Kip, D.

Rüter, C.

Smirnov, E.

Stepić, M.

Vicencio, R.

English

MPG.PuRe

Saturable discrete vector solitons in one-dimensional photonic latticesRodrigo A. Vicencio,1, 2 Eugene Smirnov,3 Christian E. Ru¨ter,3 Detlef Kip,3 and Milutin Stepic´4, 51Max Planck Institute of Physics of Complex Systems, 01187 Dresden, Germany,2Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile3Institute of Physics and Physical Technologies, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany4Department of Optical Technologies, National Institute of Metrology, 38116 Braunschweig, Germany,5Vincˇa Institute of Nuclear Sciences, P.O.B. 522, 11001 Belgrade, SerbiaLocalized vectorial modes, with equal frequencies and mutually orthogonal polarizations, are in-vestigated both analytically and experimentally in a one-dimensional photonic lattice with saturablenonlinearity. It is shown that these modes may span over many lattice elements and that energytransfer among the two components is both phase and intensity dependent. The transverse electri-cally polarized mode exhibits a single-hump structure and spreads in cascades in saturation, whilethe transverse magnetically polarized mode exhibits splitting into a two-hump structure. Experi-mentally such discrete vector solitons are observed in lithium niobate lattices for both coherent andmutually incoherent excitations.PACS numbers: 42.65.Tg, 42.82.Et, 42.65.Sf, 63.20.PwSolitons or stable strongly localized nonlinear struc-tures, which can elastically interact with linear wavesand other solitons, have been studied in various systemsin nature, ranging from astrophysics [1] and ocean waves[2], down to Josephson junctions [3] and nanowires [4].These localized structures exist due to an exact balancebetween two or more counteracting effects such as, forexample, dispersion and nonlinearity in the temporal do-main [5]. In the optical domain, solitons may exist inspecific materials, such as Kerr and photorefractive ones[6, 7]. On the other hand, solitons occur in different formslike incoherent, discrete, and vector solitons, which arenot directly related to a particular material [5]. Vectorsolitons are composite structures that consist of two ormore components which mutually self-trap in a nonlinearmedium. Importantly, the individual components decayin isolation. The existence of vector solitons was firstsuggested by Manakov in 1974 [8]. Later on, vector soli-tons have been, for example, studied in carbon disulfide[9] and photorefractive crystals [10]. In periodic nonlin-ear systems, the so-called discrete solitons exist due tothe balance between nonlinearity and discreteness [11].They have been observed in diverse physical configura-tions such as biological systems, charge-transfer solids,Josephson junctions, micromechanical oscillator arrays,and photonic lattices. The latter have overtaken an im-portant role in the study of nonlinear periodic systems[12, 13] due to compact setups with good control of rel-evant parameters like diffraction coefficients and nonlin-earities. Such photonic crystals have periodic distribu-tions of the refractive index and light propagation is as-sociated with allowed bands and forbidden gaps, analogto propagation of electrons in crystalline lattices [14].One-dimensional (1D) discrete vector solitons (DVS)originating from the first band have been already in-vestigated both analytically [15, 16] and experimentally[17] in nonlinear cubic waveguide arrays (WA). The two-dimensional case was studied, too [18]. Finally, multi-band vector solitons, in which individual componentsstem from different bands, were also recently suggestedand demonstrated [19]. The aim of the present study isto investigate DVS in media with saturable nonlinear-ity. Prime examples of such media are photonic latticesin photorefractive crystals. Additionally, semiconductorsat higher light intensities also exhibit saturation [20]. Ithas been shown that saturation, which may occur in acascade manner in discrete systems [21], is responsiblefor the existence of multiple zeros of the Peierls-Nabarropotential, leading to free steering of large amplitude soli-tons, and stable propagation of inter-site modes in 1Dand 2D systems [22]. Various species of two-componentsaturable DVS have been investigated recently, where itwas assumed that both components have the same po-larization and different frequencies [23]. In what followswe are interested in the situation where the componentshave the same frequency but differ in polarization.By following the procedures outlined in Refs. [10, 24],assuming only nearest neighbor interactions, and by us-ing the slowly varying envelope approximation, one mayobtain the following model equations:i∂un∂ξ+ Luun − β(un + s2Bvn)(1 + |un|2 + s1|vn|2)= 0,i∂vn∂ξ+ Lvvn − β(s1Bun + s3Avn)(1 + |un|2 + s1|vn|2)= 0, (1)where Luun = (C0 − ∆k)un + V0(un+1 + un−1) andLvvn = [C0vn + V0(vn+1 + vn−1)]/s1. Here ξ is the nor-malized propagation coordinate (y in the experiment).The normalized envelopes un and vn correspond to trans-verse electrically (TE) and magnetically (TM) polarizedfields, respectively. The parameter β represents the non-linear coefficient, the normalized coupling constant isdenoted by V0, ∆k is the normalized difference of TEand TM wave numbers, whereas C0 can be regarded asa normalized propagation constant. By defining bire-fringence ∆n = nx − nz and average refractive index2n0 = (nx+nz)/2, we write the function sj ≈ 1+j∆n/n0.Finally, A = rxxz/rzzz and B = rxzx/rzzz, where rijkdenote the components of the Pockels tensor [10]. Onemay notice that in general, alike DVS in Kerr media[16–18], in our situation there exists no possibility toseparate cross-phase and four-wave mixing effects. Aconserved quantity of this model is the total power,P =∑n(|un|2 + s1|vn|2)= Pu + s1Pv. By using (1),it can be shown that ∂P/∂ξ = 0 implies:∂Pu∂ξ= −s1∂Pv∂ξ= 2βs2B∑nIm(vnu∗n)(1 + |un|2 + s1|vn|2).(2)This expression gives us information on the energy(power) exchange among the TE and TM componentswhich clearly will depend on the total level of powerin each waveguide. By considering a one-channelconstant-amplitude propagation of the form u0(ξ) =u0 exp[i(λuξ + φu)] and v0(ξ) = v0 exp[i(λvξ + φv)],where λi and φi correspond to the respective propaga-tion constants and initial phases, respectively, we ob-tain the following expression for the power transfer:∂Pu/∂ξ ∼ sin(∆λξ + ∆φ), where ∆λ = λv − λu and∆φ = φv − φu. By assuming only a linear and local de-pendence of the propagation constants, from (1) we get:λu ≈ C0 − ∆k, λv ≈ C0/s1, which results in ∆λ ≈ ∆k.If the components are initially in phase (∆φ = 0), thepower transfer will be initially towards the TE mode pro-vided that ∆k > 0, and towards the TM mode otherwise[16].-12 -10 -8 -6 -4 -2 0 2Λ-4-202468lnPI IIHaL21 23 25 27 29waveguide number0.07.0unHbL21 23 25 27 29waveguide number0.00.5vnHcLFIG. 1: (Color online). (a) TE (dashed line) and TM (con-tinuous line) powers as a function of the propagation con-stant. Vertical lines separate regions. (b) and (c) TE andTM profiles, respectively: Diamonds, squares, triangles andstars correspond to λ = −5,−1, 0, and 1.1, respectively.C0 = ∆k = β = 10, V0 = 1.To gain a theoretical background of the model (1), weuse a Newton-Raphson method to find coupled localizedstationary solutions of the form: un(ξ) = un exp(iλuξ)and vn(ξ) = vn exp(iλvξ), with un, vn ∈ ℜ. For thesake of simplicity we assume λu = λv ≡ λ, which inturn disables the power exchange between DVS compo-nents [Im(vnu∗n) = 0 in (2)]. These solutions may beregarded as the final stage of mode profiles after theDVS is formed. The power dependence of the two com-ponents on propagation constant, for the chosen set ofexperimentally achievable parameters [25], is shown inFig. 1(a) in a logarithmic scale. The region of exis-tence of localized modes is between the low-amplitudeand high-amplitude limits for the upper band edge planewave [22] of the composed system of equations (1). In thepresent case, this region corresponds to ∼ λ ∈ {−13, 2}.Power of the TE mode always exceeds that of TM po-larization and, interestingly, grows in a similar fashionas the power of the on-site mode A in Ref. [21]. Forany λ, the TE mode is always a one-hump structure [seeFig. 1(b)] which spreads transversally in the region ofsaturation [21]. We may separate the Pλ diagram insmaller regions depending on the shape of the TM mode.In region I [∼ λ ∈ {−13,−0.6}], the TM mode corre-sponds to a one-hump structure [diamonds and squaresin Fig. 1(c)]. In region II [∼ λ ∈ {−0.6, 1}], the TM modecorresponds to a two-hump structure separated by onlyone site [triangles in Fig. 1(c)]. In the next regions, theTM mode increases its distance between the two humpsin an odd number of waveguides [as an example, see starsin Fig. 1(c)].While the total power increases, local saturation takesplace [22]. As ∆k > 0, the TE mode is the one whichgains power. Therefore the TE mode starts to increaseits power locally together with the TM mode [region I inFig. 1(a), diamonds and squares in Fig. 1(b,c)]. However,above some critical level of power [region II in Fig. 1(a),triangles in Fig. 1(b,c)] the local power in the center siteis too high and the only possibility for the TM modeto exist is by exploring the neighborhood looking for amore stable configuration. Then, the TE mode furtherincreases its power but now, due to saturation, by in-creasing the amplitudes in the next sites [see stars inFig. 1(b)]. Again, the TM mode finds a new configu-ration which is initially stable, but now the separationbetween peaks consists of three sites [stars in Fig. 1(c)].If we continue increasing the power we observe that theTE mode preserves its one-hump structure, by increas-ing its width, while the TM mode has a two-hump struc-ture where the separation between peaks continuouslyincreases. Therefore, the DVS is mostly TE polarized,except at tails which have a dominating TM polariza-tion. The linear stability analysis of solutions coincideswith the Vakhitov-Kolokolov criterion [5]: modes are sta-ble for ∂P/∂λ > 0, and unstable otherwise. This impliesthat in region I solutions are always stable and, in the3FIG. 2: Scheme of the experimental setup.next regions, there exist both stable and unstable sub-regions [see Fig. 1(a)].To verify our theoretical predictions we use the ex-perimental setup sketched in Fig. 2. A cw laser withwavelength 532nm is split into two orthogonally polar-ized (TE and TM) mutually coherent waves with the helpof a polarizing beam splitter PBS. Optionally, to allowfor mutually incoherent interaction of the two compo-nents, a TM polarized wave can be provided by a secondlaser of the same wavelength. Input power is adjustedwith a combination of half wave plate λ/2 and polarizerP. The two input beams are used to excite narrow single-channel TE and TM polarized modes of the WA by usinga 40× microscope lens ML. This nonlinear WA is fabri-cated in x-cut lithium niobate doped with copper. Thelength of our sample along the propagation y-directionis 11mm and the array consists of 250 parallel titaniumin-diffused waveguide channels that are 4µm wide with aseparation of 4.4µm (grating period Λ = 8.4µm) [26]. Asecond microscope lens ML images the intensity on theoutput face onto a CCD camera, where an additional po-larizer P allows for independent observation of both TEand TM components of the DVS.The nonlinear dynamics of TE only, TM only, and bothTE and TM modes (mutually coherent from the samelight source) is presented in Fig. 3. Here we make use ofthe fact that, in photorefractives crystals, the nonlinear-ity grows exponentially in time, β(t) = β (1−exp[−t/τ ]),where τ is the dielectric response time [27]. Initially, af-ter switching on the light (t = 0), discrete diffraction ismonitored for each situation. Although one may observeinitial focusing of the TE mode within the first minutesin Fig. 3(a) (an even weaker effect is observed for TM),both modes alone are incapable to form a localized struc-ture due to the low individual level of power [Fig. 3(a,b)][28]. On the other hand, when both input polarizationsare present [Fig. 3(c)] a very well localized five-channelDVS is formed after t = 30min and does not change itsshape for another two hours of observation.Stationary images of DVS collected from the outputfacet of the sample for a fixed value of TE power anddifferent values of TM power are presented in Fig. 4(a).As can be seen, the shape of the DVS slightly changesfor different power ratios Pu/Pv. TE and TM polar-FIG. 3: (Color online). Experimentally observed DVS for mu-tually coherent input beams. Temporal nonlinear evolution ofoutput intensity for (a) TE component alone (Pu = 150µW);(b) TM component alone (Pv = 300µW); and (c) both com-ponent together (Pu = 150µW, Pv = 300µW).FIG. 4: (Color online). Mode analysis of stationary DVS.Stationary output intensity for (a) DVS (TE + TM) for dif-ferent power ratios Pu/Pv ; (b) TE and TM (amplified 40×)polarized components for Pu/Pv = 1; and (c) TE and TM po-larized components for Pu/Pv = 1 immediately taken whenTE beam is blocked after formation of the DVS.ized components for a power ratio Pu/Pv = 1 (steadystate) are shown in Fig. 4(b). As predicted, a dominat-ing single-hump TE polarized component and a weakerdouble-humped TM component are observed. The roleof the TM input polarization can be further analyzedby blocking the TE input after stable formation of theDVS in Fig. 4(c). Obviously the TM polarized inputlight has transferred most of its energy to the TE com-ponent, forming a single-hump solution, while the re-maining power is trapped in form of a two-hump solu-4tion [within several minutes, this structure decays intothe one of Fig. 3(b)]. The energy transfer, from ordi-nary to extraordinary polarization (TM → TE), is dueto a specific anisotropic nonlinearity in LiNbO3 [in model(1), this corresponds to consider ∆k > 0]. The mecha-nism of coupling of orthogonally polarized modes is ex-plained by writing holographic gratings due to photo-voltaic currents. Light is anisotropically diffracted fromthese shifted gratings with polarization conversion, whichleads to an energy exchange among the modes [29].FIG. 5: (Color online). Experimentally observed DVS formutually incoherent input beams. Stationary output inten-sity for (a) DVS [TE + TM] for Pu/Pv = 1.5; (b) TE polar-ized component; and (c) TM polarized components (amplified8×).Energy coupling of orthogonally polarized waves canbe prevented by using mutually incoherent input beams(B = 0). Experimentally this is realized by a second laserof the same wavelength (see Fig. 2), which now providesthe TM polarized input beam, and corresponding resultsfor the steady-state DVS formation are shown in Fig. 5.Again a two-hump structure is observed for the TM com-ponent, which now guides a significant part of the totalpower of the soliton.In conclusion, we suggested a rather general theoreticalmodel to describe saturable discrete vector solitons hav-ing orthogonally polarized components. Power transferand coupling between TE and TM components is inves-tigated as well as the corresponding localized stationarysolutions. 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Saturable discrete vector solitons in one-dimensional photonic lattices

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Saturable discrete vector solitons in one-dimensional photonic lattices

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