Disorder-induced soliton transmission in nonlinear photonic lattices
DDisorder-induced soliton transmission in nonlin-ear photonic lattices
Yaroslav V. Kartashov, Victor A. Vysloukh, and Lluis Torner
ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain *Corresponding author: [email protected]
Received Month X, XXXX; revised Month X, XXXX; accepted Month X, XXXX; posted Month X, XXXX (Doc. ID XXXXX); published Month X, XXXX We address soliton transmission and reflection in nonlinear photonic lattices embedded into uniform Kerr nonlinear media. We show that by introducing disorder into the guiding lattice channels, one may achieve soliton transmission even under conditions where regular lattices reflect the input beam completely. In contrast, in the parameter range where the lattice is almost transparent for incoming solitons, disorder may induce a significant reflection.
OCIS Codes: 190.5940, 190.6135
Soliton propagation in quasi-periodic and random refractive index landscapes has been extensively discussed over the last years [1-4]. Under appropriate conditions, disorder may completely change the evolution of light. It may cause the formation of spatially localized states even in linear lattices [5-9]. In the fully nonlinear regime, disorder may result in trapping of strongly localized walking solitons [10] or, vice versa, it may facilitate the transport of such states in quasi-periodic lattice [11]. Disorder leads to Brownian soliton mo-tion [12,13] and percolation [14]. Most of previous studies address the effect of disorder on light evolution in infinite periodic systems, while the effects that may become avail-ab sion even un er conditions where when the regu-r lattice is transparent. We use the nonlinear Schrödinger equation to describe the propagation of light beams inside a disordered optical lattice imprinted in a Kerr nonlinear medium: le in finite disordered systems have not been studied yet. In this Letter we consider soliton transmission and reflec-tion by disordered nonlinear photonic lattices of finite width embedded into a uniform medium. Transmission in such lattices may be obtained by tailoring solutions in the uni-form medium and in the lattice using continuity conditions. Thus, waves in the uniform medium should be matched with Bloch-like lattice eigenfunctions. Since disorder drasti-cally affects the topology of lattice eigenfunctions one ex-pects considerable disorder-induced modifications in the transmission and reflection properties. Backscattering ex-periments have been already used to study Anderson local-ization in strongly scattering media, such as semiconductor powders [15,16]. Here we show that addition of disorder into finite lattices may lead to considerable transmisder conditions where the corresponding regular lattice reflects the input beam completely, or, vice-versa, one may increase reflection undla q qi pQ q q q ηηξ ∂ ∂= − − −∂∂ (1) Here η and ξ are the normalized transverse and longitudi-he second transverse coordi-nate is omitted b ly elliptical or guided by a slab waveguide); the parameter p he linear refractive index modulation depth; the lattice shape is described by the function ( 1)/2( 1)/2 ( ) ( ) n mm n Q G η η η −=− − = − ∑ , where the profiles of indi-vidual guides are given by ) exp( / ) G a η η = − . The coor-dinates m η of waveguide centers, or scattering centers, are ized such that m m md s η = + , where d c-ing of wavegui gular lattice, while m s stan the random shift of the wavenal coordinates, respectively (tecause the beam is supposed to be high characterizes t the spdsly distri ( random des in the reut − at e width of μ anddes may bemplex amplit ρ . Withi a as exp( nn ρ ρ = − ∑ efle isguide center uniform a mean spaciude reflection ction of eak scatt fo ) i ϕ , where ϕ a forb- ϕ α ed within the segment d d [ , ] S S − + . Such array with a fi-nite number of guides n is embedded into an unbounded uniform Kerr nonlinear medium. We set p = , a = , and d = that correspond to a refractive index contrast n δ ≈ the wavelength λ μ = and for a waveguid ng of
16 m μ . To gain intuitive insight, we first conducted a qualitative analysis where gui considered as scatterers characterized by co ients n the Born approxim w rs the net amplitude reflection coefficien exp( ) i αη interacting with an array of n guides can be es-timated mm = m is the phase difference for waves backscattered by the first and m -th waveguides. In the regular array where m d the energy r ction coefficient is therefore given by oefficerer plane waves m = t sin ( )/ sin ( ) n n r r n d d ρ α α = = , with
21 1 r = the Bragg condition d M α π = ( 1, 2,...) M = is satisfied one gets a considerable reflection with n r n ∼ . In contrast, in a dis-ordered array the mean value of energy reflection coefficient is given by ρ . If exp[ ( )] nn l mm l m l r r n r i ϕ ϕ = ≠ = + − ∑ , where the angular brackets stand for statistical averaging and , m l ϕ ϕ are random phases. In particular, in the strong dis-order lime it, when for different array realizatioan reflection coeffici l m fluctuates between ns, the lastent ϕ ϕ − π − rm vanishes a and π nd tethe m n r r (cid:17) le approach works well for p for localized beamam are involvelinearity further co reshapittices accurately. n d n (1) directly to study be decreases sub-stantially. Such a simwaves but it iscatterers covered by the bprocess. Ththe picture Therefore, pof we integrate Eq. lane nly ion licates am s s, since oreflecte presence non mpbeam g upon propaga-propagation in disordered la not accurateesince it causes in th e tion. Figure 1. The averaged (over realizations) propagation dynam-ics of broad solitons = colliding with disordered arrays of waveguidesat (a , S , ( , S , , S , (d) .80 d , d , and (f) , d S . In (a)-(e) n while in (f) n = . Dashed lines indicate the mean positions of the cen d α = = with ) b)(c) χ = α = α = rray. , S α = = , (e) = d = α = , tral α = = veguide in t d = he a S wa Figure 2. (a) Reflection coefficient in the regular arrays versus input angle for different number of ides. (b) Reflection and trans-mission coe ve wavegufficients rsus number of waveguides at . In al
In the simula α = l cases the input soliton has form-factor χ = . tions we use as the input the soliton sech[ ( )]exp[ ( )] i ξ χ χ η η α η η = = − − having a form-factor χ and launched into a uniform nonlinear medium far from the finite array. Here α stands for the incidence angle m e. We calcu-late statistically averaged reflection q easured from the array edge in the plan ( , ) η ξ ref in / R U U = and transmission tr in U part / T U = coelow for a iculars fficients, where the re-flected energy f der realization is define disord a ( , ) q d η ξ η −∞ , while the transmitted energy flow is defined as U = ∫ ( , ) U q d η ξ η +∞ = ∫ , where the final distance end 0 ξ η α = was selected to ensure sufficiently large separation f the output reflected andtr o ir approximthe bandgap lattice e value m ddle of the third finite gap at The function h different maxansmitted beams from the array edges. in U χ = in the input soliton energy flow. The level of disorder is controlled by the parameter d S . In regular lattices the transmission and reflection coeff -cients for broad low-power solitons depend strongly on the incidence angle α [17]. While for certain angles total reflec-tion occurs [see propagation dynamics in Fig. 1(a) for α = ], even moderate detuning of α from the resonant values results in almost complete soliton transmission [see Fig. 1(d) fo . This is illustrated in Fig. 2(a), that shows the angular dependencies of the reflection coefficient for different number of waveguides n . Note the qualitative similarity of these dependencies and the expression for the reflection coefficient sin ( )/ sin ( n d d α α ∼ obtained in the Born ation. The maxima in ( ) R α are caused by spectrum. Th flection occurs when the propagation constant of the input beam, given by /2 b α = − at χ → falls into one of the finite gaps of th = α ] ) u. For the parameters of Fig. 2 the = falls exactly into the [ 14.45 13.20] b ∈ − exhibits several maxima associated witflection c waveguides in the lattice [Fig. s, re , − oefficient inlattice spectrucorresponding to ( ) R α gaps. Thenumber of m α imum re b icreases with the .2(b)]. Figure 3. (a) Reflection coefficient versus disorder level for n = at α = (curve 1) an α (curve 2). (b) Reflection coeffi-cient versus disorder level for n = at α = (curve 1) and α = (curve 2). (c) Reflection and transmission coefficients vs n at α = , d S = . In all cases χ = . The important finding repo d = rted latticeics for = plete co Ficident angle solitons propagate through the rs, reflection increases too anin this Letter is that the ablargely incthe identical inumove picture may change drastically in the presence of dis-order. Thus, addition of even weak disorder with d S = reases the mean transmission coefficient T of a lattice that otherwise reflects all light launched at the reso-nant angle α = [compare Fig. 1(b) showing the aver-aged reflection dynamics of disorder s with ) corresponding to regular lattices]. Interestingly, larger disorder does not necessarily result in larger transmission: Fig. 1(c) illustrates that a further growth of disorder up to d S = increases the average reflection on the lattice. Under appropriate conditions, dis-order may also result in the opposite effect. Namely, it can suppress soliton transmission completely. Thus, Fig. 1(e) shows the averaged dynam ed d S ntrast toeciable refl Fi α = where g. 1(a and appearance of considerable reflection. This is in com d), for ection. When the d so where one can clearly see the nregular lattice without any appber of waveguides grow g. 1(oes the width of the statistically averaged reflected beam [Fig. 1(f)]. The beam becomes asymmetric and it may have long exponentially decaying tails which are typical of the diffusive regime of light scattering. Figure 4. Histograms of the reflection coefficient calculated for 130re icient are e ne smg.pFigures 4(a) and 4(b) show histograms of the reflection co-efficient calculated over random lattice realizations. Both his ng a con-n and most probable val-uemoof the even inn of refra e inde ntrast and to the overall de-cr tested in uide arrays that ricated (see, e.g., [8]) or created by optical lattice induction, a technique spe-cially suited to address statistical problems [6]. It is worth ics beca r-induced effects einrich, ) (2008).
V. Folli and C. Conti, Phys. Re 4, 193901 (2010).
Y. Kartashov, V. A. Vys ner, Opt. Express 15, 12409 (2007).
D. S. Wiersma, P. Bartolini, A. Lagendij, and R. Righini, Na-390, 671 (1997).
F. J. P. Schuurmans, M. Me ns, D. Vanmaekelbergh, and A gendijk, Phys. Rev. 99). 0 s alizations of disordered waveguide arrays for χ = , n = , α = at d S = (a) and (b). (c) Reflection coefficient ver-sus disorder level for n = , α = at χ = (curve 1), (curve 2), and (curve 3). The statistical properties of the soliton transmission are analyzed in Figs. 3(a) and 3(b). For incident angles corre-sponding to resonant reflection (curves 1) the statistically averaged reflection coeff first decreases, thus indi-cating disorder-induced partial transparency, and then starts performing decaying oscillations, gradually approach-ing a limiting value at large values of d S . We did not in-crease disorder beyond / 2 d because for d S > the neighboring guides may overlap (note that already at d S > the width of eigenmodes in the disordered lattice becomes smaller than the array width). In c t, for α values detuned from the resonant angle (curves 2) one ob-es an almost monotonic growth of the reflection coeffi-cient with increasing d S , indicating disorder-induced partial reflection. For all values of the input angle α the reflection coefficient R approaches the same limiting value at strong diso uch limiting value grows as the number of waveguides increases. For example, for n = the limiting value amounts to about , while for n = it approaches t R Fim ontrasdisorder d S = ission coeffi- The plots show a arison to regular lattices servrders. d ( ) R S isThe dependencients on mu[see Fig. 2(b)]. S attai n ch slower growth of . This indicates that the largest variations of the reflec-tion coefficien available in arrays with small n , while the largest variations of the transmission are possible in arrays with large n . In any case, th first deep minimum of tograms are strongly asymmetric, indicatisiderable difference between mead at thecies of the re are presented in R relativelyfl in co all . ection and transm 3(c). s of R . At the small disorder d S = reflection scenar-ios dominate, but already at d S = transmission becomes re probable. Nonlinearity substantially affects the reflection properties ice [see Fig. 4(c) that depicts the average reflection coefficient versus d S for different form-factors χ of input soliton]. The impact of nonlinearity is most pronounced in weakly disordered arrays with d S < where addition of nonlinearity results in drastic reduction of reflectio the regular case. This occurs due to the reduction of reflec-tion on individual guides due to nonlinearity-mediated di-mi ish lattingsets of waveg V. ture . La nctiv x couse loukh Lett ease of the number of scatterers covered by the soliton. Interestingly, at d S = when disorder results in maximal diminishing of reflection coefficient in the linear case, the coefficient R is almost insensitive to variations in χ . Summarizing, the disorder-induced soliton transmission and reflection reported here illustrate another example of the new phenomena that appear due to the interplay be-tween disorder and nonlinearity. Here we focused on predic-tions for the averaged quantities, which may be may be fabthe disorde v. Lett. 10, and L. Torge. 83, 2183 (19 tressing, however, that our predictions have key implica-tions beyond statistmay occur for every sample. References F. Abdullaev, Theory of Solitons in Inhomogeneous Media (Wiley, New York, 1994). F. Abdullaev and J. Garnier, Prog. Opt. 48, 35 (2005). F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. As-santo, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008). Y. V. Kartashov, V. A. Vysloukh and L. Torner, Prog. Opt. 52, 63 (2009). T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004). T. Schwartz, G. Bartal, S. Fishman and M. Segev, Nature 466, 52 (2007). Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N.Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008). A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. HR. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010). G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
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