WABNITZ, Stefan

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 21, NO. 13, JULY 1, 2009 875Cross-Polarization Modulation Domain Wall Solitonsfor WDM Signals in Birefringent Optical FibersStefan Wabnitz, Member, IEEEAbstract—Nonlinear cross-polarization modulation betweentwo wavelength multiplexed channels propagating in an ellipti-cally birefringent optical fiber leads to polarization domain wallsolitons. These solitons represent a switching of the polarizationof both waves between two stable orthogonal states. The potentialapplication of polarization domains to nonlinear loss-free polar-izers is described.Index Terms—Nonlinear optics, optical fiber polarization, op-tical propagation in nonlinear media, optical solitons, wavelength-division multiplexing (WDM).I. INTRODUCTIONT HE instantaneous change of the state of polarizationof an optical signal due to its cross-interaction withadjacent wavelength channels may have a significant impacton wavelength-division-multiplexed (WDM) transmissions.Indeed, cross-polarization modulation (CPM) degrades theperformance of polarization-multiplexed WDM systems [1],[2] and polarization-mode dispersion compensators [3]. CPMwas earlier theoretically and experimentally investigated fortwo and three continuous-wave (CW) or intensity modulatedchannels propagating in randomly birefringent fibers [4], [5].In this letter, we study CPM among two WDM channels infibers with elliptical birefringence, and obtain novel polariza-tion domain wall soliton solutions [6]. These solitons describetemporal switching of both waves among orthogonal, stable po-larization states, in analogy with domain walls along the longi-tudinal axis of the fiber from counterpropagating waves [7], [8].An interesting and novel application of these temporal polar-ization domains is the lossless polarization attraction [9] of aninitially depolarized probe beam to the same polarization stateof a copropagating pump wave.II. THEORYLet us express two optical waves at nearby frequencies and, copropagating in a spun birefringent optical fiber, in terms ofthe normal modes and . These linearly polarized modes arealigned with the local axes of birefringence, and rotate along thefiber with a rate (rad/m). The mode amplitudesobey the coupled equations(1)Manuscript received January 04, 2009; revised February 17, 2009. First pub-lished May 19, 2009; current version published June 12, 2009.The author is with the Dipartimento di Elettronica per l’Automazione, Uni-versità di Brescia, 25123 Brescia, Italy (e-mail: stefano.wabnitz@ ing.unibs.it).Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LPT.2009.2019848where is the distance along the fiber, is a retarded timein a reference frame traveling with the average speed, , and are the groupvelocities of pulses at and . We neglect the effect of stim-ulated Raman scattering, polarization-mode dispersion, and in-trachannel group-velocity dispersion (GVD); i.e., we supposethat the walk-off distance between two pulses at frequenciesand is much shorter than the dispersion distance. The totalHamiltonian in (1) is , where ,, and describe the action of linear birefringence, non-linear self-polarization modulation, and CPM, respectively [6],[10]. The contribution (for and , see [10]) reads as(2)where , , ,, , is the inten-sity-dependent refractive index, is the fiber mode effectivearea, and depends upon the nonlinearity; e.g., forelectrostriction, or for electronic distortion. Since, henceforth we shall suppose that ,where and are the linear propagation constants ofthe modes and . Let us consider the elliptically po-larized (with ellipticity ) eigenstates of , i.e.,and , where[10], [11]. We assume , so that . One obtains(3)where . Typically,the linear beat length of a birefringent fiber, where the nonlinear length , andis the wave power. As a result, nonlinear self-polarization modu-lation and CPMs lead to that are slowly varying inwith respect to . By expressing in terms of and , onehas since the two elliptical eigenmodes propagate un-coupled in the absence of nonlinearity. Whereas the terms pro-portional to in (2) contain the rapidly oscillating factors. When considering the nonlinear polariza-tion changes of the beams over the relatively long scale , we1041-1135/$25.00 © 2009 IEEE876 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 21, NO. 13, JULY 1, 2009may set , where brackets denote -average. Addition-ally, we obtainand . Letus express the nonlinear polarization evolution of the two wavesat frequencies and by means of their respective Stokes pa-rameters, defined asSubstitution in (1) leads to , and(4)where , ,andare the self-polarization mod-ulation and CPM tensors (note that a differentis obtained with counterpropagating beams [10]).Moreover, ,and we introduced the characteristic coordinates. Note that for, or , theself-induced polarization tensor vanishes [10], [11]. Such acondition is particularly significant, as it corresponds to thecase of a long, randomly birefringent optical fiber. Indeed,by setting , where is a uniformlydistributed random angle, one obtainsand , so that (4) reduces to the principal chiralfield equations [6](5)where , and we have set as electronicdistortion is the dominant nonlinear index mechanism in fibers.Analytical polarization soliton solutions in the special case of(5) were discussed by Tratnik and Sipe in 1987 [12].We shall consider here the general case of (4), and restrictour analysis for simplicity to the case of equal power waves;i.e., we set . To find the stationary (i.e., suchthat ) solutions of (4), we set , where. Based on the symmetries of (4), we find that theirsolutions admit the two conserved quantities(6)which, in addition to the two conserved wave powers, permitfor the integration of (4) by quadratures. Let us consider thespecial case whereand , so that and . The timeevolution of may be expressed in terms of Jacobian ellipticfunctions, and is represented on the Poincaré sphere as illus-trated in Fig. 1, where we show the two extreme cases with ei-ther , (unspun highly birefringent fiber), or ,(highly spun fiber). As can be seen in Fig. 1, the non-linear eigenpolarizations of (4) (i.e., the Stokes vectors such thatFig. 1. Trajectories of Stokes vector   on the Poincaré sphere representingstationary solutions for the self-polarization modulation and CPM in a linearly     and circularly       birefringent fiber. Thick red curves representpolarization kink solitons.) are directed along the principal axes of ,i.e., , , or axis. For , so that , the vectorsare saddle points emanating separatrix trajec-tories (thick red curves) that represent the kink solitons(7)with ,,, and is an arbi-trary time shift.The polarization domain wall (7) describes a switchingof polarization between the two stable (with respect tospatio-temporal perturbations) parallel polarization arrange-ments and . Fig. 1 also showsthat for , so that , the saddles coincide with. The associated CPM solitons read as(8)with ,, and.Note that for [equation (5)], all trajectories reduce tocircles along the parallels and the steady-state solutions may beexpressed in terms of elementary trigonometric functions [12].III. SIMULATIONSWith reference to a specific example, we demonstrate by nu-merical integration of (4) the propagation stability of the polar-ization domain wall solutions (7), whose -invariant time pro-files are shown in Fig. 2 for the case with negative . We con-sidered two 1-mW beams at nm spaced by 50 GHz,copropagating in a linearly birefingent fiber withmW , m , and the GVD ofps/(nm km). Note that km, and the walk-offdistance of the hyperbolic secant pulse components in Fig. 2is km. Fig. 2 shows that full switching between twoWABNITZ: CPM DOMAIN WALL SOLITONS FOR WDM SIGNALS IN BIREFRINGENT OPTICAL FIBERS 877Fig. 2. Time profile of polarization domain wall soliton (7) components inwaves at frequency   and   for a linearly birefringent fiber    .Fig. 3. Stable propagation of the cross-polarization soliton of Fig. 2 over1000 km of linearly birefringent fiber.pairs of orthogonal polarization arrangements (these are ini-tially corotating circularly polarized waves, since forand one has and ) occurs on atimescale of about 60 ps. Fig. 3, which shows the propagationthe cross-polarization soliton of Fig. 2 over 1000 km, was ob-tained by numerically solving (4) along the characteristics and, and it confirms the spatio-temporal stability of the domainwall solutions (7). Fig. 3 shows that the nonlinear CPM fullycompensates for the GVD induced temporal walk-off betweenthe two channels. Clearly, at low powers the two pulses walkaway from each other after .The existence of stable domains of parallel mutual polariza-tion arrangements for two wavelength channels in birefringentfibers leads to possibility of implementing a nonlinear, loss-freepolarizer in a copropagating configuration. A numerical proofof this principle is provided by the result of Fig. 4: in the leftwe show the power in the two orthogonal polarization compo-nents (defined as ) of an initially randomlypolarized in time, 1-mW CW probe wave. This probe copropa-gates in a short fiber loop (so that resonance effects can be ne-glected) containing linearly birefringent fiber (with parametersas given above) along with a 50-GHz-spaced circularly polar-ized CW pump mW . As shown in Fig. 4, right,after recirculating for 50 ms, the probe virtually acquires thesame polarization state of the pump.Fig. 4. Power in the two orthogonal polarization components of an initiallydepolarized 1-mW probe, at the fiber input and after copropagation with a po-larized 3-mW pump.IV. CONCLUSIONWe theoretically described CPM between two WDM chan-nels in optical fibers with elliptical birefringence. We obtainedanalytical domain wall soliton solutions that represent thelocked temporal switching of the state of polarization of bothbeams. Lossless polarization attraction in a copropagatinggeometry was also predicted.REFERENCES[1] L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scat-tering by soliton-soliton collisions,” Opt. 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Cross-polarization modulation domain wall solitons for WDM signals in birefringent optical fibers

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