Abstract

Using the integrable nonlinear Schrödinger equation (NLSE) as a channel model, we describe the application of nonlinear spectral management for effective mitigation of all nonlinear distortions induced by the fiber Kerr effect. Our approach is a modification and substantial development of the so-called “eigenvalue communication” idea first presented in A. Hasegawa, T. Nyu, J. Lightwave Technol. 11, 395 (1993). The key feature of the nonlinear Fourier transform (inverse scattering transform) method is that for the NLSE, any input signal can be decomposed into the so-called scattering data (nonlinear spectrum), which evolve in a trivial manner, similar to the evolution of Fourier components in linear equations. We consider here a practically important weakly nonlinear transmission regime and propose a general method of the effective encoding/modulation of the nonlinear spectrum: The machinery of our approach is based on the recursive Fourier-type integration of the input profile and, thus, can be considered for electronic or all-optical implementations. We also present a novel concept of nonlinear spectral pre-compensation, or in other terms, an effective nonlinear spectral pre-equalization. The proposed general technique is then illustrated through particular analytical results available for the transmission of a segment of the orthogonal frequency division multiplexing (OFDM) formatted pattern, and through WDM input based on Gaussian pulses. Finally, the robustness of the method against the amplifier spontaneous emission is demonstrated, and the general numerical complexity of the nonlinear spectrum usage is discussed.

© 2013 OSA

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2013 (2)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
[CrossRef]

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
[CrossRef]

2012 (6)

2011 (6)

2010 (3)

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol.28, 423–433 (2010).
[CrossRef]

D. J. Richardson, “Filling the light pipe,” Science330, 327–328 (2010).
[CrossRef] [PubMed]

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
[CrossRef]

2008 (8)

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
[CrossRef]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express16, 841–859 (2008).
[CrossRef] [PubMed]

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

R.-J. Essiambre, G. Foschini, G. Kramer, and P. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101, 163901 (2008).
[CrossRef] [PubMed]

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A78, 063819 (2008).
[CrossRef]

W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking7, 234–255 (2008).
[CrossRef]

S. A. Derevyanko and J. E. Prilepsky, “Soliton generation from randomly modulated return-to-zero pulses,” Opt. Comm.281, 5439–5443 (2008).
[CrossRef]

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrodinger equation,” Phys. Rev. E78, 046610 (2008).
[CrossRef]

2007 (3)

2006 (1)

2005 (1)

2004 (1)

S. Oda, A. Maruta, and K. Kitayama, “All-Optical Quantization Scheme Based on Fiber Nonlinearity,” IEEE Photon. Technol. Lett.16, 587–589 (2004).
[CrossRef]

2002 (1)

M. Klaus and J. K. Shaw, “Purely imaginary eigenvalues of Zakharov-Shabat systems,” Phys. Rev. E65, 036607 (2002).
[CrossRef]

2001 (2)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411, 1027–1030 (2001).
[CrossRef] [PubMed]

M. Van Barel, G. Heinig, and P. Kravanja, “A stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. Appl.23, 494–510 (2001).
[CrossRef]

1999 (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quant. Electron.35, 1105–1115 (1999).
[CrossRef]

1994 (1)

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol.12, 1993–2000 (1994).
[CrossRef]

1993 (2)

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication, 1993, paper MoC2.4.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave 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]

1990 (1)

1976 (1)

D. J. Kaup, “Closure of the squared Zakharov-Shabat eigenstates,” J. Math. Anal. Appl.54, 849–864 (1976).
[CrossRef]

1974 (3)

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).

J. Satsuma and N. Yadjima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Progr. Theor. Phys. Suppl.55, 284–306 (1974).
[CrossRef]

S.V. Manakov, “On nonlinear Fraunhofer diffraction,” Soviet Physics-JETP38, 693–696 (1974).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Soviet Physics-JETP34, 62–69 (1972).

Ablowitz, M. J.

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

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

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, 2006).

Bao, H.

Bayer, S.

S. Herbst, S. Bayer, H. Wernz, and H. Griesser, “21-GHz single-band OFDM transmitter with QPSK modulated subcarriers,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference, IEEE, 2011, paper OMS3.

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
[CrossRef]

Behrens, C.

R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt.58, 1–10 (2011).
[CrossRef]

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
[CrossRef]

Belai, O. V.

Benlachtar, Y.

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]

Bouziane, R.

Chandrasekaran, S.

S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
[CrossRef]

Chandrasekhar, S.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
[CrossRef]

Cheng, H.

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
[CrossRef]

Cho, P.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
[CrossRef]

Chraplyvy, A. R.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
[CrossRef]

Cotter, D.

Derevyanko, S.

Derevyanko, S. A.

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett.108, 183902 (2012).
[CrossRef] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Lattice approach to the dynamics of phase-coded soliton trains,” J. Phys. A: Math. Theor.45, 025202 (2012).
[CrossRef]

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A78, 063819 (2008).
[CrossRef]

S. A. Derevyanko and J. E. Prilepsky, “Soliton generation from randomly modulated return-to-zero pulses,” Opt. Comm.281, 5439–5443 (2008).
[CrossRef]

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrodinger equation,” Phys. Rev. E78, 046610 (2008).
[CrossRef]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasilossless and lossy optical fiber spans,” J. Opt. Soc. Am. B24, 1254–1261 (2007).
[CrossRef]

Desurvire, E. B.

Djordjevic, I.

S. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2010).

Du, L. B.

Du, L. B. Y.

Ellis, A. D.

Essiambre, R.-J.

A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudo-linear coherent systems,” J. Lightwave Technol.30, 2011–2024 (2012).
[CrossRef]

R.-J. Essiambre, G. Foschini, G. Kramer, and P. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101, 163901 (2008).
[CrossRef] [PubMed]

Fan, Y.

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quant. Electron.35, 1105–1115 (1999).
[CrossRef]

Foschini, G.

R.-J. Essiambre, G. Foschini, G. Kramer, and P. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101, 163901 (2008).
[CrossRef] [PubMed]

Frumin, L. L.

Glick, M.

Gnauk, A.H.

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett.15, 1351–1353 (1990).
[CrossRef] [PubMed]

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, 2006).

Griesser, H.

S. Herbst, S. Bayer, H. Wernz, and H. Griesser, “21-GHz single-band OFDM transmitter with QPSK modulated subcarriers,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference, IEEE, 2011, paper OMS3.

Gu, M.

S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
[CrossRef]

Hasegawa, A.

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

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

Heinig, G.

M. Van Barel, G. Heinig, and P. Kravanja, “A stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. Appl.23, 494–510 (2001).
[CrossRef]

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
[CrossRef]

Herbst, S.

S. Herbst, S. Bayer, H. Wernz, and H. Griesser, “21-GHz single-band OFDM transmitter with QPSK modulated subcarriers,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference, IEEE, 2011, paper OMS3.

Hoe, J. C.

Ip, E.

Kahn, J.

Karagodsky, V.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Closure of the squared Zakharov-Shabat eigenstates,” J. Math. Anal. Appl.54, 849–864 (1976).
[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).

Khurgin, J.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
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D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
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H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
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X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
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D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
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M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
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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).

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M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
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S. Oda, A. Maruta, and K. Kitayama, “All-Optical Quantization Scheme Based on Fiber Nonlinearity,” IEEE Photon. Technol. Lett.16, 587–589 (2004).
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G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys.102, 252–264 (1992).
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A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication, 1993, paper MoC2.4.

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V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method(Colsultants Bureau, 1984).

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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Lattice approach to the dynamics of phase-coded soliton trains,” J. Phys. A: Math. Theor.45, 025202 (2012).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett.108, 183902 (2012).
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S. A. Derevyanko and J. E. Prilepsky, “Soliton generation from randomly modulated return-to-zero pulses,” Opt. Comm.281, 5439–5443 (2008).
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S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrodinger equation,” Phys. Rev. E78, 046610 (2008).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasilossless and lossy optical fiber spans,” J. Opt. Soc. Am. B24, 1254–1261 (2007).
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D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
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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).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).
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V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Soviet Physics-JETP34, 62–69 (1972).

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M. Klaus and J. K. Shaw, “Purely imaginary eigenvalues of Zakharov-Shabat systems,” Phys. Rev. E65, 036607 (2002).
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M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
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Splett, A.

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication, 1993, paper MoC2.4.

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P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411, 1027–1030 (2001).
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S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
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Tang, Y.

Terauchi, H.

H. Terauchi and A. Maruta, “Eigenvalue Modulated Optical Transmission System Based on Digital Coherent Technology,” in 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching (OECC/PS), IEICE, 2013, paper WR2-5.

Tkach, R. W.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett.108, 183902 (2012).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Lattice approach to the dynamics of phase-coded soliton trains,” J. Phys. A: Math. Theor.45, 025202 (2012).
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S. Vergeles and S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A83, 061801(R) (2011).
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S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A78, 063819 (2008).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasilossless and lossy optical fiber spans,” J. Opt. Soc. Am. B24, 1254–1261 (2007).
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M. Van Barel, G. Heinig, and P. Kravanja, “A stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. Appl.23, 494–510 (2001).
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S. Vergeles and S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A83, 061801(R) (2011).
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M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008).
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R.-J. Essiambre, G. Foschini, G. Kramer, and P. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101, 163901 (2008).
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Winzer, P. J.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
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S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
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W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking7, 234–255 (2008).
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Yang, Z.

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
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Yi, X.

W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking7, 234–255 (2008).
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M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” http://arxiv.org/abs/1202.3653 , submitted to IEEE Trans. Inf. Theory, Feb. 2012; “Part II: Numerical methods,” Apr. 2012, online: http://arxiv.org/abs/1204.0830 , submitted to IEEE Trans. Inf. Theory, Feb. 2012; “Part III: Spectrum modulation,” Feb. 2013, online: http://arxiv.org/abs/1302.2875 , submitted to IEEE Trans. Inf. Theory, Feb. 2013.

Yu, S.

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Soviet Physics-JETP34, 62–69 (1972).

V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method(Colsultants Bureau, 1984).

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quant. Electron.35, 1105–1115 (1999).
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Zhang, Z.

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
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Zhao, J.

Zhu, J.

S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
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IEEE J. Quant. Electron. (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quant. Electron.35, 1105–1115 (1999).
[CrossRef]

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

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16, 1217–1226 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Oda, A. Maruta, and K. Kitayama, “All-Optical Quantization Scheme Based on Fiber Nonlinearity,” IEEE Photon. Technol. Lett.16, 587–589 (2004).
[CrossRef]

J. Comput. Phys. (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]

J. Lightwave Technol. (7)

J. Math. Anal. Appl. (1)

D. J. Kaup, “Closure of the squared Zakharov-Shabat eigenstates,” J. Math. Anal. Appl.54, 849–864 (1976).
[CrossRef]

J. Mod. Opt. (1)

R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt.58, 1–10 (2011).
[CrossRef]

J. Opt. Networking (1)

W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking7, 234–255 (2008).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Theor. (1)

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Lattice approach to the dynamics of phase-coded soliton trains,” J. Phys. A: Math. Theor.45, 025202 (2012).
[CrossRef]

Nat. Photonics (1)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics7, 560–568 (2013).
[CrossRef]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411, 1027–1030 (2001).
[CrossRef] [PubMed]

Opt. Comm. (2)

H. Cheng, W. Li, Y. Fan, Z. Zhang, S. Yu, and Z. Yang, “A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module,” Opt. Comm.290, 152–157 (2013).
[CrossRef]

S. A. Derevyanko and J. E. Prilepsky, “Soliton generation from randomly modulated return-to-zero pulses,” Opt. Comm.281, 5439–5443 (2008).
[CrossRef]

Opt. Express (6)

Opt. Lett. (4)

Phys. Rev. A (2)

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A78, 063819 (2008).
[CrossRef]

S. Vergeles and S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A83, 061801(R) (2011).
[CrossRef]

Phys. Rev. E (2)

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrodinger equation,” Phys. Rev. E78, 046610 (2008).
[CrossRef]

M. Klaus and J. K. Shaw, “Purely imaginary eigenvalues of Zakharov-Shabat systems,” Phys. Rev. E65, 036607 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

R.-J. Essiambre, G. Foschini, G. Kramer, and P. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101, 163901 (2008).
[CrossRef] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett.108, 183902 (2012).
[CrossRef] [PubMed]

Progr. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yadjima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Progr. Theor. Phys. Suppl.55, 284–306 (1974).
[CrossRef]

Science (1)

D. J. Richardson, “Filling the light pipe,” Science330, 327–328 (2010).
[CrossRef] [PubMed]

SIAM J. Matrix Anal. Appl. (1)

M. Van Barel, G. Heinig, and P. Kravanja, “A stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. Appl.23, 494–510 (2001).
[CrossRef]

SIAM J. Matrix. Anal. Appl. (1)

S. Chandrasekaran, M. Gu, X. Sun, J. Xia, and J. Zhu, “A superfast algorithm for Toeplitz systems of linear equations,” SIAM J. Matrix. Anal. Appl.29, 1247–1266 (2007).
[CrossRef]

Soviet Physics-JETP (2)

S.V. Manakov, “On nonlinear Fraunhofer diffraction,” Soviet Physics-JETP38, 693–696 (1974).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Soviet Physics-JETP34, 62–69 (1972).

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).

Tech. Digest of European Conference on Optical Communication (1)

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication, 1993, paper MoC2.4.

Other (12)

S. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2010).

see Impact of nonlinearities on fiber-optic communication systems, Ed. S. Kumar, (Springer, 2011).
[CrossRef]

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

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, 2006).

A. C. Newell, Solitons in Mathematics and Physics (SIAM, 1985).
[CrossRef]

V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method(Colsultants Bureau, 1984).

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

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” http://arxiv.org/abs/1202.3653 , submitted to IEEE Trans. Inf. Theory, Feb. 2012; “Part II: Numerical methods,” Apr. 2012, online: http://arxiv.org/abs/1204.0830 , submitted to IEEE Trans. Inf. Theory, Feb. 2012; “Part III: Spectrum modulation,” Feb. 2013, online: http://arxiv.org/abs/1302.2875 , submitted to IEEE Trans. Inf. Theory, Feb. 2013.

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, 2006).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

S. Herbst, S. Bayer, H. Wernz, and H. Griesser, “21-GHz single-band OFDM transmitter with QPSK modulated subcarriers,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference, IEEE, 2011, paper OMS3.

H. Terauchi and A. Maruta, “Eigenvalue Modulated Optical Transmission System Based on Digital Coherent Technology,” in 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching (OECC/PS), IEICE, 2013, paper WR2-5.

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Figures (7)

Fig. 1
Fig. 1

The flowchart of the precompensation scheme for the equalization of the linear and nonlinear spectra up to ε5 and the subsequent recovery of the informational content encoded at the transmitter.

Fig. 2
Fig. 2

Dependencies of the absolute value of NS spectral function |N(ω)|, Eq. (12) [red line] and linear spectrum |Q(ω)|, Eq. (33) [black line], calculated for a single tone of OFDM with Ωk = Ω2 = 2π and (normalized) base extent T = 1, for the different values of ck: (a) ck = 0.5; (b) ck = 1.5; (c) ck = 2.5.

Fig. 3
Fig. 3

Nonlinear (a,b) and linear (c,d) spectra for 3 slots of 10-subcarrier QPSK-OFDM before and after the propagation. Panes (a) and (b): the real and imaginary parts of NS N(ω), for the initial distribution (red) and distribution after the propagation at z = 1.2 (i.e. L ≈ 5000km), blue line. The red and blue parts for N(ω) are almost indistinguishable insofar as the initial and final NS coincide. Panes (c) and (d): the real and imaginary parts of linear spectrum Q(ω) for the initial distribution (black) and after the propagation at z = 1.2 (i.e. L ≈ 5000km), green line. For both spectra at z = 1.2 (blue and green), the accumulated linear dispersion was removed.

Fig. 4
Fig. 4

Pane (a): Real (magenta) and imaginary (purple) parts of the input profile corresponding to a single second OFDM tone with T = 1, Ωk = Ω2 = 2π, amplitude of the input ck = 0.5. Pane (b): real (dark red) and imaginary (green) parts of the second OFDM tone nonlinear pre-distortion profile s(t) ∼ ε3, given by the backward FT of Eq. (25) with the expression (37) inserted for r1(ξ). Pane (c): The absolute errors, obtained by the direct integration of NLSE (4) and dispersion removal at at distance z = 1 (L = 4000km) [black], the usage of the pre-processed NS at the same distance z = 1 (L = 4000km) [blue] and the (expected) error of the associated with the pre-distorted signal at the input z = 0 (red dots, almost indistinguishable). The more detailed explanations of the error definitions are given in the text, see Subsection 5.1.

Fig. 5
Fig. 5

The absolute errors for the propagation of 3 slots (duration of each slot T = 1) of the 10-mode QPSK-OFDM (with the amplitudes of individual coefficients |cαk| = 0.15), obtained by the direct dispersion removal at distance z = 1 (L = 4000km) [black]; the usage of the nonlinear pre-distorted NS at the same distance z = 1 (L = 4000km) [blue]; and the (expected) error associated with the pre-processed signal at the input z = 0 (red, almost indistinguishable). The error definitions are given in Subsection 5.1.

Fig. 6
Fig. 6

The absolute errors for the propagation of 3 slots of the 10-mode Gaussian-based QPSK-WDM input (with the amplitudes of individual coefficients |cαk| = 0.15), with extent ρ = 0.3 (see Eq. (47)), obtained by the direct dispersion removal at distance z = 1 (L = 4000km) [black]; the usage of the nonlinear precompensated NS at the same distance z = 1 (L = 4000km) [blue]; and the (expected) error associated with the pre-processed signal at the input z = 0 (red, almost indistinguishable). The error definitions are given in Subsection 5.1, normalizations are the same as in Figs. 4, 5.

Fig. 7
Fig. 7

The average integral absolute errors for the propagation of 3 slots of the 10-mode QPSK-OFDM input (the parameters are the same as in Fig. 5) in the noisy NLSE, Eq. (48), versus the noise intensity 2Γ, obtained (a) by the direct dispersion removal [black] and the usage of the nonlinear precompensated NS at distance L = 2000km [red]; (b) - the same for the propagation distance L = 4000km. The averaging for each point was made over 50 realizations.

Equations (60)

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Q ( ω ) = q ( t ) e i ω t d t , q ( t ) = 1 2 π Q ( ω ) e i ω t d ω .
i q z β 2 2 q t t + γ q | q | 2 = 0 ,
t T s t , z Z s z , q γ Z s q .
i q z + 1 2 q t t + q | q | 2 = 0 ,
d ϕ 1 d t = q ( t ) ϕ 2 i ζ ϕ 1 , d ϕ 2 d t = q ¯ ( t ) ϕ 1 + i ζ ϕ 2 .
Φ ( t , ζ ) | t = ( 1 0 ) exp ( i ζ t ) .
a ( ζ ) = lim t ϕ 1 ( t , ζ ) , exp ( i ζ t ) , b ( ζ ) = lim t ϕ 2 ( t , ζ ) exp ( i ζ t ) ,
r ( ξ ) = b ( ξ ) a ( ξ ) = lim t ϕ 2 ( ζ , t ) ϕ 1 ( ζ , t ) exp ( 2 i ζ t ) .
Σ = [ r ( ξ ) , ξ R , { ζ n , C n b ( ζ n ) a ( ζ n ) } ] ,
Σ = [ r ( ξ ) , ξ Re ] ,
R ( L , ξ ) = r ( ξ ) exp ( 2 i ξ 2 L ) ,
r ( ξ ) = Q ¯ ( 2 ξ ) ,
N ( ω ) = r ¯ ( ξ ) | ξ = ω 2 .
r ( ξ ) = lim t φ 2 ( ξ , t ) φ 1 ( ξ , t ) .
d φ 1 d t = e 2 i ξ t q ( t ) φ 2 , d φ 2 d t = e 2 i ξ t q ¯ ( t ) φ 1 ,
φ 2 ( t , ξ ) = T / 2 t d t 1 e 2 i ξ t 1 q ¯ ( t 1 ) φ 1 ( t 1 , ξ ) ,
φ 1 ( t , ξ ) = 1 + T / 2 t d t 2 e 2 i ξ t 2 q ( t 2 ) φ 2 ( t 2 , ξ ) .
φ 2 ( t , ξ ) = T / 2 t d t 1 e 2 i ξ t 1 q ¯ ( t 1 ) + T / 2 t d t 1 T / 2 t 1 d t 2 0 t 2 d t 3 e 2 i ξ ( t 2 t 1 t 3 ) q ¯ ( t 1 ) q ( t 2 ) q ¯ ( t 3 ) ,
φ 1 ( t , ξ ) = 1 T / 2 t d t 1 T / 2 t 1 d t 2 e 2 i ξ ( t 1 t 2 ) q ( t 1 ) q ¯ ( t 2 ) ,
r 0 ( ξ ) = T / 2 T / 2 d t 1 e 2 i ξ t 1 q ¯ ( t 1 ) ,
r 1 ( ξ ) = T / 2 T / 2 d t 1 t 1 T / 2 d t 2 T / 2 t 2 d t 3 e 2 i ξ ( t 2 t 1 t 3 ) q ¯ ( t 1 ) q ( t 2 ) q ¯ ( t 3 ) .
R ( L , ξ ) = R 0 ( L , ξ ) + R 1 ( L , ξ ) = [ r 0 ( ξ ) + r 1 ( ξ ) ] e 2 i ξ 2 L ,
q s ( t ) = q ( t ) + s ( t ) .
r ( ξ ) = r 0 ( ξ ) + r 1 ( ξ ) + r s ( ξ ) + O ( ε 5 ) ,
r s ( ξ ) = T / 2 T / 2 d t 1 e 2 i ξ t 1 s ¯ ( t 1 ) .
r s ( ξ ) = r 1 ( ξ ) .
S ( ω ) = r ¯ 1 ( ξ ) | ξ = ω 2 ,
q ( t ) = α = k = 0 N s c 1 c α k s k ( t α T ) e i Ω k t .
δ k l = 1 T 0 T s k ( t ) s ¯ l ( t ) e i ( Ω k Ω l ) t d t ,
Ω k Ω l = ( 2 π / T ) m
c α k = 1 T α T ( α + 1 ) T q ( t ) e i Ω k t d t .
Ω k = ( 2 π / T ) ( k 1 ) ,
Q ( ω ) = 2 α = k = 1 N s c c α k exp [ i ω α T + ( i / 2 ) ( Ω k ω ) T ] sin [ ( Ω k ω ) T 2 ] Ω k ω .
Arg { c α k } = 2 π p / 4 ,
q ( t ) = { c k exp ( i Ω k t ) if t [ T 2 , T 2 ] , 0 otherwise ,
Φ ( t + T / 2 , ξ ) = ( cos ( ξ k t ) i ξ k / 2 Ξ k sin ( Ξ k t ) ) ( e i Ω k t / 2 0 ) c ¯ k Ξ k sin ( Ξ k t ) ( 0 e i Ω k t / 2 ) ,
ξ k ( ξ ) = 2 ξ + Ω k , Ξ k ( ξ ) = ( ξ k / 2 ) 2 + | c k | 2 .
r ( ξ ) = 2 c ¯ k sin Ξ k T 2 Ξ k cos Ξ k T i ξ k sin Ξ k T exp [ i T ξ k / 2 ] .
| r ( ξ ) | 2 = cos 2 ψ k sin 2 Ξ k T cos 2 Ξ k T + sin 2 ψ k sin 2 Ξ k T , ψ k = arctan ξ k 2 | c k | .
| Q ( ω ) | 2 = 4 | c k | 2 sin 2 [ ( Ω k ω ) / 2 ] ( Ω k ω ) 2 .
| c k | t h T = π / 2 , ξ t h = Ω k 2 .
r 0 ( ξ ) = 2 c ¯ k sin ξ k T / 2 ξ k ,
R 0 = 2 c ¯ k sin ξ k T / 2 ξ k e 2 i ξ 2 L .
r 1 ( ξ ) = 2 c ¯ k | c k | 2 exp [ i ξ k T / 2 ] sin T ξ k T ξ k ( ξ k / 2 ) 3 .
q ( t ) = α = 0 N α k = 1 N s c c α k Π ( τ α , t ) exp ( i Ω k t ) ,
Π ( τ α , t ) = { 1 if t [ T 2 + τ α , T 2 + τ α ] , 0 otherwise .
r 0 OFDM ( ξ ) = α r 0 α ( ξ ) ,
r 0 α ( ξ ) = 2 k c ¯ α k exp ( i τ α ξ k ) sin ξ k T / 2 ξ k .
r 1 OFDM ( ξ ) = α r 1 α ( ξ ) + α β > α γ > β r 0 α ( ξ ) r ¯ 0 β ( ξ ) r 0 γ ( ξ ) + α γ < α r 12 α ( ξ ) r 0 γ ( ξ ) + α β > α r 0 α ( ξ ) r 21 β ( ξ ) ,
r 1 α ( ξ ) = i j k exp ( i τ α F i j k ) r i j k ic ( ξ ) .
r i j k ic ( ξ ) = 2 i c α i c ¯ α j c α k ξ i ξ j ξ k F i j k ( Ω i Ω j ) ( Ω k Ω j ) [ E ¯ k E j E ¯ i ξ j ( Ω k Ω j ) ( Ω i Ω j ) 2 E k E ¯ j E i ξ k ξ i ( Ω k + Ω i 2 Ω j ) + E k E j E ¯ i ξ j F i j k ( Ω k Ω j ) + E k E j E i F i j k ( Ω i Ω j ) ( Ω k Ω j ) / 2 + E ¯ k E j E i ξ j F i j k ( Ω i Ω j ) ] ,
E i ( ξ ) = e i ξ i T , F i j k ( ξ ) = 2 ξ + Ω i Ω j + Ω k ,
r 12 α ( ξ ) = i i , j c ¯ α i c α j e i τ α ( Ω j Ω i ) ξ j ( T δ i j 2 exp ( i T ξ j / 2 ) sin T ξ i / 2 ξ i ) ,
r 21 β ( ξ ) = i j , k c β j c ¯ β k e i τ β ( Ω j Ω k ) ξ k ( T δ j k 2 exp ( i T ξ k / 2 ) sin T ξ j / 2 ξ j ) .
q ( t ) = α = 0 N α k = 1 N c h c α k G ( τ α , ρ , t ) exp ( i Ω k t ) ,
G ( τ α , ρ , t ) = exp [ ( t τ α ) 2 ρ 2 ] ,
i q z β 2 2 q t t + γ q | q | 2 = η ( t , z ) .
η ( t , z ) η ¯ ( t , z ) = 2 Γ δ ( t t ) δ ( z z ) .
2 Γ = h ν 0 n s p K T χ ,
K ( t , t ) + F ¯ ( t + t ) t t K ( t , λ ) F ( λ + σ ) F ¯ ( σ + t ) d σ d λ = 0 .

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