## Intra-channel nonlinearity in differentially phase-modulated transmission |

Optics Express, Vol. 19, Issue 5, pp. 3990-3995 (2011)

http://dx.doi.org/10.1364/OE.19.003990

Acrobat PDF (648 KB)

### Abstract

The mechanisms responsible for nonlinear impairments in single-channel phase modulated system employing differential detection are investigated. The role of dispersion precompensation is discussed. It is shown that precompensation may be designed as to minimize the in-phase components of the fluctuations thus reducing nonlinear impairments. In differential-phase-shift-keying the effect of precompensation is stronger than in differential-quadrature-phase-shift-keying. The results of an analytic theory are compared with split-step based computer simulations using realistic system parameters.

© 2011 Optical Society of America

## 1. Introduction

1. P. J. Winzer and R-J. Essiambre, “Advanced Optical Modulation Formats,” Proc. IEEE **94**, 952–985 (2006). [CrossRef]

2. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of Filtering on RZ-DPSK Reception,” IEEE Photon. Technol. Lett. **15**, 840–842 (2003). [CrossRef]

3. A. Mecozzi, “A unified theory of intra-channel nonlinearity in pseudo-linear phase-modulated transmission,” IEEE Photon. J. **2**, 728–735 (2010). [CrossRef]

## 2. Analysis

3. A. Mecozzi, “A unified theory of intra-channel nonlinearity in pseudo-linear phase-modulated transmission,” IEEE Photon. J. **2**, 728–735 (2010). [CrossRef]

4. X. Wei and X. Liu, “Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion,” Opt. Lett. **18**, 2300–2302, Dec. 1, 2003. [CrossRef]

*f*

_{j}_{,}

*= 2 always, except*

_{l}*f*

_{j}_{,}

*= 1 and*

_{j}*f*

_{j}_{,0}=

*f*

_{0,}

*=*

_{l}*f*

_{0,0}= 0;

*q*= 4 except

_{j}*q*

_{1}=

*q*

_{−1}= 2 and

*q*

_{0}= 0; and finally

*s*= 2 except

_{j}*s*

_{0}=

*s*

_{1}= 0.

*Q*factor at the receiver as

*I*〉 =

*eη*/(

*h̄ω*

_{0})Re[exp(−

*iφ*)

_{d}*E*

_{0}], where

*P*

_{av}the average launched power. Using

*φ*= 0 for DBPSK and

_{d}*φ*= ±

_{d}*π*/4 for DQPSK, we obtain

*Q*factor at the receiver the expressions

*I*〉 ∝

*P*

_{av}and

*Q*factor is inversely proportional to

*P*

_{av}.

*N*– 1 lumped locations equally spaced along the line, which we assume coincident with the amplifier locations, one

*N*th of the total line dispersion is compensated. Compensation at the transmitter and receiver takes care of the residual one

*N*th of the total dispersion such that the full line dispersion is compensated. Such a line may always be represented as the concatenation of

*N*identical sections of length

*L*made of precompensation, line propagation and postcompensation with the sum of pre and postcompensation of each section equal to the total dispersion of the section. In this case, 〈Δ

*I*

^{2}〉 is

*N*

^{2}times the photocurrent fluctuations for a single section, because the fluctuations, in the linear regime, are additive. The total system length is now

*L*

_{tot}=

*NL*.

## 3. Discussion

5. T. J. Ellingham, J. D. Ania-Castañón, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Quasi-Lossless Optical Links for Broad-Band Transmission and Data Processing,” IEEE Photon. Technol. Lett. **18**, 268–230, Jan. 1, 2006. [CrossRef]

*f*(

*z*) = 1. This case is fully equivalent to a transmission through an ideal lossless fiber, so that in the following we will refer to it as such. In this case, the imaginary part of

*G*(

*T*,

_{j}*T*;

_{l}*z*) is an antisymmetric function of

*z*so that, if we chose

*z*

^{*}such that the integration interval in Eq. (1) is itself symmetric, the real part of

*J*

_{j}_{,}

*, proportional to the integral of the antisymmetric imaginary part of*

_{l}*G*(

*T*,

_{j}*T*;

_{l}*z*) over a symmetric interval, becomes zero, and

*J*

_{j}_{,}

*purely imaginary. The value of*

_{l}*z*

^{*}that makes the integration interval symmetric is half of the span length

*z*

^{*}=

*L*/2, corresponding to a symmetric profile where in each fiber sections between dispersion compensating stations half of the dispersion is compensated at the input, half at the output of the section. Let us now analyze separately the effect of a symmetric compensation profile on DBPSK and DQPSK system performance.

*φ*= 0 and

_{d}*a*= ± exp(−

_{j}*iφ*), where

*φ*is an arbitrary bit-independent phase added by the link, so that the products

*I*only the values ±1. With symmetric compensation,

*J*

_{j,l}is purely imaginary, so that both Δ

*I*

_{1}and Δ

*I*

_{0}are purely imaginary hence Δ

*I*= 0. Physically, the absence of nonlinear fluctuations with a symmetric dispersion profile reflects the fact that in this case the nonlinear fluctuations are in quadrature with the field. With symmetric compensation, therefore, the photocurrent is not affected by the nonlinear fluctuations in the lossless case, within our first order perturbation theory.

*a*are now four, namely ±exp(−

_{j}*iφ*) and ±

*i*exp(−

*iφ*), and the products

*I*the four values ±1 and ±

*i*, with equal probability for random codes with equally probable symbols. Therefore, Δ

*I*

_{1}and Δ

*I*

_{0}are with equal probability purely real or purely imaginary. This implies that in the symmetric case the nonlinear fluctuations of a DQPSK pulse are either pure amplitude noise or pure phase noise. The effects of the fluctuations on the receiver are however identical for pure in quadrature or in phase fluctuations. This is because if we represent the electric field of the four symbols at the center of the four quadrants of a complex plane, the photo-currents detected after the two interferometers, whose arms are unbalanced by

*φ*= ±

_{d}*π*/4, measure the projection of the field on the real and imaginary axis and are therefore sensitive to both amplitude and phase noise of the field. The reduction of one of the two components of the nonlinear fluctuations is however still beneficial because the total fluctuations become smaller, although they do not become zero in the lossless case as they do in DBPSK.

## 4. Results

*Q*factors for DBPSK and DQPSK systems with the parameters listed in Tab. 1 vs. the precompensated fiber dispersion

*z*

^{*}. We assume full dispersion compensation, the sum of precompensation and postcompensation, after each of the

*N*= 7 amplifier spans. For instance,

*z*

^{*}= 0 corresponds to full span dispersion compensation at each amplifier station and at the receiver;

*z*

^{*}=

*L*to full span dispersion compensation at the line input and at each amplifier station;

*z*

^{*}=

*L*/2 to half span dispersion compensation at the line input, full span dispersion compensation at each amplifier station, and half span dispersion compensation at the receiver. Notice that, although the baud rate 40Gbaud/s is the same in both cases, the bit rate of the DQPSK system (two bit/symbol) is 80Gbit/s whereas that of the DBPSK (one bit/symbol) is 40Gbit/s. Lines are a plot of the theory. Circles are the results of a split-step based simulation program, using at receiver a third order Butterworth optical filter of 1.5 nm bandwidth, and an electrical filter of approximately 30 GHz bandwidth compliant with the G.957 ITU-T Recommendation, with a bandwidth 0.8 the baud rate. We used in the simulations De Brujin sequences of 2

^{9}symbols for DBPSK and of 4

^{4}symbols, each encoding two bits, for DQPSK. The number of pulses was sufficient in our case where full compensation is performed every

*L*= 100 km, and the number of overlapping pulses, approximately (

*L*/

*z*)(

_{d}*τ*

_{FWHM}/

*T*), is about 45. The results are represented with circles in figs. 1 and 2.

_{s}*Q*factors vs. the precompensated fiber dispersion

*z*

^{*}for a lossless fiber system with the parameters listed in table 1. The input average power was

*P*

_{av}= 3 dBm. Fig. 1a refers to DBPSK, the solid line is the plot of the theory for DBPSK, circles are the results of the computer simulations. The theoretical curves have been obtained by truncating the sums in Eq. (4), (5) and (6) up to |

*j*|, |

*l*| ≤ 200. This means that we include the interaction with the 200 pulses preceding and the 200 following any given pulse. We have verified that increasing this number does not change the first few digits of the results. The benefits of the cancellation of the in-phase components of the nonlinear fluctuations with symmetric compensation show up clearly [6

6. X. Wei, X. Liu, S. H. Simon, and C. J. McKinstrie, “Intrachannel four-wave mixing in highly dispersed return-to-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map,” Opt. Lett. **31**, 29–31, 2006. [CrossRef] [PubMed]

*Q*, that is zero nonlinear fluctuations, whereas the numerical solution of the complete equations show a large, but finite,

*Q*, about one order of magnitude larger than the

*Q*in the absence of precompensation. Fig. 1b refers to DQPSK, the solid line being again the result of the theory, the circles of the simulations. It shows up clearly that in the lossless case, although the benefits of a symmetric compensation are still measurable, the impact of a symmetric compensation is significantly smaller than in DBPSK [7].

*Q*is smaller than in the lossless case, because the efficiency of the nonlinear effects in the final part of each span is negligible, so that the effective span length is smaller than the actual length. With loss, the in-phase components of the nonlinear fluctuations are never zero hence, at optimum precompensation, the theory does not predict for DBPSK an infinite

*Q*. With loss the benefit of precompensation, still larger for DBPSK than for DQPSK, becomes comparable in the two transmission formats.

*μ*m. The

*Q*factor for DBPSK vs. the input power

*P*

_{av}are reported in the insets of Fig. 2, for zero dispersion precompensation. Dots are the results of the simulations, dot-dashed red lines the

*Q*caused by the ASE noise alone,

*Q*

^{2}∝

*P*

_{av}and nonlinear intrachannel effects alone,

*Q*factor, assuming the variance of the noise as the sum of the variances caused by the ASE noise and intrachannel effects alone, i.e. neglecting their interaction (i.e. the nonlinear noise coupling,

*Q*

^{2}∝ 1/

*P*

_{av}). It is apparent that nonlinear noise coupling, if present, does not modify the qualitative and even quantitative behavior of the two curves.

## 5. Conclusions

6. X. Wei, X. Liu, S. H. Simon, and C. J. McKinstrie, “Intrachannel four-wave mixing in highly dispersed return-to-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map,” Opt. Lett. **31**, 29–31, 2006. [CrossRef] [PubMed]

## References and links

1. | P. J. Winzer and R-J. Essiambre, “Advanced Optical Modulation Formats,” Proc. IEEE |

2. | P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of Filtering on RZ-DPSK Reception,” IEEE Photon. Technol. Lett. |

3. | A. Mecozzi, “A unified theory of intra-channel nonlinearity in pseudo-linear phase-modulated transmission,” IEEE Photon. J. |

4. | X. Wei and X. Liu, “Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion,” Opt. Lett. |

5. | T. J. Ellingham, J. D. Ania-Castañón, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Quasi-Lossless Optical Links for Broad-Band Transmission and Data Processing,” IEEE Photon. Technol. Lett. |

6. | X. Wei, X. Liu, S. H. Simon, and C. J. McKinstrie, “Intrachannel four-wave mixing in highly dispersed return-to-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map,” Opt. Lett. |

7. | A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, “Dispersion Management in Phase Modulated Optical Transmission Systems,” Proceedings of the 36th European Conference on Optical Communication, (ECOC 2010, Torino, Italy), Vol. 1, 311–313, September 19–23, 2010. |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

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

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 30, 2010

Revised Manuscript: January 12, 2011

Manuscript Accepted: January 13, 2011

Published: February 15, 2011

**Citation**

A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, "Intra-channel nonlinearity in
differentially phase-modulated
transmission," Opt. Express **19**, 3990-3995 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3990

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### References

- P. J. Winzer, and R.-J. Essiambre, “Advanced Optical Modulation Formats,” Proc. IEEE 94, 952–985 (2006). [CrossRef]
- P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of Filtering on RZ-DPSK Reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003). [CrossRef]
- A. Mecozzi, “A unified theory of intra-channel nonlinearity in pseudo-linear phase-modulated transmission,” IEEE Photon. J. 2, 728–735 (2010). [CrossRef]
- X. Wei, and X. Liu, “Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion,” Opt. Lett. 18, 2300–2302 (2003). [CrossRef]
- T. J. Ellingham, J. D. Ania-Castañón, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Quasi-Lossless Optical Links for Broad-Band Transmission and Data Processing,” IEEE Photon. Technol. Lett. 18, 268–270 (2006). [CrossRef]
- X. Wei, X. Liu, S. H. Simon, and C. J. McKinstrie, “Intrachannel four-wave mixing in highly dispersed returnto-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map,” Opt. Lett. 31, 29–31 (2006). [CrossRef] [PubMed]
- A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, “Dispersion Management in Phase Modulated Optical Transmission Systems,” Proceedings of the 36th European Conference on Optical Communication, (ECOC 2010, Torino, Italy), Vol. 1, 311–313, September 19–23, 2010.

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