Abstract

In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.

© 2017 Optical Society of America

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References

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2017 (2)

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(3), 307–322 (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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

2016 (3)

S. Hari and F. R. Kschischang, “Bi-Directional Algorithm for Computing Discrete Spectral Amplitudes in the NFT,” IEEE J. Lightwave Technol. 34(15), 3529 (2016).

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]

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

2015 (1)

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

2014 (4)

O. V. Yushkoa, A. A. Redyuk, M. P. Fedoruk, and S. K. Turitsyn, “Coherent Soliton Communication Lines,” Exper. Theor. Phys. 119(5), 787–794 (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(7), 4312–4328 (2014).
[Crossref]

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

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

2013 (2)

2005 (1)

2003 (1)

2001 (1)

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601 (2001).

1997 (1)

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent Progress in Dispersion-Managed Soliton Transmission Technologies,” Opt. Fiber Technol. 3, 197–213 (1997).

1993 (1)

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

1986 (1)

1974 (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53(4), 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, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53(4), 249–315 (1974).
[Crossref]

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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

Derevyanko, S. A.

Dong, Z.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Ellis, A. D.

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

Falkovich, G. E.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601 (2001).

Fedoruk, M. P.

O. V. Yushkoa, A. A. Redyuk, M. P. Fedoruk, and S. K. Turitsyn, “Coherent Soliton Communication Lines,” Exper. Theor. Phys. 119(5), 787–794 (2014).
[Crossref]

Frumin, L. L.

Gordon, J. P.

Gui, 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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Hari, S.

S. Hari and F. R. Kschischang, “Bi-Directional Algorithm for Computing Discrete Spectral Amplitudes in the NFT,” IEEE J. Lightwave Technol. 34(15), 3529 (2016).

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Harper, P.

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

Hasegawa, A.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent Progress in Dispersion-Managed Soliton Transmission Technologies,” Opt. Fiber Technol. 3, 197–213 (1997).

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

Haus, H. A.

Kamalian, M.

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53(4), 249–315 (1974).
[Crossref]

Kodama, Y.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent Progress in Dispersion-Managed Soliton Transmission Technologies,” Opt. Fiber Technol. 3, 197–213 (1997).

Kolokolov, I.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601 (2001).

Kschischang, F. R.

S. Hari and F. R. Kschischang, “Bi-Directional Algorithm for Computing Discrete Spectral Amplitudes in the NFT,” IEEE J. Lightwave Technol. 34(15), 3529 (2016).

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

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

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

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

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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Le, S. T.

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(3), 307–322 (2017).
[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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

Lebedev, V.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601 (2001).

Lu, C.

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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Maruta, A.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent Progress in Dispersion-Managed Soliton Transmission Technologies,” Opt. Fiber Technol. 3, 197–213 (1997).

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53(4), 249–315 (1974).
[Crossref]

Nyu, T.

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

Phillips, I. D.

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

Prilepsky, J. E.

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(3), 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]

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

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

Redyuk, A. A.

O. V. Yushkoa, A. A. Redyuk, M. P. Fedoruk, and S. K. Turitsyn, “Coherent Soliton Communication Lines,” Exper. Theor. Phys. 119(5), 787–794 (2014).
[Crossref]

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53(4), 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).

Turitsyn, S. K.

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(3), 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]

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,” IEEE J. Lightwave Technol. 34(10), 2459–2466 (2016).
[Crossref]

O. V. Yushkoa, A. A. Redyuk, M. P. Fedoruk, and S. K. Turitsyn, “Coherent Soliton Communication Lines,” Exper. Theor. Phys. 119(5), 787–794 (2014).
[Crossref]

E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38(20), 4186–4188 (2013).
[Crossref] [PubMed]

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

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-Plank equation approach to the description of soliton statistics in optical fiber transmission systems,” J. Opt. Soc. Am. B 22(4), 743 (2005).
[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Non-Gaussian Statistics of an Optical Soliton in the Presence of Amplified Spontaneous Emission,” Opt. Lett. 28(21), 2097–2099 (2003).
[Crossref] [PubMed]

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601 (2001).

Turitsyna, E. G.

Wahls, S.

Wai, P. K. A.

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,” IEEE J. Lightwave Technol. 35(9), 1542–1550 (2017).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Yakushev, D. A.

Yousefi, M.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear Frequency Division Multiplexed Transmissions based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Yousefi, M. I.

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

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

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

Yushkoa, O. V.

O. V. Yushkoa, A. A. Redyuk, M. P. Fedoruk, and S. K. Turitsyn, “Coherent Soliton Communication Lines,” Exper. Theor. Phys. 119(5), 787–794 (2014).
[Crossref]

Zakharov, V. E.

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Figures (7)

Fig. 1
Fig. 1

DSP structure and experimental setup. AWG: arbitrary waveform generator; OBPF: optical band-pass filter; PC: polarization controller. NZ-DSF: non-zero dispersion shifted fiber.

Fig. 2
Fig. 2

4GBaud square 16-QAM transmission on eigenvalue λ rt =0.3j. (a) Received signal distributions of q ˜ ( λ rt ), b( λ rt ) and b( λ rt ) after the LMMSE filter and their corresponding BER performance versus transmission distance. (b) BER versus the number of training symbols for estimating the LMMSE filter coefficients for an 800-km system.

Fig. 3
Fig. 3

BER versus distance of 4GBaud transmission on λ rt =0.3j for (a) different 4 bit/symbol modulation formats (received signal distributions after 1000 km transmission are shown in the insets); (b) BER versus distance for 4GBaud 16-APSK transmissions with different d between the 2 rings.

Fig. 4
Fig. 4

Transmissions of 16-APSK (with d = 3) signals on λ rt =0.3j with different baud rates; (a) Time-domain waveform Q( τ ); (b) Frequency spectra; (c) BER versus transmission distance; (d) BER versus launched power for specific transmission distances

Fig. 5
Fig. 5

BER versus distance of 4 GBaud transmissions of joint 16-APSK spectral amplitude modulation and eigenvalue modulation on λ rt ={0.075+0.3j,0.075+0.3j} and λ rt ={0.3j,0.45j}, frequency spectra and signal distributions over 600 km are shown as insets.

Fig. 6
Fig. 6

(a) A 1-soliton pulse with eigenvalue λ rt = λ R +j λ I where λ R 0 will drift in time after propagation. (b) Pre-shifting the original pulse at the transmitter can enhance the amount of acceptable pulse drifting, thus increase the transmission distance. (c) 16-APSK constellation on the spectral amplitude q ˜ ( λ rt ) of pre-shifted pulse for Re{ λ rt }>0 and Re{ λ rt }<0.

Fig. 7
Fig. 7

Four GBaud 16-APSK transmission over 1000 km with 2-bit eigenvalue modulation from the set λ rt ={1.5δ+0.3j,0.5δ+0.3j,0.5δ+0.3j,0.5δ+0.3j} (a) BER versus eigenvalue spacing δ between Re{ λ rt } (received eigenvalue distributions shown in inset); (b) signal launched power versus BER; (c) BER comparison between 6 GBaud 16-APSK and 4 GBaud APSK with 2-bit eigenvalue modulation (total 24 Gb/s) system as a function of transmission distance.

Equations (7)

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

dv dt =( jλ q * (t) q(t) jλ )v, v( T 1 ,λ)=( v 1 ( T 1 ,λ) v 2 ( T 1 ,λ) )=( 1 0 ) e jλ T 1
q( t )=2j λ I e j q ˜ ( λ rt ) sech( 2 λ I ( t t 0 ) ) e 2j λ R t
t 0 ( z )=4 λ R z+ 1 2 λ I ln( | q ˜ ( λ rt )| 2 λ I ).
Q= | β 2 | γ T 0 2 q,    τ= T 0 t,    l= T 0 2 | β 2 | z
l max { Δτ T 0 2 4 λ R | β 2 | + T 0 2 8 λ I λ R | β 2 | ln( min{| q ˜ (λ) |} 2 λ I ) Δτ T 0 2 4| λ R β 2 | T 0 2 8 λ I | λ R β 2 | ln( min{| q ˜ (λ) |} 2 λ I ) λ R > 0 λ R < 0
α={ 2 λ I e 2 λ I Δτ max{ | q ˜ (λ) | } 2 λ I e 2 λ I Δτ min{ | q ˜ (λ) | } λ R >0 λ R <0
l max = Δτ T 0 2 2| λ R β 2 | T 0 2 8 λ I | λ R β 2 | ln( max{| q ˜ ( λ ) |} min{| q ˜ ( λ ) |} )