Abstract

In this work, we study the performance of polarization division multiplexing nonlinear inverse synthesis transmission schemes for fiber-optic communications, expected to have reduced nonlinearity impact. Our technique exploits the integrability of the Manakov equation—the master model for dual-polarization signal propagation in a single mode fiber—and employs nonlinear Fourier transform (NFT) based signal processing. First, we generalize some algorithms for the NFT computation to the two- and multicomponent case. Then, we demonstrate that modulating information on both polarizations doubles the channel information rate with a negligible performance degradation. Moreover, we introduce a novel dual-polarization transmission scheme with reduced complexity which separately processes each polarization component and can also provide a performance improvement in some practical scenarios.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Corrections

Stella Civelli, Sergei K. Turitsyn, Marco Secondini, and Jaroslaw E. Prilepsky, "Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing: erratum," Opt. Express 27, 3617-3617 (2019)
http://proxy.osapublishing.org/oe/abstract.cfm?uri=oe-27-3-3617

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  33. S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in 2015 Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.
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    [Crossref]
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    [Crossref]
  38. A. Aricò, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).
    [Crossref]
  39. S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
    [Crossref]
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  41. S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
    [Crossref] [PubMed]
  42. I. Tavakkolnia and M. Safari, “Dispersion pre-compensation for NFT-based optical fiber communication systems,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2016).
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    [Crossref] [PubMed]

2018 (4)

V. Aref, S. T. Le, and H. Buelow, “Modulation over nonlinear Fourier spectrum: Continuous and discrete spectrum,” J. Lightw. Technol. 36, 1289–1295 (2018).
[Crossref]

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 5, 263–270 (2018).
[Crossref]

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

S. Civelli, E. Forestieri, and M. Secondini, “Decision-feedback detection strategy for nonlinear frequency-division multiplexing,” Opt. Express 26, 12057–12071 (2018).
[Crossref] [PubMed]

2017 (8)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

T. Gui, T. H. Chan, C. Lu, A. P. T. Lau, and P.-K. A. Wai, “Alternative decoding methods for optical communications based on nonlinear Fourier transform,” J. Lightw. Technol. 35, 1542–1550 (2017).
[Crossref]

T. Gui, C. Lu, A. P. T. Lau, and P. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express 25, 20286–20297 (2017).
[Crossref] [PubMed]

J.-W. Goossens, M. I. Yousefi, Y. Jaouën, and H. Hafermann, “Polarization-division multiplexing based on the nonlinear Fourier transform,” Opt. Express 25, 26437–26452 (2017).
[Crossref] [PubMed]

I. Tavakkolnia and M. Safari, “Capacity analysis of signaling on the continuous spectrum of nonlinear optical fibers,” J. Lightw. Technol. 35, 2086–2097 (2017).
[Crossref]

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

J. C. Cartledge, F. P. Guiomar, F. R. Kschischang, G. Liga, and M. P. Yankov, “Digital signal processing for fiber nonlinearities,” Opt. Express 25, 1916–1936 (2017).
[Crossref]

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
[Crossref]

2016 (3)

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communication, part II: Eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).
[Crossref] [PubMed]

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. 34, 2459–2466 (2016).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
[Crossref] [PubMed]

2015 (2)

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P.-K. A. Wai, F. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 991 (2015).

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
[Crossref]

2014 (3)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inf. Theory 60, 4312–4369 (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, 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, 26720–26741 (2014).
[Crossref] [PubMed]

2013 (1)

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

2012 (1)

2011 (1)

A. Aricò, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).
[Crossref]

2010 (2)

D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).
[Crossref] [PubMed]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28, 662–701 (2010).
[Crossref]

2006 (1)

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

1993 (1)

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

1992 (1)

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[Crossref]

1974 (1)

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

1972 (1)

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

Ablowitz, M. J.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 (SIAM, 1981).
[Crossref]

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Nonlinear Schroedinger Systems (Cambridge University, 2004).

Agrawal, G. P.

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

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Alvarado, A.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

Antonelli, C.

Aref, V.

V. Aref, S. T. Le, and H. Buelow, “Modulation over nonlinear Fourier spectrum: Continuous and discrete spectrum,” J. Lightw. Technol. 36, 1289–1295 (2018).
[Crossref]

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

Aricò, A.

A. Aricò, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).
[Crossref]

Barletti, L.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in 2015 Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

Bayvel, P.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

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

Boffetta, G.

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[Crossref]

Buchali, F.

S. T. Le, K. Schuh, F. Buchali, and H. Bülow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), pp. W1G–6.

Buelow, H.

V. Aref, S. T. Le, and H. Buelow, “Modulation over nonlinear Fourier spectrum: Continuous and discrete spectrum,” J. Lightw. Technol. 36, 1289–1295 (2018).
[Crossref]

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

Bülow, H.

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
[Crossref]

S. T. Le, K. Schuh, F. Buchali, and H. Bülow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), pp. W1G–6.

Cartledge, J. C.

Chan, T. H.

T. Gui, T. H. Chan, C. Lu, A. P. T. Lau, and P.-K. A. Wai, “Alternative decoding methods for optical communications based on nonlinear Fourier transform,” J. Lightw. Technol. 35, 1542–1550 (2017).
[Crossref]

Civelli, S.

S. Civelli, E. Forestieri, and M. Secondini, “Decision-feedback detection strategy for nonlinear frequency-division multiplexing,” Opt. Express 26, 12057–12071 (2018).
[Crossref] [PubMed]

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

S. Civelli, E. Forestieri, and M. Secondini, “Precompensation and windowing for nonlinear frequency-division multiplexing,” in Progress In Electromagnetics Research Symposium - Spring (PIERS) 2017 (2017).

S. Civelli, E. Forestieri, and M. Secondini, “A novel detection strategy for nonlinear frequency-division multiplexing,” in Optical Fiber Communication Conference (Optical Society of America, 2018), pp. W1G–5.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in 2015 Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

Da Ros, F.

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 5, 263–270 (2018).
[Crossref]

S. Gaiarin, A. M. Perego, E. P. da Silva, F. Da Ros, and D. Zibar, “Experimental demonstration of dual polarization nonlinear frequency division multiplexed optical transmission system,” in 43rd European Conference on Optical Communications (ECOC, 2017), p. W.3.C.2.

da Silva, E. P.

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 5, 263–270 (2018).
[Crossref]

S. Gaiarin, A. M. Perego, E. P. da Silva, F. Da Ros, and D. Zibar, “Experimental demonstration of dual polarization nonlinear frequency division multiplexed optical transmission system,” in 43rd European Conference on Optical Communications (ECOC, 2017), p. W.3.C.2.

Derevyanko, S. A.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
[Crossref] [PubMed]

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

Dong, Z.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P.-K. A. Wai, F. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 991 (2015).

Ellis, A.

S. Le, I. Philips, J. Prilepsky, M. Kamalian, A. Ellis, P. Harper, and S. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

Ellis, A. D.

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. 34, 2459–2466 (2016).
[Crossref]

S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

Essiambre, R.-J.

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

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28, 662–701 (2010).
[Crossref]

Forestieri, E.

S. Civelli, E. Forestieri, and M. Secondini, “Decision-feedback detection strategy for nonlinear frequency-division multiplexing,” Opt. Express 26, 12057–12071 (2018).
[Crossref] [PubMed]

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

S. Civelli, E. Forestieri, and M. Secondini, “A novel detection strategy for nonlinear frequency-division multiplexing,” in Optical Fiber Communication Conference (Optical Society of America, 2018), pp. W1G–5.

S. Civelli, E. Forestieri, and M. Secondini, “Precompensation and windowing for nonlinear frequency-division multiplexing,” in Progress In Electromagnetics Research Symposium - Spring (PIERS) 2017 (2017).

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Gemechu, W. A.

T. Gui, W. A. Gemechu, J.-W. Goossens, M. Song, S. Wabnitz, M. I. Yousefi, H. Hafermann, A. P. T. Lau, and Y. Jaouën, “Polarization-division-multiplexed nonlinear frequency division multiplexing,” in CLEO: Science and Innovations (Optical Society of America, 2018), pp. STu4C–3.

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R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28, 662–701 (2010).
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J.-W. Goossens, M. I. Yousefi, Y. Jaouën, and H. Hafermann, “Polarization-division multiplexing based on the nonlinear Fourier transform,” Opt. Express 25, 26437–26452 (2017).
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T. Gui, C. Lu, A. P. T. Lau, and P. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express 25, 20286–20297 (2017).
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Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P.-K. A. Wai, F. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 991 (2015).

T. Gui, W. A. Gemechu, J.-W. Goossens, M. Song, S. Wabnitz, M. I. Yousefi, H. Hafermann, A. P. T. Lau, and Y. Jaouën, “Polarization-division-multiplexed nonlinear frequency division multiplexing,” in CLEO: Science and Innovations (Optical Society of America, 2018), pp. STu4C–3.

Guiomar, F. P.

Hafermann, H.

J.-W. Goossens, M. I. Yousefi, Y. Jaouën, and H. Hafermann, “Polarization-division multiplexing based on the nonlinear Fourier transform,” Opt. Express 25, 26437–26452 (2017).
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Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P.-K. A. Wai, F. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 991 (2015).

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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. 34, 2459–2466 (2016).
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S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

S. Le, I. Philips, J. Prilepsky, M. Kamalian, A. Ellis, P. Harper, and S. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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J.-W. Goossens, M. I. Yousefi, Y. Jaouën, and H. Hafermann, “Polarization-division multiplexing based on the nonlinear Fourier transform,” Opt. Express 25, 26437–26452 (2017).
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S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
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S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28, 662–701 (2010).
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Lau, A. P. T.

T. Gui, T. H. Chan, C. Lu, A. P. T. Lau, and P.-K. A. Wai, “Alternative decoding methods for optical communications based on nonlinear Fourier transform,” J. Lightw. Technol. 35, 1542–1550 (2017).
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S. Le, I. Philips, J. Prilepsky, M. Kamalian, A. Ellis, P. Harper, and S. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communication, part II: Eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).
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S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

Liga, G.

Lu, C.

T. Gui, C. Lu, A. P. T. Lau, and P. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express 25, 20286–20297 (2017).
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S. Le, I. Philips, J. Prilepsky, M. Kamalian, A. Ellis, P. Harper, and S. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

Philips, I. D.

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. 34, 2459–2466 (2016).
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S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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S. Le, I. Philips, J. Prilepsky, M. Kamalian, A. Ellis, P. Harper, and S. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
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N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

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Song, M.

T. Gui, W. A. Gemechu, J.-W. Goossens, M. Song, S. Wabnitz, M. I. Yousefi, H. Hafermann, A. P. T. Lau, and Y. Jaouën, “Polarization-division-multiplexed nonlinear frequency division multiplexing,” in CLEO: Science and Innovations (Optical Society of America, 2018), pp. STu4C–3.

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I. Tavakkolnia and M. Safari, “Capacity analysis of signaling on the continuous spectrum of nonlinear optical fibers,” J. Lightw. Technol. 35, 2086–2097 (2017).
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N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bayvel, and S. K. Turitsyn, “Capacity lower bounds of the noncentral chi-channel with applications to soliton amplitude modulation,” IEEE Trans. Commun. 66280 (2018).

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
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S. T. Le, I. D. Phillips, 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 Proceedings of ECOC 2016, 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.

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T. Gui, T. H. Chan, C. Lu, A. P. T. Lau, and P.-K. A. Wai, “Alternative decoding methods for optical communications based on nonlinear Fourier transform,” J. Lightw. Technol. 35, 1542–1550 (2017).
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R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28, 662–701 (2010).
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Figures (6)

Fig. 1
Fig. 1 Details of NFT processing in PDM-NIS (above) and PDM-NISNLS (below).
Fig. 2
Fig. 2 (a) Basic PDM-NIS scheme; (b) Performance (Q2-factor) of single polarization modulation over the ME channel model (symbols only), where both polarization components are corrupted by noise, compared to that over the NLSE channel model (solid lines) (similar to systems considered in Refs. [18,32]).
Fig. 3
Fig. 3 Performance for different burst lengths with same color: (a) PDM-NIS performance compared with single polarization NIS; and (b) PDM-NIS in the noisy and noise-free scenarios, with actual (4 samples per symbol) and increased (8 samples per symbol) accuracy for NFTs.
Fig. 4
Fig. 4 (a) NMSE on the nonlinear spectrum after INFT and NFT as a function of the optical power, for different oversampling factors; (b) PDM-NISNLS scheme.
Fig. 5
Fig. 5 Performance Vs power per symbol for different burst lengths with same color: (a) PDM-NIS (solid lines) compared with reduced complexity PDM-NISNLS (dashed lines); (b) PDM-NISNLS compared with PDM-NISNLS n.f. with actual (4 samples per symbol) and increased (8 samples per symbol) accuracy, and with back-to-back performance.
Fig. 6
Fig. 6 (a) Performance Vs power per symbol for Nb = 32 for PDM-NIS, PDM-NISNLS, and conventional systems; (b) Optimal performance as a function of the rate efficiency.

Equations (40)

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j Q X = β 2 2 2 Q T 2 γ 8 9 Q 2 Q ,
j q x = 2 q t 2 + 2 σ q 2 q ,
ρ ( λ ) = b ( λ ) / a ( λ )
C i = b ( λ i ) / a ( λ i ) .
K ( x , y ) σ F ( x + y ) + σ x x K ( x , r ) F ( r + s ) F ( s + y ) d s d r = 0 ,
F ( x ) = 1 2 π + ρ ( λ ) e j λ x d λ j i = 1 N C i e j λ i x .
b ( λ ) = ( b 1 ( λ ) , b 2 ( λ ) )
ρ ( λ ) = ( ρ 1 ( λ ) , ρ 2 ( λ ) ) = ( b 1 ( λ ) / a ( λ ) , b 2 ( λ ) / a ( λ ) )
C i = ( b 1 ( λ i ) / a ( λ i ) , b 2 ( λ i ) / a ( λ i ) ) .
| a ( λ ) | 2 + σ | b 1 ( λ ) | 2 + σ | b 2 ( λ ) | 2 = 1 .
+ | q 1 ( t ) | 2 + | q 2 ( t ) | 2 d t = 4 k = 1 N { λ k } + σ π + log ( 1 + σ | ρ 1 ( λ ) | 2 + σ | ρ 2 ( λ ) | 2 ) d λ .
NFT M ( R q ) = R * NFT M ( q ) .
P = ( j λ q 1 ( t ) q M ( t ) σ q 1 * ( t ) j λ σ q M * ( t ) j λ )
ϕ ( t , λ ) ~ ( 1 0 M × 1 ) e j λ t M + 1 × 1 , ϕ ¯ ( x , λ ) ~ ( 0 1 × M I M ) e j λ t M + 1 × M as t ,
ψ ( t , λ ) ~ ( 0 1 × M I M ) e j λ t M + 1 × M , ψ ¯ ( x , λ ) ~ ( 1 0 M × 1 ) e j λ t M + 1 × 1 as t + .
ϕ ( t , λ ) = ψ ( t , λ ) b ( λ ) + ψ ¯ ( t , λ ) a ( λ ) .
a ( λ ) = lim t + ϕ 1 ( t , λ ) e + j λ t , b m ( λ ) = lim t + ϕ m + 1 ( t , λ ) e j λ t ,
{ ϕ t = P ( n ) ϕ for t ( t n δ / 2 , t n + δ / 2 ] , ϕ ( t n δ / 2 ) = ϕ ( n ) boundary condition ,
ϕ ( 1 ) = ( 1 0 M × 1 ) e j λ ( T + δ / 2 ) .
U ( n ) = ( c 0 j λ s 0 q 1 ( n ) s 0 q M ( n ) s 0 σ q 1 ( n ) * s 1 c 1 + j λ s 1 σ q M ( n ) * s M c M + j λ s M ) ,
d 0 = λ 2 σ k = 1 M | q k ( n ) | 2 and d k = λ 2 σ | q k ( n ) | 2
{ a ( λ ) = Σ 1 e j λ ( 2 T + δ ) , b m ( λ ) = Σ m + 1 , for m = 1 , , M , where Σ = U ( N t + 1 ) U ( 1 ) ( 1 0 M × 1 ) M + 1 × 1
a ( λ ) = ϕ 1 ( N t + 1 ) e j λ 2 T
ϕ ( n + 1 ) = U ( n ) ϕ ( n ) + U ( n ) ϕ ( n ) ,
U ( n ) = ( j λ 2 δ / d 0 2 c 0 Γ 0 s 0 q 1 ( n ) λ / d 0 2 ( δ c 0 s 0 ) q M ( n ) λ / d 0 2 ( δ c 0 s 0 ) σ q 1 ( n ) * λ / d 1 2 ( δ c 1 s 1 ) j λ 2 δ / d 1 2 c 1 + Γ 1 s 1 σ q M ( n ) * λ / d M 2 ( δ c M s M ) j λ 2 δ / d M 2 c M + Γ M s M ) ,
u : = v = ( 1 0 0 0 R 11 * R 12 * 0 R 21 * R 22 * ) v ,
ρ R ( λ ) = b R ( λ ) / a R ( λ ) = R * b ( λ ) / a ( λ ) = R * ρ ( λ ) ,
= ( h ) = ( h 1 h 2 h 3 h N h 2 h 3 h N 0 h 3 h N h N h N 0 0 ) ,
𝒞 ( c ) = ( c 1 c 2 c 3 c N 𝒞 c N 𝒞 c 1 c 2 c N 𝒞 1 c N 𝒞 c 1 c 3 c 2 c 2 c N 𝒞 1 c N 𝒞 c 1 ) ,
{ B 1 ( t , α ) σ 0 + B 2 ( t , β ) F ( α + β + 2 t ) d β = 0 scalar equation B 2 ( t , α ) σ F ( α + 2 t ) + 0 + B 1 ( t , β ) F ( α + β + 2 t ) d β = 0 M coupled equations ,
a b f ( x ) d x Δ 3 [ f ( x 1 ) + 4 j = 1 n / 2 f ( x 2 j ) + 2 j = 1 n / 2 1 f ( x 2 j + 1 ) + f ( x n + 1 ) ] ,
( I L σ H 1 D σ H M D H 1 * D I LM H M * D ) L ( M + 1 ) × L ( M + 1 ) ( b 1 b 2 × 1 b 2 , M ) L ( M + 1 ) × 1 = ( 0 L × 1 σ f 1 * σ f M * ) L ( M + 1 ) × 1 ,
( I L σ HD LM H D I LM ) L ( M + 1 ) × L ( M + 1 ) ( b 1 b 2 ) L ( M + 1 ) × 1 = ( 0 L × 1 σ f ) , L ( M + 1 ) × 1
( D LM H D HD LM + σ D LM ) b 2 = D LM f * ,
A m , 1 b 2 , 1 + + A m , M b 2 , M + σ D b 2 , m = D f m * ,
E s = { ( 2 N b ) 1 + ( | Q 1 ( T ) | 2 + | Q 2 ( T ) | 2 ) d T dual pol ( N b ) 1 + ( | Q 1 ( T ) | 2 ) d T single pol . .
NMSE ( m = 1 , 2 k = 1 N sa | ρ k , m ρ ˜ k , m | 2 ) ( m = 1 , 2 k = 1 N sa | ρ k , m | 2 ) 1
η = { N b / 2 ( N z + N b ) single polarization , N b / ( N z + N b ) dual polarization . .
M 𝒞 ( 1 , N sa ) < 𝒞 ( M , N sa ) .
[ NFT NLS ( R 11 q 1 + R 12 q 2 ) NFT NLS ( R 21 q 1 + R 22 q 2 ) ] [ R 11 * NFT NLS ( q 1 ) + R 12 * NFT NLS ( q 2 ) R 21 * NFT NLS ( q 1 ) + R 22 * NFT NLS ( q 2 ) ] ,

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