Abstract

New services and applications are causing an exponential increase in Internet traffic. In a few years, the current fiber optic communication system infrastructure will not be able to meet this demand because fiber nonlinearity dramatically limits the information transmission rate. Eigenvalue communication could potentially overcome these limitations. It relies on a mathematical technique called “nonlinear Fourier transform (NFT)” to exploit the “hidden” linearity of the nonlinear Schrödinger equation as the master model for signal propagation in an optical fiber. We present here the theoretical tools describing the NFT for the Manakov system and report on experimental transmission results for dual polarization in fiber optic eigenvalue communications. A transmission of up to 373.5 km with a bit error rate less than the hard-decision forward error correction threshold has been achieved. Our results demonstrate that dual-polarization NFT can work in practice and enable an increased spectral efficiency in NFT-based communication systems, which are currently based on single polarization channels.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

2016 (5)

2015 (2)

2014 (3)

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. 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]

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]

2012 (1)

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (1)

2006 (2)

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).
[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]

2004 (2)

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, “Integrable nonlinear Schrödinger systems and their soliton dynamics,” Dyn. Partial Differ. Equ. 1, 239–299 (2004).

T. P. Horikis and J. N. Elgin, “Nonlinear optics in a birefringent optical fiber,” Phys. Rev. E 69, 016603 (2004).
[Crossref]

2003 (1)

O. C. Wright, “The Darboux transformation of some Manakov systems,” Appl. Math. Lett. 16, 647–652 (2003).
[Crossref]

2002 (1)

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).
[Crossref]

2000 (2)

Y. Chen and H. Haus, “Manakov solitons and polarization mode dispersion,” Chaos 10, 529–538 (2000).
[Crossref]

R. G. Docksey and J. N. Elgin, “Closure of the Manakov system,” SIAM J. Math. Anal. 32, 54–79 (2000).
[Crossref]

1999 (1)

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[Crossref]

1997 (1)

T. Lakoba and D. Kaup, “Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[Crossref]

1993 (1)

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

1991 (1)

1990 (1)

1974 (2)

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

N. 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]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-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, B. Prinari, and A. D. Trubatch, “Integrable nonlinear Schrödinger systems and their soliton dynamics,” Dyn. Partial Differ. Equ. 1, 239–299 (2004).

Ablowitz, N. J.

N. 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]

Agrawal, G. P.

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

Agrell, E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Al-Khateeb, M. A. Z.

Andrekson, P. A.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

Aref, V.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 11, 570–576 (2017).
[Crossref]

V. Aref, “Control and detection of discrete spectral amplitudes in nonlinear Fourier spectrum,” arXiv:1605.06328 (2016).

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC) (2015), paper Tu.1.1.2.

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 42st European Conference on Optical Communications (ECOC) (2016), paper Th.3.B.2.

Baronio, F.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Bayvel, P.

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]

Bowers, J. E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Brandt-Pearce, M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Buelow, H.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 11, 570–576 (2017).
[Crossref]

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 42st European Conference on Optical Communications (ECOC) (2016), paper Th.3.B.2.

Bülow, H.

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

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC) (2015), paper Tu.1.1.2.

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. Lightwave Technol. 35, 1542–1550 (2017).
[Crossref]

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (IEEE, 2015), pp. 455–459.

Q. Zhang and T. H. Chan, “Noise models in the nonlinear spectral domain for optical fibre communications,” arXiv:1702.06226 (2017).

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory, June2015, pp. 1660–1664.

Chen, H. H.

Chen, Y.

Y. Chen and H. Haus, “Manakov solitons and polarization mode dispersion,” Chaos 10, 529–538 (2000).
[Crossref]

Chraplyvy, A. R.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Conforti, M.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Da Ros, F.

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

Degasperis, A.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Derevyanko, S. A.

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]

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]

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).
[Crossref]

Docksey, R. G.

R. G. Docksey and J. N. Elgin, “Closure of the Manakov system,” SIAM J. Math. Anal. 32, 54–79 (2000).
[Crossref]

Doran, N.

Eggleton, B. J.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Elgin, J. N.

T. P. Horikis and J. N. Elgin, “Nonlinear optics in a birefringent optical fiber,” Phys. Rev. E 69, 016603 (2004).
[Crossref]

R. G. Docksey and J. N. Elgin, “Closure of the Manakov system,” SIAM J. Math. Anal. 32, 54–79 (2000).
[Crossref]

Ellis, A. D.

Essiambre, R. J.

Fabbri, S.

Fischer, J. K.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Foschini, G. J.

Frumin, L. L.

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, 013901 (2014).
[Crossref]

Gaiarin, S.

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

Gisin, N.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Goebel, B.

Goossens, J.-W.

Gordienko, V.

Gui, T.

Hafermann, H.

Hall, T. J.

N. Nabavi and T. J. Hall, “Demultiplexing by independent component analysis in coherent optical transmission: the polarization channel alignment problem,” in Photonics North (2015).

Hari, S.

Harper, P.

Hasegawa, A.

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

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[Crossref]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, 7th ed. (Oxford University, 1995).

Haus, H.

Y. Chen and H. Haus, “Manakov solitons and polarization mode dispersion,” Chaos 10, 529–538 (2000).
[Crossref]

Hoffmann, S.

Horikis, T. P.

T. P. Horikis and J. N. Elgin, “Nonlinear optics in a birefringent optical fiber,” Phys. Rev. E 69, 016603 (2004).
[Crossref]

Idler, W.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC) (2015), paper Tu.1.1.2.

Ip, E.

Iqbal, M. A.

Jaouën, Y.

Kahn, J.

Kamalian, M.

Karlsson, M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

Kaup, D.

T. Lakoba and D. Kaup, “Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[Crossref]

Kaup, D. J.

N. 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]

Kikuchi, K.

Kodama, Y.

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[Crossref]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, 7th ed. (Oxford University, 1995).

Kramer, P. J.

Krummrich, P. M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Kschischang, F. R.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).
[Crossref]

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).
[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 III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).
[Crossref]

Lakoba, T.

T. Lakoba and D. Kaup, “Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[Crossref]

Lau, A. P. T.

Lavery, D.

Le, S. T.

Li, J.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

Liga, G.

Lord, A.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Lu, C.

Maher, R.

Manakov, S. V.

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

Marks, B. S.

Maruta, A.

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

Matsuda, Y.

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

Matveev, V. B.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, 1991).

McCarthy, M. E.

Menyuk, C. R.

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Mondaca, G. S.

Nabavi, N.

N. Nabavi and T. J. Hall, “Demultiplexing by independent component analysis in coherent optical transmission: the polarization channel alignment problem,” in Photonics North (2015).

Newell, A. C.

N. 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]

Noé, R.

Nyu, T.

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

Perego, A. M.

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

Perentos, A.

Pfau, T.

Phillips, I. D.

Porto da Silva, E.

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

Prat, J.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Prilepsky, J. E.

Prinari, B.

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, “Integrable nonlinear Schrödinger systems and their soliton dynamics,” Dyn. Partial Differ. Equ. 1, 239–299 (2004).

Richardson, D. J.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Roberts, K.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Salle, M. A.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, 1991).

Savory, S. J.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Schuh, K.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC) (2015), paper Tu.1.1.2.

Secondini, M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Segur, H.

N. 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]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Srinivasan, S.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Stephens, M. F. C.

Sunnerud, H.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

Sygletos, S.

Tan, M.

Tomkos, I.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Trubatch, A. D.

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, “Integrable nonlinear Schrödinger systems and their soliton dynamics,” Dyn. Partial Differ. Equ. 1, 239–299 (2004).

Turitsyn, S. K.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “On the design of NFT-based communication systems with lumped amplification,” J. Lightwave Technol. 35, 5464–5472 (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]

A. D. Ellis, M. Tan, M. A. Iqbal, M. A. Z. Al-Khateeb, V. Gordienko, G. S. Mondaca, S. Fabbri, M. F. C. Stephens, M. E. McCarthy, A. Perentos, I. D. Phillips, D. Lavery, G. Liga, R. Maher, P. Harper, N. Doran, S. K. Turitsyn, S. Sygletos, and P. Bayvel, “4  Tb/s transmission reach enhancement using 10 × 400  Gb/s super-channels and polarization insensitive dual band optical phase conjugation,” J. Lightwave Technol. 34, 1717–1723 (2016).
[Crossref]

S. T. Le, I. D. Phillips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).
[Crossref]

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]

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]

Wabnitz, S.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Wahls, S.

Wai, P. K.

Wai, P. K. A.

Winzer, P.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

Wright, O. C.

O. C. Wright, “The Darboux transformation of some Manakov systems,” Appl. Math. Lett. 16, 647–652 (2003).
[Crossref]

Xie, C.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

Yakushev, D. A.

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).
[Crossref]

Yang, J.

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[Crossref]

Yousefi, M. I.

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]

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).
[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]

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]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Zhang, Q.

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (IEEE, 2015), pp. 455–459.

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory, June2015, pp. 1660–1664.

Q. Zhang and T. H. Chan, “Noise models in the nonlinear spectral domain for optical fibre communications,” arXiv:1702.06226 (2017).

Zibar, D.

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

Appl. Math. Lett. (1)

O. C. Wright, “The Darboux transformation of some Manakov systems,” Appl. Math. Lett. 16, 647–652 (2003).
[Crossref]

Chaos (1)

Y. Chen and H. Haus, “Manakov solitons and polarization mode dispersion,” Chaos 10, 529–538 (2000).
[Crossref]

Dyn. Partial Differ. Equ. (1)

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, “Integrable nonlinear Schrödinger systems and their soliton dynamics,” Dyn. Partial Differ. Equ. 1, 239–299 (2004).

IEEE J. Sel. Top. Quantum Electron. (1)

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, “Influences of polarization-mode dispersion on soliton transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8, 575–590 (2002).
[Crossref]

IEEE Trans. Inf. Theory (2)

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]

J. Lightwave Technol. (12)

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

S. T. Le, I. D. Phillips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).
[Crossref]

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

A. D. Ellis, M. Tan, M. A. Iqbal, M. A. Z. Al-Khateeb, V. Gordienko, G. S. Mondaca, S. Fabbri, M. F. C. Stephens, M. E. McCarthy, A. Perentos, I. D. Phillips, D. Lavery, G. Liga, R. Maher, P. Harper, N. Doran, S. K. Turitsyn, S. Sygletos, and P. Bayvel, “4  Tb/s transmission reach enhancement using 10 × 400  Gb/s super-channels and polarization insensitive dual band optical phase conjugation,” J. Lightwave Technol. 34, 1717–1723 (2016).
[Crossref]

E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[Crossref]

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

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “On the design of NFT-based communication systems with lumped amplification,” J. Lightwave Technol. 35, 5464–5472 (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. Lightwave Technol. 35, 1542–1550 (2017).
[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. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).
[Crossref]

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).
[Crossref]

T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for MQAM constellations,” J. Lightwave Technol. 27, 989–999 (2009).
[Crossref]

J. Opt. (1)

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18, 063002 (2016).
[Crossref]

J. Phys. A (1)

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).
[Crossref]

Nat. Photonics (1)

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 11, 570–576 (2017).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optica (1)

Phys. Rev. E (3)

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[Crossref]

T. Lakoba and D. Kaup, “Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[Crossref]

T. P. Horikis and J. N. Elgin, “Nonlinear optics in a birefringent optical fiber,” Phys. Rev. E 69, 016603 (2004).
[Crossref]

Phys. Rev. Lett. (2)

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[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]

SIAM J. Math. Anal. (1)

R. G. Docksey and J. N. Elgin, “Closure of the Manakov system,” SIAM J. Math. Anal. 32, 54–79 (2000).
[Crossref]

Sov. Phys. JETP (2)

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

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

Stud. Appl. Math. (1)

N. 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 (12)

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

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC) (2015), paper Tu.1.1.2.

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in 42st European Conference on Optical Communications (ECOC) (2016), paper Th.3.B.2.

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

S. Gaiarin, A. M. Perego, E. Porto 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), paper W.3.C.2.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, 1991).

V. Aref, “Control and detection of discrete spectral amplitudes in nonlinear Fourier spectrum,” arXiv:1605.06328 (2016).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, 7th ed. (Oxford University, 1995).

N. Nabavi and T. J. Hall, “Demultiplexing by independent component analysis in coherent optical transmission: the polarization channel alignment problem,” in Photonics North (2015).

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (IEEE, 2015), pp. 455–459.

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory, June2015, pp. 1660–1664.

Q. Zhang and T. H. Chan, “Noise models in the nonlinear spectral domain for optical fibre communications,” arXiv:1702.06226 (2017).

Supplementary Material (1)

NameDescription
» Supplement 1       Numerical methods, constellations selection, noise analysis

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the DT. The S-node is the signal update operation corresponding to Eq. (14), and the E-node is the eigenvector update operation corresponding to Eq. (12). At the step i=1,2,3 the auxiliary solution v¯(i) for λ=λi (red arrow) modifies the signal qj(i1) and all the other auxiliary solutions (blue arrows). The seed null signal qj(0), j=1,2 entering the first S-node is transformed after each step in such a way that its discrete spectrum has a new eigenvalue added as shown in the four insets in upper part of the figure.
Fig. 2.
Fig. 2. Ideal normalized constellations are illustrated schematically: in (a) the discrete eigenvalues λ1=i0.3 and λ2=i0.6 are depicted. The scattering coefficients b1,2(λi),i=1,2, associated with the two orthogonal polarization components of the signal, are shown in (b) and (c), respectively. Polarization 1 and Polarization 2 are on a violet and green background, respectively. The scattering coefficients associated with λ1 are chosen from a QPSK constellation of radius 5 and rotated by π/4, while those associated with λ2 from a QPSK constellation and radius 0.14.
Fig. 3.
Fig. 3. Experimental setup with transmitter and receiver DSP chain. Abbreviations not defined in the main text: balanced photodetector (BPD), direct current (DC).
Fig. 4.
Fig. 4. System performance in terms of BER as a function of the OSNR in a back-to-back configuration. The BER of the individual constellations are shown by the violet (Polarization 1) and green (Polarization 2) curves and are grouped per eigenvalue (λ1=i0.3,λ2=i0.6). The black curve represents the average BER over the four constellations.
Fig. 5.
Fig. 5. System performance in terms of BER as a function of the transmission distance for the four individual constellations for (a) L=41.5  km and (b) L=83  km spans. The violet (Polarization 1) and green (Polarization 2) curves are grouped per eigenvalue (λ1=i0.3,λ2=i0.6). (c) Comparison of the average BER versus transmission distance between links of the two different span lengths. The error bars represent the standard deviation over five processed blocks of 105 DP-NFDM symbols.
Fig. 6.
Fig. 6. The four experimental constellations of the scattering coefficients b1,2(λi),i=1,2 associated with the two eigenvalues (λ1=i0.3,λ2=i0.6) are shown at the transmitter side (left) and after 373.5 km transmission with 41.5 km spans (right). Polarization 1 and Polarization 2 are on a violet and green background, respectively.

Equations (23)

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{E1=iβ222E1τ2+i8γ9(|E1|2+|E2|2)E1E2=iβ222E2τ2+i8γ9(|E1|2+|E2|2)E2,
qj=EjP,t=τT0,z=L,
{iq1z=2q1t2+2(|q1|2+|q1|2)q1iq2z=2q2t2+2(|q1|2+|q2|2)q2,
γeff=γ(1eαL)/(αL),
vt=(λA+B)v
A=(i000i000i)B=(0q1q2q1*00q2*00),
ϕN(t,λ)(100)eiλt;ϕ¯N(t,λ)(001001)eiλtt,
ϕP(t,λ)(001001)eiλt;ϕ¯P(t,λ)(100)eiλtt+.
ϕN(t,λ)=ϕP(t,λ)b(λ)+ϕ¯P(t,λ)a(λ),
ϕ¯N(t,λ)=ϕP(t,λ)a¯(λ)+ϕ¯P(t,λ)b¯(λ),
Qc(λ)=b(λ)a(λ)1λR,
Qd,i(λi)=b(λi)a(λi)1λ1,nC\R,
a(λ,z)=a(λ,0)a¯(λ,z)=a¯(λ,0),
b(λ,z)=b(λ,0)e4iλ2zb¯(λ,z)=b¯(λ,0)e4iλ2z.
a(λ)=limt+[ϕ1N(t,λ)ϕ¯1P(t,λ)1]
b1(λ)=limt+[ϕ2N(t,λ)ϕ2,1P(t,λ)1],
b2(λ)=limt+[ϕ3N(t,λ)ϕ3,2P(t,λ)1],
a(λ)=limt+[ϕ1N(t,λ)]eiλt],
b(λ)=(b1(λ)b2(λ))=limt+[(ϕ2N(t,λ)ϕ3N(t,λ))eiλt].
v^=(λI3G0)v,
Θ=(v¯1v¯2*v¯3*v¯2v¯1*0v¯30v¯1*),
q^j=qj+2i(λ0*λ0)uj*1+s=12|us|2(j=1,2),
(v¯^1(k)v¯^2(k)v¯^3(k))=(λkI3G0i1)(λkI3G01)(A(k)eiλktB(k)eiλktC(k)eiλkt),

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