Abstract

An advanced split-step method is employed for the digital backward-propagation (DBP) method using the coupled nonlinear Schrodinger equations for the compensation of inter-channel nonlinearities. Compared to the conventional DBP, cross-phase modulation (XPM) can be efficiently compensated by including the effect of the inter-channel walk-off in the nonlinear step of the split-step method (SSM). While self-phase modulation (SPM) compensation is inefficient in WDM systems, XPM compensation is able to increase the transmission reach by a factor of 2.5 for 16-QAM-modulated signals. The advanced SSM significantly relaxes the step size requirements resulting in a factor of 4 reduction in computational load.

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References

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2009 (7)

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

F. Yaman and G. Li, “Nonlinear Impairment Compensation for Polarization-Division Multiplexed WDM Transmission Using Digital Backward Propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Evaluation of the computational effort for chromatic dispersion compensation in coherent optical PM-OFDM and PM-QAM systems,” Opt. Express 17(3), 1385–1403 (2009).
[CrossRef] [PubMed]

G. Li, “Recent advances in coherent optical communication,” Adv. Opt. Photon. 1(2), 279–307 (2009).
[CrossRef]

E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt. 48(25), F6–F10 (2009).
[CrossRef] [PubMed]

R. Waegemans, S. Herbst, L. Holbein, P. Watts, P. Bayvel, C. Fürst, and R. I. Killey, “10.7 Gb/s electronic predistortion transmitter using commercial FPGAs and D/A converters implementing real-time DSP for chromatic dispersion and SPM compensation,” Opt. Express 17(10), 8630–8640 (2009).
[CrossRef] [PubMed]

X. Liu, F. Buchali, and R. W. Tkach, “Improving the Nonlinear Tolerance of Polarization-Division-Multiplexed CO-OFDM in Long-Haul Fiber Transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009).
[CrossRef]

2008 (5)

2007 (2)

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007).
[CrossRef] [PubMed]

2004 (1)

M. G. Taylor, “Coherent Detection Method using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

2003 (2)

J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003).
[CrossRef]

O. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21(1), 61–68 (2003).
[CrossRef]

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

1990 (1)

Bayvel, P.

Buchali, F.

Carena, A.

Chen, X.

Chen, Z.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Curri, V.

Dou, L.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Forghieri, F.

Fürst, C.

Gao, Y.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Goldfarb, G.

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental Demonstration of Fiber Impairment Compensation Using the Split-Step Finite-Impulse-Response Filtering Method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

Gordon, J. P.

Herbst, S.

Holbein, L.

Holzlohner, R.

Inuzuka, F.

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

Ip, E.

Kahn, J. M.

Kikuchi, K.

Killey, R. I.

Kim, I.

Koga, M.

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

Leibrich, J.

J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003).
[CrossRef]

Li, G.

Li, X.

Liu, X.

Lowery, A. J.

Mateo, E.

Mateo, E. F.

Menyuk, C. R.

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Mollenauer, L. F.

Poggiolini, P.

Rosenkranz, W.

J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003).
[CrossRef]

Sinkin, O.

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Takada, A.

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

Taylor, M. G.

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental Demonstration of Fiber Impairment Compensation Using the Split-Step Finite-Impulse-Response Filtering Method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

M. G. Taylor, “Coherent Detection Method using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

Tkach, R. W.

Waegemans, R.

Watts, P.

Xu, A.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Yaman, F.

F. Yaman and G. Li, “Nonlinear Impairment Compensation for Polarization-Division Multiplexed WDM Transmission Using Digital Backward Propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

Yamazaki, E.

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

Yonenaga, K.

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

Zhang, F.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Zhu, L.

Zweck, J.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

IEEE Photon. J. (1)

F. Yaman and G. Li, “Nonlinear Impairment Compensation for Polarization-Division Multiplexed WDM Transmission Using Digital Backward Propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003).
[CrossRef]

M. G. Taylor, “Coherent Detection Method using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

E. Yamazaki, F. Inuzuka, K. Yonenaga, A. Takada, and M. Koga, “Compensation of interchannel crosstalk induced by optical fiber nonlinearity in carrier phase-locked WDM system,” IEEE Photon. Technol. Lett. 19(1), 9–11 (2007).
[CrossRef]

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental Demonstration of Fiber Impairment Compensation Using the Split-Step Finite-Impulse-Response Filtering Method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

J. Lightwave Technol. (3)

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Opt. Commun. (1)

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Electrical post-compensation of intrachannel nonlinearities in 10GBaud coherent QPSK transmission systems,” Opt. Commun. 282(5), 992–996 (2009).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Other (3)

G. P. Agrawal, Nonlinear fiber optics, (Academic Press, 2007).

V. Oppenheim, and R. V. Schafer, Digital Signal Processing, (Prentice-Hall, 1975).
[PubMed]

J. G. Proakis, Digital Communications, (McGraw-Hill, 2001).

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Figures (6)

Fig. 1
Fig. 1

Block diagram with the implementation of a SSM step. The illustration is also valid for the conventional SSM by removing the first FFT and IFFT blocks and using the filters W m q given by Eq. (10).

Fig. 2
Fig. 2

Performance results for: (A) 12, (B) 24 and (C) 36 channels respectively. For the 12 channels case, results are shown for channel spacing of 50 GHz (solid) and channel spacing of 100 GHz (dotted).

Fig. 4
Fig. 4

Step size for the advanced and conventional implementation of the split-step method: (A) 12, (B) 24 and (C) 36 channels respectively. For the 12 channels case, results are shown for channel spacing of 50 GHz (solid) and channel spacing of 100 GHz (dotted).

Fig. 3
Fig. 3

Performance comparison of CD and XPM compensation for different transmission lengths. Results are obtained for the 24 channel WDM system.

Fig. 5
Fig. 5

12 channel results for 50 GHz and 100 GHz. Q-value (left axis) and step size (right axis) as a function of the optical power per channel.

Fig. 6
Fig. 6

Computation results as a function of the Q-improvement provided by the advanced SSM for XPM compensation. The power values per channel are also shown for each point. Results are shown for the 50 GHz spacing case.

Tables (1)

Tables Icon

Table 1 Summary of results for 10 × 100 km WDM-(16 QAM) systems

Equations (18)

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E m z + α 2 E m + K 1 m E m t + K 2 m 2 E m t 2 + K 3 m 3 E m t 3 + i γ ( | E m | 2 + 2 q m | E q | 2 ) E m = 0 ,
E m ( t , z + h ) = F 1 { F [ E m ( t , z ) ] H m ( ω , h ) } ,
H m ( ω , h ) = exp [ ( i β 2 ( ω m Δ ω ) 2 2 + i β 3 ( ω m Δ ω ) 3 6 ) h ] .
E m ( t , z + h ) = E m ( t , z ) exp ( i ϕ m , S P M + i ϕ m , X P M ) ,
ϕ m , S P M ( t , z + h ) = γ z z + h | E m ( t , z ^ ) | 2 e α z ^ d z ^
ϕ m , X P M ( t , z + h ) = 2 γ z z + h ( q m | E q ( t , z ^ ) | 2 )  e α z ^ d z ^ .
ϕ m , S P M ( t , z + h ) = γ h eff | E m ( t , z ) | 2 ,
ϕ m , X P M ( t , z + h ) = 2 γ h eff q m | E q ( t , z ) | 2 ,
ϕ m ( t , z + h ) = q | E q ( t , z ) | 2 W m q ( h ) ,
W m q ( h ) = { γ h eff    for   q = m 2 γ h eff    for   q m .  
ϕ m , X P M ( t , z + h ) = 2 γ z z + h ( q m | E q ( t d m q z ^ , z ^ ) | 2 )  e α z ^ d z ^ ,
ϕ m , X P M ( ω , z + h ) = 2 γ z z + h [ q m F ( | E q ( t , z ^ ) | 2 ) e i d m q ω z ^ ]  e α z ^ d z ^ ,
ϕ m , X P M ( ω , z + h ) = 2 γ q m F ( | E q ( t , z ^ ) | 2 ) exp ( α h + i d m q ω h ) 1 α + i d m q ω ,
ϕ m ( t , z + h ) = F 1 [ q F   ( | E q ( t , z ) | 2 ) W m q ( ω , h ) ]
W m q ( ω , h ) = { γ h eff                   for   q = m 2 γ e ( α + i d m q ω ) h 1 α + i d m q ω    for   q m .  
W m q ( ω , h ) = { γ h eff e α h / 2                            for   q = m 2 γ e ( α + i d m q ω ) h 1 α + i d m q ω e ( α + i d m q ω ) h / 2    for   q m .  
O P X P M a = n a [ 4 ( M a + P a ) log 2 ( M a + P a ) + ( N + 1 ) ( M a + P a ) + 8 M a ] / M a ,
O P X P M c = n c [ 2 ( M c + P c ) log 2 ( M c + P c ) + ( M c + P c ) + 9 M c ] / M c ,

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