Polarization Changes in Optical Fibers induced by Self

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Polarization Changes in Optical Fibers induced bySelf-Phase Modulation and Cross-Phase Modulation in

Conjunction with BirefrigenceLutz Rapp

To cite this version:Lutz Rapp. Polarization Changes in Optical Fibers induced by Self-Phase Modulation and Cross-Phase Modulation in Conjunction with Birefrigence. 2008. hal-00301019

https://hal.archives-ouvertes.fr/hal-00301019

https://hal.archives-ouvertes.fr

Polarization Changes in Optical Fibers induced bySelf–Phase Modulation and Cross–Phase

Modulation in Conjunction with BirefrigenceLutz Rapp

July 20, 2008

Abstract— Polarization dependence of various fiber effects affect-ing signal quality in wavelength division multiplexing (WDM) sys-tems has become a major field of research activities. Polarization–mode dispersion (PMD) and polarization–dependent loss (PDL)are quite well understood today, but there are still major openquestions with respect to the interaction with nonlinear fibereffects, although the picture has already become clearer. The aimof this paper is to provide a better understanding of the impactof self–phase modulation (SPM), cross–phase modulation (XPM),and birefrigence on the state of polarization during propagationin a transmission fiber. For several configurations, closed–formexpressions describing the dependence of different quantities onpropagation distance are presented.

Both phase modulating effects do not induce a power trans-fer among the channels of WDM system. However, power isexchanged among the components of the Jones vector of theindividual channels. Without attenuation, this power exchange isa periodic function of the propagation distance. Furthermore, theanalysis reveals that the sum of the ellipticities weighted by thecorresponding fiber input power is preserved during propagationas long as the wavelength dependence of fiber attenuation canbe neglected.

In the presence of SPM only, the ellipticity of the channelunder consideration is maintained during propagation and thepowers of the different components of the Jones vector changesinusoidally as a function of a normalized propagation distance.Closed–form expressions describing the dependence of the mag-nitude of the power variation and the period length on initialconditions at the fiber input are derived. Conservation of theellipticity is canceled out by the interaction with birefrigence.Neglecting fiber attenuation, a different quantity is identified thatis conserved during propagation. Ellipticity as well as the powerof the two components of the Jones vector change periodicallyversus propagation distance, but no longer sinusoidally. Further-more, it is shown that there is a separatrix that cannot be crossedby the traces embracing the states of polarization adopted duringpropagation. Thus, the Poincare sphere is split into up to threesegments.

The ellipticity of a channel influenced by XPM has a sinusoidaldependence on the normalized propagation distance with a periodlength depending on the ellipticity of the first channel only. Powervariation of both components of the Jones vector of the secondchannel can be mathematically described by a superposition ofthree sine waves. The magnitude of the power exchange on initialconditions at the fiber input is pointed out. In addition, thisanalysis is extended for the case that a channel is affected by

This work has been done at the Communication Technology Laboratory,Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, in 1996.It has now been updated and translated into English.

c© by Dr. Lutz Rapp, Jagerstr. 16, D–82041 Deisenhofen, Germany, Email:[email protected]

SPM, XPM, and birefrigence simultaneously.

Index Terms— Optical communications, optical networks, wave-length division multiplexing, self–phase modulation, cross–phasemodulation, birefrigence, polarization–mode dispersion, state ofpolarization

I. INTRODUCTION

S INGLE–mode fibers, also called monomode fibers, are theonly kind of transmission medium used today for long–

haul communications. The name of these fibers suggests thatthey support the propagation of a single fundamental modeonly [1]. But in fact, there are two modes of propagation [2]which are mutually orthogonally polarized. Under weaklyguiding conditions, they can be assumed to be linearly polar-ized. In an ideal cylindrical waveguide, these two modes aredegenerate, which means, that there is no difference betweentheir propagation constants. Thus, they propagate with thesame phase–velocity.

Real fibers are neither completely circular nor perfectlystraight. In addition, the fiber material is slightly anisotropic.As a consequence, the propagation constants of the two modesbecomes different, which is referred to as birefrigence [3].The axis with maximum propagation velocity is called fastaxis, whereas the axis with minimum propagation velocity isnamed slow axis. Birefrigence leads to a periodic change ofthe state of polarization during propagation. Furthermore, fiberemployed in field environments are exposed to mechanicalstress, temperature variation, twists and bends [4] causing un-stable fluctuations in the polarization state of the propagatinglight.

Random variations of birefrigence give rise to effects summa-rized under the term polarization–mode dispersion (PMD) [5].The interest in these effects has grown with increasing bitrateof the transmitted wavelength division multiplexing (WDM)channels. In particular when using older fibers, PMD maybecome one of the most serious impairments in high–bitratesystem. Not surprisingly, the impact of PMD on systemperformance has been studied widely [6]. As an example,quite early results on optical transmission system penalties aredescribed in [7]. A large number of concepts to compensateor at least to reduce the impact of these effects have beendeveloped in the last years [8][9]. Polarization–dependent loss

2 L. RAPP: POLARIZATION CHANGES IN OPTICAL FIBERS INDUCED BY SELF–PHASE MODULATION AND CROSS–PHASE MODULATION. . . JULY 20, 2008

(PDL) or gain (PDG) [10] is another effect related to the stateof polarization that may affect system performance.

Nonlinear fiber effects constitute another class of effectshaving the potential to severely degrade signal quality.Performance degradation caused by self–phase modulation(SPM) [11][12] and four–wave mixing (FWM)[13] has beeninvestigated from the beginning of the commercial deploymentof WDM systems, both, theoretically and experimentally.However, the effect of cross–phase modulation (XPM) hasbeen overlooked for a while [14]. Although the influence ofpolarization was apparent from some early experiments [15],polarization has either been completely neglected in theoreticalinvestigations [16] or only worst case situations have beenconsidered [17] for several years.

Already in 1995, it has been shown that performance degra-dation resulting from PMD can be mitigated by means of theKerr effect for nonreturn–to–zero signals [18]. Since around1999, the interaction of PMD with nonlinear fiber effects isinvestigated more thoroughly.

Many of the investigations are devoted to XPM and degrada-tion of the degree of polarization (DOP). It has been shownexperimentally that, from a statistical point of view, PMDis exacerbated in presence of XPM, as long as polarizationinterleaving is not employed. In addition, XPM–induced de-polarization can lead to a degradation of PMD compensatorefficiency [19][20]. A closed–form approximate expressionfor DOP degradation of a signal degraded by XPM of anonlinearly interfering pump is presented in [21]. Furthermore,it turned out that channel depolarization occurs on a time scalecomparable with the bit period [22].

Significant progresses have also been achieved in the field ofmodeling the interaction of PMD with nonlinearity. Funda-mental aspects are reviewed in a tutorial [23] with focus onthe derivation the Manakow equation. A vector theory of XPMin optical fibers has been developed [24][25] that is mainlyuseful for pump–probe configurations and allowed to derive ananalytical expression for the amplitude of probe fluctuationsby a copropagating pump channel. In addition, the theoryrevealed that PMD helps to reduce the XPM–induced crosstalkin WDM systems, as long as polarization interleaving is notused. This result is in good agreement with findings based onsystem simulations and laboratory experiments [26], leadingto the statement, that evaluations making use of the scalarnonlinear Schrodinger equation tend to overestimate the XPMinterchannel coupling of WDM transmission. Models includ-ing the effect of FWM are also available [27]. In a pump–probeconfiguration, states of polarization different from the carrierhave been observed for optical spectral components generatedby XPM [28].

In presence of large dispersion, the mean field approach isuseful [29]. It provides results with sufficient accuracy atsignificantly reduced computational effects as compared withtechniques based on the Manakov equation. The calculationsare based on the Stokes parameters of the WDM channelsonly and they ignore the detailed temporal behavior when

determining the evolution of the polarization.

First, different representations used to illustrate the evolutionof the state of polarization are described. Next, a mathematicaldescription is presented and some fundamental properties ofthe evolution of the state of polarization are derived. Insection IV, the effect of self–phase modulation on the stateof polarization is investigated, whereas its interaction withbirefrigence is considered in section V. Section VI dealswith the effect of XPM and its interplay with SPM. Subjectof section VII is the interaction of XPM with birefrigence.Finally, the results are summarized and some conclusions aredrawn.

II. REPRESENTATION OF STATES OF POLARIZATION

Light phenomena in optical fibers can be described by usingthe notion of electromagnetic fields propagating as transversewaves [30]. Conventionally, when considering polarization, theelectric field vector is described only, since the magnetic fieldis perpendicular to the electric field and the amplitudes of bothfields are proportional to each other.

Any arbitrary state of polarization can be created by super-imposing two linearly polarized waves, which are oriented inorthogonal directions of a Cartesian coordinate system. Fora simple harmonic wave, where the amplitude of the electricvector varies in a sinusoidal manner, the two components haveexactly the same frequency. In general, the amplitude and thephase of these waves are different.

Typically, a right–oriented coordinate system is used to de-scribe the propagation of a lightwave in an optical fibers,where the z–axis is oriented in the direction of propagation.Thus, polarized light can be represented by a two–elementcomplex vector, the elements of which represent the complexenvelopes of the two linearly polarized waves. This so calledJones vector has the form

~Eell = E0

[ux

uy

]= E0

[ξx · eϕx

ξy · eϕy

]= E0 · eϕx

[ξx√

1− ξ2x · e∆ϕ

](1)

with 0 ≤ ξx ≤ 1 and ∆ϕ = ϕy − ϕx. All parameters onthe right side of the last equal sign represent real quantities.The real quantity E0 represents the magnitude of the electricalfield vector, whereas the complex quantities ux and uy arenormalized complex envelopes with |ux|2 + |uy|2 = 1. Thevariables ϕx and ϕy stand for the respective phase terms.

The pattern traced out by the electrical field has ellipticalshape. In general, the principal axes of the ellipse are tiltedagainst the x–axis and the y–axis. The angle between thex–axis and the larger principal axis of the ellipse will bedenoted by αrot in the following, as shown in Fig. 1. Anotherimportant parameter characterizing the polarization ellipse isthe ellipticity

ε = 2 · ξx ·√

1− ξ2x · sin(∆ϕ) . (2)

L. RAPP: POLARIZATION CHANGES IN OPTICAL FIBERS INDUCED BY SELF–PHASE MODULATION AND CROSS–PHASE MODULATION. . . JULY 20, 2008 3

Fig. 1. Schematic representation of the polarization ellipse.

The magnitude of this parameter indicates the size of the areacovered by the ellipse, whereas the sign of this parameterspecifies the sense of rotation of the electrical field vector.Positive numbers indicate that the field vector is rotatingclockwise as seen by an observer from whom the wave ismoving away (0 ≤ ∆ϕ < 180), whereas the electric field isrotating counter clockwise if the parameter value is negative(180 ≤ ∆ϕ < 360).

Stokes parameters constitute another common way to describethe state of polarization of a lightwave. For the followingconsideration, the three standard Stokes parameters

S1 = 2 · ξ2x − 1 (3)

S2 = 2 · ξx ·√

1− ξ2x · cos(∆ϕ) (4)

S3 = 2 · ξx ·√

1− ξ2x · sin(∆ϕ) = ε (5)

withS2

1 + S22 + S2

3 = 1 (6)

are sufficient. The parameter S3 is identical to the alreadyintroduced ellipticity ε.

Commonly, the state of polarization of a lightwave is repre-sented on the Poincare sphere. However, this three dimensionalrepresentation does not always reveal clearly all importantfeatures. Therefore, two additional kinds of representationwill be used. Both make use of polar coordinates. One ofthem is directly based on the Jones representation of the stateof polarization and can always be found on the right sideof figures describing the state of polarization. The radius isidentified with the parameter ξx and the angle with referenceto the x–axis corresponds to ∆ϕ. As shown by the templatein Fig. 2, all linear states of polarization are found on the x–axis, whereas the data points (0,

√2) and (0,−

√2) represent

Fig. 2. Stokes and Jones based representation of polarization states.

circular states of polarization. Traces with constant magnitudeof the ellipticity have been marked by closed curves.

Representations on the left side of the figures are related to theStokes representation. In this case, the angle with reference tothe x–axis in the plot indicates the rotation of the transverseaxis of the polarization axis, whereas the radius is a functionof the magnitude of the ellipticity and is given by r = 1−|ε|.Thus, linear states of polarization are located on the outercircle with radius one and circular polarization is found atthe origin or coordinates. In both representations, clockwiserotating states of polarization are found in the upper half–plane, whereas the lower half–plane comprises all states ofpolarization with counter clockwise rotation.

III. MATHEMATICAL DESCRIPTION

Let us consider a wavelength division multiplexing systemwith N channels. Since arbitrary elliptical polarizations areallowed, two linear lightwaves with orthogonal polarizationshave to be considered per channel. In total, there are 2Nlightwaves propagating within the optical fiber.

Four–wave mixing (FWM) is an important effect causing inter-action between different channels transmitted simultaneouslyin an optical fiber. However, this effect is only effective if thephase matching condition is fulfilled. Therefore, it is generallyassumed that this effect can be neglected in standard single–mode fibers (SSMFs) due to their large dispersion leading tophase mismatch already at quite small channels spacings.

In the present case, there is a complementary lightwave toeach lightwave having the same wavelength. Therefore, phasematching occurs for some of the four–wave mixing productsdespite the large dispersion, as long as this is not preventedby birefrigence or polarization mode dispersion (PMD). In thefollowing, we will limit our investigations to fibers havinglarge dispersion, so that only four–wave mixing mediatedinteraction among lightwaves having identical wavelengthsplays an important role.

The mathematical description of the wave propagation withinthe optical fiber is based on Cartesian coordinates. The axisof propagation coincides with the z–axes. The temporal andspatial evolution of the complex envelope is characterized


by the normalized parameters uxi(τ, z) and uyi(τ, z), respec-tively. In addition, it is assumed that lightwaves with the samefrequency have identical modal field distribution. Furthermore,the typically small wavelength dependence of attenuation isneglected. This assumptions lead to the system of differentialequations (7)1. Due to space limitations, the dependence ofthe different parameters on time and space is not indicatedexplicitly. Average power levels of the different channelsat the fiber input are denoted by Pi and Pj , respectively.The parameters βi stem from an expansion of the mode–propagation constant β in a Taylor series about the centerfrequency. The extent of nonlinear fiber effects is governedby the nonlinear–index coefficient n2 [31].

Typically, all channels are located within one transmissionwindow of the optical fiber. Thus, the difference with respect tothe modal field distributions are small and the overlap integralsgii and gij , respectively, can be assumed to be constant andcan be replaced by the inverse of the effective area Aeff.Replacing the individual wavelengths λi and λj by the averagewavelength λ allows to introduce a normalized effective length

ζ = 2πn2P0

λAeff

1α

[1− exp (−αz)

]

=1

zeff

1α

[1− exp (−αz)

](8)

with

zeff =λAeff

2πn2P0.

The parameter P0 denotes an arbitrary power level greaterthan zero. It is a good choice to set this parameter value tothe maximum channel power at the fiber input. Please notethat the parameter ζ tends towards a finite value

ζmax =1

zeff · α(9)

when increasing the coordinate z continuously in the presenceof fiber attenuation (α > 0). As a consequence, the range ofvalues of ζ is limited to the interval [0 ζmax].

For the following investigations on polarization changes in-duced by nonlinear fiber effects, it will be assumed that thecomplex envelopes of the electrical field are time independent.Changes of the amplitude and phase of the complex envelopeare described by separate differential equations. Since the stateof polarization depends on the phase difference between thetwo components uxi and uyi only, the propagation of alllightwaves within the transmission fiber can be described by3N coupled differential equations containing real numbersonly. The resulting system of coupled differential equationsis given by (10). It follows from the first two equations thatthe sum ξ2

xi + ξ2yi is constant during propagation.

1In this work, the propagation of a plane wave with wave vector ~k isdescribed by the term exp

nΩ · τ − ~k · ~r

o, where ~r stands for the

position vector and Ω denotes the angular frequency. When using the completeconjugate description, as done in [31], the sign of terms with leadingimaginary symbol changes.

Self–phase modulation and cross–phase modulationdo not induce power exchange among the differentchannels of a WDM system, but power is exchangedbetween the two components of the Jones vectors ofthe channels.

From the already mentioned set of differential equations (10)with real quantities, an additional equation can be derived forthe ellipticity of the lightwaves:

∂

∂ζ

2ξxi

√1− ξ2

xi sin (∆ϕi)

=83

∑j 6=i

Pj

P0

[ (1− 2ξ2

xi

)ξxj

√1− ξ2

xj cos (∆ϕj)

−(1− 2ξ2

xj

)ξxi

√1− ξ2

xi cos (∆ϕi)

](11)

Weighting the ellipticity of all channels with the correspondingfiber input powers and summing up all these terms leads toan additional figure that is preserved during propagation.

N∑i=1

Pj

P0· 2ξxi

√1− ξ2

xi sin (∆ϕi)

=N∑

i=1

Pj

P0· εi = const. (12)

The sum of ellipticities weighted by the correspond-ing fiber input power is preserved during propaga-tion.

In the following, the powers Pxi = ξ2xi of the waves that

are linearly polarized in direction of the x–axis will be usedinstead of the amplitudes ξxi.

For interpretation of the results presented in the following, itis of importance to be aware of the effect of birefrigence onpolarization. Fig. 3 shows closed curves representing traces ofthe state of polarization that are adopted during propagationdue to birefrigence in a fiber without nonlinearity. In theJones based representation, we get circles that are centeredat the origin of the system of coordinates. The Stokes basedrepresentation indicates that the magnitude of the elliptic-ity changes almost continuously. In addition, the curves aresymmetrical with respect to the x–axis and the y–axis. Theimportant point is that significant parts of the curves can beapproximated by straight lines. Thus, we can describe thebehavior qualitatively by stating that the orientation of thepolarization ellipse switches between two directions. The anglewith the x–axis corresponds either to its initial value αinit atthe fiber input or to 180 − αinit.

Remark: It is common practice to denote all nonlinearinteractions between polarization components of one channelby SPM. However, this wording is not correct in a strict sensesince the nonlinear effects do no only induce modulation of


∂uxi

∂z+ ∆β1(λi)

∂uxi

∂τ−

β2(λi)2

∂2uxi

∂τ2− β3(λi)

6∂3uxi

∂τ3

= −2πn2

λiexp (−αz)

giiPiuxi

[|uxi|2 +

23|uyi|2

]+

13giiPiu

∗xiu

2yi

+N∑

j 6=i

gijPjuxi

[2 |uxj |2 +

23|uyj |2

]+

23

N∑j 6=i

gijPjuyi

[uxju

∗yj + u∗xjuyj

]

∂uyi

∂z+ ∆β1(λi)

∂uyi

∂τ−

β2(λi)2

∂2uyi

∂τ2− β3(λi)

6∂3uyi

∂τ3

= −2πn2

λiexp (−αz)

giiPiuyi

[|uyi|2 +

23|uxi|2

]+

13giiPiu

∗yiu

2xi

+N∑

j 6=i

gijPjuyi

[2 |uyj |2 +

23|uxj |2

]+

23

N∑j 6=i

gijPjuxi

[uyju

∗xj + u∗yjuxj

](7)

∂ξxi

∂ζ=

13

[Pi

P0

]ξxiξ

2yi sin (2∆ϕi) + 4ξyi sin (∆ϕi)

∑j 6=i

[Pj

P0

]ξxjξyj cos (∆ϕj)

∂ξyi

∂ζ= −1

3

[Pi

P0

]ξyiξ

2xi sin (2∆ϕi) + 4ξxi sin (∆ϕi)

∑j 6=i

[Pj

P0


∂∆ϕi

∂ζ= −1

3

[Pi

P0

] (ξ2yi − ξ2

xi

)(1− cos (2∆ϕi)) + 4

∑j 6=i

[Pj

P0

] (ξ2yj − ξ2

xj

)

+4[ξxi

ξyi− ξyi

ξxi

]cos (∆ϕi)

∑j 6=i

[Pj

P0


(10)

the phase. There is a power exchange also in the absence ofdispersion caused by degenerate FWM terms.

IV. THE EFFECT OF SELF–PHASE MODULATION

In this section, the propagation of a single channel affectedby self–phase modulation is considered. According to equa-tion (12), the ellipticity ε1 = 2ξx1

√1− ξ2

x1 sin (∆ϕ1) ismaintained during propagation. With growing value of ζ, thestate of polarization follows the curves shown in figure 4.The shape of the polarization ellipse as well as the sense ofrotation of the electrical field vector remain unchanged, onlythe principal axes are rotated.

The power of the electrical field vector in direction of thex–axis as a function of the parameter ζ can be described

analytically by

Px1 =12

[1 +

√1− ε2

1 · sin(

ζ + ζ01

ζc1

)]. (13)

Two new figures are introduced that depend on the initialcondition at the fiber input as indicated by the followingequations:

ζc1 =3

2 |ε1|· P0

P1

ζ01 = ζc1 · arcsin

(2Px1(0)− 1√

1− ε21

)(14)

The normalized power of the component in direction of the x–axis has a sinusoidal dependence on the normalized distanceζ. The average value always equals 1

2 , irrespectively of thestate of polarization at the fiber input. In contrast, the period


Fig. 3. Changes of the state of polarization induced by birefrigence.

length as well as the extent of the power exchange depend onthe initial condition. Smaller magnitude of the ellipticity goesalong with a larger period length and increasing extent of thepower exchange. In the borderline case of linear polarization(vanishing ellipticity), the period length tends toward infinity,so that there is no power exchange.

Of particular interest are states of polarization that do notundergo changes during propagation. In the Jones based rep-resentation, this applies to all polarization states on the unitcircle and all polarization states on the dividing line betweenupper and lower semicircle. By the way, this dividing lineconstitutes a separatrix, since all states of polarization that alightwaves takes during propagation are either above of belowthis line. In addition, all points in this plane with a radius of1√2

and a phase difference of ±90 (circular polarization) arestationary.

In case of linear and circular polarization, self–phasemodulation does not induce any change of the stateof polarization. In addition, self–phase modulationalone does not induce changes of the sense of rotationof the electrical field vector.

In figure 5, the evolution of the power in the x–axis as wellas the ellipticity are shown versus the normalized propagation

Fig. 4. Changes of the polarization state induced by self-phase modulationonly. The polarization state at the fiber input has been marked bya dot.

distance ζ for the polarization states that have been markedby dots in figure 4.

V. INTERPLAY OF SELF–PHASE MODULATION ANDBIREFRIGENCE

When considering the propagation of lightwaves with timeindependent complex envelopes of the electrical field, only dif-ferences with respect to the phase velocity are of importance.The different propagation constants can be taken into accountby modifying the set of differential equations (7). The result

Fig. 5. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input dueto self–phase modulation (no birefrigence).


can be taken from (15). The parameter ∆βi stands for thedifference of the propagation constants for the two componentsof the Jones vector of channel i.

Again, it is helpful to describe changes of the magnitudeof the complex parameters and their phase difference byseparate differential equations. The resulting set of differentialequations can be derived directly from (10) by replacing thelast equation describing the evolution of the phase differenceby equation (16).

A. without attenuation

First, the situation with vanishing attenuation is analyzed. Inthe Jones based representation and on the Poincare sphere, weagain get closed curves that are characterized by the equation

23ξ2x

(1− ξ2

x

)sin2(∆ϕ) + ∆β · zeff · ξ2

x = const . (17)

This equation indicates that the quantity

ε2 + 6∆β · zeff · ξ2x

is conserved during propagation. For different initial states ofpolarization but constant birefrigence, the closed curves areshown in figure 6. The power of the component in directionof the x–axis versus normalized propagation distance and theellipticity along the fiber axis can be taken from figure 7.

As discussed before, self–phase modulation alone does notalter the ellipticity of a lightwave and induces a sinusoidalpower exchange among the two elements of the Jones vector.The average power in directions of the x–axis and the y–axisequals 1

2 . Combined with birefrigence, there is still a periodicpower exchange among the two element, but it is no longersinusoidal and the average power values of the two powers arein general no longer equal, as shown in figure 9 for differentvalues of the birefrigence. The ellipticity also shows a periodicbehavior versus propagation distance. At small birefrigence,the sign of the ellipticity does not changes, which implies thatthe corresponding traces on the Poincare sphere (see figure 8)are either completely in the lower or the upper hemisphere. Inaddition, the amount of the exchanged power as well as theperiod length of the power exchange increase with growingbirefrigence until a certain value has been reached at whichthe trace of polarization states comprises states with positiveand negative ellipticity. If the birefrigence is further increased,the amount of exchanged power as well as the period lengthdecrease. In the borderline case of infinite birefrigence orinfinite product ∆β · zeff, the traces degenerate into circles inthe Jones based representation or straight lines in the Stokesbased representation. In addition, there is no power exchangeanymore.

Whether the orientation of the rotation of the electrical fieldvector changes during propagation depends on the magnitudeof the product ∆β ·zeff and the state of polarization at the fiberinput. For a given value of this product, we can distinguishbetween initial states of polarization that lead to a changeof the sign of the ellipticity and states for which such a

Fig. 6. Changes of the polarization state induced by self–phase modula-tion and birefrigence without fiber attenuation. The polarizationstate at the fiber input has been marked by a dot (constantbirefrigence).

change does not happen. According to this criterion, the areacontaining all possible states of polarization in the Jones basedrepresentation can be segmented into different areas that areseparated by separatrices. For the fast axis in direction of thex–axis, the separatrix is shown in Fig. 6 for ∆β · zeff = 0.15.In general, the separatrix is described by

PsA · sin2 (∆ϕ) =32|∆β| zeff ,

where PsA = (1− ξ2xi) stand for the power in direction of the

Fig. 7. Evolution of power in direction of the x–axis and ellipticity versusnormalized fiber position for different polarization states at thefiber input (no fiber attenuation, with birefrigence).


∂uxi

∂z−

∆βi

2uxi = −

2πn2

λiexp (−αz)

giiPiuxi

[|uxi|2 +

23|uyi|2

]+

13giiPiu

∗xiu

2yi

+N∑

j 6=i

gijPjuxi

[2 |uxj |2 +

23|uyj |2

]+

23

N∑j 6=i

gijPjuyi

[uxju

∗yj + u∗xjuyj

]

∂uyi

∂z+

∆βi

2uyi = −

2πn2

λiexp (−αz)

giiPiuyi

[|uyi|2 +

23|uxi|2

]+

13giiPiu

∗xiu

2yi

+N∑

j 6=i

gijPjuyi

[2 |uyj |2 +

23|uxj |2

]+

23

N∑j 6=i

gijPjuxi

[uyju

∗xj + u∗yjuxj

](15)

∂∆ϕi

∂ζ= − ∆βizeff

1− αzeffζ− 1

3

[Pi

P0

] (ξ2yi − ξ2

xi

)(1− cos (2∆ϕi))

+4∑j 6=i

[Pj

P0

] (ξ2yj − ξ2

xj

)+ 4

[ξxi

ξyi− ξyi

ξxi

]cos (∆ϕi)

∑j 6=i

[Pj

P0


(16)

slow axis. Obviously, the separatrix can be decomposed intofour segments. Two of them are circular arcs of the unit circle,whereas the other two are horizontal lines with distance√

32|∆β| zeff

to the x–axis. Starting from states of polarization within thearea shaded in light blue, the ellipticity will change sign duringpropagation. Since in the absence of absorption each state ofpolarization adopted during propagation can be considered asinitial state and the process itself does not have a memory, theseparatrix constitutes a borderline, that will not be crossed byany possible trace.

Figure 8 shows the polarization states adopted during prop-agation for a single initial state of polarization but variablemagnitude of the birefrigence. Power exchange and evolutionof the ellipticity are illustrated in figure 9. In this figure, thecurves for |∆β| zeff = 10 are not shown, since the period is inthis case significantly smaller as compared to the other onesso that this curve would cover all the other ones.

The red curve shows the trace for |∆β| zeff = 0, i. e. forvanishing birefrigence. It is the already presented curve withconstant ellipticity. In case of the green curve, representingthe trace for |∆β| zeff = 0.150, the initial state of polarizationis outside of the area encompassed by the separatrix. Thus,the orientation of rotation of the electrical field vector doesnot change during propagation and the trace on the Poincaresphere is completely on the upper hemisphere. The othercurves represent traces for |∆β| zeff > 1/6, so that theinitial state of polarization is within the area embraced by

the separatrix. Therefore, the sign of the ellipticity changesperiodically. For increasing values of |∆β| zeff, the traces inthe Jones based representation tend towards a circle. The traceson the Poincare sphere also converge towards a cercle. It canbe generated by rotating the initial state of polarization aroundthe S1–axis. The constant value of S1 implies that there is nolonger a power exchange between the two components of theJones vector.

B. with attenuation

The influence of fiber attenuation becomes clear when consid-ering the first term after the equals sign in equation (16). Withincreasing propagation diestance, the normalized effectivelength ζ tends towards its maximum value 1/(α · zeff). Asa consequence, the denominator converges against zero, whatcan be interpreted as a continuous increase of the effectivebirefrigence

∆βeff =∆β

1− α · zeff · ζ.

Therefore, the trace in the Jones based representation followsfirst the trace that has been determined for the case withoutfiber attenuation, but finally tends more and more towards acircle, as shown in figure 10. This implies, that the amount ofpower exchanged among the two components of the Jonesvector decreases continuously and tends towards zero (seefigure 11). The period length assigned to this power exchangealso decreases with growing ζ, whereas it is constant ifexpressed in terms of the physical propagation distance z.


Fig. 8. Changes of the polarization state induced by self–phase modula-tion and birefrigence without fiber attenuation. The polarizationstate at the fiber input has been marked by a dot.

VI. THE EFFECT OF CROSS–PHASE MODULATION

A. Cross–phase modulation only

The propagation of two channels with arbitrary polarizationis investigated in order to study the effect of cross–phasemodulation. One channel, named first channel in the following,is launched at power P1 into the fiber, whereas the input powerof the second channel is assumed to be negligible (P2 ≈ 0). Inthis way, there is no interaction between the two componentsof the second channel, but they are influenced by the first

Fig. 9. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input (nofiber attenuation, with birefrigence).

Fig. 10. Changes of the polarization state induced by self–phase modula-tion and birefrigence with fiber attenuation. The polarization stateat the fiber input has been marked by a dot.

channel. Thus, the polarization of the second channel is alteredby XPM only. Fiber attenuation will be neglected in thissection.

Polarization changes of the first channel induced by SPM arevisualized by the blue curve in figure 12. It is the curve withconstant ellipticity. The structure of the polarization traces forthe second channel is significantly more complex. In addition,this curve crosses the equator, separating the Poincare sphereinto an upper and lower hemisphere. Its shape depends onthe initial polarizations of both channels. The traces shown in

Fig. 11. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input(with fiber attenuation, with birefrigence).


Fig. 12. Changes of the polarization state induced by cross–phase mod-ulation without fiber attenuation and without birefrigence. Thepolarization state at the fiber input has been marked by a dot.Blue curve: channel at large input power, red curve: channel withnegligible input power.

the Stokes based representation indicate that the magnitude ofthe ellipticity changes significantly faster than the direction ofthe polarization ellipse. The segments in the upper part of thediagram are very similar to the ray–like traces that we havealready seen as a result of birefrigence (see Fig. 3). However,the significant difference is that the orientation does not changebetween two values, but rather changes in small steps.

As shown in figure 13, the ellipticity is a sinusoidal function

Fig. 13. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input(without attenuation, without birefrigence). Blue curve: channel atlarge input power, red curve: channel with negligible input power.

of the normalized fiber position. It can be described by theequation

ε2 = 2ξx2

√1− ξ2

x2 sin (∆ϕ2)

= C2 + ∆ε2 cos(

ζ + ζ02

ζc2

). (18)

The period length

ζc2 =34

1√1− 3

4ε21

· P0

P1(19)

is governed solely by the ellipticity of the first channel. It canbe shown that the power of the x–component of the secondchannel can be described by

Px2 =12

+43

[P1

P0

]√1− ε2

1

C2ζc1 sin

(ζ + ζ01

ζc1

)

+∆ε2

2ζc1ζc2

ζc2 + ζc1sin(

ζc2 + ζc1

ζc1ζc2ζ +

ζ01

ζc1+

ζ02

ζc2

)+

∆ε2

2ζc1ζc2

ζc2 − ζc1sin(

ζc2 − ζc1

ζc1ζc2ζ +

ζ01

ζc1− ζ02

ζc2

).

(20)

Obviously, the power of the x–component of the secondchannel versus normalized propagation distance ζ can bedescribed by the superposition of a constant value and threesine functions with different period lengths. The averagevalue of the power of each component equals one half. Weastain here from describing the constants ∆ε2, C2, and ζ02

in an analytical form. However, the peak–to–peak variation2∆ε2 of the ellipticity of the second channel is illustrated infigures 14 and 15 versus the initial state of polarization of thefirst channel for two initial states of polarization of the firstchannel.

First, the case of linear polarization of the first channel atthe input of the fiber is considered. According to figure 14,the XPM induced change of the ellipticity is minimum ifthe principal axis of the initial states of polarization of bothchannels are either oriented in identical directions or it theyare orthogonal, whereas maximum variation occurs if the anglebetween both transverse axes equals 45 + N × 90 with Nbeing an entire number. The variation vanishes completely ifthe direction of the polarization of both channels is identical.In addition, the magnitude of the variation is independent ofsign of the ellipticity.

A different behavior is observed if the first channel is ellip-tically polarized at the input of the fiber. Again, maximumvariation is observed in case the angle between both principalaxis equals 45 + N × 90. Minimum variation also takesnow place if the sign of the initial ellipticity is identical forboth channels and the principal axes have the same directionor if the principal axis are orthogonal for different signs ofthe initial ellipticities. But the values of the two other localminimum are now significantly larger.


Fig. 14. Color coded plot showing the peak–to–peak variation of theellipticity 2∆ε2 versus the initial state of polarization of thesecond channel induced by XPM only. The first channel is linearlypolarized at the input of the fiber. The angle with the x–axis equals30.

B. Combined with self–phase modulation

So far, the effect of cross–phase modulation without in-teraction with other effects has been studied. In order toinvestigate the interplay of self–phase modulation and cross–phase modulation, we will now assume that both interactingchannels are launched at identical power into the fiber. Thestates of polarization of the first channel are found on the bluecurves in Fig. 16, whereas the red line represents the trace ofthe polarization states of the second channel.

Not surprisingly, the shape of the traces is very similar forboth channels. This statement is also valid with respect tothe evolution of power and ellipticity versus fiber axes, asillustrated in Fig. 17. Both curves are periodic and the sum ofthe ellipticities of the curves is constant. This is in agreementwith equation (12).

As before, the total variation of the ellipticity of the secondchannel has been determined for linear polarization of the firstchannel at the fiber input. As shown in Fig. 18, there are againfour local minima in the Stokes based representation.

Corresponding results for elliptical polarization of the firstchannel at the fiber input are represented in Fig. 19. The most

Fig. 15. Color coded plot showing the peak–to–peak variation of theellipticity 2∆ε2 versus the initial state of polarization of thesecond channel. The first channel is elliptically polarized at theinput of the fiber.

remarkable aspect of this representation is that the patterns inthe two hemisphere are no longer identical. In contrast to theformer cases, minimum variation only occurs if the electricalfield vector of the second channel is rotating clockwise, asthe electrical field vector of the first channel does. In contrast,maximum variation can only be achieved for contrarian sensesof the rotation.

VII. INTERPLAY OF CROSS–PHASE MODULATION ANDBIREFRIGENCE

Evolution of the state of polarization for the combined effectof XPM and birefrigence without fiber attenuation is shownin Fig. 20. The structure of the trace for the second channelis rather complex. However, there are still several segmentswhere the magnitude of the ellipticity changes very rapidly,whereas the orientation of the polarization ellipse is ratherconstant. Power and ellipticity are shown in Fig. 21 versusnormalized propagation distance for both channels. The twofigures are periodic functions. For comparison purposes, thetrace propagation in the presence of fiber attenuation is shownin Fig. 22.

Adding now the effect of SPM but neglecting again fiberattenuation, we get the plot shown in Fig. 23 for the magnitude


Fig. 16. Changes of the polarization state induced by cross–phase modu-lation without fiber attenuation and birefrigence. The polarizationstate at the fiber input has been marked by a dot. Both channelsare launched at equal input power into the fiber.

of the changes of the ellipticity for elliptical polarization ofthe first channel at the fiber input. As compared with theillustration for the case without birefrigence in Fig. 19, thepatterns are quite similar. However, deeply red areas cover alarger part of the total area in the Stokes based representation.In addition, there is now also an area with large magnitude ofthe variation of the ellipticity for identical sense of rotation.

Fig. 17. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input(without attenuation, without birefrigence). Both channels arelaunched at equal input power into the fiber.

Fig. 18. Color coded plot showing the peak–to–peak variation of theellipticity 2∆ε2 versus the initial state of polarization of thesecond channel induced by the interplay of SPM and XPM. Thefirst channel is linearly polarized at the input of the fiber indirection of the x–axis.

VIII. CONCLUSION

The effect of self–phase modulation (SPM), cross–phase mod-ulation (XPM) and birefrigence on the state of polarization hasbeen investigated for continuous wave (cw) configurations.

Starting from the nonlinear Schrodinger equation, it has beenshown that self–phase (SPM) modulation and cross–phasemodulation (SPM) do not induce a power transfer amongthe channels of a wavelength division multiplexing (WDM)system. However, power is exchanged among the componentsof the Jones vector of the individual channels. Without at-tenuation, this power exchange is a periodic function of thepropagation distance. Furthermore, the analysis revealed thatthe sum of the ellipticities weighted by the corresponding fiberinput power is preserved during propagation as long as thewavelength dependence of fiber attenuation can be neglectedand there is no birefrigence.

First, the effect of self–phase modulation without interactionof birefrigence has been investigated. It has been shown thatthe ellipticity of the channel under consideration is maintainedduring propagation even under the influence of SPM. Inaddition, the power of the different components of the Jones


Fig. 19. Color coded plot showing the peak–to–peak variation of theellipticity 2∆ε2 versus the initial state of polarization of thesecond channel induced by the interplay of SPM and XPM. Thefirst channel is elliptically polarized at the input of the fiber.

vector change sinusoidally as a function of a normalizedpropagation distance. Closed–form expressions describing thedependence of the magnitude of the power variation and theperiod length on initial conditions at the fiber input have beenderived. The results indicate that the state of polarization is notaffected by SPM for the special cases of linear and circularpolarization at the fiber input. When averaging over completeperiods, the average power is identical for both componentsof the Jones vector.

Conservation of the ellipticity is canceled out by the interac-tion with birefrigence. Neglecting fiber attenuation, a differentquantity has been identified that is conserved during propaga-tion. Ellipticity as well as the power of the two componentsof the Jones vector change periodically versus propagationdistance, but no longer sinusoidally. Furthermore, it has beenshown that there is a separatrix that cannot be crossed bythe traces embracing the states of polarization adopted duringpropagation. Thus, the Poincare sphere is split into up to threesegments. With fiber attenuation, the separatrix does no longerrepresent a fix borderline and the magnitude of the powerexchange decreases continuously with propagation.

To determine the effect of cross–phase modulation (XPM)without interference of other effects, a configuration with a

Fig. 20. Changes of the polarization state induced by cross–phase modu-lation without fiber attenuation, but with birefrigence. The polar-ization state at the fiber input has been marked by a dot.

first channel launched at high power into the fiber and asecond channel with small input power has been analyzed.The ellipticity of the low–power channel shows a sinusoidaldependence on the normalized propagation distance with aperiod length depending on the ellipticity of the first channelonly. Power variation of both components of the Jones vectorof the second channel can be mathematically described bya superposition of three sine waves. For elliptical and linearpolarizations at the fiber input, maximum variation of theellipticity is observed if the angle between the principal axis ofthe polarization ellipses of both channels equals approximately

Fig. 21. Evolution of power in the x–axis and ellipticity versus normalizedfiber position for different polarization states at the fiber input.


Fig. 22. Changes of the polarization state induced by cross–phase modula-tion with fiber attenuation and birefrigence. The polarization stateat the fiber input has been marked by a dot.

45 + N × 90 with N being an entire number.

Launching both channels at identical powers into the fibergives rise to interaction of SPM and XPM. In case of linearpolarization of the first channel at the fiber input, maximumvariation of the ellipticity is observed if the angle betweenthe principal axis of the polarization ellipses of both channelsequals approximately 45 + N × 90. However, in case ofelliptical polarization of the first channel at the fiber input, ithas been shown that — at least in some cases — maximumvariation of the ellipticity can now only be achieved if therotation senses of both channels are contrarian. If small bire-frigence is added, the fundamental behavior does not change,but the relative portion of initial polarization states leading tolarge variation of the ellipticity increases.

IX. APPENDIX

Birefrigence leads to a wave–vector mismatch. In thefrequently used representation of the coupled nonlinearSchrodinger equations [31], the effect of phase mismatch onthe nonlinear interaction is taken into account by a factore±2∆βiz multiplied with the product of three slowly varyingamplitudes. This factor does not appear in equation (15). How-ever, we will show in this section that the two representations

Fig. 23. Color coded plot showing the peak–to–peak variation of theellipticity 2∆ε2 versus the initial state of polarization of thesecond channel induced by the interplay of SPM and XPM inthe presence of birefrigence (∆β · zeff = 0.15). The first channelis elliptically polarized at the input of the fiber.

are equivalent.

To simplify representation, the case of self–phase modulationonly will be considered. In this case, equation (15) takes theform

∂ux

∂z−

∆β

2ux = −

e−αz

zeff

ux

[|ux|2 +

23|uy|2

]

+13u∗xu2

y

∂uy

∂z+

∆β

2uy = −

e−αz

zeff

uy

[|uy|2 +

23|ux|2

]

+13u∗xu2

y

. (21)

The normalized complex amplitudes ux and uy stem from therepresentation

Ex(τ, z) = <

E(0)x Ψx(x, y)ux(τ, z) ·

exp(−α

2z)

exp(Ωτ − β(0)z

)


Ey(τ, z) = <

E(0)y Ψy(x, y)uy(τ, z) ·

exp(−α

2z)

exp(Ωτ − β(0)z

)(22)

for the two components of the electrical field, where < standsfor the real part of the argument in brackets. The constantE(0) accounts for the initial electrical field at the fiber input,Ω stands for the angular frequency, and the modal fielddistribution is represented by Ψ. The important difference tothe usually employed mathematical description is that there isa common mode–propagation constant

β(0) =β

(0)x + β

(0)y

2(23)

used for both waves. It is defined as the average of thepropagation constants β

(0)x and β

(0)y for the two components.

In this way, the differences with respect to the propagationconstant affect directly the complex amplitudes ux and uy .

However, it is also possible to define the complex amplitudesfor the different waves with respect to their respective mode–propagation constants. This approach is very suitable if theevolution of lightwaves is considered separately. However, thedetermination of the state of polarization is more difficult. Theelectrical field components are now given by

Ex(τ, z) = <

E(0)x Ψx(x, y)Ux(τ, z) ·

exp(−α

2z)

exp(Ωτ − β(0)

x z)

Ey(τ, z) = <

E(0)y Ψy(x, y)Uy(τ, z) ·

exp(−α

2z)

exp(Ωτ − β(0)

y z)

(24)

with the normalized complex amplitudes Ux and Uy , whichare again related by the equations

ux = Ux · e− ∆β2 z (25)

uy = Uy · e ∆β2 z (26)

to the normalized amplitudes used before. As before, theparameter

∆β = β(0)x − β(0)

y

denotes the difference of the propgation constants. Substitutingux and uy in equation (21) by making use of these relationsgives

∂Ux

∂z= −

e−αz

zeff

Ux

[|Ux|2 +

23|Uy|2

]

+13U∗xU2

y e2∆βiz︸︷︷︸PMA

∂Uy

∂z= −

e−αz

zeff

uy

[|Uy|2 +

23|Ux|2

]

+13U∗xU2

y · e−2∆βiz︸︷︷︸PMB

, (27)

where the efect of the phase mismatch due to birefrigence isintroduced by the terms PMA and PMB in the well–knownway used in [31].

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