## Polarization scattering by intra-channel collisions |

Optics Express, Vol. 20, Issue 2, pp. 1213-1218 (2012)

http://dx.doi.org/10.1364/OE.20.001213

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### Abstract

We show that polarization modulated and polarization multiplexed transmission may be significantly impaired by the polarization scattering induced by intra-channel cross-phase modulation and four-wave mixing. In polarization multiplexed transmission, channel interleaving may be used to mitigate the effect when two-pulse collisions are dominant.

© 2011 OSA

## 1. Introduction

1. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. **20**, 2060–2062 (1995). [CrossRef] [PubMed]

2. A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, “Intra-channel nonlinearity in
differentially phase-modulated transmission,”
Opt. Expr. **19**, 3990–3995
(2011). [CrossRef]

1. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. **20**, 2060–2062 (1995). [CrossRef] [PubMed]

## 2. First order perturbation theory

**E**of a polarized optical field in a nonlinear dispersive fiber, averaged over the fast polarization evolution, is well described by the Manakov equation [1

1. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. **20**, 2060–2062 (1995). [CrossRef] [PubMed]

3. D. Marcuse, C. R. Menyuk, and P. K. A. Wai“Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **15**, 1735–1746 (1997). [CrossRef]

*β*″ is the group velocity dispersion (negative in the anomalous dispersion region),

*γ*= 2

*πn*

_{2}/(

*λA*

_{eff}) is the fiber nonlinear coefficient,

*n*

_{2}is the nonlinear refractive index, and

*A*

_{eff}is the effective area of the fiber. The function

*f*(

*z*) rescales the fiber nonlinearity to include the effect of a non-uniform power profile caused by the fiber loss. When equally spaced lumped Erbium amplifiers are used, this function is

*f*(

*z*) = exp[−

*α*mod(

*z,z*)], where mod is the modulus function,

_{s}*α*is the power attenuation coefficient, and

*z*is the span length. Let us now assume that the field

_{s}**E**= Σ

_{n}**E**

*(*

_{n}*z,t*) is a sequence of well separated pulses

**E**

*centered at*

_{n}*T*=

_{n}*nT*, with

*T*the symbol duration, and that the message is encoded by a different polarization, phase and amplitude of the individual pulses. A large dispersion adds on pulses a strong time-frequency correlation, because each frequency component is delayed proportionally to its detuning from the center frequency. Consequently, it is possible to show that the well-known selection rules of four-wave mixing in frequency reflect into similar rules in time domain. As a result of these rules, the evolution of the pulse centered at

*T*

_{0}= 0 is described by the equation where the triple sum in Eq. (2) is extended to terms where the four-wave mixing interaction is resonant with the pulse

**E**

_{0}, which are those with

*T*=

_{k}*T*+

_{j}*T*hence

_{l}*k*=

*j*+

*l*. In a three dimensional space (the Stokes space) spanned by the unit vectors (

*e⃗*

_{1},

*e⃗*

_{2},

*e⃗*

_{3}), let us now introduce the Pauli spin vector as

*σ⃗*= (|

*x*〉〈

*x*| – |

*y*〉〈

*y*|)

*e⃗*

_{1}+ (|

*y*〉〈

*x*| + |

*x*〉〈

*y*|)

*e⃗*

_{2}+

*i*(|

*y*〉〈

*x*| – |

*x*〉〈

*y*|)

*e⃗*

_{3}, where |

*x*〉 and |

*y*〉 are unit polarization vectors parallel to the

*x*and

*y*axis respectively. Let us then define

**E**

*=*

_{n}*E*|

_{n}*s*〉 where |

*s*〉 is the unit polarization state, and the Stokes vectors of the field is

*s⃗*= 〈

*s*|

*σ⃗*|

*s*〉. After multiplying by

*t*we obtain An integration by parts

*s⃗*

_{0}(

*z*) = ∫d

*t*|

*E*

_{0}|

^{2}

*s⃗*

_{0}(

*z,t*)/

*U*

_{0}, where

*U*

_{0}= ∫d

*t*|

*E*

_{0}|

^{2}is the total energy of the pulse centered at

*T*

_{0}= 0, we obtain Equation (5) may be solved using a perturbation approach, by entering at right hand side for

*E*,

_{n}*n*= 0,

*k,l, j*and |

*s*〉, their expressions in the absence of nonlinearity. To this purpose, let us assume that the input pulses have a well-defined polarization, uniform across the pulses, that is

_{n}**E**

*(*

_{n}*z*= 0

*,t*) =

*E*(0

_{n}*,t*)|

*s*〉, where the unit polarization vector |

*s*〉 is independent of

*t*. A purely dispersive propagation preserves the uniformity of the polarization hence, within the perturbation approach, 〈

*s*

_{0}|

*σ⃗*|

*s*〉〈

_{l}*s*

_{j}_{+}

*|*

_{l}*s*〉 at right hand side of Eq. (5) may be assumed as time independent quantities, and pulled out of the integral.

_{j}*j*= 0. Using that

*i*[

*σ⃗*(

*σ⃗*·

*s⃗*) – (

_{l}*σ⃗*·

*s⃗*)

_{l}*σ⃗*]/2 =

*s⃗*×

_{l}*σ⃗*[4

4. J. P. Gordon and H. Kogelnik“PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA97, 4541–4550 (2000). [CrossRef] [PubMed]

*t*=

_{n}*t*−

*nT*, the solution in the absence of nonlinearity may be analytically expressed as

*ρ*(

*z*) = −

*i*(|

*β*″|/

*β*″)(

*z*−

*z*

^{*})/

*z*. Here, we defined the dispersion length

_{d}*z*= |

_{d}*β*″|/

*τ*

^{2}, and we included a possible pre-dispersion of the pulses of the amount

*β*

_{pre}= −

*β*″

*z*, in ps

^{*}^{2}if

*β*″ is in ps

^{2}/km and

*z*in km. For Gaussian pulses, Eqs. (5) and (6) become where we defined the normalized bivariate normal distribution, albeit with an imaginary correlation coefficient

^{*}*ρ*(

*z*) = −

*i*(|

*β*″|/

*β*″)(

*z*–

*z*)/

^{*}*z*, as

_{d}*s⃗*

_{0}at the right hand side of Eqs. (6) and (8) is zero, hence dln

*U*

_{0}/d

*z*= 0. This means that two-pulse collisions does not produce exchange of energy between the two pulses but only a change in their polarization. In addition, there is no polarization scattering if the pulses are either co-polarized (

*s⃗*

_{0}parallel to

*s⃗*) or cross-polarized (

_{l}*s⃗*

_{0}anti-parallel to

*s⃗*). Note once again the strong analogy between the two-pulse case and the polarization scattering caused by inter-channel XPM in wavelength division multiplexed and polarization multiplexed soliton transmission [1

_{l}**20**, 2060–2062 (1995). [CrossRef] [PubMed]

*j*≠ 0, on the other hand, correspond to non-degenerate intra-channel four-wave mixing. In this case, intra-pulse power transfer may take place. This is reflected by the fact that 〈

*s*

_{0}|

*σ⃗*|

*s*〉 may have a component parallel to

_{l}*s⃗*

_{0}. Furthermore, polarization scattering may also occur between three co-polarized pulses and one cross-polarized. To see this, let us assume without loss of generality that

*s⃗*

_{0}=

*e⃗*

_{2}. In this case, in Eqs. (5) or (7) the rate of power change dln

*U*

_{0}(

*z*)/d

*z*is proportional to |〈

*s*

_{0}|

*σ⃗*|

*s*〉 ·

_{l}*e⃗*

_{2}|

^{2}= (1 +

*s⃗*·

_{l}*s⃗*

_{0})/2, whereas the rate of depolarization d

*s⃗*

_{0}/d

*z*have components along

*e⃗*

_{1}and

*e⃗*

_{3}proportional to |〈

*s*

_{0}|

*σ⃗*|

*s*〉 ·

_{l}*e⃗*

_{1,3}|

^{2}= (1 –

*s⃗*·

_{l}*s⃗*

_{0})/2. Therefore, when

*s⃗*·

_{l}*s⃗*

_{0}= −1, hence when the pulses at

*l*and 0 are orthogonally polarized, d

*s⃗*

_{0}/d

*z*may have non-zero depolarizing components parallel to

*e⃗*

_{1}and

*e⃗*

_{3}if the pulse at

*j*is co-polarized with the pulse at

*j*+

*l*so that |〈

*s*

_{j}_{+}

*|*

_{l}*s*〉| = 1.

_{j}## 3. Numerical example: polarization multiplexed QPSK signal

*A*=

_{k}*A*and the phases of the pulses on each polarization are 0,

*π*/2,

*π*and 3

*π*/2, assuming synchronous pulse streams on the two polarizations. To evaluate the effect of two-pulse collisions, we notice that in this case the possible pulse polarizations are the two linear polarizations

*s⃗*= (0,±1,0) and the two circular polarizations

*s⃗*= (0,0,±1). Let us assume that the pulse at

*T*

_{0}= 0 has the linear polarization

*s⃗*

_{0}= (0, 1,0). The interacting pulses will be with equal probability linearly polarized, in which case

*s⃗*

_{0}(0) ×

*s⃗*(0) = 0, or circularly polarized, in which case

_{l}*s⃗*

_{0}(0) ×

*s⃗*(0) = (±1,0,0). Assuming a random message, the polarization spread Δ

_{l}*s⃗*

_{0}=

*s⃗*

_{0}(

*z*) –

*s⃗*

_{0}(0) is on average zero, 〈Δ

*s⃗*

_{0}〉 = 0, whereas its variance along

*e⃗*

_{1}is where the sum is over all pulses that overlap with the pulse 0 during propagation up to

*z*. Notice that, if the number of interacting pulses is large, because of the central limit theorem Δ

*s⃗*

_{0}·

*e⃗*

_{1}is Gaussian distributed with zero average and variance given by Eq. (10).

*s*

_{2}〉, ±|

*s*

_{±}〉, ±

*i*| ±

*s*

_{2}〉, and ±

*i*|

*s*

_{±}〉, where

*s*

_{0}〉 = | +

*s*

_{2}〉. We have 〈

*s*

_{2}|

*σ⃗*|

*s*

_{2}〉 =

*e⃗*

_{2}, 〈

*s*

_{2}|

*σ⃗*| –

*s*

_{2}〉 =

*e⃗*

_{1}+

*ie⃗*

_{3}, 〈

*s*

_{2}|

*σ⃗*|

*s*

_{+}〉 = (1 –

*i*)

*e⃗*

_{1}/2+ (1+

*i*)(

*e⃗*

_{2}+

*e⃗*

_{3})/2, 〈

*s*

_{2}|

*σ⃗*|

*s*

_{−}〉 = (1+

*i*)

*e⃗*

_{1}/2+ (1 –

*i*)(

*e⃗*

_{2}–

*e⃗*

_{3})/2, and 〈

*s*

_{2}|

*s*

_{±}〉 = (1 ±

*i*)/2 〈 −

*s*

_{2}|

*s*

_{±}〉 = (1 ∓

*i*)/2, 〈

*s*

_{2}| –

*s*

_{2}〉 = 〈

*s*

_{+}|

*s*

_{−}〉 = 0.

*l*circularly polarized or co-polarized with the pulse in 0, the four wave mixing interaction has a component parallel to

*s⃗*

_{0}, which produces a change of the pulse energy

*U*

_{0}. This means that only the intra-channel two-pulse interaction is a real soliton-like collision, where there is no power transfer between the two pulses but only a polarization change. Four-pulse interaction (not unexpectedly) produces a polarization rotation of the pulse at 0 together with a phase dependent and polarization dependent power variation.

*s*

_{j}_{+}

*|*

_{l}*s*〉 over all pulse is zero, so that the average polarization displacement is also zero. Squaring the expression for Δ

_{j}*s⃗*

_{0}(

*z*) ·

*e⃗*

_{1,3}, we obtain a four-fold sum of terms proportional to product of the real part of

*a*

_{1}= 〈

*s*

_{0}|

*σ⃗*|

*s*〉〈

_{l}*s*

_{j}_{+}

*|*

_{l}*s*〉 and

_{j}*a*

_{2}= 〈

*s*

_{0}|

*σ⃗*|

*s*

_{l}_{′}〉〈

*s*

_{j}_{′+}

_{l}_{′}|

*s*′

*〉. Let us now use the expansion*

_{j}*s*

_{0}|

*σ⃗*|

*s*〉〈

_{l}*s*

_{j}_{+}

*|*

_{l}*s*〉 · 〈

_{j}*s*

_{j}_{′}|

*s*

_{j}_{′+}

_{l}_{′}〉〈

*s*

_{l}_{′}|

*σ⃗*|

*s*

_{0}〉, which for isotropic polarization distribution has non-zero average only if

*j*=

*j*′ and

*l*=

*l*′. Using now that 〈|

*s*

_{j}_{+}

*|*

_{l}*s*〉|

_{j}^{2}〉 = 1/2 and 〈|〈

*s*

_{0}|

*σ*

_{1,3}|

*s*〉|

_{l}^{2}〉 = 1/2, and the statistical independence of the two terms, we obtain 〈Re[2

*iG*〈

*s*

_{0}|

*σ*

_{1,3}|

*s*〉〈

_{l}*s*

_{j}_{+}

*|*

_{l}*s*〉] Re [2

_{j}*iG*〈

*s*

_{0}|

*σ*

_{1,3}|

*s*

_{l}_{′}〉〈

*s*

_{j}_{′+}

_{l}_{′}|

*s*

_{j}_{′}〉]〉 =

*δ*

_{j,j}_{′}

*δ*

_{l,l}_{′}|

*G*|

^{2}/2. Notice that, being the polarization of the pulse at 0 parallel to

*e⃗*

_{2}, depolarization of this pulse is caused by the fluctuations of the first and third component of the Stokes vector only.

*e⃗*

_{1}and

*e⃗*

_{3}is the sum of the average squared of the individual elements of the sum, that is For the first component of

*s⃗*

_{0}, to this term should be added the one deriving from two-pulse collisions. The term with

*j*= 0 has been excluded because it comes from two-pulse collisions, the term

*l*= 0 has been excluded because it is zero at the right side of Eq. (7).

*z*= 100 km each hence total length

_{s}*L*= 700 km, with precompensation −

*β*″

*z*

^{*}, full compensation after each span, and post-compensation at the receiver −

*β*″(

*z*–

_{s}*z*

^{*}) so that 100% of the total fiber dispersion is compensated. Figure 2 has been obtained with no pre-compensation,

*z*

^{*}= 0. The fiber dispersion is 16 ps/nm/km, loss is 0.25 dB/km, and nonlinear coefficient

*γ*= 1.3 W

^{−1}km

^{−1}. The average input power is (left to right) 3, 7 and 13 dBm. The polarization rotation caused by intra-channel collisions is evident.

*s*

_{0}(

*L*) ·

*e⃗*

_{1}]

^{2}〉

^{1/2}, for 7 dBm input power vs. the pre-compensated length of fiber. The red dot-dashed line is the contribution of the two-pulse collisions alone, the purple dashes line is the contribution of the four pulse collisions alone, the blue solid line is the combination of the two, so it is the observed value of the standard deviation of the polarization jitter along

*e⃗*

_{1}. Along

*e⃗*

_{3}, the contribution of the two-pulse collisions to polarization jitter is zero, and only the contribution of the four pulse collisions is effective, and its variance is equal to the variance of the four pulse collisions along

*e⃗*

_{1}. Consequently, the purple dashed curve represents the observed standard deviation of the fluctuations of the third component of

*s⃗*

_{0}, that is 〈[ Δ

*s⃗*

_{0}(

*L*) ·

*e⃗*

_{3}]

^{2}〉

^{1/2}. The same expression may be obtained for the standard deviation of the term parallel to

*e⃗*

_{2}, hence being

*s⃗*

_{0}parallel to

*e⃗*

_{2}, to the fluctuations of the pulse amplitude [the second term at left hand side of Eq. (7)]. Scattered symbols represent the results directly obtained from simulations, with the exception of the symbols over the two-pulse collision curve, which have been inferred using

^{1/2}, and results within 10% if the fiber PMD did not exceed 0.1 ps/km

^{1/2}.

*s⃗*

_{0}(

*L*) ·

*e⃗*

_{1}]

^{2}〉

_{2–pulse}vs. the dispersion pre-compensation

*z*

^{*}obtained using Eq. (10), with no interleaving. Points show the results of simulations. The three curves refer to an average input power of 3 (lower solid curve, triangles), 7 (intermediate solid curve, squares) and 13 (upper solid curve, circles) dBm. In the same figures, dashed black lines connect triangles, squares and circles representing the result of simulations with pulse interleaving. Again, triangles, squares and circles refer to 3, 7 and 13 dBm input power. The curves show a reduction of the polarization jitter by more than one order of magnitude, down to a level compatible with that expected from the sole contribution of four-pulse collisions. The reduction of depolarization obtained with pulse interleaving relies on the validity of the Manakov equation, which neglects PMD, hence the benefit of this technique may become less significant as PMD increases.

## 4. Conclusions

## References and links

1. | L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. |

2. | A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, “Intra-channel nonlinearity in
differentially phase-modulated transmission,”
Opt. Expr. |

3. | D. Marcuse, C. R. Menyuk, and P. K. A. Wai“Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

4. | J. P. Gordon and H. Kogelnik“PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA97, 4541–4550 (2000). [CrossRef] [PubMed] |

**OCIS Codes**

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

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

**ToC Category:**

Transmission Systems and Network Elements

**History**

Original Manuscript: September 26, 2011

Revised Manuscript: November 22, 2011

Manuscript Accepted: November 25, 2011

Published: January 5, 2012

**Virtual Issues**

European Conference on Optical Communication 2011 (2011) *Optics Express*

**Citation**

Antonio Mecozzi and Francesco Matera, "Polarization scattering by intra-channel collisions," Opt. Express **20**, 1213-1218 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1213

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### References

- L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett.20, 2060–2062 (1995). [CrossRef] [PubMed]
- A. Mecozzi, M. Tabacchiera, F. Matera, and M. Settembre, “Intra-channel nonlinearity in differentially phase-modulated transmission,” Opt. Expr.19, 3990–3995 (2011). [CrossRef]
- D. Marcuse, C. R. Menyuk, and P. K. A. Wai“Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997). [CrossRef]
- J. P. Gordon and H. Kogelnik“PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA97, 4541–4550 (2000). [CrossRef] [PubMed]

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