Abstract

Most of the nonlinear Fourier transform (NFT) based optical communication systems studied so far deal with the burst mode operation that substantially reduce achievable spectral efficiency. The burst mode requirement emerges due to the very nature of the commonly used version of the NFT processing method: it can process only rapidly decaying signals, requires zero-padding guard intervals for processing of dispersion-induced channel memory, and does not allow one to control the time-domain occupation well. Some of the limitations and drawbacks imposed by this approach can be rectified by the recently introduced more mathematically demanding periodic NFT processing tools. However, the studies incorporating the signals with cyclic prefix extension into the NFT transmission framework have so far lacked the efficient digital signal processing (DSP) method of synthesizing an optical signal, the shortcoming that diminishes the approach flexibility. In this paper, we introduce the Riemann–Hilbert problem (RHP) based DSP method as a flexible and expandable tool that would allow one to utilize the periodic NFT spectrum for transmission purposes without former restrictions. First, we outline the theoretical framework and clarify the implementation underlying the proposed new DSP method. Then we present the results of numerical modelling quantifying the performance of long-haul RHP-based transmission with the account of optical noise, demonstrating the good performance quality and potential of RHP-based optical communication systems.

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2018 (4)

X. Yangzhang, D. Lavery, P. Bayvel, and M. I. Yousefi, “Impact of perturbations on nonlinear frequency-division multiplexing,” J. Lightw. Technol., vol. 36, no. 2, pp. 485–494, 2018.

S. Gaiarin, A. M. Perego, E. P. da Silva, F. Da Ros, and D. Zibar, “Dual-polarization nonlinear Fourier transform-based optical communication system,” Optica, vol. 5, no. 3, pp. 263–270, 2018.

S. T. Le, V. Aref, and H. Buelow, “High speed precompensated nonlinear frequency-division multiplexed transmissions,” J. Lightw. Technol., vol. 36, no. 6, pp. 1296–1303, 2018.

H. Bülow, V. Aref, and L. Schmalen, “Modulation on discrete nonlinear spectrum: Perturbation sensitivity and achievable rates,” IEEE Photon. Tech. Lett., vol. 30, no. 5, pp. 423–426,  2018.

2017 (10)

I. T. Lima, T. D. S. DeMenezes, V. S. Grigoryan, M. O’Sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol., vol. 35, no. 23, pp. 5056–5068, 2017.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Techol. Lett., vol. 29, no. 16, pp. 1332–1335,  2017.

T. Gui, C. Lu, A. P. T. Lau, and P. K. A. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express, vol. 25, no. 17, pp. 20286–20297, 2017.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, p. 570, 2017.

S. T. Le and H. Buelow, “$\text{64} \times \text{0.5}$ Gbaud nonlinear frequency division multiplexed transmissions with high order modulation formats,” J. Lightw. Technol., vol. 35, no. 17, pp. 3692–3698, 2017.

S. K. Turitsynet al., “Nonlinear Fourier transform for optical data processing and transmission: Advances and perspectives,” Optica, vol. 4, no. 3, pp. 307–322, 2017.

J. C. Cartledge, F. P. Guiomar, F. R. Kschischang, G. Liga, and M. P. Yankov, “Digital signal processing for fiber nonlinearities,” Opt. Express, vol. 25, no. 3, pp. 1916–1936, 2017.

A. Calini and C. M. Schober, “Characterizing JONSWAP rogue waves and their statistics via inverse spectral data,” Wave Motion, vol. 71, pp. 5–17, 2017.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “On the design of NFT-based communication systems with lumped amplification,” J. Lightw. Technol., vol. 35, no. 24, pp. 5464–5472, 2017.

V. Kotlyarov and D. Shepelsky, “Planar unimodular Baker– Akhiezer function for the nonlinear Schrödinger equation,” Ann. Math. Sci. Appl., vol. 2, pp. 343–384, 2017.

2016 (5)

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: Eigenvalue communication,” Opt. Express, vol. 24, pp. 18370–18381, 2016.

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nature Commun., vol. 7, 2016, Art. no. .

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol., vol. 34, no. 10, pp. 2459–2466, 2016.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: Theory and numerical methods,” Opt. Express, vol. 24, pp. 18353–18369, 2016.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov– Shabat type,”  2016, arXiv:1607.01305v2.

2015 (1)

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory, vol. 61, no. 12, pp. 6957–6974,  2015.

2014 (4)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I – III,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4312–4369,  2014.

J. E. Prilepsky. S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett., vol. 113, 2014, Art. no. .

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express, vol. 22, no. 22, pp. 26720–26741, 2014.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” IEEE J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

2013 (1)

A. O. Smirnov, “Periodic two-phase rogue waves”,” Math. Notes, vol. 94, pp. 897–907, 2013.

2012 (1)

S. Olver, “A general framework for solving Riemann– Hilbert problems numerically,” Numerische Mathematik, vol. 122, pp. 305–340, 2012.

2010 (1)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol., vol. 28, no. 4, pp. 662–701, 2010.

2009 (1)

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A, vol. 373, pp. 675–678, 2009.

2008 (1)

V. B. Mateev, “30 years of finite-gap integration theory,” Philos. Trans. Royal Soc. London A, Math., Phys. Eng. Sci., vol. 366, pp. 837–875, 2008.

2006 (1)

J. D. Ania-Castañónet al., “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett., vol. 96, no. 2, 2006, Art. no. .

2004 (1)

B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput., vol. 73, pp. 1417–1442, 2004.

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature, vol. 411, no. 6842, pp. 1027–1030, 2001.

1997 (1)

A. Kamchatnov, “New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability,” Phys. Rep., vol. 286, pp. 199–270, 1997.

1993 (1)

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightw. Technol., vol. 11, no. 3, pp. 395–399, 1993.

1989 (2)

A. I. Bobenko and L. A. Bordag, ” Periodic multiphase solutions of the Kadomsev-Petviashvili equation,” J. Phys. A: Math. General, vol. 22, no. 9, pp. 1259–1274, 1989.

I. M. Krichever, “The spectral theory of two-dimensional periodic operators and applications,” Russian Math. Surv., vol. 44, pp. 121–184, 1989.

1988 (1)

E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: an exactly solvable model,” Phys. Rev. A, vol. 37, pp. 815–839, 1988.

1982 (1)

H. C. Yuen and B. M. Lake, “Nonlinear dynamics of deep-water gravity waves,” Adv. Appl. Mech., vol. 22, pp. 67–229, 1982.

1981 (1)

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math., vol. 65, pp. 113–158, 1981.

1974 (1)

S. P. Novikov, “The periodic problem for the Korteweg—de vries equation,” Functional Anal. Appl., vol. 8, no. 3, pp. 236–246, 1974.

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media,” Soviet Physics-J. Exp. Theor. Phys., vol. 34, pp. 62–69, 1972.

Ablowitz, M. J.

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math., vol. 65, pp. 113–158, 1981.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform.Philadelphia, PA, USA: SIAM, 1981.

Agrell, E.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” IEEE J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

Akhmediev, N.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A, vol. 373, pp. 675–678, 2009.

Alvarado, A.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” IEEE J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

Ania-Castañón, J. D.

J. D. Ania-Castañónet al., “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett., vol. 96, no. 2, 2006, Art. no. .

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A, vol. 373, pp. 675–678, 2009.

Aref, V.

S. T. Le, V. Aref, and H. Buelow, “High speed precompensated nonlinear frequency-division multiplexed transmissions,” J. Lightw. Technol., vol. 36, no. 6, pp. 1296–1303, 2018.

H. Bülow, V. Aref, and L. Schmalen, “Modulation on discrete nonlinear spectrum: Perturbation sensitivity and achievable rates,” IEEE Photon. Tech. Lett., vol. 30, no. 5, pp. 423–426,  2018.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, p. 570, 2017.

Bayvel, P.

X. Yangzhang, D. Lavery, P. Bayvel, and M. I. Yousefi, “Impact of perturbations on nonlinear frequency-division multiplexing,” J. Lightw. Technol., vol. 36, no. 2, pp. 485–494, 2018.

Belokolos, E. D.

E. D. Belokolos, A. I. Bobenko, V. Z. Enolski, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach in the Theory of Integrable Equations (Springer Series in Nonlinear Dynamics), Berlin, Germany: Springer, 1994.

Blow, K. J.

J. E. Prilepsky. S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett., vol. 113, 2014, Art. no. .

Bobenko, A.

B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput., vol. 73, pp. 1417–1442, 2004.

Bobenko, A. I.

A. I. Bobenko and L. A. Bordag, ” Periodic multiphase solutions of the Kadomsev-Petviashvili equation,” J. Phys. A: Math. General, vol. 22, no. 9, pp. 1259–1274, 1989.

E. D. Belokolos, A. I. Bobenko, V. Z. Enolski, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach in the Theory of Integrable Equations (Springer Series in Nonlinear Dynamics), Berlin, Germany: Springer, 1994.

Bordag, L. A.

A. I. Bobenko and L. A. Bordag, ” Periodic multiphase solutions of the Kadomsev-Petviashvili equation,” J. Phys. A: Math. General, vol. 22, no. 9, pp. 1259–1274, 1989.

Buchali, F.

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Proc. Opt. Fiber Commun. Conf., 2018, Paper W1G.6.

Buelow, H.

S. T. Le, V. Aref, and H. Buelow, “High speed precompensated nonlinear frequency-division multiplexed transmissions,” J. Lightw. Technol., vol. 36, no. 6, pp. 1296–1303, 2018.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, p. 570, 2017.

S. T. Le and H. Buelow, “$\text{64} \times \text{0.5}$ Gbaud nonlinear frequency division multiplexed transmissions with high order modulation formats,” J. Lightw. Technol., vol. 35, no. 17, pp. 3692–3698, 2017.

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Proc. Opt. Fiber Commun. Conf., 2018, Paper W1G.6.

Bülow, H.

H. Bülow, V. Aref, and L. Schmalen, “Modulation on discrete nonlinear spectrum: Perturbation sensitivity and achievable rates,” IEEE Photon. Tech. Lett., vol. 30, no. 5, pp. 423–426,  2018.

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