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Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 5953–5959
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Folded digital backward propagation for dispersion-managed fiber-optic transmission

Likai Zhu and Guifang Li  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 5953-5959 (2011)
http://dx.doi.org/10.1364/OE.19.005953


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Abstract

In periodically dispersion managed long-haul transmission systems, waveform distortion is dominated by chromatic dispersion. As a result of the periodic waveform evolution, the nonlinear behavior also repeats itself in every dispersion period. It is shown that, under the weakly nonlinear assumption, nonlinear effects accumulated in a large number (K) of spans can be approximated by nonlinear effects accumulated in a single span with the same dispersion map and K times the nonlinearity. Thus, significant savings in computational load can be achieved in digital compensation of fiber nonlinearity using folded digital backward propagation (DBP). Simulation results show that the required computation for DBP of dispersion managed transoceanic transmission systems can be reduced by up to 2 orders of magnitude with negligible penalty using folded DBP.

© 2011 OSA

1. Introduction

Optical signal is distorted by the joint effects of dispersion and nonlinearity during its propagation in optical fiber. In most of the installed long-haul systems, dispersion is compensated by periodically cascading two or more kinds of fiber with inverse dispersion parameters. With the advent of new inverse dispersion fibers (IDF), wide-band dispersion flatness has been obtained by compensating for both dispersion and dispersion slope while minimizing the total polarization mode dispersion (PMD) [1

1. K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. 3(5), 292–339 (2006). [CrossRef]

]. In the emerging coherent communication systems, dispersion can also be compensated using digital signal processing (DSP) [2

2. C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26(1), 64–72 (2008). [CrossRef]

]. As the technology of dispersion compensation matured, fiber nonlinearity effects including self phase modulation (SPM), cross phase modulation (XPM) and four-wave mixing (FWM) become the limiting factor to further increase the spectral efficiency and transmission distance of long-haul fiber communication systems [3

3. A. Pilipetski, “Nonlinearity management and compensation in transmission systems,” in Proceedings of OFC/NFOEC 2010, paper OTuL5.

5

5. Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004). [CrossRef]

].

In order to mitigate nonlinear effects, methods such as optimized dispersion management [6

6. B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. 5(10), 1250–1253 (1993). [CrossRef]

], large effective area fiber [1

1. K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. 3(5), 292–339 (2006). [CrossRef]

] and new modulation formats [7

7. T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. 21(9), 1933–1943 (2003). [CrossRef]

] have been investigated and employed. In addition to the above methods that mitigate nonlinearity, methods of compensating nonlinear impairments have been proposed. Nonlinear phase shift in dispersion-shifted fibers can be compensated with lumped nonlinear phase de-rotation based on the assumption that the intensity waveform remains unchanged throughout fiber propagation [8

8. K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004). [CrossRef]

]. However, lumped nonlinearity compensation performs poorly where there is significant interaction between nonlinearity and dispersion. Additionally, nonlinearity pre-compensation at the transmitter side was proposed for direct-detection systems [9

9. K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006). [CrossRef]

] and coherent optical OFDM systems [10

10. L. B. Du and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” Proceedings of OFC/NFOEC 2009, paper OTuO1.

].

Enabled by coherent detection, nonlinearity post-compensation via digital backward propagation (DBP) has attracted significant attention [11

11. 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]

13

13. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010). [CrossRef]

]. Recently, experimental demonstration of DBP for polarization-division multiplexed (PDM) wavelength-division multiplexing (WDM) transmission was reported [14

14. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed wdm transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

].

DBP is typically implemented using the split-step method. In order for the split-step method to be accurate, a large number of steps are needed especially for inter-channel nonlinearity compensation of WDM systems, resulting in a very heavy computational load. Some efforts have been devoted to increase the computational efficiency of DBP. In comparison with solving the nonlinear Schrodinger equation (NLSE) for the total field of the WDM signal, solving the coupled NLSE was suggested because it requires a smaller step number and lower sampling rate [15

15. E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]

,16

16. 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]

]. The step number can be further reduced by factorizing the dispersive walk-off effects in the DBP algorithm [17

17. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

] and using variable step size [18

18. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]

].

In this paper, we propose an efficient method of nonlinearity compensation for dispersion-managed fiber-optic transmission systems, taking advantage of the periodic behavior of the optical signal. In section 2, the theory of folded DBP is presented. In section 3, the effectiveness of this method is demonstrated with simulation of a long-haul fiber transmission system. The effect of residual dispersion is also investigated.

2. Theory

We focus on dispersion-managed fiber-optic transmission systems. Without loss of generality, we assume that each fiber span with a length of L is one period of the dispersion map. For long-haul fiber-optic transmission, an optimum power exists as a result of the tradeoff between optical signal to noise ratio (OSNR) and nonlinear impairments. The total nonlinear phase shift at the optimum power level is on the order of 1 radian [19

19. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

]. Therefore, for transoceanic fiber transmission systems which consist of many (>100) amplified spans, the nonlinear effects in each span is weak. As a result, chromatic dispersion is the dominant factor that determines the evolution of the waveform within each span.

We can analyze the nonlinear behavior of the optical signal using a perturbation approach. The NLSE governing the propagation of the optical field, Aj(z,t), in the j th fiber span can be expressed as
Aj(z,t)z=[D+εN(|Aj(z,t)|2)]Aj(z,t),
(1)
where0<z<Lis the propagation distance within each span, Dis the linear operator for dispersion, fiber loss and amplifier gain, N(|Aj(z,t)|2) is the nonlinear operator, ε(to be set to unity) is a parameter indicating that the nonlinear perturbation is small for the reasons given above. The boundary conditions are
A1(0,t)=a(0,t),
(2)
Aj(0,t)=Aj1(L,t) for j2,
(3)
where a(0,t)is the input signal at the beginning of the first span. We assume that the solution of Eq. (1) can be written as,
Aj(z,t)=Aj,l(z,t)+εAj,nl(z,t).
(4)
Substituting Eq. (4) into Eq. (1) and expanding the equation in power series of ε, we have
Aj,l(z,t)zDAj,l(z,t)+ε[Aj,nl(z,t)zDAj,nl(z,t)N(|Aj,l(z,t)|2)Aj,l(z,t)]+O(ε2)=0,
(5)
Equating to zero the successive terms of the series, we have
Aj,l(z,t)z=DAj,l(z,t),
(6)
Aj,nl(z,t)z=DAj,nl(z,t)+N(|Aj,l(z,t)|2)Aj,l(z,t).
(7)
The boundary conditions are
A1,l(0,t)=a(0,t),
(8)
Aj,l(0,t)=Aj1(L,t) for j2,
(9)
and
Aj,nl(0,t)=0.
(10)
First, we assume that dispersion is completely compensated in each span. As a result, at the end of the first span,
A1,l(L,t)=a(0,t),
(11)
and
A2(0,t)=A1(L,t)=a(0,t)+εA1,nl(L,t),
(12)
where A1,nl(z,t)is the solution of Eq. (7) with j=1. In the second span,
A2,l(z,t)=A1,l(z,t)+εA¯(z,t),
(13)
where the first and second terms are solutions to Eq. (6) with boundary conditions A2,l(0,t)=a(0,t) and A2,l(0,t)=εA1,nl(L,t), respectively, as a result of the principle of superposition. At the end of the second span, because of complete dispersion compensation,
A2,l(L,t)=a(0,t)+εA1,nl(L,t).
(14)
The nonlinear distortion in the second span is governed by Eq. (7) withj=2. Since
|A2,l|2=|A1,l+εA¯|2=|A1,l|2+O(ε),
(15)
the differential equation and the boundary conditions for A2,nl(z,t) and A1,nl(z,t) are identical, so
A2,nl(L,t)=A1,nl(L,t).
(16)
As a result, the optical field at the end of the second span is given by
A2(L,t)=A2,l(L,t)+εA2,nl(L,t)=a(0,t)+ε2A1,nl(L,t).
(17)
That is, the nonlinear distortion accumulated in 2 spans is approximately the same as the nonlinear distortion accumulated in 1 span with the same dispersion map and twice the nonlinearity. It follows that, assuming weak nonlinearity and periodic dispersion management, the optical field after K spans of propagation can be written as
AN(L,t)=a(0,t)+εKA1,nl(L,t),
(18)
which is the solution of the NLSE
Aj(z,t)z=[D+εKN(|Aj(z,t)|2)]Aj(z,t).
(19)
The nonlinear term N(|Aj(z,t)|2) in Eq. (1) is proportional to the fiber nonlinear parameter γ. So the NLSE describing the optical propagation in a fiber span with the same dispersion map and K times the nonlinearity can be written as
Aj(z,t)z=[D+εKN(|Aj(z,t)|2)]Aj(z,t).
(20)
The equivalence of Eqs. (19) and (20) suggests that DBP for K spans can be folded into a single span with K times the nonlinearity. Assuming that the step size for the split-step implementation of DBP is unchanged, the computational load for the folded DBP can be saved by the folding factor K.

The above derivation is based on the assumption that waveform distortion due to fiber nonlinearity and the residual dispersion per span (RDPS) are negligible, and consequently the nonlinear behavior of the signal repeats itself in every span. This assumption is not exactly valid because first, fiber nonlinearity also changes the waveform, and secondly, dispersion is not perfectly periodic if the RDPS is non-zero or the dispersion slope is not compensated. These effects accumulate and as a result, the waveform evolutions are not identical between two spans that are far away from each other.

3. Simulation

The DBP was performed as illustrated in Fig. 1. Without loss of generality, we solved the coupled scalar NLSE with the non-iterative asymmetric split-step Fourier method (SSFM) [13

13. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010). [CrossRef]

]. After matched filtering, phase estimation and clock recovery, the Q-value averages of the WDM channels were estimated.

We first simulated the transmission with full inline dispersion compensation, i.e., RDPS = 0. The Q-value as a function of the launching power is shown in Fig. 3(a)
Fig. 3 (a) Q-value vs. launching power per channel and (b) Q-value at optimum power vs. step number per span with RDPS = 0 ps/nm.
. Without nonlinearity compensation, the maximum Q-value is 10.8 dB. With conventional DBP in all spans, the Q-value is increased to 13.3 dB as a result of nonlinearity compensation. With folded DBP with a folding factor of 140 (i.e., M = 1, K = 140), the maximum Q-value was 13.1 dB. The 0.2 dB Q-value penalty was due to the accumulated nonlinear waveform distortion which reduced the accuracy of nonlinearity compensation. There was almost no penalty when the folding factor was 70 (i.e. M = 2, K = 70).

In the split-step implementation DBP, the step size has to be small enough so that the dispersion and nonlinear effects can be properly de-coupled. In long-haul WDM fiber links, the step size is usually limited by dispersion [17

17. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

,20

20. O. V. 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]

]. Therefore in each fiber span, we use the same number of steps in SLA fiber and IDF fiber, so that the dispersion in each step is approximately the same. Figure 3(b) shows the Q-value as a function of step number per span. The folded DBP method does not require increased step number per span.

4. Conclusion and discussion

In this paper, we proposed an efficient DBP method for periodically dispersion managed fiber system. With periodic dispersion management, the linear and nonlinear behavior of the signal repeats itself in every dispersion period. Taking advantage this periodic behavior, DBP of many fiber spans can be folded into one span. Polarization-mode dispersion does impact the effectiveness of DBP. We expect the effect of PDM on folded DBP to be similar to regular DBP described in [14

14. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed wdm transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

].

We demonstrated the folded DBP at the receiver. However, this method of folding nonlinearity compensation of many spans into one span can also be applied to pre-compensation of fiber nonlinearity at the transmitter. In the current work, each amplified span contains one period of the dispersion map. The folding factor can be further increased by shortening the dispersion period, i.e., each amplified span may consist of several dispersion periods. Folded DBP can also be applied when each dispersion period consists of several amplified spans.

References and links

1.

K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. 3(5), 292–339 (2006). [CrossRef]

2.

C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26(1), 64–72 (2008). [CrossRef]

3.

A. Pilipetski, “Nonlinearity management and compensation in transmission systems,” in Proceedings of OFC/NFOEC 2010, paper OTuL5.

4.

R. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 1018–1026 (1999). [CrossRef]

5.

Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004). [CrossRef]

6.

B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. 5(10), 1250–1253 (1993). [CrossRef]

7.

T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. 21(9), 1933–1943 (2003). [CrossRef]

8.

K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004). [CrossRef]

9.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006). [CrossRef]

10.

L. B. Du and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” Proceedings of OFC/NFOEC 2009, paper OTuO1.

11.

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]

12.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

13.

L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010). [CrossRef]

14.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed wdm transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

15.

E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]

16.

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]

17.

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

18.

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]

19.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

20.

O. V. 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]

21.

K. Nakkeeran, A. B. Moubissi, and P. Tchofo Dinda, “Analytical design of dispersion-managed fiber system with map strength 1.65,” Phys. Lett. A 308(5-6), 417–425 (2003). [CrossRef]

22.

V. A. J. M. Sleiffer, D. van den Borne, M. S. Alfiad, S. L. Jansen, and H. de Waardt, “Dispersion management in long-haul 111-Gb/s POLMUX-RZ-DQPSK transmission systems,” in Proceedings of LEOS Annual Meeting Conference 2009, pp.569–570.

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 7, 2011
Revised Manuscript: February 7, 2011
Manuscript Accepted: February 21, 2011
Published: March 16, 2011

Citation
Likai Zhu and Guifang Li, "Folded digital backward propagation for dispersion-managed fiber-optic transmission," Opt. Express 19, 5953-5959 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5953


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References

  1. K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. 3(5), 292–339 (2006). [CrossRef]
  2. C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26(1), 64–72 (2008). [CrossRef]
  3. A. Pilipetski, “Nonlinearity management and compensation in transmission systems,” in Proceedings of OFC/NFOEC 2010, paper OTuL5.
  4. R. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 1018–1026 (1999). [CrossRef]
  5. Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004). [CrossRef]
  6. B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. 5(10), 1250–1253 (1993). [CrossRef]
  7. T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. 21(9), 1933–1943 (2003). [CrossRef]
  8. K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004). [CrossRef]
  9. K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006). [CrossRef]
  10. L. B. Du and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” Proceedings of OFC/NFOEC 2009, paper OTuO1.
  11. 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]
  12. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]
  13. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010). [CrossRef]
  14. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed wdm transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]
  15. E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]
  16. 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]
  17. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]
  18. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]
  19. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
  20. O. V. 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]
  21. K. Nakkeeran, A. B. Moubissi, and P. Tchofo Dinda, “Analytical design of dispersion-managed fiber system with map strength 1.65,” Phys. Lett. A 308(5-6), 417–425 (2003). [CrossRef]
  22. V. A. J. M. Sleiffer, D. van den Borne, M. S. Alfiad, S. L. Jansen, and H. de Waardt, “Dispersion management in long-haul 111-Gb/s POLMUX-RZ-DQPSK transmission systems,” in Proceedings of LEOS Annual Meeting Conference 2009, pp.569–570.

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