Abstract

Amplitude and phase noise correlation matrices are of fundamental importance for studying noise properties of frequency combs. They include information about the origin of noise sources as well as the scaling and correlation of the noise across the comb lines. These matrices provide an insight that is essential for obtaining low-noise performance which is important for, e.g., applications in optical communication, low–noise microwave signal generation, and distance measurements. Estimation of amplitude and phase noise correlation matrices requires highly–accurate measurement technique which can distinguishes between noise sources coming from the frequency comb and the measurement system itself. Bayesian filtering provides a theoretically optimum approach for filtering of measurement noise and thereby, the most accurate measurement of phase and amplitude noise. In this paper, a novel Bayesian filtering based framework for joint estimation of amplitude and phase noise of multiple frequency comb lines is proposed, and demonstrated for phase noise characterization. Compared to the conventional approaches, that do not employ any measurement noise filtering, the proposed approach provides significantly more accurate measurements of correlation matrices, operates over a wide range of signal–to–noise–ratios and gives an insight into comb’s dynamics at short scales (<10−8 s).

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
    [Crossref]
  2. R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
    [Crossref]
  3. X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
    [Crossref]
  4. P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years,” Opt. Express 26(18), 24190–24239 (2018).
    [Crossref]
  5. L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
    [Crossref]
  6. L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
    [Crossref]
  7. A. Schlatter, S. Zeller, R. Paschotta, and U. Keller, “Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser,” Appl. Phys. B 88(3), 385–391 (2007).
    [Crossref]
  8. D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
    [Crossref]
  9. I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
    [Crossref]
  10. V. Durán, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” Opt. Express 23(23), 30557–30569 (2015).
    [Crossref]
  11. G. Brajato, L. Lundberg, V. Torres-Company, and D. Zibar, “Optical frequency comb noise characterization using machine learning,” Presented at the European Conference on Optical Communication (ECOC), Dublin, Ireland, 22–26 Sept. 2019.
  12. A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, A. Takada, T. Sogawa, and M. Koga, “Phase-noise characteristics of a 25-GHz-spaced optical frequency comb based on a phase-and intensity-modulated laser,” Opt. Express 21(24), 29186–29194 (2013).
    [Crossref]
  13. S. Särkkä, Bayesian Filtering and Smoothing, vol. 3 (Cambridge University, 2013).
  14. C. J. Wu, “On the convergence properties of the em algorithm,” Ann. Statist. 11(1), 95–103 (1983).
    [Crossref]
  15. L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.
  16. D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
    [Crossref]
  17. F. Riehle, Frequency Standards: Basics and Applications (John Wiley & Sons, 2006).

2020 (1)

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

2019 (1)

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

2018 (3)

P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years,” Opt. Express 26(18), 24190–24239 (2018).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

2017 (1)

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

2016 (2)

J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
[Crossref]

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
[Crossref]

2015 (1)

2014 (1)

R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
[Crossref]

2013 (1)

2007 (1)

A. Schlatter, S. Zeller, R. Paschotta, and U. Keller, “Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser,” Appl. Phys. B 88(3), 385–391 (2007).
[Crossref]

1983 (1)

C. J. Wu, “On the convergence properties of the em algorithm,” Ann. Statist. 11(1), 95–103 (1983).
[Crossref]

Alexandre, C.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Andersen, U.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Andrekson, P.

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Andrekson, P. A.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

Bouchand, R.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Bowers, J.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Brajato, G.

G. Brajato, L. Lundberg, V. Torres-Company, and D. Zibar, “Optical frequency comb noise characterization using machine learning,” Presented at the European Conference on Optical Communication (ECOC), Dublin, Ireland, 22–26 Sept. 2019.

Carlson, D. R.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Chang, L.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Chin, H.-M.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Chraplyvy, A. R.

Coddington, I.

Datta, S.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Diddams, S. A.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Durán, V.

Fabre, C.

R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
[Crossref]

Foo, B.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

Fülöp, A.

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

Gehring, T.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Giunta, M.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Guo, J.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Hänsel, W.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Hickstein, D. D.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Holzwarth, R.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Ishizawa, A.

Jain, N.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Joshi, A.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Karlsson, M.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

Keller, U.

A. Schlatter, S. Zeller, R. Paschotta, and U. Keller, “Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser,” Appl. Phys. B 88(3), 385–391 (2007).
[Crossref]

Kim, J.

J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
[Crossref]

Koga, M.

Le Coq, Y.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Lezius, M.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Lorences-Riesgo, A.

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Lours, M.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Lundberg, L.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

G. Brajato, L. Lundberg, V. Torres-Company, and D. Zibar, “Optical frequency comb noise characterization using machine learning,” Presented at the European Conference on Optical Communication (ECOC), Dublin, Ireland, 22–26 Sept. 2019.

Mazur, M.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

Metcalf, A. J.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Mirani, A.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

Mizutori, A.

Neilson, D. T.

Newbury, N.

Nicolodi, D.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Nishikawa, T.

Papp, S. B.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Paschotta, R.

A. Schlatter, S. Zeller, R. Paschotta, and U. Keller, “Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser,” Appl. Phys. B 88(3), 385–391 (2007).
[Crossref]

Quinlan, F.

D. R. Carlson, D. D. Hickstein, W. Zhang, A. J. Metcalf, F. Quinlan, S. A. Diddams, and S. B. Papp, “Ultrafast electro-optic light with subcycle control,” Science 361(6409), 1358–1363 (2018).
[Crossref]

Riehle, F.

F. Riehle, Frequency Standards: Basics and Applications (John Wiley & Sons, 2006).

Roslund, J.

R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
[Crossref]

Santarelli, G.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Särkkä, S.

S. Särkkä, Bayesian Filtering and Smoothing, vol. 3 (Cambridge University, 2013).

Schlatter, A.

A. Schlatter, S. Zeller, R. Paschotta, and U. Keller, “Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser,” Appl. Phys. B 88(3), 385–391 (2007).
[Crossref]

Schmeissner, R.

R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
[Crossref]

Schröder, J.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Sogawa, T.

Song, Y.

J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
[Crossref]

Swann, W.

Tainta, S.

Takada, A.

Takara, H.

Tong, Y.

D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Torres-Company, V.

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

V. Durán, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” Opt. Express 23(23), 30557–30569 (2015).
[Crossref]

G. Brajato, L. Lundberg, V. Torres-Company, and D. Zibar, “Optical frequency comb noise characterization using machine learning,” Presented at the European Conference on Optical Communication (ECOC), Dublin, Ireland, 22–26 Sept. 2019.

L. Lundberg, M. Mazur, A. Fülöp, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase correlation between lines of electro-optical frequency combs,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2018), p. JW2A.149.

Tremblin, P. A.

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
[Crossref]

Treps, N.

R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, “Spectral noise correlations of an ultrafast frequency comb,” Phys. Rev. Lett. 113(26), 263906 (2014).
[Crossref]

Winzer, P. J.

Wu, C. J.

C. J. Wu, “On the convergence properties of the em algorithm,” Ann. Statist. 11(1), 95–103 (1983).
[Crossref]

Xie, X.

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[Crossref]

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[Crossref]

Appl. Sci. (1)

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency comb-based wdm transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
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D. Zibar, H.-M. Chin, Y. Tong, N. Jain, J. Guo, L. Chang, T. Gehring, J. Bowers, and U. Andersen, “Highly–sensitive phase and frequency noise measurement technique using bayesian filtering,” IEEE Photonics Technol. Lett. 31(23), 1866–1869 (2019).
[Crossref]

Nat. Commun. (1)

L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schröder, V. Torres-Company, M. Karlsson, and P. A. Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nat. Commun. 11, 201 (2020).
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Nat. Photonics (1)

X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
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Science (1)

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Figures (13)

Fig. 1.
Fig. 1. (a) System set-up for numerical and experimental investigations. Two optical frequency combs are heterodyned together, and the donwconverted frequency comb is sampled and digitized, obtaining the sequence $y_k$. (b) Typical power spectral density of the downconverted comb. (c) Bayesian filtering framework for joint estimation of static, $\mathbf {Q}$, and dynamic parameters, $[{\phi }_k^1,\ldots ,{\phi }_k^M]$ and $[{a}_k^1,\ldots , {a}_k^M]$
Fig. 2.
Fig. 2. (Numerical) (a) Evolution of the estimated $\mathbf {Q}_{\boldsymbol {\phi}}$ matrix during the EM learning algorithm. Each matrix is estimated after the $n$-th EM iteration. (b) Negative log–likelihood of the matrix $\mathbf {Q}_{\boldsymbol {\phi}}$ plotted for every EM iteration number.
Fig. 3.
Fig. 3. Illustration of the conventional method for phase noise estimation of the donwconverted. The resulting phases are denoted as ${\phi }^{1,\textrm {conv}}_k,\ldots ,{\phi }^{M,\textrm {conv}}_k$.
Fig. 4.
Fig. 4. (Numerical) (a) True correlation matrices (left) shown together with the the estimated matrices obtained by the conventional method (center) and the Bayesian filtering method (right). The true correlation matrix is the same for all the simulations. Each row is a simulation with different value of average $\textrm {SNR}_{\textrm {avg}}$, obtained by averaging the SNR per line over all lines. (b) The RMSE between the true and the estimated phase noise traces, for all $M=49$ comb lines, as function of the $\textrm {SNR}_{\textrm {avg}}$. The linewidth under consideration is 1 kHz and 1 MHz.
Fig. 5.
Fig. 5. (Numerical) Variance of the differential phase, $\sigma ^2_{\Delta \phi _T^m}$, using Bayesian filtering (blue curve) and a conventional method (black curve). The red curve represents the ground truth.
Fig. 6.
Fig. 6. (Numerical) Normalized eigenvectors corresponding to the two principal eigenvalues for the longest observation time $k=T$. The eigenvectors are extracted using a conventional, (Black curve), and a Bayesian filtering, (Blue curve), method. The true eigenvectors that associated with the CW laser and the RF phase noise are depicted in red.
Fig. 7.
Fig. 7. (Numerical) Evolution of two main eigenvalues as a function of the observation time ($\tau _{\textrm {obs}}$). We also show the evolution of the variance of the CW laser and the RF oscillator phase noise, $\sigma ^2_{\textrm {L},\tau _{\textrm {obs}}}$ and $\sigma ^2_{\textrm {RF},\tau _{\textrm {obs}}}$, respectively.
Fig. 8.
Fig. 8. (Experimental) Evolution of the estimated phase noise as a function of time using for comb-lines with different relative powers. (a) Comb line number $m=3$ with 0 dBm of relative power. (b) Comb line number $m=7$ with -5 dBm of relative power.
Fig. 9.
Fig. 9. (Experimental) Variance of the differential phase, $\sigma ^2_{\Delta \phi _T^m}$, using Bayesian filtering (blue curve) and a conventional method (black curve). (b) estimated correlation matrices.
Fig. 10.
Fig. 10. (Experimental) Correlation matrices of the phase noise traces estimated for different Fourier frequencies. (a) and (c) show the correlation corresponding to $f_{\textrm {obs}} = 5$ GHz for the conventional and Bayesian filtering framework respectively. (b) and (d) show the correlation corresponding to $f_{\textrm {obs}} = 4$ kHz for the conventional and Bayesian filtering framework respectively.
Fig. 11.
Fig. 11. (Experimental) Normalized eigenvectors corresponding to the two principal eigenvalues for the longest observation time $k=T$. The eigenvectors are extracted using a conventional, (Black curve), and a Bayesian filtering, (Blue curve), method.
Fig. 12.
Fig. 12. (Experimental) Evolution of two main eigenvalues as a function of the observation time $\tau _{\textrm {obs}}$.
Fig. 13.
Fig. 13. (Experimental) Power spectral density of the extracted phase noise traces using the Eigenvector projection with a conventional method (black curve) and with the Bayesian filtering framework (blue curve). (a) shows the extracted laser source phase noise and (b) the RF oscillator phase noise.

Equations (19)

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ϕ k m = ϕ k 1 m + q ϕ , k 1 m with  m = 1 M ,
a k m = a k 1 m + q A , k 1 m with  m = 1 ,
y k = m = 1 M a k m sin ( Δ ω m T s k + ϕ k m ) + n k ,
Q := [ Q ϕ Q A , ϕ Q A , ϕ Q A ] .
x k opt = E [ x k | y 1 : k ] = x k p ( x k | y 1 : k ) d x k .
v k = y k m = 1 M a k 1 opt , m sin ( Δ ω m T s k + ϕ k 1 opt , m ) , s k = h x ( x k 1 opt ) [ Σ k 1 opt + Q ] h x ( x k 1 opt ) + σ n 2 , k k = [ Σ k 1 opt + Q ] h x ( x k opt ) s k 1 , x k opt = x k 1 opt + k k v k , Σ k opt = Σ k 1 opt + Q k k s k k k .
h ϕ ( x k ) = [ a k 1 cos ( Δ ω 1 T s k + ϕ k 1 ) , , a k M cos ( Δ ω M T s k + ϕ k M ) ] , h A ( x k ) = [ sin ( Δ ω 1 T s k + ϕ k 1 ) , , sin ( Δ ω M T s k + ϕ k M ) ] .
Q opt = argmax Q [ p ( Q | y 1 : T ) ] = argmax Q [ p ( y 1 : T | Q ) p ( Q ) ] ,
LL ( Q ) 1 2 k = 1 T [ log 2 π s k ( Q ) + v k 2 ( Q ) s k ( Q ) 1 ] ,
Q opt = argmin Q [ LL ( Q ) log ( p ( Q ) ) ] .
G k = Σ k opt [ Σ k opt + Q ( n ) ] 1 , x k s = x k opt + G k ( x k + 1 s x k opt ) , Σ k s = Σ k opt + G k ( Σ k + 1 s x k opt ) G k ,
Q ( n + 1 ) = P C C + B ,
P = 1 T k = 1 K Σ k s + x k s [ x k s ] ,
C = 1 T k = 1 K Σ k s G k 1 s + x k s [ x k 1 s ] ,
B = 1 T k = 1 K Σ k 1 s + x k 1 s [ x k 1 s ] .
Σ u , l : n = 1 K 1 k = l n ( u k u ¯ l : n ) ( u k u ¯ l : n ) = ( σ u , 11 2 σ u , 12 2 σ u , 1 M 2 σ u , 21 2 σ u , 22 2 σ u , 2 M 2 σ u , M 1 2 σ u , M 2 2 σ u , M M 2 )
ρ u , l : n = [ diag ( Σ u , l : n ) ] 1 / 2 Σ u , l : n [ diag ( Σ u , l : n ) ] 1 / 2 = ( 1 ρ u , 12 ρ v , 1 M ρ u , 21 1 ρ v , 2 M ρ u , M 1 ρ u , M 2 1 )
ϕ k m = ϕ k L + m r ϕ k RF ,
Φ = ϕ 1 : T L v 1 + ϕ 1 : T RF v 2

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