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Defect-controlled Anderson localization of light in photonic lattices

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2013 Phys. Scr. 2013 014001

(http://iopscience.iop.org/1402-4896/2013/T157/014001)

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Phys. Scr. T157 (2013) 014001 (4pp) doi:10.1088/0031-8949/2013/T157/014001

Defect-controlled Anderson localization oflight in photonic latticesDragana Jovic1,2

1 Institute of Physics, PO Box 68, University of Belgrade, Serbia2 Institut fur Angewandte Physik and Center for Nonlinear Science (CeNoS), WestfalischeWilhelms-Universitat Munster, D-48149 Munster, Germany

E-mail: jovic@ipb.ac.rs

Received 12 December 2012Accepted for publication 11 February 2013Published 15 November 2013Online at stacks.iop.org/PhysScr/T157/014001

AbstractThe transverse localization of light in a disordered photonic lattice with a central defect isanalyzed numerically. The effect of different input beam widths on various regimes ofAnderson localization is investigated. The inclusion of a defect enhances the localization ofboth narrow and broad beams, as compared to the lattice with no defect. But, in the case of abroad beam a higher disorder level is needed to reach the same localization as for a narrowinput beam. It is also investigated how the transverse localization of light in such geometriesdepends on both the strength of disorder and the strength of nonlinearity in the system. Whilein the linear regime the localization is most pronounced in the lattice with the defect, in thenonlinear regime this is not the case.

PACS numbers: 42.25.Dd, 72.15.Rn

(Some figures may appear in color only in the online journal)

1. Introduction

The localization of waves in disordered media, known asAnderson localization, is a universal phenomenon predictedand observed in a variety of classical and quantumwave systems [1–3]. Originally proposed for electrons andone-particle excitations in solids [1], it was soon extendedto many other fields such as acoustics, Bose–Einsteincondensates and optics. Anderson localization of lighthas become an important part of recent investigationsof random photonic lattices. In particular, the transverselocalization of light in a disordered, two-dimensionalphotonic lattice was realized experimentally [4]. The effectof a medium’s nonlinearity on light localization wasexperimentally demonstrated with the observation that thenonlinearity in a disordered medium favors localizationwithin a shorter distance [5]. Interesting issues concerningthe effect of nonlinearity on Anderson localization of lighthave been reported theoretically in photonic lattices withdimensionality crossover [6], near boundaries of disorderedphotonic lattices [7] and at the interface between linear andnonlinear dielectric media [8]. The past decade has also

witnessed progress in random lasing, which is based on thelight localization in random gain media [9].

Defect states in photonic lattices [10, 11] and theformation of defect channels [12, 13] have attracted increasedinterest in recent years. Generally, defects can be createdby changing the spacing of two adjacent waveguides inan otherwise uniform array, by variation of the effectiveindex or the width of a single channel, or by opticalinduction techniques. These defects disrupt the periodicityof the crystal, creating optical states within the otherwiseforbidden bandgap frequencies. That has not only a direct linkto technologically important systems of periodic structuressuch as photonic crystal fibers, but also brings about thepossibility of studying, in an optical setting, many novelphenomena in periodic systems beyond optics such as edgedislocation, defect healing, eigenmode splitting and nonlinearmode coupling. In addition, the insertion of point-like orline-like defects into photonic crystals or waveguide arraysmay provide an additional physical mechanism for lightconfinement, and a possibility to control light propagation inphotonic lattices. Light can be localized within defect regionsand manipulated by engineering the defect geometry and

0031-8949/13/014001+04$33.00 1 © 2013 The Royal Swedish Academy of Sciences Printed in the UK

Phys. Scr. T157 (2013) 014001 D Jovic

Figure 1. Lattice geometry: sketch of the triangular photonic latticewithout a defect (a) and with a central defect (b). The crosses markthe locations of input Gaussian beams.

placement. This topic has recently attracted special attention,owing to its novel physics and various potential applications,such as high-quality waveguide structures, complex integratedcircuits such as beam splitters and optical routers, and lasercavities [12, 14].

In this paper, these two concepts are extended to thetransverse light localization in a disordered photonic latticewith different lattice geometries (with and without defects). Itis analyzed numerically how lattice defects affect and modifythe Anderson localization of light, considering also a casewith the lattice without defects (see figure 1). The effectof different input beam widths on transverse localization isinvestigated, in both geometries. In the linear regime, thepresence of defects enhances localization for both narrow andbroad beams, as compared to the lattice with no defects. Butfor the broad input beam, a higher disorder level is needed toreach the same localization as for the narrow input beam, i.e.to observe the same localization lengths.

Also, it is investigated how the transverse localizationof light depends on both the strength of the disorder andthe strength of the nonlinearity in the system. The characterof such localization is non-trivial, depending on the jointinfluence of all parameters: the lattice geometry, the levelof disorder and the strength of nonlinearity. Two nonlinearregimes are considered: focusing and defocusing. However,compared with the case with no lattice defects, the presenceof defects in the focusing regime enhances the localization.On the other hand, in the defocusing nonlinear regime, thesuppression of localization in the presence of defects isdemonstrated, as compared to the lattice without defects.

2. Theoretical modeling of light localization inphotonic lattices

The localization of light in optically induced disorderedphotonic lattices with central lattice defects has been studied,and compared with the case without defects. The propagationof a light beam along the z-axis is described using thescaled nonlinear Schrodinger equation for the electric fieldamplitude E :

i∂ E

∂z= −1E − γ |E |

2 E − V E, (1)

where 1 is the transverse Laplacian, γ is the dimensionlessstrength of the nonlinearity and V (x, y) is the transverselattice potential, with the peak intensity V0. A scaling

x/x0 → x , y/x0 → y, z/LD → z is utilized for thedimensionless equation, where x0 is the typical full-widthat half maximum (FWHM) beam waist and LD is thecorresponding diffraction length. The propagation equation issolved numerically by employing a beam propagation methoddeveloped earlier [15]. Narrow Gaussian beams positionedat the defect center are launched perpendicular to the lattice,and it is observed how they spread in propagation.

To study Anderson localization effects, one has to usea random lattice potential Vr, instead of the perfect periodicpotential V . The disorder is realized using a randomizedlattice peak intensity, V0r, which takes values from the intervalV0(1 − Nr) < V0r < V0(1 + Nr), where r is the randomnumber generator and N determines the degree of disorder.They both may take values from the interval [0,1]. Thedisorder level is quantified by the ratio between the intensityof the random lattice and the intensity of the periodic lattice.Thus, the position of randomized peaks is regular; only theirintensities vary. In the end, Gaussian beams are launched. Thestudy of Anderson localization involves statistical descriptionof different realizations of a randomized system. In numericsthey are realized by starting each simulation with a differentseed of the random number generator and then formingensemble averages.

3. Effect of input beam width on localization in thelinear regime

First, localization effects of various input beams for differentlattice geometries are investigated in the linear regime(γ = 0). To estimate how the lattice geometry affects thelocalization process, the corresponding localized modes in thedisordered photonic lattice with and without lattice defects arecompared. Localization effects for narrow (FWHM = 10 µm)and broader input beams (FWHM = 50 µm) are analyzedin some detail. It is observed how the input beamwidth influences the localization process in differentlattice geometries. To compare Anderson localizationin different regimes, quantitative analysis is performedby measuring the standard quantities characterizingthe localization: the inverse participation ratio, P =∫

|E |4(x, y, L) dx dy/

{∫|E |

2(x, y, L) dx dy}2

, the effectivebeam width ωeff = P−1/2 and the localization length. Tomeasure ensemble averages for the quantities of interest,many realizations of disorder are needed. For each disorderlevel, the averaged quantities are taken over 100 realizationsof disorder. In both lattice geometries with and withoutdefects, the effect of Anderson localization is observed byincreasing the level of disorder. Figure 2 presents typicalexamples of localized broad beam modes in both geometrieswith and without defects.

The effective beam width is measured at the latticeoutput for different lattice geometries and various input beamwidths in the linear regime, for different disorder levels. Thegeneral trend for both cases (with and without defects) issimilar: the effective beam width decreases as the level ofdisorder is increased, or, in other words, disorder alwayssuppresses transport. But, for the lattice with the defect,the reduction in the effective width is much steeper than inthe case with no defect. Similar conclusions follow for the

2

Phys. Scr. T157 (2013) 014001 D Jovic

Figure 2. Examples of localized modes for a broad input beam inthe linear regime. Physical parameters are: the crystal lengthL = 20 mm, input lattice intensity V0 = 1, lattice period 11.2 µm,input beam intensity |E0|

2= 0.5 and input beam FWHM = 10 µm.

Figure 3. Comparison between transverse localization of narrowand broad beams in the linear regime for different latticegeometries. The averaged effective beam width at the lattice outputis shown as a function of the disorder level, for different input beamwidths. Points are ensemble averages and lines are the least-squarefits through the points. Error bars depict the spread in values comingfrom statistics. Parameters are as in figure 2.

localization of both narrow and broad beams. To compare thelocalization in different regimes, averaged effective widths arenormalized to the corresponding values without disorder. Ifsuch a normalized effective beam width is used to indicatethe strength of localization (figure 3), it is observed that thelocalization is more pronounced for the lattice with defectsthan with no lattice defects. Such a normalized quantity is ageometric criterion, in that if the effective beam width is morereduced, the localization is more pronounced.

To compare the localization of narrow and broad inputbeams, the averaged intensity profiles obtained at differentdisorder levels are also used, in the geometry with a latticedefect (figure 4). Increasing the level of disorder, the output

Figure 4. Ensemble-averaged intensity profiles (on a logarithmicscale) at the lattice output in the linear regime for narrow (red line)and broad (black line) beams for a 60% disorder level. Parametersare as in figure 2.

intensity beam profile narrows down and the exponentiallydecaying tails become a direct indication of the level oflocalization. It is seen that for a 60% disorder level, thelocalization regime for the broad beam is not yet reached,as the average beam profile is non-exponential. This meansthat a higher disorder level is necessary to observe thelocalization of the broad beam under the same conditions (forthe same parameters). If the localization length ξ is measured,by fitting the averaged intensity profile to an exponentiallydecaying profile I∼ exp(−2|x |/ξ), the value ξ = 35.7 µmis determined. For a broader beam, such a value of thelocalization length is observed at the ≈95% disorder level.

4. Transverse localization of light in the nonlinearregime

It is of great interest to know how the nonlinearity affects thelocalization process in different lattice geometries. Next, theinfluence of the medium nonlinearity on the localization isstudied, and how the simultaneous existence of nonlinearity,defects and disorder affects Anderson localization. As before,a quantitative analysis is performed, measuring the standardquantities and comparing Anderson localization in differentlattice geometries. Again, the effective beam width ismeasured at the lattice output but, to compare the localizationin different regimes, averaged effective widths are normalizedto the corresponding values without disorder. The strengthof the nonlinearity γ = 5 is used as representative for thefocusing, and γ = −5 for the defocusing case.

While in the linear regime the localization is morepronounced in the lattice with a defect than without a defect,the inclusion of the medium’s nonlinearity changes theseconclusions. In the defocusing nonlinear regime (figure 5(a))the localization is more pronounced in the geometry without adefect, at lower levels of disorder. As the strength of disorderis further increased, the localization with a defect becomesmore pronounced than without a defect. In the focusingnonlinear regime (figure 5(b)), the conclusions are the sameas in the linear case.

3

Phys. Scr. T157 (2013) 014001 D Jovic

Figure 5. Comparison between Anderson localization in differentnonlinear regimes. (a) Defocusing nonlinearity and (b) focusingnonlinearity. The normalized effective beam width at the latticeoutput is depicted as a function of the disorder level. The widths arenormalized to their values without disorder. Points are ensembleaverages and lines are least-squares fits through the points.Parameters are as in figure 2.

For the narrow input beam, the localization depends onthe joint influence of the defect and nonlinearity. However,in the case of the broad input beam the inclusion of defectsalways enhances localization in both the linear and nonlinearregimes.

5. Conclusions

In conclusion, it is analyzed numerically how the presence ofa defect modifies Anderson localization of light in photoniccrystals. The effect of input beam width on Anderson

localization is considered for various lattice geometries. Fora broad input beam, a higher disorder level is needed toreach the same localization as for the narrow input beam.The presence of a defect could suppress or enhance lightlocalization, depending on the strength of nonlinearity, aswell as on the strength of disorder in the system. In thelinear regime, the defect enhances the localization of light.But in the nonlinear regime, localization effects depend onthe strength of the nonlinearity differently, for different latticegeometries. Such studies on the light localization in photoniccrystals with defects might be of interest in the realizationof novel light-manipulating optical circuits for future opticaltelecommunication and sensing networks.

Acknowledgments

This work was supported by the Ministry of Educationand Science of the Republic of Serbia (project no. OI171036). DMJ expresses gratitude to the Alexander vonHumboldt foundation for the Fellowship for PostdoctoralResearchers.

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