## Analytical modeling of cross-phase modulation in coherent fiber-optic system |

Optics Express, Vol. 22, Issue 2, pp. 1426-1439 (2014)

http://dx.doi.org/10.1364/OE.22.001426

Acrobat PDF (1040 KB)

### Abstract

An analytical model to calculate the variance of cross-phase modulation (XPM) distortion in a wavelength-division multiplexed (WDM) fiber-optic system is developed. The method is based on the first order perturbation technique and it is applicable for both dispersion managed and dispersion uncompensated systems. For dispersion managed systems, it is shown that the variance of XPM distortion scales as *N ^{x}* where

*N*is the number of spans and

*x*∈ [

© 2014 Optical Society of America

## 1. Introduction

1. R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. **10**, 1271–1273 (1998). [CrossRef]

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

3. M. Eiselt, M. Shtaif, and L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. **11**, 748–750 (1999). [CrossRef]

1. R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. **10**, 1271–1273 (1998). [CrossRef]

13. A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. **30**, 2011–2024 (2012). [CrossRef]

4. D. Marcuse, A.R. Chraplyvy, and R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. **12**, 885–890 (1994). [CrossRef]

1. R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. **10**, 1271–1273 (1998). [CrossRef]

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

5. A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. **17**, 178–190 (1999). [CrossRef]

6. Z. Jiang and C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. **21**, 953–960 (2003). [CrossRef]

7. S. Kumar and D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. **23**, 2073–2080 (2005). [CrossRef]

8. J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. **20**, 1095–1101 (2002). [CrossRef]

13. A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. **30**, 2011–2024 (2012). [CrossRef]

8. J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. **20**, 1095–1101 (2002). [CrossRef]

9. Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, and J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. **29**, 974–986 (2011). [CrossRef]

10. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**, 15777–15810 (2008). [CrossRef] [PubMed]

11. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. **30**, 1524–1539 (2012). [CrossRef]

12. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. **23**, 742–744 (2011). [CrossRef]

13. A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. **30**, 2011–2024 (2012). [CrossRef]

11. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. **30**, 1524–1539 (2012). [CrossRef]

12. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. **23**, 742–744 (2011). [CrossRef]

*N*

^{2}where

*N*is the number of spans. This is because the XPM field generated in each span is identical and they add up linearly leading to quadratic dependance for the variance or power. In general, the XPM variance scales as

*N*where

^{x}*x*∈ [1

**10**, 1271–1273 (1998). [CrossRef]

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

*x*being close to 1 when there is no inline-dispersion compensation.

## 2. SPM and XPM analytical model

*q*is the electric field envelope,

*β*

_{2}(

*z*) is the dispersion profile,

*γ*

_{0}is nonlinear coefficient and

*α*(

*z*) is the fiber loss/gain profile. Using the transformation, where

*γ*(

*z*) =

*γ*

_{0}exp[−

*w*(

*z*)]. We consider the interaction between two channels of a WDM system. We spilt the field

*u*into two parts, where

*u*is the field of channel

_{j}*j*,

*j*= 1, 2. Substituting Eq. (4) into Eq. (3) and ignoring the four wave mixing terms, we obtain Without loss of generality, we consider the interaction between a pulse of channel 1 in symbol slot 0 and a pulse train of channel 2 consisting of random symbol pattern. We assume that the leading order solution of Eq. (5) is linear and treat the nonlinear terms appearing on the right hand side as perturbation. Assuming Gaussian pulse and quadrature amplitude modulation (QAM), we have where

*n*shows the symbol location in time domain,

*T*is the symbol interval,

_{s}*P*is the power, Ω is the channel separation in radians,

*p*(0,

*T*) is Gaussian pulse shape at

*z*= 0,

*T*

_{0}is the half-width at 1/e- intensity point [14] and

*x*and

_{n}*y*are random variables that take values ±1, ±3,...,±(

_{n}*X*− 1) and ±1, ±3,...,±(

*Y*− 1) (

*X*and

*Y*are the number of amplitude levels of the in-phase and quadrature components, respectively.) with equal probability, respectively. Using the perturbation technique [7

7. S. Kumar and D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. **23**, 2073–2080 (2005). [CrossRef]

15. S. Kumar, S. N. Shahi, and D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express **20**, 27740–27755 (2012). [CrossRef] [PubMed]

*k*can be expanded as, where

*γ*

_{0}is a small parameter and

*m*th order solution. Here

*order solution which satisfies Solving Eq. (11), the field*

^{th}7. S. Kumar and D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. **23**, 2073–2080 (2005). [CrossRef]

*S*(

*z*) is the accumulated dispersion given by Our reference frame is fixed to channel 1. The pulse in channel 2 moves with an inverse group speed of

*β*

_{2}Ω relative to channel 1.

*γ*

_{0}, we obtain To solve Eq. (18), we first derive the following identity. Consider a differential equation, where the forcing function

*F*(

*z*,

*T*) is of the form The solution of Eq. (19) is given by (see the Appendix) where To find the first order correction for

*u*

_{1}due to XPM term,

*F*(

*z*,

*T*) in this case is where Using Eq. (21), the first order correction for

*u*

_{1}due to XPM is given by where The distortion due to the XPM is obtained by summation over all nonlinear distortions caused by nonlinear interaction between the pulses located in time intervals

*mT*and

_{s}*nT*.

_{s}*u*

_{1}due to SPM term,

*τ*and

_{m}*τ*to 0 and by replacing the XPM factor 2 with SPM factor 1, i.e., Total first order solution for

_{n}*u*

_{1}is obtained by adding the SPM and XPM contributions,

## 3. Variance calculation

*E*{.} denotes the ensemble average. In this paper, we mainly focus on the variance due to XPM and we ignore the first term in Eq. (35). So, we have Using Eq. (31), the mean XPM distortion can be found as For QAM signals, we have where and

*δ*is Kronecker delta function. Using Eq. (39), Eq. (38) becomes So the absolute square of the mean is obtained by Now, let us find the mean of absolute square of XPM nonlinearity. Using Eq. (31), we have We can consider four cases:

*K*

_{1}=

*K*

_{2}= 1. So, for QPSK, Eq. (53) reduces to

## 4. Results and discussion

*L*= 80 km, channel spacing = 50 GHz, symbol rate = 10 Gbaud, pulse width

*T*(full width at half maximum) = 50 ps, QPSK modulation, nonlinear coefficient and loss of transmission fiber are 1.1 W

_{FWHM}^{−1}km

^{−1}and 0.2 dB/km, respectively. A random bit sequence of 100 pulses is used for the pump channel (channel 2) and a single pulse is launched to channel 1. Number of Monte-Carlo simulation is 2000.

*γ*

_{0}= 0. The XPM distortion is obtained by subtracting the SPM distortion from the output field of channel 1 when channel 2 is present.

*P*= 0 dBm, fiber dispersion

_{peak}*β*

_{2}= −10 ps

^{2}/km, and number of spans

*N*= 10. As can be seen in the Figs. 2 and 3, the discrepancy between the analytical and numerical is more at the center of the pulse (higher power) than at the edges of the pulse (lower power). It is due to the fact that the first order perturbation technique is less accurate at higher powers. The maximum discrepancy between the analytical model and numerical simulations for the variance is less than 10%.

**23**, 2073–2080 (2005). [CrossRef]

^{2}/km, it decays slowly. When the dispersion is very large, the XPM effect is reduced since pulses walk-off quickly. The maximum discrepancy between the model and the simulation is less than 10% in this example.

*β*

_{2}= −22 ps

^{2}/km,

*γ*= 1.1 W

^{−1}km

^{−1}and loss

*α*= 0.2 dB/km. The pulse width

*T*= 35.71 ps. The rest of the parameters are the same as before. Figs. 6(a) and 6(b) show the variances for 2-channel and 5-channel WDM system, respectively. For the 5-channel case, the variance of XPM distortion of the central channel due to other channels is calculated analytically and numerically. Analytical modeling is done by adding the variances of XPM distortion of the central channel due to each channel. This approach ignores the four wave mixing (FWM) among channels, but the numerical simulation includes the FWM impairments. Due to the good agreement between analytical and numerical simulations, we conclude that the FWM is negligible in this example.

_{FWHM}### 4.1. Dispersion-managed system

*α*= 0.2 dB/km, fiber nonlinearity

_{TF}*γ*= 1.1 W

_{TF}^{−1}km

^{−1}, fiber dispersion

*β*

_{2,}

*= −10 ps*

_{TF}^{2}/km, and fiber length

*L*= 80 km. Dispersion compensating fiber (DCF) is used for inline compensation. The parameters of DCF are as follows: fiber loss

_{TF}*α*= 0.5 dB/km, fiber nonlinearity

_{DCF}*γ*= 4.4 W

_{DCF}^{−1}km

^{−1}, fiber dispersion

*β*

_{2,}

*= 150 ps*

_{DCF}^{2}/km. The virtual DCFs are used in digital domain for pre- and post- compensation. The parameters of the virtual DCF are the same as the real DCF except that its nonlinear coefficient is set to zero. Pre- and post-compensation fibers’ lengths are calculated based on the pre-compensation ratio defined as, where

*L*and

_{pre}*L*are pre- and post-compensation fiber lengths, respectively. The pre-, inline-, and post-compensation fiber lengths are chosen so that the total accumulated dispersion at the receiver is zero.

_{post}*η*when the pre-compensation ratio is 50%. When

*η*= 100%, it corresponds to a resonant dispersion map in which the residual dispersion per span is zero. As can be seen, this is the worst case since the XPM distortion in each span is identical (up to the first order) and they add up coherently. When

*η*= 0%, dispersion is compensated equally at the transmitter and receiver with no inline-compensation, which is the same as the dispersion uncompensated case except for the fact that pre- and post-compensating fibers have nonlinearity. From Fig. 9, we find that

*η*= 70% is the optimum value which corresponds to a residual dispersion

*D*of 185 ps/nm per span. The maximum discrepancy between the model and simulations in this example is 10%.

_{res}*N*. When

*η*= 100%, a curve fitting to the simulation data shows that variance scales as

*N*

^{1.95}while it scales as

*N*

^{1.75}when

*η*= 90% and

*N*

^{1.54}when

*η*= 80%. Typically, it varies as

*N*where

^{x}*x*∈ [1

**10**, 1271–1273 (1998). [CrossRef]

**17**, 1018–1026 (1999). [CrossRef]

*η*= 100%), XPM distortion in each span is identical and the coherent addition of XPM fields leads to

*N*

^{2}dependence. For a non-resonant map (

*η*= 0%), the XPM distortion in different spans are nearly independent and hence, the variances due to each span can be added which leads to a linear scaling of XPM variance with

*N*. For the other values of

*η*, there is some correlation between the XPM distortions arising from different spans and hence, the variance scales as

*N*where

^{x}*x*∈ (1, 2). It may be noted that variance builds up slowly for the cases of

*η*= 80% and

*η*= 90% as compared to the case of

*η*= 100%. The maximum discrepancy between the model and simulations in this example is less than 10%.

## 5. Conclusions

*N*where

^{x}*x*∈ [1

**10**, 1271–1273 (1998). [CrossRef]

**17**, 1018–1026 (1999). [CrossRef]

## 6. Appendix: Differential equation solution

*f̃*(

*z*,

*ω*) is the Fourier transform of

*f*(

*z*,

*T*) and where The solution of Eq. (58) with the initial condition

*f̃*(0,

*ω*) = 0, is where Using Eq. (59) in Eq. (63) and inverse Fourier transforming, we obtain where After evaluating the inner integral in Eq. (65), we obtain Using Eqs. (60)–(62) and Eq. (67), and after some algebra, we arrive at Eq. (21).

## References and links

1. | R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. |

2. | R. Hui, K. Demarest, and C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. |

3. | M. Eiselt, M. Shtaif, and L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. |

4. | D. Marcuse, A.R. Chraplyvy, and R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. |

5. | A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. |

6. | Z. Jiang and C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. |

7. | S. Kumar and D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. |

8. | J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. |

9. | Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, and J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. |

10. | M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express |

11. | A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. |

12. | P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. |

13. | A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. |

14. | G. P. Agrawal, |

15. | S. Kumar, S. N. Shahi, and D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express |

16. | D. Yang and S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2310) Fiber optics and optical communications : Fiber optics

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Optical Communications

**History**

Original Manuscript: August 19, 2013

Revised Manuscript: October 16, 2013

Manuscript Accepted: November 6, 2013

Published: January 15, 2014

**Citation**

Sina Naderi Shahi, Shiva Kumar, and Xiaojun Liang, "Analytical modeling of cross-phase modulation in coherent fiber-optic system," Opt. Express **22**, 1426-1439 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1426

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### References

- R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998). [CrossRef]
- R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999). [CrossRef]
- M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999). [CrossRef]
- D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994). [CrossRef]
- A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. 17, 178–190 (1999). [CrossRef]
- Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003). [CrossRef]
- S. Kumar, D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. 23, 2073–2080 (2005). [CrossRef]
- J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. 20, 1095–1101 (2002). [CrossRef]
- Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011). [CrossRef]
- M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16, 15777–15810 (2008). [CrossRef] [PubMed]
- A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012). [CrossRef]
- P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]
- A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic Press, 2007).
- S. Kumar, S. N. Shahi, D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20, 27740–27755 (2012). [CrossRef] [PubMed]
- D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

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