.

D. Mermelstein, M. Biton, S. Sternklar, and E. Granot, “Fiber-optic range sensing based on amplified spontaneous emission noise radar with Kramers-Kronig phase retrieval,” in CLEO 2011 - Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper JThB135.
[Crossref]

Bosch, V.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Burla, M.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
[Crossref] [PubMed]

Chen, J.

Coddington, I.

Culshaw, B.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. 18, 690–693 (1982).
[Crossref]

Dahnoun, N.

Derickson, D.

D. Derickson, Fiber optics test and measurement(Prentice Hall, 1998).

Deschênes, J.-D.

Djevarhidjian, L.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Durán, V.

Fernández-Pousa, C. R.

Foster, S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Giffard, R. P.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Giles, I. P.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. 18, 690–693 (1982).
[Crossref]

Giorgetta, F. R.

Glastre, W.

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

Goldman, R.

Granot, E.

D. Mermelstein, M. Biton, S. Sternklar, and E. Granot, “Fiber-optic range sensing based on amplified spontaneous emission noise radar with Kramers-Kronig phase retrieval,” in CLEO 2011 - Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper JThB135.
[Crossref]

Guillet de Chatellus, H.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
[Crossref] [PubMed]

V. Durán, C. Schnébelin, and H. Guillet de Chatellus, “Coherent multi-heterodyne spectroscopy using acousto-optic frequency combs,” Opt. Express 26, 13800–13809 (2018).
[Crossref] [PubMed]

L. Romero-Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self imaging effects on periodic objects,” Opt. Lett. 41, 340–343 (2016). Erratum, Opt. Lett. 41, 5748 (2016).
[Crossref]

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Hariyama, T.

Hasegawa, T.

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique-proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

Hirschbrand, L.

N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. 8, 1–11 (2016).
[Crossref]

Hotate, K.

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique-proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

Hugon, O.

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

Jacquin, O.

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

Kassi, S.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Kaylor, B.

Lacot, E.

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

Langer, T.

LaRochelle, S.

L. Wang and S. LaRochelle, “Talbot laser with tunable GHz repetition rate using an electro-optic frequency shifter,” in CLEO 2017 - Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2017), paper JW2A.66.

Levanon, N.

N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. 8, 1–11 (2016).
[Crossref]

N. Levanon and E. Mozeson, Radar signals (John Wiley & Sons, 2004).
[Crossref]

London, Y.

A. Zadok, E. Preter, and Y. London, “Phase-coded and noise-based Brillouin optical correlation-domain analysis,” Appl. Sci.-Basel 8, 1482 (2018).
[Crossref]

Long, X.

Matthey, R.

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 45, 557–571 (2005).
[Crossref]

Méjean, G.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Mermelstein, D.

D. Mermelstein, M. Biton, S. Sternklar, and E. Granot, “Fiber-optic range sensing based on amplified spontaneous emission noise radar with Kramers-Kronig phase retrieval,” in CLEO 2011 - Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper JThB135.
[Crossref]

Mitev, V.

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 45, 557–571 (2005).
[Crossref]

Moberly, D. S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Motil, A.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Mozeson, E.

N. Levanon and E. Mozeson, Radar signals (John Wiley & Sons, 2004).
[Crossref]

Muriel, M. A.

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron.,  7, 728–744 (2001).
[Crossref]

Nazarathy, M.

R. Goldman, A. Agmon, and M. Nazarathy, “Direct detection and coherent optical time-domain reflectometry with Golay complementary codes,” J. Lightwave Technol. 31, 2207–2222 (2013).
[Crossref]

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Newbury, N. R.

Newton, S. A.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Nithyanandan, K.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Nock, R.

Preter, E.

A. Zadok, E. Preter, and Y. London, “Phase-coded and noise-based Brillouin optical correlation-domain analysis,” Appl. Sci.-Basel 8, 1482 (2018).
[Crossref]

Randall Babbitt, Wm.

Rarity, J. G.

Reibel, R.R.

Romanini, D.

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Romero Cortés, L.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
[Crossref] [PubMed]

Romero-Cortés, L.

Roos, P. A.

Sakurai, K.

Sandborn, P. A. M.

Schnébelin, C.

V. Durán, C. Schnébelin, and H. Guillet de Chatellus, “Coherent multi-heterodyne spectroscopy using acousto-optic frequency combs,” Opt. Express 26, 13800–13809 (2018).
[Crossref] [PubMed]

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9, 2438 (2018).
[Crossref] [PubMed]

K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in Frontiers in Optics/Laser Science, OSA Technical Digest (Optical Society of America, 2018), paper LTh1F.3.

Sischka, F.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Sternklar, S.

D. Mermelstein, M. Biton, S. Sternklar, and E. Granot, “Fiber-optic range sensing based on amplified spontaneous emission noise radar with Kramers-Kronig phase retrieval,” in CLEO 2011 - Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper JThB135.
[Crossref]

Sugimoto, N.

Swann, W. C.

Takeuchi, N.

Trutna, W. R.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7, 24–38 (1989).
[Crossref]

Tur, M.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

A. Bergman, T. Langer, and M. Tur, “Coding-enhanced ultrafast and distributed Brillouin dynamic gratings sensing using coherent detection,” J. Lightwave Technol. 34, 5593–5600 (2016).
[Crossref]

Wang, L.

L. Wang and S. LaRochelle, “Talbot laser with tunable GHz repetition rate using an electro-optic frequency shifter,” in CLEO 2017 - Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2017), paper JW2A.66.

Watanabe, M.

Weiss, S.

N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. 8, 1–11 (2016).
[Crossref]

Wu, M. C.

Yang, S.

Zadok, A.

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Figures (7)

Fig. 1
Fig. 1 Scheme of (a) FSL, (b) frequency comb, and (c) the fiber/free space interferometer used in the ranging experiment. In (b), the dashed trace represents the TBPF transmittance.
Fig. 2
Fig. 2 Normalized PACFs of FSL combs with N= 250 for different values of the decay parameter ρ. The arrow describes the successive values of ρ corresponding to each curve.
Fig. 3
Fig. 3 Left: FSL heterodyne signal v(t) for frequency shifts fs set to (a) 82.018 MHz, (b) 78.572 MHz, (c) 80.841 MHz, and (d) 81.327 MHz, corresponding to the dispersion of Talbot fields with indices 1, 1/4, 1/12, and 1/21, respectively. Right: FFT spectra of the corresponding heterodyne signals.
Fig. 4
Fig. 4 Heterodyne spectra of the FSL output with fs= 80.841 MHz, corresponding to dispersed Talbot 1/4 pulses (waveform (c) in Fig. 3, N = B / f s = 247) for different values of EDFA pump currents: (a) 101 mA (ρ = 0.992, N e f f = 150), (b) 47 mA (ρ = 0.974, N e f f = 70), and (c) 31 mA (ρ = 0.857, N e f f = 13). The spikes at multiples of 5 GHz are due to the DSO’s internal sampling clock.
Fig. 5
Fig. 5 (a-c) Normalized PACF of the field E c ( t ) extracted from the single-sided heterodyne power spectrum of Fig. 4 (blue traces), and normalized PACF of the corresponding FSL envelope E ( t ) (orange traces). The peak-to-sidelobe power levels are (a) 32 dB, and (b) 21 dB. The peak-to-noise power level in (c) is 18 dB.
Fig. 6
Fig. 6 PACF of the experimental complex field E c ( t ) (blue traces) and simulated PACF, Eq. (5), of the FSL envelope E ( t ) (orange traces) corresponding to (a) dispersed Talbot 1 of Fig. 3(a), (b) dispersed Talbot 1/12 of Fig. 3(c), and (c) dispersed Talbot 1/21 of Fig. 3(d). The parameters used in the simulations are (a) ρ=0.990, N=243, (b) ρ=0.991, N=247, and (c) ρ=0.992, N=245. The peak-to-noise power levels are (a) 34 dB, and (c) 33 dB, whereas the peak-to-sidelobe power level in (b) is 28 dB.
Fig. 7
Fig. 7 Cross-correlation of the reference envelope and the complex field retrieved from single-detector heterodyne measurements in the fiber/free space interferometer, using the waveform of Fig. 3(d), at two different mirror positions, (a) and (b) in linear scale, and correspondingly (c) and (d) in dB scale. In (c) and (d), superimposed with orange trace is the simulation of the correlation peaks.

Tables (1)

Tables Icon

Table 1 Optical free-space path difference, in mm, corresponding to three different mirror positions, denoted as 1, 2, and 3, in the composite fiber/free space interferometer. The negative signs indicate that the mirror decreases the optical path with respect to the fiber mirror, as shown in the autocorrelation traces of Fig. 7. The ruler only provides relative readouts, so that they are referred to the initial unknown path difference, denoted by x.

Equations (10)

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E ( t ) = n = 0 N 1 E n e j 2 π f s n t = E 0 n = 0 N 1 ρ n / 2 e j π f s τ c n 2 e j 2 π f s n t ,
N e f f = ( n P n ) 2 n P n 2 .
N e f f = 1 + ρ 1 ρ 1 ρ N 1 + ρ N .
R ( τ ) = 0 T E ( t ) * E ( t + τ ) d t = T n = 0 N 1 | E n | 2 e j 2 π f s n τ = T P 0 n = 0 N 1 ρ n e j 2 π f s n τ
R ( τ ) = T P 0 1 ρ N e j 2 π N f s τ 1 ρ e j 2 π f s τ R N ( τ )
| R ( τ ) | = E p [ 1 + C N sin 2 ( π f s N τ ) 1 + C 1 sin 2 ( π f s τ ) ] 1 / 2
Δ τ e q = 1 R ( 0 ) 2 0 T | R ( τ ) | 2 d τ = T n P n 2 ( n P n ) 2 = T N e f f ,
v ( t ) = I L O + I ( t ) + 2 I L O R e [ E ( t ) e j φ ]
R + ( τ ) = 4 ρ e j 2 π f s τ R N 1 ( τ ) .
E c ( t ) = I L O E + ( t ) e j φ + I + ( t ) .