We study experimentally light localization at phase-slip waveguides and at
the intersection of phase-slips in a two-dimensional (2D) square photonic
lattice. Such system allows to observe a variety of effects, including the
existence of spatially localized modes for low powers, the generation of
strongly localized states in the form of discrete bulk and surface solitons, as
well as a crossover between one-dimensional (1D) and 2D localization.Comment: 4 double-column pages, 6 figures, submitted for publicatio

Alexander Szameit

Apalkov

Christodoulides

Fedele

Felix Dreisow

Fleischer

Itoh

Kivshar

Krolikowski

Mario I. Molina

Matthias Heinrich

Molina

Morandotti

Robert Keil

Rosberg

Stefan Nolte

Sukhorukov

Suntsov

Szameit

Trompeter

Wang

Yuri S. Kivshar

Optics Letters

English

arXiv

We experimentally study light localization at phase-slip waveguides and at the intersection of phase slips in a two-dimensional (2D) square photonic lattice. Such systems allow for the observation of a variety of effects, including the existence of spatially localized modes for low powers, the generation of strongly localized states in the form of discrete bulk and surface solitons, as well as a crossover between one-dimensional and 2D localization.This work was supported by the Leopoldina—German
Academy of Science (grant LPDS 2009–13), Fondo Nacional
de Desarrollo Cientifico y Tecnologico (grant
1080374), the Programa de Financiamiento Basal de Conicyt
(grant FB0824/2008), the Deutsche Forschungsgemeinschaft
(DFG) (Leibniz program), and by the
Australian Research Council

Szameit, Alexander

Molina, Mario I

Heinrich, Matthias

Dreisow, Felix

Keil, Robert

Nolte, Stefan

Kivshar, Yuri

The Australian National University

Observation of localized modes at phase slips in two-dimensional photonic lattices

Artículo de publicación ISIWe study experimentally light localization at phase-slip waveguides and at the intersection of
phase-slips in a two-dimensional (2D) square photonic lattice. Such system allows to observe a
variety of effects, including the existence of spatially localized modes for low powers, the generation
of strongly localized states in the form of discrete bulk and surface solitons, as well as a crossover
between one-dimensional (1D) and 2D localization.This work was supported by the Leopoldina-German
Academy of Science (grant LPDS 2009-13), Fondecyt
(grant 1080374), Programa de Financiamiento Basal de
Conicyt (grant FB0824/2008), Deutsche Forschungsge-
meinschaft (Leibniz program), and by the Australian Re-
search Council

Molina Gálvez, Mario

Kivshar, Yuri S.

Repositorio Académico de la Universidad de Chile

arXiv:1004.3728v1  [nlin.PS]  21 Apr 2010Observation of localized modes at phase slips in two-dimensional photonic latticesAlexander Szameit1, Mario I. Molina2,3, Matthias Heinrich4Felix Dreisow4, Robert Keil4, Stefan Nolte4, and Yuri S. Kivshar51Solid State Institute and Physics Department, Technion, 32000 Haifa, Israel2Departmento de F́ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile3Center for Optics and Photonics, Universidad de Concepción, Casilla 4016, Concepción, Chile4Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany5Nonlinear Physics Center, Research School of Physics and Engineering,Australian National University, Canberra ACT 0200, AustraliaWe study experimentally light localization at phase-slip waveguides and at the intersection ofphase-slips in a two-dimensional (2D) square photonic lattice. Such system allows to observe avariety of effects, including the existence of spatially localized modes for low powers, the generationof strongly localized states in the form of discrete bulk and surface solitons, as well as a crossoverbetween one-dimensional (1D) and 2D localization.The study of nonlinear dynamics in waveguide arraysand photonic lattices has attracted a great deal of atten-tion due to the possibility to observe many novel effects ofnonlinear physics and possible interesting applications inphotonics [1]. In particular, it was shown that discretenonlinear photonic systems can support different typesof spatially localized states in the form of discrete soli-tons [2]. These solitons can be controlled by the insertionof suitable defects in an array, as was suggested theoreti-cally [3] and also verified experimentally for arrays of op-tical waveguides [4]. Defects may provide an additionalphysical mechanism for light confinement, and they cansupport spatially localized modes, which have been stud-ied theoretically for different linear [5] and nonlinear [6–8] models and observed experimentally in 1D [5, 9] and2D [10] photonic lattices.Recently, a novel type of nonlinear mode has been in-troduced [11, 12] at the interface between two spatiallyshifted nonlinear waveguide arrays. These defect modesare closely linked with the phase-slip defects in 2D pho-tonic crystals [13]. In the presence of a phase-slip, thedistance between two lattice sites located at both sides ofthe phase-slip is a non-integer multiple of the lattice con-stant. As a result, the nonlinear phase-slip defect modespossess the specific properties of both discrete nonlinearsurface modes and bulk solitons.In this Letter we study experimentally, for the firsttime to our knowledge, light localization in a 2D squarephotonic lattice containing one phase-slip defect or an in-tersection of two phase-slip defects. We observe a varietyof novel effects, including linear modes localized at thecross intersection of the phase-slips [13], strongly nonlin-ear localized states in the form of discrete solitons at asingle slip [11], and discrete surface solitons at the lat-tice edges [12]. We also observe and discuss a crossoverbetween 1D and 2D localization.The considered system consists of an array of N × Nnonlinear focussing (Kerr) waveguides, originally form-ing a square lattice with periodicity a. Breaking thetranslational symmetry of the system by altering the dis-tance between two consecutive rows or columns creates aFIG. 1: (Color online) Schematics of an intersection of twodifferent phase slips (H,V 6= a). The case of a single phaseslip corresponds to either H 6= a or V 6= a. Marked waveg-uides show the excitation points.phase defect or a cross intersection of two phase-slips (seeFig. 1). The coupling across the defect in the x-direction(y-direction) is termed CH(CV ). In the framework of thecouple-mode theory, the evolution equations for the elec-tric field amplitudes En(z) can be written in the form,idEndz+∑mCn,mEm + γ|En|2En = 0 (1)where En(z) is the electric field (in units of (Watt)1/2)on waveguide n at distance z (in meters), Cn,m is thecoupling coefficient (in units of 1/meter) between guidesn and m, and γ (in units of 1/(Watt×meter)) is the non-linear coefficient, defined by γ = ω0n2/cAeff , where ω0 isthe angular frequency of light, n2 is the nonlinear refrac-tive index of the guide and Aeff is the effective area ofthe linear modes. Equation (1) is normalized by defininga dimensionless distance s ≡ Cz, where C is the cou-pling between nearest-neighbor guides outside the vicin-ity of the slip defect, and the dimensionless electric field2ABHaL8 10 12 140102030ΒPowerAHbLB6 7789ΒPowerFIG. 2: Power diagrams for the cases: (a) The mode is local-ized at a cross intersection of two phase slips with V = H =29µm. (b) Surface mode localized at a single phase slip withV =25 µm, H = a = 40 µm. Solid lines refer to stable stateswhile dashed branches refer to the unstable modes.φn = (γ/C)1/2En. With the above definitions, the con-served power (in Watts) is given by∑n |En|2 = (C/γ) P ,where P ≡∑n |φn|2 is the dimensionless power. As wasfirst shown by Apalkov and Raikh [13], a topological de-fect created by the intersection of two phase-slips, canFIG. 3: (Color online) Light diffraction and localization ina perfect square lattice, for different input powers: (a) 75kW, (b) 1600 kW, and (c) 2000 kW. Excited waveguides aremarked with white circles.support a linear localized mode. Looking for station-ary linear modes (γ = 0) of the system (1) in the form,φn(s) = φn exp(iβs), we find a linear localized mode ata fixed value of the propagation constant,β = CV +1CV+ CH +1CH, (2)provided CV > 1 and CH > 1. In the nonlinearcase, we follow the recent theoretical results [11, 12] anddemonstrate that nonlinearity can support a variety of lo-calized modes including the modes which bifurcate fromthe symmetric states and describe nonlinear asymmet-ric localized states. The corresponding bifurcation dia-gram of low-power branches of localized modes in shownin Figs. 2(a,b) for bulk and surface modes, respectively.When the mode power grows, the symmetric mode A de-scribed in the anti-continuum limit by the pattern (+ ++ + )becomes unstable, and it transforms into the asymmet-ric mode B with the pattern (+ 00 0) corresponding to thelower bifurcation branch in Fig. 2(a). Similarly, the sur-face symmetric mode (++ ) bifurcates into the asymmetricmode (+0) above a certain power threshold, as shown inFig. 2(b). Whereas in the case of Fig. 2(a), the localizedmodes extend toward vanishing powers, a finite thresholdpower is required to generate a surface mode in the caseof Fig. 2(b).For the fabrication, we employ the femtosecond direct-writing technique [14]. The specific fabrication parame-ters can be found, e.g., in [15]. Each array is 100 mmlong and consists of 7 × 7 waveguides, as shown in Fig.1(a). The spacing between the individual lattice sitesis a =40µm except in the phase-slip channels, wherethe spacing is decreased. In the experiments, light ofa Ti:sapphire chirped pulse amplification laser system(RegA, Coherent), with a pulse duration of about 180fs and a repetition rate of 150 kHz at 800 nm, was cou-pled into the central guide using a 4 × microscope objec-tive [numerical aperture (NA) of 0.1], coupled out by a10 × objective (NA = 0.25), and projected onto a CCDcamera.First, we study the light propagation in a perfect lat-tice V = H = a =40µm. Figures 3(a-c) show the ex-perimental images of the light at the lattice output fordifferent input powers. In the linear regime [Fig. 3(a)],we observe strong discrete diffraction but, when the inputpower grows, the diffraction is suppressed [Fig. 3(b)], andthe generation of a discrete 2D soliton [16] is observed forthe input power of 2000 kW [Fig. 3(c)].Next, we consider a symmetric cross-intersection of twoFIG. 4: (Color online) (a) Experimental and (b) Theoreticalresults for the generation of a linear localized mode at an in-tersection of two phase-slip waveguides for V = H =22 µmand an input power 75 kW [cf. Fig. 3(a)]. Excited waveg-uides are marked with white circles; phase slips are markedby dashed lines.phase-slip waveguides with the parameters V = H =22µm, and excite one of the waveguides near the cross, asmarked in Fig. 1. Even for a power level as low as 75kW we observe, in a sharp contrast with the diffractionpattern of Fig. 3(b), the generation of a linear modelocalized at the phase-slip intersection. Figures 4(a,b)show both experimental and numerical images of the lightintensity at the output facet that confirm the generationof a strongly localized state at the phase-slip defect, inaccord with the theory [13]. Because we excite only onesite of the lattice, the resulting four-site mode can not begenerated as a stationary mode, but it is clearly seen howthe light is bound to the phase-slip, which is reproducedby direct numerical simulations, see Fig. 4(b).3FIG. 5: (Color online) Experimental images of the light in-tensity at the output facet showing the generation of (a) a1D stripe surface soliton for an input power 1700 kW, and(b) its transformation into a 2D soliton at 2000 kW. Thewaveguide parameters are: for H =29 µm, V = a =40 µm.Excited waveguides are marked with white circles; phase slipsare marked by dashed lines.For the next series of experiments, we fabricate a singlephase slip and excite one of the centralFIG. 6: (Color online) Experimental images of the outputlight intensity showing the generation of (a,b) 2D soliton (H =V =29µm), and (c,d) generation of a surface soliton at a phaseslip (H =40 µm and V =25 µm). Excited waveguides aremarked with white circles; phase slips are marked by dashedlines.waveguides, as shown in Fig. 5(a). For low inputpower, we excite one of the linear guided modes of thisphase slip waveguides (not shown), that undergoes sym-metry breaking for larger powers and transforms into anasymmetric 1D stripe soliton, as shown in Fig. 5(a) for1700 kW. Increasing the input power further, we observea sharp crossover between 1D and 2D localization andthe generation of a 2D soliton, see Fig. 5(b).The generation of a 2D asymmetric nonlinear mode atthe intersection of two phase-slips (H = V =29 µm) isshown in Fig. 6(a,b). Even at relatively large power (1400kW), the light diffracts across a quarter of the lattice[Fig. 6(a)], but for higher powers (2500 kW) the modebecomes localized eventually [Fig. 6(b)].An example for nonlinear surface mode is presentedin Fig. 6(c,d) for a single phase-slip waveguide, H =40µm and V =25 µm. For lower powers (75 kW), the lightis repelled by the surface [Fig. 6(c)]. This behavior isvery similar to the light propagation in one-dimensionalwaveguide arrays [17]. However, when the input powerexceeds a threshold (2800 kW), we observe the generationof a localized state at the surface [Fig. 6(d)].In conclusion, we have studied experimentally the gen-eration of spatially localized modes at phase-slip waveg-uides and their intersections. We have generated bothdiscrete bulk and surface solitons near the lattice struc-tural defects, in a qualitative agreement with earlier the-oretical predictions. We have also observed an interplaybetween the effectively 1D and 2D dynamics.This work was supported by the Leopoldina-GermanAcademy of Science (grant LPDS 2009-13), Fondecyt(grant 1080374), Programa de Financiamiento Basal deConicyt (grant FB0824/2008), Deutsche Forschungsge-meinschaft (Leibniz program), and by the Australian Re-search Council.[1] See, e.g., D.N. Christodoulides, F. Lederer, and Y. Sil-berberg, Nature 424, 817 (2003).[2] D. Christodoulides and R. Joseph, Opt. Lett. 13, 794(1988).[3] W. Krolikowski and Yu.S. Kivshar, J. Opt. Soc. Am. B13, 876 (1996).[4] R. Morandotti, H.S. Eisenberg, D. Mandelik, Y. Sil-berberg, D. Modotto, M. Sorel, C.R. Stanley, and J.S.Aitchison, Opt. Lett. 28, 834 (2003).[5] H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U.Streppel, D. Michaelis, and A. Bräuer, Opt. Express 11,3404 (2003).[6] Yu.S. Kivshar, Phys. Rev. B 47, 11167 (1993).[7] A.A. Sukhorukov and Yu.S. Kivshar, Phys. Rev. Lett.87, 083901 (2001).[8] F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506(2005).[9] X. Wang, J. Young, Z. Chen, D. Weinstein and J. Yang.4Opt. Express 14, 7362 (2006).[10] A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow,T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A.Vysloukh, and L. Torner, Opt. Lett. 34, 797 (2009).[11] M.I. Molina and Yu.S. Kivshar, Opt. Lett. 33, 917(2008).[12] M.I. Molina and Yu.S. Kivshar, Phys. Rev. A 80, 063812(2009).[13] V.M. Apalkov and M.E. Raikh, Phys. Rev. Lett. 90,253901 (2003).[14] K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRSBull. 31, 620 (2006).[15] A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, and A.Tünnermann, Opt. Express 14, 6055 (2006).[16] J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N.Christodoulides, Nature 422, 147 (2003).[17] S. Suntsov, K.G. Makris, D.N. Christodoulides, G.I.Stegeman, A. Naché, R. Morandotti, H. Yang, G.Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901(2006); C.R. Rosberg, D.N. Neshev, W. Krolikowski, A.Mitchell, R.A. Vicencio, M.I. Molina, and Yu.S. Kivshar,Phys. Rev. Lett. 97, 083901 (2006).

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Observation of localized modes at phase slips in two-dimensional
  photonic lattices

Observation of localized modes at phase slips in two-dimensional photonic lattices

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