## Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation

Optics Express, Vol. 16, Issue 20, pp. 16124-16137 (2008)

http://dx.doi.org/10.1364/OE.16.016124

Acrobat PDF (607 KB)

### Abstract

The impact of cross-phase modulation (XPM) and four-wave mixing (FWM) on electronic impairment compensation via backward propagation is analyzed. XPM and XPM+FWM compensation are compared by solving, respectively, the backward coupled Nonlinear Schrödinger Equation (NLSE) system and the total-field NLSE. The DSP implementations as well as the computational requirements are evaluated for each post-compensation system. A 12×100 Gb/s 16-QAM transmission system has been used to evaluate the efficiency of both approaches. The results show that XPM post-compensation removes most of the relevant source of nonlinear distortion. While DSP implementation of the total-field NLSE can ultimately lead to more precise compensation, DSP implementation using the coupled NLSE system can maintain high accuracy with better computation efficiency and low system latency.

© 2008 Optical Society of America

## 1. Introduction

1. D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. **9**, 121–129 (1991). [CrossRef]

2. S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. **14**, 243–249 (1996). [CrossRef]

4. J. Leibrich, C. Wree, and W. Rosenkranz, “CF-RZ-DPSK for suppression of XPM on dispersion-managed long-haul optical WDMtransmission on standard single-mode fiber,” IEEE Photon. Technol. Lett. **14**, 155–157 (2002). [CrossRef]

5. S. L. Woodward, S. Huang, M.D. Feuer, and M. Boroditsky, “Demonstration of an electronic dispersion compensation in a 100-km 10-Gb/s ring network,” IEEE Photon. Technol. Lett. **15**, 867–869 (2003). [CrossRef]

6. K. Roberts, C. Li, L. Strawczynski, M. OSullivan, and I. Hardcatle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. **18**, 403–405 (2006). [CrossRef]

7. 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**, 9–11 (2007). [CrossRef]

8. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. 0, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express **16**, 880–889 (2008). [CrossRef] [PubMed]

8. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. 0, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express **16**, 880–889 (2008). [CrossRef] [PubMed]

*z*-reversed nonlinear Schrödinger equation (NLSE). Although this method has been proven effective, reducing the number of computations required and its impact on the system latency is desirable for the eventual implementation of the post-compensation method.

*Q*-factor, optimum launching power and channel spacing will be analyzed in detail. Additionally, generalized conclusions about the DSP computational requirements for each compensation scheme will be presented for WDM systems.

## 2. Theory of backward propagation compensation and digital implementation

### 2.1. Backward propagation equations and numerical procedure

*EÊ*be the envelope of the

_{m}*m*th-channel field where

*m*∈

*I*,

*I*={1,2, …,

*N*} and

*N*is the total number of WDM channels. By rewriting the field expression as,

*E*=

_{m}*Ê*exp(

_{m}*imΔωt*), where Δ

*f*=Δ

*ω*/2

*π*is the channel spacing, the expression of the full optical field can be expressed as,

*E*=∑

*. The total-field back propagation equation, i.e T-NLSE, is given by [9],*

_{m}E_{m}*β*represent the

_{j}*j*th-order dispersion, α is the absorption coefficient,

*γ*is the nonlinear parameter and

*t*is the retarded time [9]. Equation (1) governs the backward propagation of the total field including second and third order dispersion, SPM, XPM and FWM compensation.

*E*into Eq. (1), expanding the |

*E*|

^{2}term and neglecting the so-calledFWM terms, that is,

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

11. O. V. Sinkin, J. Holzlöhner, C. Zweck, and Menyuk,” “Optimization of the split-step Fourier method in modelling optical-fiber communications system,” J. Light-wave Technol. **21**, 61–68 (2003). [CrossRef]

8. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. 0, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express **16**, 880–889 (2008). [CrossRef] [PubMed]

*D*(

*x*)=𝓕

^{-1}[

*H*𝓕(

*x*)],

*P*(

*x*)=|

*x*|

^{2}and E(x)=exp(iγxh), where the transfer function H for fiber dispersion and loss is given by,

*ω*being the angular frequency and

*h*the step size.

*M*spans, the iterative procedure for SSFM backward propagation is given by the block diagram shown in Fig. 3 where

*n*is the number of steps per span (note that an attenuation element has been introduced to compensate the amplification stages).

_{s}#### 2.2. SSFM Step size and digital implementation efficiency

*n*), and hence, on the step size (

_{s}*h*=

*L*/

*n*) where

_{s}*L*is the span length. The SSFM accuracy depends fundamentally on the mutual influence of dispersion and nonlinearity within the step length. Due to the nature of the dispersion and nonlinearity operators, the step size has to make sure that: (1) the nonlinear phase shift is small enough to preserve the accuracy of the dispersion operation and (2) the optical power fluctuations due to dispersion effects are small enough to preserve the accuracy of the nonlinear operation.

*L*

_{nl}, the walk-off length

*L*

_{wo}and the four-wave mixing length

*L*

_{fwm}. The nonlinear and walk-off lengths can be defined, for a multi-channel system, as follows,

*P*=∑

_{T}*|*

_{m}*E*|

_{m}^{2}is the total launched power and

*B*is the symbol rate (effectively the inverse of the pulse width). The nonlinear length has been defined as the length after which an individual channel experiences a 1 radian phase shift due to SPM and XPM. The walk-off length is defined as the distance after which the relative delay of pulses from the edge channels is equal to the pulse width. The above characteristic lengths are well known [12] and widely used to qualitatively describe the optical field behavior through fiber propagation However, when FWM is considered, the nonlinear and walk-off lengths are not enough to qualitatively identify the range where the

*fastest*field fluctuations take place. In order to identify the

*fastest*field fluctuations due to FWM, the total optical field should be rewritten as

*E*=∑

*exp(*

_{m}E_{m}*ik*) where

_{m}z*k*is the linear propagation constant of the

_{m}*m*th-channel. By following the same procedure as for Eq. (2), the nonlinear term, now including FWM, can be expressed as follows for the

*m*th-channel,

*l*=

*r*+

*s*-

*m*, [

*m*,

*r*,

*s*] ∈

*I*and

*r*≠

*s*≠

*m*. The first condition neglects fast time-oscillating terms (

*frequency matching*). The second condition forces the new generated waves to lay within the WDM band. Finally, the third condition excludes SPM and XPM terms.

*δk*is the phase mismatch parameter, given by,

_{rslm}*fastest z*-fluctuations for the

*m*th-channel, let us set

*r*=1 and

*s*=

*N*corresponding to the indexes of the edge channels. By maximizing Eq. (5), the expression for the maximum phase-mismatch is given by,

13. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. **12**, 489–397 (2000). [CrossRef]

11. O. V. Sinkin, J. Holzlöhner, C. Zweck, and Menyuk,” “Optimization of the split-step Fourier method in modelling optical-fiber communications system,” J. Light-wave Technol. **21**, 61–68 (2003). [CrossRef]

*τ*and

_{r}*ϕ*

_{fwm}represent, respectively, the maximum inter-pulse relative delay (with respect to the pulse width) and the maximum phase-mismatch allowed within one step.

*C*represents total number of operations. Note that the the XPM characteristic step size dictates the number of operations

*per channel*; thus, the total number of operations for XPM compensation is multiplied in Eq. (9) by the total number of channels.

*τ*

_{fwm}∝

*C*

_{fwm}). However, for XPM compensation via C-NLSE, the system latency (

*τ*

_{xpm}) is proportional to the number of operations per step and per channel; hence, the following relationship can be obtained for the processing latency,

## 3. Simulation results

*B*=25 Gbaud. The total transmission distance is 1000 km, divided into

*M*=10 spans of 100 km. Two channel spacing values will be considered according to the ITU-T standards for WDM systems, i.e. Δ

*f*=50,100 GHz. The schematic of the transmission system is shown in Fig. 4 where post-compensation is performed in the digital domain. The 16-QAM/WDM signals are transmitted over multiple amplified fiber spans; after transmission, the received signals are mixed in a 90° optical hybrid with a set of co-polarized local oscillators (LOs). The in-phase and quadrature components of each WDM channel are obtained by balanced photo-detectors.

*f*=50/100 GHz) and the 12 channels are combined to reconstruct a final optical waveform of 800/1600 GHz bandwidth (corresponding to an upsampled transmitted bandwidth of 600/1200 GHz). The details of the DSP implementation of the up-sampling procedure can be found in [8

**16**, 880–889 (2008). [CrossRef] [PubMed]

*β*

_{2}=-5.63 ps

^{2}/km (

*D*=16 ps/km/nm),

*β*

_{3}=0.083 ps

^{3}/km,

*α*=0.046 km

^{-1}(0.2 dB/km) and

*γ*=1.46W

^{-1}km

^{-1}. The signal is amplified after each span with an EDFA with a noise figure of 5 dB. For simplicity, the laser phase noise is neglected.

*Q*-factor values for, respectively, back-to-back, dispersion compensation only and XPM compensation via C-NLSE. The results are given for one of the central channels (which presents the highest inter-channel nonlinear distortion). The

*Q*-factor is obtained from the constellation diagram as in [16

16. A. J. Lowery, L. B. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. **25**, 131–138 (2007). [CrossRef]

### A. Results for Δf=50 GHz

*Q*-factor values as a function of to the total input power for different step sizes.

*Q*-factor. Above the optimum power, the non-deterministic nonlinear distortion due to signal-ASE beat starts to offset the SNR growth with launching power.

*Q*-factor as well as on the optimum power. Lines in blue correspond to step sizes close to the respective characteristic step size. In this case, the results show that the optimum

*Q*-factor (i.e. for the characteristic step size and for the optimum power) increases by approximately 1 dB when FWM is compensated, indicating in a very small impact of FWM in the optimum operation point. Likewise, the compensation of FWM allows to increase the total launching power by 1 dBm. The improved performance with the correction ofFWM requires approximately 1.4 times more computations than XPM compensation and incurs a latency 17 times larger than for XPM compensation according to Eqs. (9 and 10).

*Q*-factor for XPM and FWM compensation can be compared for the XPM characteristic step size. For

*P*=12 dBm and

_{T}*h*=3 km the

*Q*-factor is reduced by 7 dB in the FWM case, which confirms the great distortion that is induced due to a wrong estimation of FWM, even when XPM is properly compensated.

*Q*-factor for step sizes below the characteristic step size. Such behavior (see also Fig. 9) sets the values of

*τ*and

_{r}*ϕ*

_{fwm}, which provide an optimum ratio between the

*Q*-factor and the computational load. Two patterns depicted in Fig. 7 are worth mentioning. First, a flat transition with respect to the step size is observed in the high

*Q*-factor region. This flatness indicates the independence of the step size with the power, which confirms that linear dispersion effects, such as walk-off and dispersive phase-mismatch, limit the step size values. On the other hand, in the upper right sides of the maps, a diagonal pattern of the iso-

*Q*s can be observed. This pattern show a correlation between the step size and the optical power, suggesting that nonlinear effects start to be compensated. Those diagonal transitions do not appear in the left hand side of the map, where nonlinearity does not play a significant role and the -factor grows with the power regardless of the step size.

#### B. Results for Δf=100 GHz

*Q*-factor values as a function of the total input power for different step sizes, including the results for the characteristic step size of the system (lines in blue).

*h*

_{wo}=1.54 km and

*h*

_{fwm}=44 m. The values

*τ*=3/2 and

_{r}*ϕ*

_{fwm}=3 are preserved, confirming that those parameters are independent on the WDM system. Note that the XPM and FWM characteristic step sizes, are respectively halved and quartered with respect to the 50 GHz channel spacing, according to the scaling of the walk-off and phase mismatch with the channel spacing. To illustrate this, Fig. 9 shows the

*Q*-factor with respect to the step size for both channel spacings. Here, the above mentioned asymptotic behavior of the

*Q*-factor is observed as well as the characteristic step size locations. Both Figs. 8 and 9 show that FWM has a negligible influence for Δ

*f*=100 GHz, having effectively the same

*Q*-factor for XPM and FWM compensation. The larger channel spacing gives rise to a higher phase-mismatch, which rapidly averages to zero the contribution of the FWM products. Because of this small contribution, the optimum power is also effectively equal for both compensation schemes. Regarding DSP requirements, and according to Eqs. (9 and 10) the correction of FWM requires approximately 2.8 times more computations than XPM compensation. In addition, FWM compensation incurs a latency 34 times larger than XPM compensation. The figure 10 show the

*Q*-factor map for the 100 GHz channel spacing.

*f*=50 GHz, confirming the well known reduction of both XPM and FWM effects when the channel spacing grows [17

17. M. Shtaif, M. Eiselt, and L. D. Garret, “Cross-phase modulation distortion measurements in multispan WDM systems,” IEEE Photon. Technol. Lett. **12**, 88–90 (2000). [CrossRef]

*Q*-factor values, which are increased 2.4 and 1.5 dB respectively for the XPM and FWM compensation case (as it is also depicted in Fig. 9).

#### C. Enhanced DSP implementation for XPM compensation.

*T*=

_{m}*d*/2, where

_{m}h_{xpm}*d*=2

_{m}*πβ*Δ

_{2}m*f*is the walk-off parameter. Therefore, a DSP delay operator will shift each channel data array by

*K*samples, where

_{m}*K*=[

_{m}*S*/

_{r}*T*] being

_{m}*S*the sampling rate and [

_{r}*x*] the nearest integer of

*x*.

*Q*-factor for the DSP implementation of Fig.(2) with dispersion operators and the same implementation with delay operators. The results show that almost no penalty is incurred by using delay operators indicating that this approach can be applied to reduce the number of operations required for XPM compensation. Since the delay operators do not contribute to the total number of operations, the ratio

*C*

_{fwm}/

*C*

_{xpm}is now increased by a factor of 3/2 according to the elimination of one dispersion operator. Consequently, for 50(100) GHz channel spacing, XPM compensation via C-NLSE requires 2.1(4.2) times less operations than FWM compensation via T-NLSE.

**16**, 880–889 (2008). [CrossRef] [PubMed]

## 4. Discussion

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

### Modulation format

19. J. M. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. **10**, 259–272 (2004). [CrossRef]

20. J. Wang and K. Petermann, “Small signal analysis for dispersive optical fiber communication systems,” J. Lightwave Technol. **10**, 96–100 (1992). [CrossRef]

#### Channel spacing

14. G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. **19**, 2015–2017 (2007). [CrossRef]

#### Number of WDM channels

*Mean Field Approach*(MFA) [21

21. T. Yu, W. M. Reimer, V. S. Grigoryan, and C. R. Menyuk, “A mean field approach for simulating wavelength-division multiplexed systems,” IEEE Photon. Technol. Lett. **12**, 443–445 (2000). [CrossRef]

*z*-variations of the channels outside the effective bandwidth. To test this approach, we monitored the received

*Q*-factor for one of the central channels, in the XPM compensation case, by reducing the number of channels included in the C-NLSE (i.e. sequentially removing the most separated channels). For the 100 GHz channel spacing and with

*P*=13 dBm, we observed that the received

_{T}*Q*-factor is consistently reduced as the effective bandwidth is reduced. The fact that no convergence is observed, means that all the channels contribute to XPM and hence, the MFA is inefficient in this system. This can be explained by considering the fact that, even though the walk-off effect averages the XPM interaction between well-separated channels, there is a cascaded effect between channels which

*propagates*the XPM distortion from edge to central channels. Such effect, might require an increase in the effective bandwidth, even for largely spaced channels.

## 5. Conclusions

*Q*-factor with respect to FWM compensation. The different physical restrictions that are imposed for the FWM and XPM characteristic step sizes give rise to a digital XPM post-compensation that requires 1.4 times less number of operations and performs 17 times faster in terms of latency. For the Δ

*f*=100 GHz case, almost no improvement in the

*Q*-factor is observed when FWM is compensated. In this case, the total-field NLSE solution requires almost 3 times more computations with a latency 34 times larger than the coupled NLSE scheme for XPM compensation. Finally, the possibility of modeling dispersive walk-off (as a pure delay) in the XPM compensation scheme, allows an additional reduction of the number of operations by a factor of 3/2.

## References and links

1. | D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. |

2. | S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. |

3. | C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” J. Lightwave Technol. |

4. | J. Leibrich, C. Wree, and W. Rosenkranz, “CF-RZ-DPSK for suppression of XPM on dispersion-managed long-haul optical WDMtransmission on standard single-mode fiber,” IEEE Photon. Technol. Lett. |

5. | S. L. Woodward, S. Huang, M.D. Feuer, and M. Boroditsky, “Demonstration of an electronic dispersion compensation in a 100-km 10-Gb/s ring network,” IEEE Photon. Technol. Lett. |

6. | K. Roberts, C. Li, L. Strawczynski, M. OSullivan, and I. Hardcatle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. |

7. | 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. |

8. | X. Li, X. Chen, G. Goldfarb, E. Mateo, I. 0, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express |

9. | G. P. Agrawal, |

10. | J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. |

11. | O. V. Sinkin, J. Holzlöhner, C. Zweck, and Menyuk,” “Optimization of the split-step Fourier method in modelling optical-fiber communications system,” J. Light-wave Technol. |

12. | T. Schneider, |

13. | G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. |

14. | G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. |

15. | X. Liu and D. A. Fishman, “A fast and reliable algorithm for electronic pre-equalization of SPM and chromatic dispersion,” in |

16. | A. J. Lowery, L. B. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. |

17. | M. Shtaif, M. Eiselt, and L. D. Garret, “Cross-phase modulation distortion measurements in multispan WDM systems,” IEEE Photon. Technol. Lett. |

18. | P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature |

19. | J. M. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. |

20. | J. Wang and K. Petermann, “Small signal analysis for dispersive optical fiber communication systems,” J. Lightwave Technol. |

21. | T. Yu, W. M. Reimer, V. S. Grigoryan, and C. R. Menyuk, “A mean field approach for simulating wavelength-division multiplexed systems,” IEEE Photon. Technol. Lett. |

**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: May 30, 2008

Revised Manuscript: June 24, 2008

Manuscript Accepted: June 25, 2008

Published: September 26, 2008

**Citation**

Eduardo Mateo, Likai Zhu, and Guifang Li, "Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation," Opt. Express **16**, 16124-16137 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16124

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### References

- D. Marcuse, A. R. Chraplyvy and R. W. Tkach, "Effect of fiber nonlinearity on long-distance transmission," J. Lightwave Technol. 9,121-129 (1991). [CrossRef]
- S. Watanabe and M. Shirasaki, "Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation," J. Lightwave Technol. 14, 243-249 (1996). [CrossRef]
- C. Kurtzke, "Suppression of fiber nonlinearities by appropriate dispersion management," J. Lightwave Technol. 5, 1250-1253 (1993).
- J. Leibrich, C. Wree and W. Rosenkranz, "CF-RZ-DPSK for suppression of XPM on dispersion-managed longhaul optical WDMtransmission on standard single-mode fiber," IEEE Photon. Technol. Lett. 14, 155-157 (2002). [CrossRef]
- S. L. Woodward, S. Huang, M.D. Feuer, and M. Boroditsky, "Demonstration of an electronic dispersion compensation in a 100-km 10-Gb/s ring network," IEEE Photon. Technol. Lett. 15, 867-869 (2003). [CrossRef]
- K. Roberts, C. Li, L. Strawczynski, M. OSullivan, and I. Hardcatle, "Electronic precompensation of optical nonlinearity," IEEE Photon. Technol. Lett. 18, 403-405 (2006). [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, 9-11 (2007). [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, 880-889 (2008). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear fiber optics, (Academic Press, 2007).
- J. Leibrich and W. Rosenkranz, "Efficient numerical simulation of multichannel WDM transmission systems limited by XPM," IEEE Photon. Technol. Lett. 15, 395-397 (2003). [CrossRef]
- O. V. Sinkin, Holzlohner, J. Zweck, and C. Menyuk," "Optimization of the split-step Fourier method in modeling optical-fiber communications system," J. Lightwave Technol. 21, 61-68 (2003). [CrossRef]
- T. Schneider, Nonlinear optics in telecommunications, (Springer, 2004).
- G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Suppression of spurious tones induced by the split-step method in fiber systems simulation," IEEE Photon. Technol. Lett. 12, 489-397 (2000). [CrossRef]
- G. Goldfarb, G. Li and M. G. Taylor, "Orthogonal wavelength-division multiplexing using coherent detection," IEEE Photon. Technol. Lett. 19, 2015-2017 (2007). [CrossRef]
- X. Liu and D. A. Fishman, "A fast and reliable algorithm for electronic pre-equalization of SPM and chromatic dispersion," in OFC 1996, paper OThD4.
- A. J. Lowery, L. B. Du, and J. Armstrong, "Performance of optical OFDM in ultralong-haul WDM lightwave systems," J. Lightwave Technol. 25, 131-138 (2007). [CrossRef]
- M. Shtaif, M. Eiselt, and L. D. Garret, "Cross-phase modulation distortion measurements in multispan WDM systems," IEEE Photon. Technol. Lett. 12, 88-90 (2000). [CrossRef]
- P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001). [CrossRef] [PubMed]
- J. M. Kahn and K. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004). [CrossRef]
- J. Wang and K. Petermann, "Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. 10, 96-100 (1992). [CrossRef]
- T. Yu, W. M. Reimer, V. S. Grigoryan, and C. R. Menyuk, "A mean field approach for simulating wavelengthdivision multiplexed systems," IEEE Photon. Technol. Lett. 12, 443-445 (2000). [CrossRef]

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