Abstract

The nonlinear inverse synthesis (NIS) method, in which information is encoded directly onto the continuous part of the nonlinear signal spectrum, has been proposed recently as a promising digital signal processing technique for combating fiber nonlinearity impairments. However, because the NIS method is based on the integrability property of the lossless nonlinear Schrödinger equation, the original approach can only be applied directly to optical links with ideal distributed Raman amplification. In this paper, we propose and assess a modified scheme of the NIS method, which can be used effectively in standard optical links with lumped amplifiers, such as, erbium-doped fiber amplifiers (EDFAs). The proposed scheme takes into account the average effect of the fiber loss to obtain an integrable model (lossless path-averaged model) to which the NIS technique is applicable. We found that the error between lossless path-averaged and lossy models increases linearly with transmission distance and input power (measured in dB). We numerically demonstrate the feasibility of the proposed NIS scheme in a burst mode with orthogonal frequency division multiplexing (OFDM) transmission scheme with advanced modulation formats (e.g., QPSK, 16QAM, and 64QAM), showing a performance improvement up to 3.5 dB; these results are comparable to those achievable with multi-step per span digital back-propagation.

© 2015 Optical Society of America

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References

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2014 (9)

X. Yi, X. Chen, D. Sharma, C. Li, M. Luo, Q. Yang, Z. Li, and K. Qiu, “Digital coherent superposition of optical OFDM subcarrier pairs with Hermitian symmetry for phase noise mitigation,” Opt. Express 22(11), 13454–13459 (2014).
[Crossref] [PubMed]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

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. 113(1), 013901 (2014).
[Crossref] [PubMed]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22(22), 26720–26741 (2014).
[Crossref] [PubMed]

E. Giacoumidis, M. A. Jarajreh, S. Sygletos, S. T. Le, F. Farjady, A. Tsokanos, A. Hamié, E. Pincemin, Y. Jaouën, A. D. Ellis, and N. J. Doran, “Dual-polarization multi-band optical OFDM transmission and transceiver limitations for up to 500 Gb/s uncompensated long-haul links,” Opt. Express 22(9), 10975–10986 (2014).
[PubMed]

S. T. Le, K. Blow, and S. Turitsyn, “Power pre-emphasis for suppression of FWM in coherent optical OFDM transmission,” Opt. Express 22(6), 7238–7248 (2014).
[Crossref] [PubMed]

L. Son Thai, K. J. Blow, V. K. Mezentsev, and S. K. Turitsyn, “Bit Error Rate Estimation Methods for QPSK CO-OFDM Transmission,” J. Lightwave Technol. 32(17), 2951–2959 (2014).
[Crossref]

2013 (3)

2011 (1)

2010 (2)

2009 (1)

2008 (1)

2006 (1)

1999 (1)

S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163(1-3), 122–158 (1999).
[Crossref]

1996 (1)

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

1993 (1)

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

1980 (1)

1974 (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Bland-Hawthorn, J.

Blow, K.

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. 113(1), 013901 (2014).
[Crossref] [PubMed]

L. Son Thai, K. J. Blow, V. K. Mezentsev, and S. K. Turitsyn, “Bit Error Rate Estimation Methods for QPSK CO-OFDM Transmission,” J. Lightwave Technol. 32(17), 2951–2959 (2014).
[Crossref]

Buryak, A.

Calabro, S.

Chandrasekhar, S.

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

Chen, X.

Chikama, T.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Chraplyvy, R. A.

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

Chugtai, M. N.

Cotter, D.

de Waardt, H.

Derevyanko, S. A.

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. 113(1), 013901 (2014).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
[Crossref] [PubMed]

Doran, N. J.

Ellis, A. D.

Essiambre, R.

Farjady, F.

Forzati, M.

Foschini, G. J.

Gabitov, I.

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. 113(1), 013901 (2014).
[Crossref] [PubMed]

Giacoumidis, E.

Goebel, B.

Hamié, A.

Hasegawa, A.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

Ip, E.

Jansen, S. L.

Jaouën, Y.

Jarajreh, M. A.

Jian, Z.

Kahn, J. M.

Kaneko, S.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Khoe, G.-D.

Kramer, G.

Krummrich, P. M.

Kschischang, F. R.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

Le, S. T.

Li, C.

Li, Z.

Liu, X.

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

Luo, M.

Mårtensson, J.

Mezentsev, V. K.

L. Son Thai, K. J. Blow, V. K. Mezentsev, and S. K. Turitsyn, “Bit Error Rate Estimation Methods for QPSK CO-OFDM Transmission,” J. Lightwave Technol. 32(17), 2951–2959 (2014).
[Crossref]

S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163(1-3), 122–158 (1999).
[Crossref]

Mussolin, M.

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Nyu, T.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

Pepper, D. M.

Pincemin, E.

Poor, H. V.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
[Crossref]

S. Wahls and H. V. Poor, “Inverse Nonlinear Fourier Transforms Via Interpolation: The Ablowitz-Ladik Case,” in Proceeding of International Symposium on Mathematical Theory of Networks and Systems (MTNS), 1848–1855 (2014).

Prilepsky, J. E.

Qiu, K.

Rafique, D.

Schäfer, T.

S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163(1-3), 122–158 (1999).
[Crossref]

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Sharma, D.

Sohler, W.

Son Thai, L.

Spatschek, K. H.

S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163(1-3), 122–158 (1999).
[Crossref]

Spinnler, B.

Steblina, V.

Suche, H.

Sygletos, S.

Tkach, W. R.

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

Tsokanos, A.

Turitsyn, S.

Turitsyn, S. K.

Turitsyna, E. G.

Van den Borne, D.

Wahls, S.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
[Crossref]

S. Wahls and H. V. Poor, “Inverse Nonlinear Fourier Transforms Via Interpolation: The Ablowitz-Ladik Case,” in Proceeding of International Symposium on Mathematical Theory of Networks and Systems (MTNS), 1848–1855 (2014).

Watanabe, S.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Winzer, P. J.

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

R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity Limits of Optical Fiber Networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

Yang, Q.

Yariv, A.

Yi, X.

Yousefi, M. I.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

IEEE Trans. Inf. Theory (3)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

J. Lightwave Technol. (6)

Nat. Photonics (1)

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

Opt. Commun. (1)

S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163(1-3), 122–158 (1999).
[Crossref]

Opt. Express (7)

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
[Crossref] [PubMed]

E. Giacoumidis, M. A. Jarajreh, S. Sygletos, S. T. Le, F. Farjady, A. Tsokanos, A. Hamié, E. Pincemin, Y. Jaouën, A. D. Ellis, and N. J. Doran, “Dual-polarization multi-band optical OFDM transmission and transceiver limitations for up to 500 Gb/s uncompensated long-haul links,” Opt. Express 22(9), 10975–10986 (2014).
[PubMed]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22(22), 26720–26741 (2014).
[Crossref] [PubMed]

S. T. Le, K. Blow, and S. Turitsyn, “Power pre-emphasis for suppression of FWM in coherent optical OFDM transmission,” Opt. Express 22(6), 7238–7248 (2014).
[Crossref] [PubMed]

X. Yi, X. Chen, D. Sharma, C. Li, M. Luo, Q. Yang, Z. Li, and K. Qiu, “Digital coherent superposition of optical OFDM subcarrier pairs with Hermitian symmetry for phase noise mitigation,” Opt. Express 22(11), 13454–13459 (2014).
[Crossref] [PubMed]

A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of Inverse Scattering Algorithms for Designing Ultrabroadband Fibre Bragg Gratings,” Opt. Express 17(3), 1995–2004 (2009).
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Figures (7)

Fig. 1
Fig. 1 (a) – A comparison of output fields obtained by using the standard NLSE and the LPA NLSE, the amplifier spacing is 80km. (b) – NMSE as a function of the transmission distance. (c) – NMSE as a function of the signal’s bandwidth for a given input power and a given input power density. (d) – NMSE as a function of the input power.
Fig. 2
Fig. 2 Level curves of NMSE (in dB) indicating the error of using LPA NLSE (5), plotted as a function of the signal’s bandwidth (in GHz) and the input power (in dBm) for the propagation distance 2000km.
Fig. 3
Fig. 3 (a) – Block diagram of NIS-based optical communication systems. (b) – Illustration of a burst mode transmission, in which neighbouring packet data are separated by a guard time.
Fig. 4
Fig. 4 Q-factor as a function of the launch power for 100 Gb/s QPSK OFDM NIS-based system in the back-to-back case and in a 2000 km optical link, the ASE is ignored.
Fig. 5
Fig. 5 (a) – Performance comparison of the 100-Gb/s QPSK OFDM systems with the NIS vs. the DBP methods for fiber nonlinearity compensation, and constellation diagrams at the optimum launch powers for the cases: (b) – without NIS and DBP, (c) DBP with 10 steps/span (d) with the NIS method, (e) DBP with 20 steps/span. The propagation distance is 2000km.
Fig. 6
Fig. 6 (a) – Performance comparison of the 200-Gb/s 16QAM OFDM systems with the NIS vs. the DBP methods for fiber nonlinearity compensation and constellation diagrams at the optimum launch powers for (b) – Without NIS and DBP, (b) with the NIS method. The transmission distance is 2000km.
Fig. 7
Fig. 7 (a) – Performance comparison of the 300-Gb/s 64QAM OFDM systems with the NIS vs. the DBP methods for fiber nonlinearity compensation and constellation diagrams at the optimum launch powers for (b) – Without NIS and DBP, (b) with the NIS method. The transmission distance is 640km.

Equations (10)

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

j q z β 2 2 q tt +γq | q | 2 =j α 2 q,
q=exp( α 2 z)A,
j A z β 2 2 A tt +γexp(αz)A | A | 2 =0
γ 1 = 1 L 0 L γexp(αz) dz=γ G1 Gln(G) ,
j A z β 2 2 A tt + γ 1 A | A | 2 =0
NMSE= | q 2 (Z,t) q 1 (Z,t) | 2 | q 1 (Z,t) | 2 ,
r(ξ)| ξ=ω/2 = S(ω)
S ¯ (ω)=r(L,ξ) e 2j ξ 2 L | ξ=ω/2
t T s t, z Z s z, s γ 1 Z s s
ΔT=2πB β 2 L,

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