Abstract

Digital backpropagation (DBP) is a promising digital-domain technique to mitigate Kerr-induced nonlinear interference. While it successfully removes deterministic signal–signal interactions, the performance of ideal DBP is limited by stochastic effects, such as polarization-mode dispersion (PMD). In this paper, we consider an ideal full-field DBP implementation and modify it to additionally account for PMD; reversing the PMD effects in the backward propagation by passing the reverse propagated signal also through PMD sections, which concatenated equal the inverse of the PMD in the forward propagation. These PMD sections are calculated analytically at the receiver based on the total accumulated PMD of the link estimated from channel equalizers. Numerical simulations show that, accounting for nonlinear polarization-related interactions in the modified DBP algorithm, additional signal-to-noise ratio gains of 1.1 dB are obtained for transmission over 1000 km.

© 2017 Optical Society of America

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References

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2016 (3)

Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightw. Technol. 34(2), 321–326 (2016).
[Crossref]

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

2015 (2)

E. Temprana, E. Myslivets, L. Liu, V. Ataie, A. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Two-fold transmission reach enhancement enabled by transmitter-side digital backpropagation and optical frequency comb-derived information carriers,” Opt. Express 23(16), 20774–83 (2015).
[Crossref] [PubMed]

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

2014 (4)

2013 (1)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (3)

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010).
[Crossref]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol. 28(6), 939–951 (2010).
[Crossref]

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Topics Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

2006 (2)

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

1997 (2)

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightw. Technol. 15(9), 1735–1746 (1997).
[Crossref]

1990 (1)

R. Haber and H. Unbehauen, “Structure identification of nonlinear dynamic systems—a survey on input/output approaches,” Automatica 26(4), 651–677 (1990).
[Crossref]

1976 (1)

W. Kabsch, “A solution for the best rotation to relate two sets of vectors,” Acta Crystallogr. Sect. A 32(5), 922–923 (1976).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1964).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 5th ed.

Agrell, E.

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun. 62(11), 3956–3968 (2014).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

Alic, N.

Alvarado, A.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

Ataie, V.

Awwad, E.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

Bayvel, P.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

Bellman, R.

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, 1960).

Borowiec, A.

Burrows, E. C.

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

Cartledge, J. C.

Chandrasekhar, S.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

Charlet, G.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, P. Tran, and G. Charlet, “Impact of polarization mode dispersion on digital nonlinear compensation algorithms in dispersion unmanaged systems,” in “Proc. of Optical Fiber Communication Conference (OFC),” (Anaheim, CA, 2016), p. Th3D.3.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

Chen, X.

Chraplyvy, A. R.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

Czegledi, C. B.

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

Dal Forno, A. O.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

de Waardt, H.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

Ellis, A. D.

Essiambre, R. J.

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

Fernandez de Jauregui Ruiz, I.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, P. Tran, and G. Charlet, “Impact of polarization mode dispersion on digital nonlinear compensation algorithms in dispersion unmanaged systems,” in “Proc. of Optical Fiber Communication Conference (OFC),” (Anaheim, CA, 2016), p. Th3D.3.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

Forestieri, E.

M. Secondini and E. Forestieri, “On XPM mitigation in WDM fiber-optic systems,” IEEE Photon. Technol. Lett. 26(22), 2252–2255 (2014).
[Crossref]

Galdino, L.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

Gao, G.

Gao, Y.

Ghazisaeidi, A.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, P. Tran, and G. Charlet, “Impact of polarization mode dispersion on digital nonlinear compensation algorithms in dispersion unmanaged systems,” in “Proc. of Optical Fiber Communication Conference (OFC),” (Anaheim, CA, 2016), p. Th3D.3.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

Gisin, N.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

Goroshko, K.

K. Goroshko, H. Louchet, and A. Richter, “Fundamental limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of Asia Communications and Photonics Conference (ACP),” (Hong Kong, China, 2015), p. ASu3F.5.

K. Goroshko, H. Louchet, and A. Richter, “Overcoming performance limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of International Conference on Transparent Optical Networks (ICTON),” (Trento, Italy, 2016), p. Mo.B1.4.

Haber, R.

R. Haber and H. Unbehauen, “Structure identification of nonlinear dynamic systems—a survey on input/output approaches,” Automatica 26(4), 651–677 (1990).
[Crossref]

Hirano, M.

Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightw. Technol. 34(2), 321–326 (2016).
[Crossref]

Ip, E.

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol. 28(6), 939–951 (2010).
[Crossref]

Irukulapati, N. V.

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun. 62(11), 3956–3968 (2014).
[Crossref]

Ives, D.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

Jansen, S. L.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

Johannisson, P.

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun. 62(11), 3956–3968 (2014).
[Crossref]

Kabsch, W.

W. Kabsch, “A solution for the best rotation to relate two sets of vectors,” Acta Crystallogr. Sect. A 32(5), 922–923 (1976).
[Crossref]

Karar, A. S.

Karlsson, M.

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

Kawaguchi, Y.

Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightw. Technol. 34(2), 321–326 (2016).
[Crossref]

Khoe, G.-D.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

Killey, R. I.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

Krummrich, P. M.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

Kuo, B. P. P.

Kuzmin, K.

H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in “Proc. of European Conference on Optical Communication (ECOC),” (Brussels, Belgium, 2008), p. Tu.1.E.6.

Laperle, C.

Laub, A. J.

A. J. Laub, Matrix Analysis for Scientists & Engineers (Society for Industrial and Applied Mathematics, 2005).
[Crossref]

Lavery, D.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

Lee, W.

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

Li, G.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010).
[Crossref]

Liga, G.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

Liu, L.

Liu, X.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

Louchet, H.

H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in “Proc. of European Conference on Optical Communication (ECOC),” (Brussels, Belgium, 2008), p. Tu.1.E.6.

K. Goroshko, H. Louchet, and A. Richter, “Overcoming performance limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of International Conference on Transparent Optical Networks (ICTON),” (Trento, Italy, 2016), p. Mo.B1.4.

K. Goroshko, H. Louchet, and A. Richter, “Fundamental limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of Asia Communications and Photonics Conference (ACP),” (Hong Kong, China, 2015), p. ASu3F.5.

Maher, R.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

Manyuk, C. R.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightw. Technol. 15(9), 1735–1746 (1997).
[Crossref]

Marcuse, D.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightw. Technol. 15(9), 1735–1746 (1997).
[Crossref]

Myslivets, E.

O’Sullivan, M.

Passy, R.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

Pereira da Silva, J. A.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

Prola, C. H.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

Radic, S.

Rafique, D.

Richter, A.

K. Goroshko, H. Louchet, and A. Richter, “Fundamental limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of Asia Communications and Photonics Conference (ACP),” (Hong Kong, China, 2015), p. ASu3F.5.

K. Goroshko, H. Louchet, and A. Richter, “Overcoming performance limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of International Conference on Transparent Optical Networks (ICTON),” (Trento, Italy, 2016), p. Mo.B1.4.

H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in “Proc. of European Conference on Optical Communication (ECOC),” (Brussels, Belgium, 2008), p. Tu.1.E.6.

Roberts, K.

Sälter, S.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

Sato, M.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

Savory, S. J.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Topics Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

Secondini, M.

M. Secondini and E. Forestieri, “On XPM mitigation in WDM fiber-optic systems,” IEEE Photon. Technol. Lett. 26(22), 2252–2255 (2014).
[Crossref]

Shi, K.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

Shieh, W.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1964).

Temprana, E.

Thomsen, B. C.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

Tkach, R. W.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

Tran, P.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, P. Tran, and G. Charlet, “Impact of polarization mode dispersion on digital nonlinear compensation algorithms in dispersion unmanaged systems,” in “Proc. of Optical Fiber Communication Conference (OFC),” (Anaheim, CA, 2016), p. Th3D.3.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

Unbehauen, H.

R. Haber and H. Unbehauen, “Structure identification of nonlinear dynamic systems—a survey on input/output approaches,” Automatica 26(4), 651–677 (1990).
[Crossref]

van den Borne, D.

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

von der Weid, J. P.

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

Wai, P. K. A.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightw. Technol. 15(9), 1735–1746 (1997).
[Crossref]

Wang, X. Q.

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

White, C. A.

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

Wiberg, A.

Winzer, P. J.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

Wymeersch, H.

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun. 62(11), 3956–3968 (2014).
[Crossref]

Xu, T.

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

Yam, S. S.-H.

Yamamoto, Y.

Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightw. Technol. 34(2), 321–326 (2016).
[Crossref]

Yaman, F.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010).
[Crossref]

Acta Crystallogr. Sect. A (1)

W. Kabsch, “A solution for the best rotation to relate two sets of vectors,” Acta Crystallogr. Sect. A 32(5), 922–923 (1976).
[Crossref]

Automatica (1)

R. Haber and H. Unbehauen, “Structure identification of nonlinear dynamic systems—a survey on input/output approaches,” Automatica 26(4), 651–677 (1990).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (2)

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Topics Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

S. L. Jansen, D. van den Borne, P. M. Krummrich, S. Sälter, G.-D. Khoe, and H. de Waardt, “Long-haul DWDM transmission systems employing optical phase conjugation,” IEEE J. Sel. Topics Quantum Electron. 12(4), 505–520 (2006).
[Crossref]

IEEE Photon. J. (1)

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (4)

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for optical fiber communications,” IEEE Photon. Technol. Lett. 28(17), 1803–1806 (2016).
[Crossref]

M. Secondini and E. Forestieri, “On XPM mitigation in WDM fiber-optic systems,” IEEE Photon. Technol. Lett. 26(22), 2252–2255 (2014).
[Crossref]

R. J. Essiambre, P. J. Winzer, X. Q. Wang, W. Lee, C. A. White, and E. C. Burrows, “Electronic predistortion and fiber nonlinearity,” IEEE Photon. Technol. Lett. 18(17), 1804–1806 (2006).
[Crossref]

C. H. Prola, J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48-Gb/s IM-DD lightwave systems,” IEEE Photon. Technol. Lett. 9(6), 842–844 (1997).
[Crossref]

IEEE Trans. Commun. (1)

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun. 62(11), 3956–3968 (2014).
[Crossref]

J. Lightw. Technol. (3)

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightw. Technol. 15(9), 1735–1746 (1997).
[Crossref]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol. 28(6), 939–951 (2010).
[Crossref]

Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightw. Technol. 34(2), 321–326 (2016).
[Crossref]

Nat. Photon. (1)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 7(7), 560–568 (2013).
[Crossref]

Nat. Sci. Rep. (2)

R. Maher, T. Xu, L. Galdino, M. Sato, A. Alvarado, K. Shi, S. J. Savory, B. C. Thomsen, R. I. Killey, and P. Bayvel, “Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation,” Nat. Sci. Rep. 5, 8214 (2015).
[Crossref]

C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Nat. Sci. Rep. 6, 21217 (2016).
[Crossref]

Opt. Express (5)

Other (12)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 5th ed.

G. Liga, C. B. Czegledi, T. Xu, E. Agrell, R. I. Killey, and P. Bayvel, “Ultra-wideband nonlinearity compensation performance in the presence of PMD,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 794–796.

K. Goroshko, H. Louchet, and A. Richter, “Fundamental limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of Asia Communications and Photonics Conference (ACP),” (Hong Kong, China, 2015), p. ASu3F.5.

A. J. Laub, Matrix Analysis for Scientists & Engineers (Society for Industrial and Applied Mathematics, 2005).
[Crossref]

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, P. Tran, and G. Charlet, “Impact of polarization mode dispersion on digital nonlinear compensation algorithms in dispersion unmanaged systems,” in “Proc. of Optical Fiber Communication Conference (OFC),” (Anaheim, CA, 2016), p. Th3D.3.

I. Fernandez de Jauregui Ruiz, A. Ghazisaeidi, E. Awwad, P. Tran, and G. Charlet, “Polarization effects in nonlinearity compensated links,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 379–381.

K. Goroshko, H. Louchet, and A. Richter, “Overcoming performance limitations of digital back propagation due to polarization mode dispersion,” in “Proc. of International Conference on Transparent Optical Networks (ICTON),” (Trento, Italy, 2016), p. Mo.B1.4.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Polarization-mode dispersion aware digital backpropagation,” in “Proc. of European Conference on Optical Communication (ECOC),” (Düsseldorf, Germany, 2016), pp. 1091–1093.

C. B. Czegledi, G. Liga, D. Lavery, M. Karlsson, E. Agrell, S. J. Savory, and P. Bayvel, “Modified digital back-propagation accounting for polarization-mode dispersion,” to appear in “Proc. of Optical Fiber Communication Conference (OFC),” (Los Angeles, CA, 2017), p. W1G.6.

H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in “Proc. of European Conference on Optical Communication (ECOC),” (Brussels, Belgium, 2008), p. Tu.1.E.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1964).

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, 1960).

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Figures (5)

Fig. 1
Fig. 1 Schematic of the proposed DBP method, where NDBP is the number of steps used by the DBP algorithm over the entire link and the equalizer is a conventional adaptive channel equalizer such as the CMA or MMA. For brevity, the blocks modeling the amplifiers and attenuation are not shown.
Fig. 2
Fig. 2 The evolution of the accumulated DGD at different frequencies versus distance in the forward and backward propagation. The PMD parameter is 0.1 ps / km and NPMD = 10.
Fig. 3
Fig. 3 Average SNR versus input power per channel for various setups: ((a)) Nch = 1 and Ns = {10, 40}; ((b)) Nch = 7 and Ns = {10, 40}. For comparison, the performance of CD compensation (CDC) only and DBP, both without PMD, are shown. The maximum (in dB) of each curve from the plots is summarized in the legend above.
Fig. 4
Fig. 4 (a): Average SNR obtained at the optimal input power by the conventional DBP and by the modified DBP scheme with varying number of PMD sections NPMD. (b): Scatter plot of the achieved SNR gain by the proposed DBP modification versus the SNR attained by the conventional DBP obtained over 500 fiber realizations at the optimum input power 11 and 10 dBm, respectively.
Fig. 5
Fig. 5 (a): Average SNR and SNR gain versus fiber PMD parameter obtained at the optimum input power for each scenario. Note the y-axis on the right for the gain. (b): Results obtained using the blind MMA equalizer compared to an ideal equalizer for both DBP schemes.

Equations (15)

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

J k ( f ) = J k ( f 0 ) + ( f f 0 ) J k ( f 0 ) 1 ! + ( f f 0 ) 2 J k ( f 0 ) 2 ! +
J k ( f 0 ) + ( f f 0 ) J k ( f 0 )
= J k ( f 0 ) ( I + ( f f 0 ) J k 1 ( f 0 ) J k ( f 0 ) )
J k ( f 0 ) exp ( ( f f 0 ) J k 1 ( f 0 ) J k ( f 0 ) )
= J k ( f 0 ) J k s ( f ) N PMD ,
J k s ( f ) = exp ( ( f f 0 ) J k 1 ( f 0 ) J k ( f 0 ) N PMD ) .
exp A = I + Α 1 ! + A 2 2 ! + ,
J k c ( f ) = J k ( f ) J k s ( f ) N PMD
J ˜ k s ( f ) = J ˜ k s ( f 0 i + 1 w / 2 ) J k s ( f 0 i + w / 2 ) 1 J k s ( f ) for i = N ch / 2 1 , , 2 , 1 ,
J ˜ k s ( f ) = J ˜ k s ( f 0 i 1 + w / 2 ) J k s ( f 0 i w / 2 ) 1 J k s ( f ) for i = N ch / 2 + 1 , , N ch ,
T k eq [ n ] = ( T k xx [ n ] T k yx [ n ] T k xy [ n ] T k yy [ n ] )
J k eq [ m ] = ( { T k xx [ n ] } { T k yx [ n ] } { T k xy [ n ] } { T k yy [ n ] } ) ,
J k r [ m ] = U k [ m ] V k H [ m ] ,
J k [ m ] 1 Δ f l = 1 L c l ( J k [ m + l ] J k [ m l ] ) ,
c l = 3 ( L + 1 l ) L ( L + 1 ) ( L + 2 ) ,

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