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Control of light propagation in one-dimensional quasi-periodic nonlinear photonic lattices

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2014 J. Opt. 16 025201

(http://iopscience.iop.org/2040-8986/16/2/025201)

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Journal of Optics

J. Opt. 16 (2014) 025201 (8pp) doi:10.1088/2040-8978/16/2/025201

Control of light propagation inone-dimensional quasi-periodicnonlinear photonic latticesAna Radosavljevic1,2, Goran Gligoric2, Aleksandra Maluckov2 andMilutin Stepic2

1 School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73,11120 Belgrade, Serbia2 P∗Group, Vinca Institute of Nuclear Sciences, University of Belgrade, POB 522, Belgrade, Serbia

E-mail: anar@vin.bg.ac.rs

Received 9 October 2013, revised 10 December 2013Accepted for publication 12 December 2013Published 15 January 2014

AbstractWe investigate light localization in quasi-periodic nonlinear photonic lattices (PLs) composedof two periodic component lattices of equal lattice potential strength and incommensuratespatial periods. By including the system parameters from the experimentally realizable setup,we confirm that the light localization is a threshold determined phenomenon in a limit ofnegligible nonlinearity. In addition, we show that self-trapping can affect the localized light inthe established setup only in the presence of strong nonlinearity. Guided by these findings weconsider the possibility of governing light propagation by proposing a composite latticesystem comprising alternating quasi-periodic parts with different potential depths andnonlinearity strengths.

Keywords: light localization, quasi-periodic photonic lattices, nonlinearity, light propagationcontrolPACS numbers: 42.65.Tg, 42.65.Wi, 42.82.Et, 63.20.Pw

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

1. Introduction

The localization of waves is a universal phenomenon observedin various non-periodic media. It has been intensively in-vestigated, both theoretically and experimentally, in differ-ent settings such as condensed matter [1, 2], Bose–Einsteincondensates in optical potentials [3, 4], quantum chaotic sys-tems [5–7], sound waves [8] and light [9–19].

In one-dimensional (1D) disordered photonic lattices(PLs) a crossover from extended to localized light modescan occur independently of the quenched disorder strength( Anderson localization [1, 20]). However, to observe local-ization experimentally or numerically in finite size systems,the corresponding localization length should be sufficientlysmall with respect to the lattice dimensions [18]. On the otherhand, for a certain class of quasi-periodic (QP) potentials,

which are characterized by the spatial ordering betweenperiodicity and disorder, localization is a threshold determinedphenomenon [21], independent of the finiteness of the systemand dimensionality. Therefore, finite size 1D QP PLs exhibitphase transition properties as well. Below the transitionthreshold, the modes of the system are extended and thereforean initially narrow wavepacket eventually spreads across theentire lattice. Above the threshold, all modes are localized andexpansion is suppressed [21, 22]. It is shown that, in the contextof the Aubry–Andre (AA) model, weak cubic nonlinearity onlyaffects the width of the localized wavepacket [22] above thetransition.

In this paper we study light localization in 1D QP PLscomposed of two periodic lattices with mutually incommen-surate spatial periods and equal optical potential strengths.Our aim is to suggest that the obtained results allow the man-agement of light beam propagation along the PL (waveguide

2040-8978/14/025201+08$33.00 1 c© 2014 IOP Publishing Ltd Printed in the UK

J. Opt. 16 (2014) 025201 A Radosavljevic et al

array). Therefore, by including system parameters from aneasily experimentally realizable setup [23, 24], we examinePLs that can be fabricated by direct laser beam recording orby titanium in-diffusion in a copper-doped lithium niobatecrystal. The light propagation through these PLs is mathemat-ically modelled by the paraxial, time-independent Helmholtzequation with an included nonlinear media response [25, 26](section 2), which can be considered as a continual limit ofthe AA equations [21, 27]. The model equation is numericallysolved by adopting the split-step Fourier method [25, 28]. Thequalitative measure of the light beam localization is the effec-tive participation number. It has been shown in section 3 that inthe linear QP PL a sharp transition between light beam diffrac-tion and localization can be detected for a critical (threshold)value of the optical potential strength [22, 29]. Additionally, weinvestigate the interplay between quasi-periodicity and nonlin-earity (sections 4 and 5). We model the nonlinear responseto the light propagation of the photorefractive media withsaturable nonlinearity by modulating accordingly the latticerefractive index. The effects of the simultaneous variation ofthe lattice potential depth and the nonlinearity strength to thelight beam dynamics are investigated numerically. The resultsare compared with those obtained for a QP cubic nonlinearSchrodinger lattice [29], disordered nonlinear 1D PLs [25, 30]and the AA model [21].

2. Model equations

The light propagation in a 1D QP, nonlinear PL is describedby the paraxial, time-independent Helmholtz equation [23]:

i∂E∂z+

12k0n0

∂2 E∂x2 + k01n(x)E = 0, (1)

where E(x, z) is the component of the light electric field inthe propagation z-direction, x is the transverse direction [23],k0 = 2π/λ is the wavenumber, n0 is the refractive index of thesubstrate, and λ represents the wavelength of the light field.A stationary transverse lattice potential 1n(x) consists of theQP term nl(x) defining the lattice and saturable nonlinear termdescribed by the function nnl(x):

1n(x)= nl(x)+ nnl(x)= εG(x)−12

n30r Epv

|E |2

1+ |E |2.(2)

In the case of a QP PL, the G(x) represents a QP functionobtained as a combination of two incommensurate spatialharmonics with equal amplitudes:

G(x)= cos2(kLx)+ cos2(αkLx + δ), (3)

where α is an irrational number, kL is the periodic latticewavenumber and δ is the additional phase in the second peri-odic lattice. Parameters in the nonlinear part of the refractiveindex denote respectively the photorefractive field Epv, theelectro-optic coefficient r and the dark irradiance Id.

By introducing the dimensionless variables ξ = k0x andη= k0z, and the dimensionless wave amplitude ψ = E/

√Id,

equation (1) can be rewritten in the dimensionless form:

i∂ψ

∂η+

12n0

∂2ψ

∂ξ2 + εG(ξ)ψ − γ|ψ |2

1+ |ψ |2ψ = 0, (4)

where γ = 0.5n30r Epv, G(ξ) = cos2(kLξ/k0)+ cos2(αkLξ/

k0+ δ) and ε denotes the lattice potential depth. Let us notethat the case G(ξ)= cos2(kLξ/k0)will be used to represent theregular (periodic) lattice in the following. Finally, the equationwhich is numerically solved is given as follows:

i∂ψ

∂η+

12n0

∂2ψ

∂ξ2 + ε[cos2(kLξ)+ cos2(αkLξ + δ)]ψ

− γ|ψ |2

1+ |ψ |2ψ = 0, (5)

where kL = kL/k0 is the dimensionless wavenumber.In order to estimate the value of the localization transition

point we start from the anticontinuum limit of the linearcounterpart of the model equation (5) assuming very deepoptical lattice potential [31]. The discrete wave amplitude ψnis defined by its value at the maxima of the first lattice potential,ξn = nπ/kL. Therefore, discrete model equations in the tightbinding approximation [31] can be written in the form:

i∂ψn

∂η+

12n0

ψn+1+ψn−1− 2ψn

(π/kL)2+ εψn

+ ε cos2(nπα)ψn = 0. (6)

We set δ = 0 without loss of generality. Multiplyingequation (6) by 2n0(π/kL)

2 and rescaling η =η

2n0( kLπ)2,

equation (6) can be rewritten as:

i∂ψn

∂η+ψn+1+ψn−1+

[3n0ε

(π

kL

)2

− 2+ n0ε

(π

kL

)2

cos(2nπα)

]ψn = 0. (7)

The last equation, by denoting β0 = 3n0ε(π/kL)2− 2 and

ζ = n0ε(π/kL)2, is equivalent to the AA model equation

iψn + [β0+ ζ cos(2παn)]ψn +C(ψn+1+ψn−1)= 0. (8)

The model (8) is characterized by self-duality, consistingin the property that the transition from localized to extendedstates in physical space occurs simultaneously with the tran-sition from extended to localized states in the correspondingmomentum space [21, 27]. The transition takes place whenthe parameter values fulfil the condition Qth = ζ/C = 2. Bycomparing equations (7) and (8), it follows that the localizationthreshold in the continual case can be estimated to be of theorder:

Qth = n0ε

(π

kL

)2

= 2. (9)

According to equation (9), the critical value of the poten-tial depth εth, in the starting model (5), is approximately equalto

εth = 2(

kL

π

)2/n0 ≈ 0.000375. (10)

It is expected that for ε > εth the states are localized in thephysical space and extended in the momentum space, while

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J. Opt. 16 (2014) 025201 A Radosavljevic et al

the situation is reversed for ε < εth [21, 22, 27]. This point willbe verified numerically.

The qualitative change from the extended to the localizedstate is observed in the behaviour of the participation number

P(z)=1k0

(∫dξ |ψ(ξ, η)|2

)2∫dξ |ψ(ξ, η)|4

, (11)

which is often used in the literature as an effective measure ofthe confinement of the light beam in the transverse direction,i.e. the effective localization length [18, 32]. To better quantifyour findings, we performed hundreds of different numericalrealizations and averaged the participation number over them.Different realizations of the QP potential are obtained byvarying the phase δ in equation (5). The saturation to finitevalues of the 〈P(z)〉 during the propagation is an indicationof localization in the physical space, and the correspondingvalue of the effective participation number is the effectivelocalization length [25].

The parameter set describing the lattice system andinput light characteristics in our study is taken from theexperimentally realizable setup presented in detail in [23, 24].In order to analyse the effect of quasi-periodicity on the lightpropagation, we use in the numerical simulation those valuesof the parameters for which in the case of regular PL thelight beam diffracts through the lattice, i.e. the wavepacketassociated with the light ballistically extends during thepropagation: n0 = 2.242, r = 30 pm V−1, Epv = 7 kV mm−1

and γ ≈ 0.001. The results presented in the following areobtained for N = 100 lattice elements (waveguides). The widthof a single regular lattice waveguide is set to be wg0 = 4 µm.A single QP PL site is effectively narrower due to the equalpotential depths of the two incommensurate constituent regularPLs. Initially, the lattice is excited at one of the central elementsby a Gaussian beam with a wavelength λ = 514.5 nm andFWHM= 4 µm for purely technical reasons, without any lossof generality.

3. Light propagation in linear QP photonic lattice

In the first stage of our study, we set γ = 0 in equation (5)with the intention of confirming the expected sharp transitionbetween diffraction and localization of the light beam for thechosen parameter setup. From the mathematical point of view,a necessary and sufficient condition for this transition is forα to be an irrational Diophantine number [33]. However, inpractice, an irrational number can be realized only with a finitenumber of digits. This indicates that, from the numerical pointof view, a sufficient degree of incommensurability between thespatial harmonics of the lattice potential is a crucial conditionfor the observation of localization phase transition. Therefore,here we set α equal to the golden mean (α = (

√5+ 1)/2),

which is, being the most irrational among the irrationalnumbers [27, 33], a common choice for the Diophantinenumber in the study of QP potentials. Figure 1 depicts therefractive index profile in the transverse direction of the QPPL with the selected α.

Figure 1. Refractive index profiles in the transverse direction x forthe QP PL (with ε= 0.0004)—black solid line and regular PL—reddashed line.

The first numerical experiment is initiated by injectingthe Gaussian light beam (section 2) in the single element ofthe QP lattices. We perform numerical simulations for QPPLs characterized with different potential depth values. Thevalue of the average participation number from hundreds ofcalculations for each lattice is monitored during the propa-gation, as well as the amplitude profiles of the light beams.It is found that 〈P(z)〉 saturates to finite values in the regionε > εth, which are of the order of the waveguide width. Thepropagation of the light beams is depicted with 2D amplitudeplots in figure 2 for two QP lattices characterized by ε < εthand ε > εth, respectively. It is obvious that localization occursin the second case, where the potential depth is above thethreshold value. This is confirmed by the saturation of themean value of 〈P(z)〉 to a finite value approximately equal1.2 µm. Therefore, we can conclude that light localization inQP lattices is a threshold determined phenomenon. The last isshown in figure 3, which illustrates the behaviour of 〈P(z)〉 notonly for QP lattices above and below the localization threshold,but for QP lattices characterized with ε≈ εth as well.

4. Nonlinearity versus quasi-periodicity

The next numerical task is to investigate how the nonlinearityaffects the localization in the QP PLs. We focus our studyon the region above the QP localization threshold. As aconsequence, there are two competing effects, the quasi-periodicity, arising from incommensurability between spatialperiods of constituent periodic lattices, and the self-trappingcaused by nonlinearity, that are taken into account for theinvestigation of the light localization. Being localized in theregime above the transition threshold, the light beam can beaffected differently by nonlinearity, depending on its strength.To estimate these effects we calculate average participationnumbers for several values of the lattice potential strengtharound and above the QP localization threshold value, and fordifferent nonlinearity strengths (see figure 4(a)). Note that thenonlinearity strength is directly related to the intensity of theinput light beam [26]. The obtained results are compared withthose in the analogous PL setup with competing quenched

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J. Opt. 16 (2014) 025201 A Radosavljevic et al

Figure 2. 2D plot of the averaged electric field amplitude profile: (a) below the light localization threshold in the QP PL, ε= 0.0002, (b)above the localization threshold in the QP PL, ε= 0.0006.

Figure 3. Average participation number for linear quasi-periodiclattices with different values of ε(γ = 0): black dashed lineε= 0.0002, red solid line ε= 0.0004 and green dash-dotted lineε= 0.0006. Saturation to finite values of 〈P(z)〉 indicateslocalization (see inset).

disorder and saturable nonlinearity [25]. In addition, wequalitatively relate the properties of the localization processin the QP nonlinear PL setup to different regimes of thewavepacket expansion in discrete nonlinear systems modelledby the discrete nonlinear Schrodinger equation [29].

We start numerical experiments by propagating the lightbeams through the QP PL while increasing the nonlinearitystrength. In figure 4(a) it is shown that the nonlinearitystrength γ ≈ 0.0001 (red solid circles), which is not highenough to cause localization in the corresponding regularnonlinear lattice, negligibly affects the characteristics of thelight propagation through the QP lattice, independently of thepotential depth value ε. The light beam diffracts below theQP localization threshold and becomes localized above it.The presence of weak nonlinearity is manifested only in aslightly higher value of the localization length in the vicinityof the εth in comparison to the case without nonlinearity(black empty circles). In the case of medium nonlinearity,γ ≈ 0.001 (figure 4(a), green solid triangles), we observe thatthe nonlinearity can affect the system behaviour below the

critical value of the lattice potential depth, causing localizationof the light beam. On the other hand, above the critical valuefor QP localization the nonlinearity reduces the sharpness ofthe 〈〈P(z)〉〉 versus ε dependence in the vicinity of the εth withrespect to the case without nonlinearity, and slightly affectsthe values of the localization length in the ε > εth region. Inaddition, we checked the effect of a very strong nonlinearity,γ ≈ 0.01 (figure 4(a), blue solid squares), on localization andproved that the self-trapping mechanism is responsible forlight beam localization independently of the distance fromthe QP localization threshold εth. The last corresponds to theconclusions derived for the self-trapping mechanism in [29].Figure 4(b) confirms the fact that by increasing the nonlinearitystrength, the self-trapping starts to take a leading role in thelocalization: in the case of a very strong nonlinearity thelocalization length reaches values of the order found in theperiodic nonlinear PL (dozens of microns).

On the other hand, we compare the results in the QPnonlinear PL with those previously obtained for disorderednonlinear lattices [25]. There the light localization was aconsequence of two competing effects: Anderson localizationand self-trapping. The first mechanism is not threshold deter-mined, in contrast to the localization in the QP PL. It is shownin [25] that the localization is governed by self-trapping in thepresence of a very strong nonlinearity (see figure 5(a) in [25]).Therefore, we conclude that in the presence of strong nonlin-earity the self-trapping prevails, independently of the natureof the competitive localization mechanisms—the Andersonlocalization or localization determined by quasi-periodicity. Inaddition, the characteristic localization lengths in all parameterranges in the presence of the disorder were approximatelytwice the localization lengths in the cases realized in thelocalization phase in the QP lattices.

5. Light management

Here we consider the possibility of controlling light propaga-tion by simultaneously manipulating the potential depth of thecomponent lattices in the composite PLs and the nonlinearitystrength, continuing the study started in our previous publi-cation [25]. There we showed the possibility to manage the

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J. Opt. 16 (2014) 025201 A Radosavljevic et al

Figure 4. (a) Averaged values of 〈〈P(z)〉〉 from the region where they saturate for different values of ε > εth are plotted for different valuesof the nonlinearity strength: black empty circles γ = 0, red solid circles γ ≈ 0.0001, green triangles γ ≈ 0.001 and blue squares γ ≈ 0.01.(b) Values of 〈P(z)〉 for γ ≈ 0.001 and different potential strengths: ε= 0.00045 (black solid line), ε= 0.00055 (red dashed line) andε= 0.00065 (green dotted line). Lines with symbols correspond to γ ≈ 0.01: black circles ε= 0.00045, red squares ε= 0.00055 and greentriangles ε= 0.00065. Values on the ordinates are displayed on a logarithmic scale.

Figure 5. 2D plot of averaged beam amplitude profiles in lattices composed of a regular lattice until (a) 3 mm, (b) 9 mm and a QP latticewith ε= 0.0006 after that. (c) 2D plot of averaged beam amplitude of a QP lattice with ε= 0.0006 until 5 mm and a regular lattice after that.(d) 〈P(z)〉 of lattices described in (a) green solid line, (b) red dashed line and (c) black dotted line.

localized beam propagation by mutually fitting the disorderand nonlinearity in composite lattice systems.

We performed calculations for composite PL systemsconsisting of the regular lattice (L1) followed by the QPPL (L2) in the propagation direction, and vice versa, forcomposite PL systems formed from the QP (L1) and periodicPL (L2) added to the end of the QP PL. The width ofthe component lattices is the same in all studied cases, andthe total lattice length in the propagation direction is keptas L1 + L2 = 20 mm. In the first setting (periodic to QP

PL) light initially diffracts through the regular part of thecomposite PL. Diffraction is interrupted by the QP latticepotential characterized by a value of the potential depth abovethe localization threshold. Increasing the length of the regularregion gives light more chance to diffract. Consequently, thenumber of localized fragments with lower energy, whichoccupy a finite number of waveguides in the QP region,increases with elongation of the regular lattice part (comparefigures 5(a) and (b)). This is also detected in the increase ofthe effective localization length with respect to the elongation

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J. Opt. 16 (2014) 025201 A Radosavljevic et al

Figure 6. (a) QP lattice (ε= 0.0006) until 7 mm and a QP lattice (ε= 0.0002) after that. (b) QP lattice (ε= 0.0002) until 7 mm and a QPlattice (ε= 0.0006) after that. (c) 〈P(z)〉 of the lattice described in (a) red solid line and the lattice described in (b) black dashed line.

Figure 7. 2D plot of averaged beam amplitude profiles in lattices composed of (a) a linear QP lattice (ε= 0.0002) until 10 mm, a QP lattice(ε= 0.0006) from 10 mm until 15 mm and a QP lattice (ε= 0.0002) lattice after that and (b) a nonlinear (γ ≈ 0.001) QP lattice (ε= 0.0002)until 10 mm, a QP lattice (ε= 0.0006) lattice from 10 mm until 15 mm and a QP lattice (ε= 0.0002) after that. (c) 〈P(z)〉 of the latticesdescribed in (a) red solid line and (b) black dashed line.

of the regular lattice part, figure 5(d). In the case with a shorterregular part, diffraction cannot widen the initial light beamenough and, as a result, the light is strongly localized to asmall number of elements, figure 5(a). Therefore, it mightbe possible to change the number of excited waveguides inthe QP part by properly adjusting the length of the regularpart, which may be applied for optical signal multiplexing inoptoelectronic devices. On the other hand, when the regularlattice is added at the end of quasi-periodic lattice, it is evidentthat localization is destroyed in the regular part, figure 5(c).

In addition, we studied light propagation in lattices com-posed of two QP PLs, one with the lattice potential depth abovethe threshold value and the other with the lattice potential depthbelow the threshold value. In these settings, the QP part of thecomposite lattice with ε < εth behaves similarly to the regularlattice in previous examples, with the exception of producinga slightly asymmetric diffraction pattern in comparison to theregular lattice. Therefore, the results for this case with twoQP PLs and the previous type of composite PLs with periodicand QP parts are qualitatively similar to each other. All thementioned conclusions can also be observed in correspondingdiagrams which illustrate the behaviour of 〈P(z)〉 along thelattice propagation length, figure 6.

Finally, we investigate the possibility to manipulate lightbeam spreading and localization by alternating three QPcomponent lattices with different potential depths. The total

lattice length in the propagation direction is set to be 30 mm.Since we are interested in studying the interplay betweennonlinearity and quasi-periodicity below and above the QPlocalization threshold, we incorporate nonlinearity as well,and compare results for linear (figure 7(a)) and nonlinear(figure 7(b)) composite QP lattices. These results confirm thatthe nonlinearity strength is a significant parameter, especiallyin the QP below or close to the localization threshold, where theself-trapping plays the main role in the light beam localization.On the other hand, the addition of nonlinearity in the QP partwith ε > εth can only induce quantitative changes in the systemdynamics, except in the case of a very strong nonlinearity.In figure 7, the medium nonlinearity strength γ ≈ 0.001 ischosen, being the closest to standard experimental conditions.For the chosen order of QP PLs comprising the compositelattice, the localization length along the linear compositelattice is evidently larger than the localization length alongthe corresponding nonlinear composite PL (figure 7(c)). Thisis due to strong correlation between the light beam localizationlength at the end of the first component lattice (with ε < εth)and the localization length in the second QP component lattice(with ε > εth) at the point z = 10 mm which dictates the lightbeam propagation along the rest of the composite PL. Thisis in agreement with the conclusions derived in the previoussections.

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J. Opt. 16 (2014) 025201 A Radosavljevic et al

We believe that the results of our study can be used asa seed for future experimental investigations and theoreticalstudies of light beam propagation management using thecombined effects of QP, disorder and nonlinearity.

6. Summary

In this paper we investigate localization phenomena in aexperimentally realizable setup which includes a QP pho-torefractive lattice with saturable nonlinearity. Ensuring theincommensurate ratio of the component lattice periods in thelinear PL, we show that localization is a threshold determinedphenomenon. The corresponding localization lengths as afunction of the lattice potential depth are estimated as well. Inaddition, we consider the effect of nonlinearity on localizationin the QP PL and find that nonlinearity affects the localizationproperties by increasing the localization length. However,the qualitative change of the process of localization occursonly in the presence of a very strong nonlinearity, when theself-trapping mechanism dominates.

The observed sensitivity of the localized structure prop-agation on the experimentally controllable parameters, suchas the strength of the QP potential and nonlinearity in the PL,provides the possibility to control the intensity, shape, positionand fragmentation of the localized structures. We test this byperforming several numerical experiments combining the QPand regular lattices with/without nonlinearity which clearlyshow that light propagation in the PL can be managed in thisway. Therefore, our study, besides improving the fundamentalunderstanding of localization phenomena in QP lattices, can beof interest in designing optical circuits in telecommunicationand sensing networks.

Acknowledgment

This work was supported by the Ministry of Education, Scienceand Technological Development, Republic of Serbia (ProjectIII 45010).

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