Abstract

Polarization-division multiplexed (PDM) transmission based on the nonlinear Fourier transform (NFT) is proposed for optical fiber communication. The NFT algorithms are generalized from the scalar nonlinear Schrödinger equation for one polarization to the Manakov system for two polarizations. The transmission performance of the PDM nonlinear frequency-division multiplexing (NFDM) and PDM orthogonal frequency-division multiplexing (OFDM) are determined. It is shown that the transmission performance in terms of Q-factor is approximately the same in PDM-NFDM and single polarization NFDM at twice the data rate and that the polarization-mode dispersion does not seriously degrade system performance. Compared with PDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized to multi-mode fibers in the strong coupling regime, paving the way for the application of the NFT to address the nonlinear effects in space-division multiplexing.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2017 (2)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 3, 1332 (2017.

I. Tavakkolnia and M. Safari, “Signaling on the continuous spectrum of nonlinear optical fiber,” Opt. Express 25, 18685–18702 (2017).

2015 (2)

2014 (4)

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

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

2013 (1)

2010 (1)

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

2008 (1)

2006 (1)

1999 (1)

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, On discretization of the vector nonlinear Schrödinger equation, Phys. Lett. A 253, 287–304 (1999).
[Crossref]

1997 (1)

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

1996 (1)

P. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

1993 (1)

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

1992 (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

1991 (4)

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[Crossref] [PubMed]

P. Wai, C. R. Menyuk, and H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

K. Blow and N. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[Crossref]

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum. Electron. 25, 2674–2682 (1989).
[Crossref]

1974 (2)

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).
[Crossref]

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Soviet Physics-JETP 38, 248–253 (1974).

1973 (2)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

1967 (1)

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, On discretization of the vector nonlinear Schrödinger equation, Phys. Lett. A 253, 287–304 (1999).
[Crossref]

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).
[Crossref]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).
[Crossref]

Agrawal, G. P.

Alvarado, A.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

Aref, V.

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, H. Buelow, and V. Aref, “Demonstration of 64×0.5Gbaud nonlinear frequency division multiplexed transmission with 32QAM,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. W3J–1.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 2015 41st European Conference on Optical Communication (ECOC), (IEEE, 2015), pp. 1–3.

Bayvel, P.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Bertolini, M.

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

Blow, K.

K. Blow and N. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[Crossref]

Bononi, A.

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

P. Serena, N. Rossi, and A. Bononi, “Nonlinear penalty reduction induced by PMD in 112 Gbit/s WDM PDM-QPSK coherent systems,” in 2009 35th European Conference on Optical Communication, (2009), pp. 1–2.

Braimiotis, C.

M. Eberhard and C. Braimiotis, Numerical Implementation of the Coarse-Step Method with a Varying Differential-Group Delay (SpringerUS, Boston, MA, 2005), pp. 530–534.

Buelow, H.

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, H. Buelow, and V. Aref, “Demonstration of 64×0.5Gbaud nonlinear frequency division multiplexed transmission with 32QAM,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. W3J–1.

Bülow, H.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 2015 41st European Conference on Optical Communication (ECOC), (IEEE, 2015), pp. 1–3.

Chen, H.

Civelli, S.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 3, 1332 (2017.

Degasperis, A.

A. Degasperis and S. Lombardo, “Integrability in action: Solitons, instability and rogue waves,” in Rogue and Shock Waves in Nonlinear Dispersive Media, (Springer, 2016), pp. 23–53.
[Crossref]

Doran, N.

K. Blow and N. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[Crossref]

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Eberhard, M.

M. Eberhard and C. Braimiotis, Numerical Implementation of the Coarse-Step Method with a Varying Differential-Group Delay (SpringerUS, Boston, MA, 2005), pp. 530–534.

Ellis, A. D.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

Essiambre, R.-J.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

Forestieri, E.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 3, 1332 (2017.

Gardner, C. S.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Greene, J. M.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Harper, P.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Hasegawa, A.

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

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

Haus, H. A.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

Idler, W.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 2015 41st European Conference on Optical Communication (ECOC), (IEEE, 2015), pp. 1–3.

Jansen, S. L.

Kamalian, M.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

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).
[Crossref]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

Kruskal, M. D.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

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, 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, 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, 4346–4369 (2014).
[Crossref]

Lavery, D.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

Le, S. T.

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
[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, 26720–26741 (2014).
[Crossref] [PubMed]

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, H. Buelow, and V. Aref, “Demonstration of 64×0.5Gbaud nonlinear frequency division multiplexed transmission with 32QAM,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. W3J–1.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Lombardo, S.

A. Degasperis and S. Lombardo, “Integrability in action: Solitons, instability and rogue waves,” in Rogue and Shock Waves in Nonlinear Dispersive Media, (Springer, 2016), pp. 23–53.
[Crossref]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Soviet Physics-JETP 38, 248–253 (1974).

Manyuk, C.

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

Marcuse, D.

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

Marks, B. S.

Maruta, A.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in 2015 International Conference on Photonics in Switching (PS), (IEEE, 2015), pp. 265–267.

Matsuda, Y.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in 2015 International Conference on Photonics in Switching (PS), (IEEE, 2015), pp. 265–267.

Menyuk, C. R.

C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
[Crossref]

P. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. Wai, C. R. Menyuk, and H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum. Electron. 25, 2674–2682 (1989).
[Crossref]

Miura, R. M.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

Morita, I.

Mumtaz, S.

Nagel, J. A.

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).
[Crossref]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

Nyu, T.

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

Ohta, Y.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, On discretization of the vector nonlinear Schrödinger equation, Phys. Lett. A 253, 287–304 (1999).
[Crossref]

Philips, I.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Philips, I. D.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

Poole, C. D.

Poor, H. V.

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transform,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).
[Crossref]

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in “Proc. IEEE Int. Symp. Inf. Theory (ISIT),” (2015), pp. 1676–1680.

Prilepsky, J.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Prilepsky, J. E.

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
[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, 26720–26741 (2014).
[Crossref] [PubMed]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

Rossi, N.

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

P. Serena, N. Rossi, and A. Bononi, “Nonlinear penalty reduction induced by PMD in 112 Gbit/s WDM PDM-QPSK coherent systems,” in 2009 35th European Conference on Optical Communication, (2009), pp. 1–2.

Safari, M.

I. Tavakkolnia and M. Safari, “Signaling on the continuous spectrum of nonlinear optical fiber,” Opt. Express 25, 18685–18702 (2017).

I. Tavakkolnia and M. Safari, “Effect of PMD on the continuous spectrum of nonlinear optical fibre,” in 2017 65th European Conference on Optical Communication (ECOC), (2017), pp. ci-p-11.

Schenk, T. C. W.

Schuh, K.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 2015 41st European Conference on Optical Communication (ECOC), (IEEE, 2015), pp. 1–3.

Secondini, M.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 3, 1332 (2017.

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).
[Crossref]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).
[Crossref]

Serena, P.

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

P. Serena, N. Rossi, and A. Bononi, “Nonlinear penalty reduction induced by PMD in 112 Gbit/s WDM PDM-QPSK coherent systems,” in 2009 35th European Conference on Optical Communication, (2009), pp. 1–2.

Takeda, N.

Tanaka, H.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

Tavakkolnia, I.

I. Tavakkolnia and M. Safari, “Signaling on the continuous spectrum of nonlinear optical fiber,” Opt. Express 25, 18685–18702 (2017).

I. Tavakkolnia and M. Safari, “Effect of PMD on the continuous spectrum of nonlinear optical fibre,” in 2017 65th European Conference on Optical Communication (ECOC), (2017), pp. ci-p-11.

Trubatch, A. D.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, On discretization of the vector nonlinear Schrödinger equation, Phys. Lett. A 253, 287–304 (1999).
[Crossref]

Turitsyn, S.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

Turitsyn, S. K.

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
[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, 26720–26741 (2014).
[Crossref] [PubMed]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

Wahls, S.

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transform,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).
[Crossref]

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in “Proc. IEEE Int. Symp. Inf. Theory (ISIT),” (2015), pp. 1676–1680.

Wai, P.

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

P. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. Wai, C. R. Menyuk, and H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

Winters, J. H.

Yangzhang, X.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

Yousefi, M.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

Yousefi, M. I.

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

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

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

IEEE J. Quantum. Electron. (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum. Electron. 25, 2674–2682 (1989).
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 3, 1332 (2017.

IEEE Photonics Technol. Lett. (1)

K. Blow and N. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[Crossref]

IEEE Trans. Inf. Theory (4)

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transform,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).
[Crossref]

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

J. Lightwave Technol. (8)

P. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

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

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

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26, 6 (2008).
[Crossref]

C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
[Crossref]

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: Generalization of the Manakov equations,” J. Lightwave Technol. 31, 398–406 (2013).
[Crossref]

Opt. Express (3)

Opt. Fiber Technol. (1)

M. Bertolini, N. Rossi, P. Serena, and A. Bononi, “Do’s and don’ts for a correct nonlinear PMD emulation in 100Gb/s PDM-QPSK systems,” Opt. Fiber Technol. 16, 274–278 (2010).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, On discretization of the vector nonlinear Schrödinger equation, Phys. Lett. A 253, 287–304 (1999).
[Crossref]

Phys. Rev. Lett. (2)

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear evolution equations of physical significance,” Phys. Rev. Lett. 31, 125–127 (1973).
[Crossref]

Soviet Physics-JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Soviet Physics-JETP 38, 248–253 (1974).

Stud. Appl. Math. (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).
[Crossref]

Other (17)

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).
[Crossref]

A. Degasperis and S. Lombardo, “Integrability in action: Solitons, instability and rogue waves,” in Rogue and Shock Waves in Nonlinear Dispersive Media, (Springer, 2016), pp. 23–53.
[Crossref]

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in “Optical Fiber Communications Conference,” (Optical Society of America, 2016), pp. 1–3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

S. T. Le, H. Buelow, and V. Aref, “Demonstration of 64×0.5Gbaud nonlinear frequency division multiplexed transmission with 32QAM,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. W3J–1.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 2015 41st European Conference on Optical Communication (ECOC), (IEEE, 2015), pp. 1–3.

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 2016 42nd European Conference on Optical Communication (ECOC), (VDE, 2016), pp. 1–3.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in 2015 International Conference on Photonics in Switching (PS), (IEEE, 2015), pp. 265–267.

X. Yangzhang, M. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu3D–1.

M. I. Yousefi, G. Kramer, and F. R. Kschischang, “Upper bound on the capacity of the nonlinear Schrödinger channel,” https://arxiv.org/abs/1502.06455 .

P. Serena, N. Rossi, and A. Bononi, “Nonlinear penalty reduction induced by PMD in 112 Gbit/s WDM PDM-QPSK coherent systems,” in 2009 35th European Conference on Optical Communication, (2009), pp. 1–2.

I. Tavakkolnia and M. Safari, “Effect of PMD on the continuous spectrum of nonlinear optical fibre,” in 2017 65th European Conference on Optical Communication (ECOC), (2017), pp. ci-p-11.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in “Proc. IEEE Int. Symp. Inf. Theory (ISIT),” (2015), pp. 1676–1680.

V. Vaibhav, “Fast inverse nonlinear Fourier transforms using exponential one-step methods, part I: Darboux transformation,” https://arxiv.org/abs/1704.00951 (2017).

M. Eberhard and C. Braimiotis, Numerical Implementation of the Coarse-Step Method with a Varying Differential-Group Delay (SpringerUS, Boston, MA, 2005), pp. 530–534.

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” https://arxiv.org/abs/1603.04389 .

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Figures (9)

Fig. 1
Fig. 1

Comparing q ^ = [ q ^ 1 , q ^ 2 ] and nft ( inft( q ^ )), where q ^ 1 and q ^ 2 are displaced Gaussians. For fixed sampling rate, the algorithm is less accurate at higher input-power.

Fig. 2
Fig. 2

System diagram for the polarization-multiplexed NFDM and OFDM transmission systems with processing steps in the transmitter and receiver DSP. Steps highlighted in purple require joint processing of both polarization components. For further explanation see text.

Fig. 3
Fig. 3

NFDM and OFDM signals at the TX (left panel) and RX (right panel) using the ideal model ((2)) at P = 0.5 dBm without noise. Only one of the two polarizations is shown at both TX and RX. The NFDM signal at the TX is broader due to the nonlinear effects in the NFT. At the RX, the time duration of both signals is about the same.

Fig. 4
Fig. 4

BER as a function of OSNR for NFDM, for back-to-back, lossless, lossy and transformed-lossless models. For these simulations we only performed noise-loading at the receiver, to exclude signal-noise interaction from the comparison.

Fig. 5
Fig. 5

Comparison between NFT transmission based on the NLSE (single polarization) and the Manakov equation (polarization-multiplexed). P denotes the total power of the signal. At low power the offset between the curves is 3dB, as the signal power doubles when two polarization components propagate. At higher power signal-noise interactions decrease this gap.

Fig. 6
Fig. 6

Comparison of polarization-multiplexed OFDM to NFDM transmission in an idealized setting. We use the Manakov equation without PMD-effects for 25 spans of 80 km for 16QAM. NFDM performs as well as OFDM with 10step/span DBP. The full potential of NFDM is leveraged in a multi-channel scenario, where DBP is less efficient.

Fig. 7
Fig. 7

Received constellations for OFDM (left panel, without DBP) and NFDM (right panel) at the respective optimal launch power. The noise in NFDM is clearly non-Gaussian.

Fig. 8
Fig. 8

The effect of the number of taps on the Q-factor for different values of DPMD. The power of the signal in the simulation is 0.5 dBm.

Fig. 9
Fig. 9

The effect of PMD on PDM-NFDM. We used 13, 25 and 61 taps in the equalizer for DPMD = 0, 0.2 and 0.5 ps / km, respectively.

Tables (1)

Tables Icon

Table 1 The system parameters used in the simulations.

Equations (28)

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

A Z = j Δ β 0 2 σ ( Z ) A Δ β 1 2 σ ( Z ) A T α 2 A + j β 2 2 2 A T 2 j γ 8 9 A 2 A .
A Z = j β 2 2 2 A T 2 j γ 8 9 A 2 A .
T 0 = ( | β 2 | Z 0 ) / 2 , A 0 = 2 / ( 8 9 γ Z 0 ) .
j q z = 2 q t 2 2 s q 2 q ,
L ^ = j ( t q 1 q 2 s q 1 * t 0 s q 2 * 0 t ) .
L ^ v = λ v
j ± ( 0 ) e ( 0 ) exp ( j λ t ) , j ± ( i ) e ( i ) exp ( j λ t ) , i = 1 , 2 , as t ± ,
j ( 0 ) ( t , λ ) = a ( λ ) j + ( 0 ) ( t , λ ) + b 1 ( λ ) j + ( 1 ) ( t , λ ) + b 2 ( λ ) j + ( 2 ) ( t , λ ) ,
NFT ( q ) ( λ ) = { q ^ i = b i ( λ ) a ( λ ) , λ , q ˜ i = b i ( λ j ) a ( λ j ) , λ j + , i = 1 , 2 ,
H ( λ ) = e 4 j s λ 2 L .
| a ( λ ) | 2 + | b 1 ( λ ) | 2 + | b 2 ( λ ) | 2 = 1 ,
P = ( j λ q 1 ( t ) q 2 ( t ) q 1 * ( t ) j λ 0 q 2 * ( t ) 0 j λ ) .
v [ k + 1 , λ ] = c k ( z 1 / 2 Q 1 [ k ] Q 2 [ k ] Q 1 [ k ] * z 1 / 2 0 Q 2 [ k ] * 0 z 1 / 2 ) v [ k , λ ] ,
v [ 0 , λ ] = j ( 0 ) ( T 0 , λ ) = e ( 0 ) z T 0 / 2 Δ T .
a [ λ ] = z N 2 T 0 2 Δ T v 0 [ N , λ ] ,
b i [ λ ] = z N 2 + T 0 2 Δ T v i [ N , λ ] , i = 1 , 2 .
V [ k + 1 , λ ] = c k ( 1 Q 1 [ k ] z 1 Q 2 [ k ] z 1 Q 1 * [ k ] z 1 0 Q 2 * [ k ] 0 z 1 ) V [ k , λ ] ,
A [ k , λ ] = a [ k , λ ] B i [ k , λ ] = z N T 0 Δ T + 1 2 b i [ k , λ ] , i = 1 , 2 .
A ˜ [ k + 1 , l ] = c k ( A ˜ [ k , l ] + Q 1 [ k ] shift [ B ˜ 1 [ k ] ] [ l ] + Q 2 [ k ] shift [ B ˜ 2 [ k ] ] [ l ] ) , B ˜ 1 [ k + 1 , l ] = c k ( Q 1 [ k ] * A ˜ [ k , l ] + shift [ B ˜ 1 [ k ] ] [ l ] ) , B ˜ 2 [ k + 1 , l ] = c k ( Q 2 [ k ] * A ˜ [ k , l ] + shift [ B ˜ 2 [ k ] ] [ l ] ) .
| a ( λ ) | = 1 / 1 + | q ^ 1 ( λ ) | 2 + | q ^ 2 ( λ ) | 2 .
v [ k , λ ] = c k ( z 1 / 2 Q 1 [ k ] Q 2 [ k ] Q 1 * [ k ] z 1 / 2 0 Q 2 * [ k ] 0 z 1 / 2 ) v [ k + 1 , λ ] .
A ˜ [ k + 1 , 0 ] = c k A ˜ [ k , 0 ] , B ˜ 1 [ k + 1 , 0 ] = c k Q 1 [ k ] * A ˜ [ k , 0 ] , B ˜ 2 [ k + 1 , 0 ] = c k Q 2 [ k ] * A ˜ [ k , 0 ] .
U i ( λ ) = log ( 1 + | q ^ i ( λ ) | 2 ) e j q ^ i ( λ ) , i = 1 , 2 .
Δ T = 2 π | β 2 | L B ,
γ eff ( L ) = 1 L 0 L γ e α z d z = γ ( 1 e α L ) / ( α L ) .
p Δ t ( Δ t ) = 32 π 2 Δ t 2 Δ t 3 exp ( 4 Δ t 2 π Δ t 2 ) .
Δ t ~ Δ t 2 = D PMD L ,
A Z = j β ¯ 2 2 2 A T j γ κ A 2 A ,

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