Abstract

Nonlinear impairments induced by the opto-electronic components are one of the fundamental performance-limiting factors in high-speed optical short-reach communications, significantly hindering capacity improvement. This paper proposes to employ a kernel mapping function to map the signals in a Hilbert space to its inner product in a reproducing kernel Hilbert space, which has been successfully demonstrated to mitigate nonlinear impairments in optical short-reach communication systems. The operation principle is derived. An intensity modulation/direct detection system with 1.5-µm vertical cavity surface emitting laser and 10-km 7-core fiber achieving 540.68-Gbps (net-rate 505.31-Gbps) has been carried out. The experimental results reveal that the kernel mapping based schemes are able to realize comparable transmission performance as the Volterra filtering scheme even with a high order.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

2018 (5)

2013 (1)

B. Chen, S. Zhao, P. Zhu, and J. C. Principe, “Quantized kernel recursive least squares algorithm,” IEEE Trans. Neural Netw. Learning Syst. 24(9), 1484–1491 (2013).
[Crossref]

1995 (1)

P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A Practical Discrete Multitone Transceiver Loading Algorithm for Data Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

1991 (1)

J. Platt, “A resource - allocating network for function interpolation,” Neu. Comp. 3(2), 213–225 (1991).
[Crossref]

1950 (1)

N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. Soc. 68(3), 337–404 (1950).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, Wiley, Hoboken, USA, 2002.

Agrellet al., E.

E. Agrellet al., “Information-Theoretic Tools for Optical Communications Engineers,” in Proc. IPC (2018).

Altenhain, L.

Amann, M. C.

Amann, MC

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Aronszajn, N.

N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. Soc. 68(3), 337–404 (1950).
[Crossref]

Bauwelinck, J.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Bhatia, V.

U. Singh, R. Mitra, V. Bhatia, and A. Mishra, “Kernel LMS based Estimation Techniques for Radar Systems,” IEEE Transactions on Aerospace and Electronic Systems, 2019.

Bingham, J. A. C.

P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A Practical Discrete Multitone Transceiver Loading Algorithm for Data Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Chen, B.

B. Chen, S. Zhao, P. Zhu, and J. C. Principe, “Quantized kernel recursive least squares algorithm,” IEEE Trans. Neural Netw. Learning Syst. 24(9), 1484–1491 (2013).
[Crossref]

Chen, G.

Chen, J.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

X. Pang, O. Ozolins, L. Zhang, A. Udalcovs, R. Lin, R. Schatz, U. Westergren, G. Jacobsen, S. Popov, and J. Chen, “Beyond 200 Gbps per Lane Intensity Modulation Direct Detection (IM/DD) Transmissions for Optical Interconnects: Challenges and Recent Developments,” in Proc. OFC (2019).

Chen, X.

Chien, H. C.

J. Zhang, J. Yu, J. Shi, and H. C. Chien, “Digital Dispersion Pre-compensation and Nonlinearity Impairment Pre- and Post-processing for C-band 400G PAM-4 Transmission over SSMF Based on Direct Detection,” in Proc. ECOC (2017).

Chow, J. M.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

Chow, P. S.

P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A Practical Discrete Multitone Transceiver Loading Algorithm for Data Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Cioffi, J. M.

P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A Practical Discrete Multitone Transceiver Loading Algorithm for Data Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Coe, T.

M. Scholten, T. Coe, and J. Dillard, “Continuously-interleaved BCH (CIBCH) FEC delivers best in class NECG for 40G and 100G metro applications,” in Proc. OFC (2010).

Córcoles, A. D.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

Dillard, J.

M. Scholten, T. Coe, and J. Dillard, “Continuously-interleaved BCH (CIBCH) FEC delivers best in class NECG for 40G and 100G metro applications,” in Proc. OFC (2010).

Djupsjöbacka, A.

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

Du, J.

G. Chen, J. Du, L. Sun, W. Zhang, K. Xu, X. Chen, G. T. Reed, and Z. He, “Nonlinear Distortion Mitigation by Machine learning of SVM classification for PAM-4 and PAM-8 modulated optical interconnection,” J. Lightwave Technol. 36(3), 650–657 (2018).
[Crossref]

L. Sun, J. Du, and Z. He, “Machine learning for nonlinearity mitigation in CAP modulated optical interconnect system by using K-nearest neighbour algorithm,” in Proc. ACP (2016).

Engenhardt, K. M.

Fu, S.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Gambetta, J. M.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

Gan, L.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Guo, C.

Harrow, A. W.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

Hashimoto, T.

H. Yamazaki, M. Nagatani, H. Wakita, M. Nakamura, S. Kanazawa, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “160-GBd (320-Gb/s) PAM4 Transmission Using 97-GHz Bandwidth Analog Multiplexer,” IEEE Photonics Technol. Lett. 30(20), 1749–1751 (2018).
[Crossref]

H. Yamazaki, M. Nagatani, H. Wakita, Y. Ogiso, M. Nakamura, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “Transmission of 400-Gbps Discrete Multi-tone Signal Using >100-GHz-Bandwidth Analog Multiplexer and InP Mach-Zehnder Modulator,” in Proc. ECOC (2018).

Havlícek, V.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

He, Z.

G. Chen, J. Du, L. Sun, W. Zhang, K. Xu, X. Chen, G. T. Reed, and Z. He, “Nonlinear Distortion Mitigation by Machine learning of SVM classification for PAM-4 and PAM-8 modulated optical interconnection,” J. Lightwave Technol. 36(3), 650–657 (2018).
[Crossref]

L. Sun, J. Du, and Z. He, “Machine learning for nonlinearity mitigation in CAP modulated optical interconnect system by using K-nearest neighbour algorithm,” in Proc. ACP (2016).

Herbrich, R.

B. Schölkopf, R. Herbrich, A. Smola, and R. Williamson, “A generalized representer theorem,” in Proc. of the Annual Conference on Computational Learning Theory, 416 (2001).

Hong, X.

Hu, W.

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

Huo, J.

Ida, M.

H. Yamazaki, M. Nagatani, H. Wakita, M. Nakamura, S. Kanazawa, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “160-GBd (320-Gb/s) PAM4 Transmission Using 97-GHz Bandwidth Analog Multiplexer,” IEEE Photonics Technol. Lett. 30(20), 1749–1751 (2018).
[Crossref]

H. Yamazaki, M. Nagatani, H. Wakita, Y. Ogiso, M. Nakamura, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “Transmission of 400-Gbps Discrete Multi-tone Signal Using >100-GHz-Bandwidth Analog Multiplexer and InP Mach-Zehnder Modulator,” in Proc. ECOC (2018).

Jacobsen, G.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

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J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

temme, K.

V. Havlíček, A. D. Córcoles, K. temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567(7747), 209–212 (2019).
[Crossref]

Tong, W.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Torfs, G.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Udalcovs, A.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

X. Pang, O. Ozolins, L. Zhang, A. Udalcovs, R. Lin, R. Schatz, U. Westergren, G. Jacobsen, S. Popov, and J. Chen, “Beyond 200 Gbps per Lane Intensity Modulation Direct Detection (IM/DD) Transmissions for Optical Interconnects: Challenges and Recent Developments,” in Proc. OFC (2019).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

Urata, R.

X. Zhou, R. Urata, and H. Liu, “Beyond 1Tb/s datacenter interconnect technology: challenges and solutions,” in Proc. OFC (2019).

Van Kerrebrouck, J.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Van Steenberge, G.

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Van Vaerenbergh, S.

S. Van Vaerenbergh and I. Santamaría, “A comparative study of kernel adaptive filtering algorithms,” in Digital Signal Processing and Signal Processing Education Meeting, 181–186 (2013).

Wakita, H.

H. Yamazaki, M. Nagatani, H. Wakita, M. Nakamura, S. Kanazawa, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “160-GBd (320-Gb/s) PAM4 Transmission Using 97-GHz Bandwidth Analog Multiplexer,” IEEE Photonics Technol. Lett. 30(20), 1749–1751 (2018).
[Crossref]

H. Yamazaki, M. Nagatani, H. Wakita, Y. Ogiso, M. Nakamura, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “Transmission of 400-Gbps Discrete Multi-tone Signal Using >100-GHz-Bandwidth Analog Multiplexer and InP Mach-Zehnder Modulator,” in Proc. ECOC (2018).

Wei, J.

N. Stojanovic, C. Prodaniuc, L. Zhang, and J. Wei, “210/225 Gbit/s PAM-6 transmission with BER bellow KP4-FEC/EFEC and at least 14 dB link budget,” in Proc. ECOC (2018).

L. Zhang, J. Wei, N. Stojanovic, C. Prodaniuc, and C. Xie, “Beyond 200-Gb/s DMT Transmission over 2-km SMF Based on A Low-cost Architecture with Single-wavelength, Single-DAC/ADC and Single-PD,” in Proc. ECOC (2018).

Westergren, U.

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

X. Pang, O. Ozolins, L. Zhang, A. Udalcovs, R. Lin, R. Schatz, U. Westergren, G. Jacobsen, S. Popov, and J. Chen, “Beyond 200 Gbps per Lane Intensity Modulation Direct Detection (IM/DD) Transmissions for Optical Interconnects: Challenges and Recent Developments,” in Proc. OFC (2019).

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

Williamson, R.

B. Schölkopf, R. Herbrich, A. Smola, and R. Williamson, “A generalized representer theorem,” in Proc. of the Annual Conference on Computational Learning Theory, 416 (2001).

Wolf, S.

Xiao, S.

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Xie, C.

L. Zhang, J. Wei, N. Stojanovic, C. Prodaniuc, and C. Xie, “Beyond 200-Gb/s DMT Transmission over 2-km SMF Based on A Low-cost Architecture with Single-wavelength, Single-DAC/ADC and Single-PD,” in Proc. ECOC (2018).

Xu, K.

Yamazaki, H.

H. Yamazaki, M. Nagatani, H. Wakita, M. Nakamura, S. Kanazawa, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “160-GBd (320-Gb/s) PAM4 Transmission Using 97-GHz Bandwidth Analog Multiplexer,” IEEE Photonics Technol. Lett. 30(20), 1749–1751 (2018).
[Crossref]

H. Yamazaki, M. Nagatani, H. Wakita, Y. Ogiso, M. Nakamura, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “Transmission of 400-Gbps Discrete Multi-tone Signal Using >100-GHz-Bandwidth Analog Multiplexer and InP Mach-Zehnder Modulator,” in Proc. ECOC (2018).

Yin, X.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

Yu, C.

Yu, J.

J. Zhang, J. Yu, J. Shi, and H. C. Chien, “Digital Dispersion Pre-compensation and Nonlinearity Impairment Pre- and Post-processing for C-band 400G PAM-4 Transmission over SSMF Based on Direct Detection,” in Proc. ECOC (2017).

Zhang, J.

Zhang, L.

J. Van Kerrebrouck, X. Pang, O. Ozolins, R. Lin, A. Udalcovs, L. Zhang, H. Li, S. Spiga, M. C. Amann, L. Gan, M. Tang, S. Fu, R. Schatz, G. Jacobsen, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, J. Chen, and X. Yin, “High-speed PAM4-based optical SDM interconnects with directly modulated long-wavelength VCSEL,” J. Lightwave Technol. 37(2), 356–362 (2019).
[Crossref]

L. Zhang, X. Hong, X. Pang, O. Ozolins, A. Udalcovs, R. Schatz, C. Guo, J. Zhang, F. Nordwall, K. M. Engenhardt, U. Westergren, S. Popov, G. Jacobsen, S. Xiao, W. Hu, and J. Chen, “Nonlinearity-aware 200 Gbit/s DMT transmission for C-band short-reach optical interconnects with a single packaged electro-absorption modulated laser,” Opt. Lett. 43(2), 182–185 (2018).
[Crossref]

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

L. Zhang, O. Ozolins, R. Lin, A. Udalcovs, X. Pang, L. Gan, R. Schatz, A. Djupsjöbacka, J. Mårtensson, U. Westergren, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, W. Hu, S. Xiao, and J. Chen, “Kernel Adaptive Filtering for Nonlinearity-Tolerant Optical Direct Detection Systems,” in Proc. ECOC (2018).

L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

L. Zhang, J. Wei, N. Stojanovic, C. Prodaniuc, and C. Xie, “Beyond 200-Gb/s DMT Transmission over 2-km SMF Based on A Low-cost Architecture with Single-wavelength, Single-DAC/ADC and Single-PD,” in Proc. ECOC (2018).

N. Stojanovic, C. Prodaniuc, L. Zhang, and J. Wei, “210/225 Gbit/s PAM-6 transmission with BER bellow KP4-FEC/EFEC and at least 14 dB link budget,” in Proc. ECOC (2018).

X. Pang, O. Ozolins, L. Zhang, A. Udalcovs, R. Lin, R. Schatz, U. Westergren, G. Jacobsen, S. Popov, and J. Chen, “Beyond 200 Gbps per Lane Intensity Modulation Direct Detection (IM/DD) Transmissions for Optical Interconnects: Challenges and Recent Developments,” in Proc. OFC (2019).

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Zhou, X.

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L. Zhang, J. Van Kerrebrouck, O. Ozolins, R. Lin, X. Pang, A. Udalcovs, S. Spiga, MC Amann, G. Van Steenberge, L. Gan, M. Tang, S. Fu, R. Schatz, S. Popov, D. Liu, W. Tong, G. Torfs, J. Bauwelinck, X. Yin, S. Xiao, and J. Chen, “Experimental Demonstration of 503.61-Gbit/s DMT over 10-km 7-Core Fiber with 1.5-µm SM-VCSEL for Optical Interconnects,” in Proc. ECOC (2018).

B. Schölkopf, R. Herbrich, A. Smola, and R. Williamson, “A generalized representer theorem,” in Proc. of the Annual Conference on Computational Learning Theory, 416 (2001).

A. Udalcovs, R. Lin, O. Ozolins, L. Gan, L. Zhang, X. Pang, R. Schatz, A. Djupsjöbacka, M. Tang, S. Fu, D. Liu, W. Tong, S. Popov, G. Jacobsen, and J. Chen, “Inter-core crosstalk in multicore fibers: impact on 56-Gbaud/λ/Core PAM-4 transmission,” in Proc. ECOC (2018).

H. Yamazaki, M. Nagatani, H. Wakita, Y. Ogiso, M. Nakamura, M. Ida, T. Hashimoto, H. Nosaka, and Y. Miyamoto, “Transmission of 400-Gbps Discrete Multi-tone Signal Using >100-GHz-Bandwidth Analog Multiplexer and InP Mach-Zehnder Modulator,” in Proc. ECOC (2018).

N. Stojanovic, C. Prodaniuc, L. Zhang, and J. Wei, “210/225 Gbit/s PAM-6 transmission with BER bellow KP4-FEC/EFEC and at least 14 dB link budget,” in Proc. ECOC (2018).

L. Zhang, J. Wei, N. Stojanovic, C. Prodaniuc, and C. Xie, “Beyond 200-Gb/s DMT Transmission over 2-km SMF Based on A Low-cost Architecture with Single-wavelength, Single-DAC/ADC and Single-PD,” in Proc. ECOC (2018).

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of kernel mapping.
Fig. 2.
Fig. 2. Geographic interpretation of the signal processing space with kernel mapping.
Fig. 3.
Fig. 3. (a) Experimental setup, and (b) measured system frequency response.
Fig. 4.
Fig. 4. The BER versus training overhead NL
Fig. 5.
Fig. 5. BER for optical back-to-back case as a function of the received optical power.
Fig. 6.
Fig. 6. BER after 10km 7-core fiber for different cores with 7-dBm received optical power.

Tables (1)

Tables Icon

Table 1. Comparison of number of multiplications of Volterra filtering, KLMS and KRLS algorithm.

Equations (36)

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

κ ( x i , x j ) = p = 1 λ p θ p ( x i ) θ p ( x j ) .
φ ( . ) = [ λ 1 θ 1 ( . ) , λ 2 θ 2 ( . ) , ] | H F ,
κ ( x i , x j ) = φ ( x i ) T φ ( x j ) .
κ ( x i , p , x i , q ) = exp ( ( x i , p x i , q ) 2 / 2 σ 2 ) = exp ( ( x i , p 2 + x i , q 2 ) / 2 σ 2 ) m = 0 ( x i , p x i , q ) m m ! σ m = m = 0 { exp ( x i , p 2 2 σ 2 ) exp ( x i , q 2 2 σ 2 ) 1 m ! 1 m ! x i , p m σ m / 2 x i , q m σ m / 2 } , = φ ( x i , p ) T φ ( x i , q )
φ ( x i , p ) = exp ( x i , p 2 2 σ 2 ) ( 1 , 1 1 ! x i , p σ , 1 2 ! x i , p 2 σ 2 , ) .
| | f ρ T φ ( . ) | | < ε , ε > 0.
e = d k = 1 N w ( k ) φ ( x ( k ) ) .
e ( k ) = d ( k ) w ( k 1 ) T φ ( x ( k ) ) ,
w ( k ) = w ( k 1 ) + η e ( k ) φ ( x ( k ) ) ,
w ( k ) = w ( k     1 ) + η e ( k ) φ ( x ( k ) ) = w ( k     2 ) + η ( e ( k     1 ) φ ( x ( k     1 ) + e ( k ) φ ( x ( k ) ) ) = = η i = 1 k e ( i ) φ ( x ( i ) ) ( w ( 0 ) = 0 ) .
w ( k ) T φ ( x ) = ( η i = 1 k e ( i ) φ ( x ( i ) ) ) φ ( x ) = η i = 1 k e ( i ) κ ( x ( i ) , x ) .
f k 1 ( x ( k ) ) = η i = 1 k 1 e ( i ) κ ( x ( i ) , x ( k ) ) ,
e ( k ) = d ( k ) f k 1 ( x ( k ) ) ,
f k ( ) = f k 1 ( ) + η e ( k ) κ ( x ( k ) , ) .
{ D ( k ) = [ d ( 1 ) , d ( 2 ) , , d ( k ) ] T Φ ( k ) = [ φ ( x ( 1 ) ) , φ ( x ( 2 ) ) , , φ ( x ( k ) ) ] T ,
w ( k ) = [ η I + Φ ( k ) Φ ( k ) T ] 1 Φ ( k ) D ( k ) .
w ( k ) = Φ ( k ) [ η I + Φ ( k ) T Φ ( k ) ] 1 D ( k ) .
P ( k ) 1 = [ P ( k 1 ) 1 Φ ( k 1 ) T φ ( x ( k ) ) φ ( x ( k ) ) T Φ ( k 1 ) η + φ ( x ( k ) ) T φ ( x ( k ) ) ] .
{ z ( k ) = P ( k 1 ) Φ ( k 1 ) T φ ( x ( k ) ) r ( k ) = η + φ ( x ( k ) ) T φ ( x ( k ) ) z ( k ) T Φ ( k 1 ) φ ( x ( k ) ) ,
P ( k ) = r ( k ) 1 [ P ( k 1 ) r ( k ) + z ( k ) z ( k ) T z ( k ) z ( k ) T 1 ] .
β ( k ) = r ( k ) 1 [ P ( k 1 ) r ( k ) + z ( k ) z ( k ) T z ( k ) z ( k ) T 1 ] [ D ( k 1 ) x ( k ) ] . = [ β ( k 1 ) z ( k ) r ( k ) 1 e ( k ) r ( k ) 1 e ( k ) ]
f k 1 ( x ( k ) ) = φ ( x ( k ) ) T Φ ( k 1 ) β ( k 1 ) = i k 1 β i ( k 1 ) κ ( x ( i ) , x ( k ) ) ,
e ( k ) = d ( k ) f k 1 ( x ( k ) ) ,
f k ( ) = f k 1 ( ) + r ( k ) 1 [ κ ( x ( k ) , ) i = 1 k 1 z i ( k ) κ ( x ( i ) , ) ] e ( k ) .
x = i = 1 a i x i b ,
i   =   m   +   1 n a i 2 0 , | m , n R   +   , m < n   &   m , n   +   ,
i   =   1 n a i 2 < ,
min | | d ( k ) W ( k ) x ( k ) | | ,
K = [ κ ( x 1 , x 1 ) κ ( x 1 , x m ) κ ( x m , x 1 ) κ ( x m , x m ) ] , | m N ,
i = 1 m j = 1 m a i a j κ ( x i , x j ) 0 , | m N , x 1 x m H , a 1 a m R .
κ ( x i , x j ) = exp ( | | x i x j | | 2 / 2 σ 2 ) .
f = i = 1 n a i κ ( x i , ) , g = j = 1 m β j κ ( x j , ) , n , m N ,
f   ,   g = i = 1 n j = 1 m a i β j κ ( x i , x j ) .
f   ,   κ ( , x j ) = i = 1 n a i κ ( x i , x j ) = f ( x j ) .
min | | d ( k ) f ( x ( k ) ) | | .
f = k = 1 N a k κ ( , x ( k ) ) ,

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