Manipulation of light beam propagation in one-dimensional photoniclattices with linear refractive index profile

Ana Radosavljević a,b,n, Goran Gligorić b, Aleksandra Maluckov b, Milutin Stepić b

a School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbiab P* Group, Vinča Institute of Nuclear Sciences, University of Belgrade, POB 522, Belgrade, Serbia

a r t i c l e i n f o

Article history:Received 17 June 2014Received in revised form5 September 2014Accepted 11 September 2014Available online 23 September 2014

Keywords:Bloch oscillationsLight localizationLight propagation controlNonlinearity

a b s t r a c t

We use Bloch oscillations to demonstrate potential light beam propagation control in photonic latticeswith linear refractive index change in transverse direction. It is qualitatively investigated how compositephotonic lattice systems with varied values of refractive index gradient and nonlinearity strength alongthe propagation direction can be used to perform different functions, such as beam splitting andswitching, in future components of all-optical networks. Additionally, we demonstrate the robustness ofBloch oscillations to nonlinearity and disorder in the proposed setup.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The question of how an external linear field affects the behaviour ofparticles in periodic media was first theoretically addressed in con-densed matter settings and later experimentally observed for electronsin semiconductor superlattices [1]. Already in 1930s [2,3], Bloch andZener predicted that electrons experience oscillatory motion whenexternal dc electric field is applied to one-dimensional (1D) periodiccrystals, as opposed to the uniform motion in bulk materials. Even-tually, the idea was extended to optical periodic media, owing to theadvantageous ability these systems offer—to directly visualize variouseffects. Optical Bloch oscillations have been recently experimentallyobserved as well as theoretically investigated in both Bose–Einsteincondensates in optical lattices [4–9] and waveguide arrays, i.e. photoniclattices (PLs) [10–15]. In PLs, optical Bloch oscillations are possible solong as the refractive index distribution varies linearly with waveguidenumber. They can be understood in terms of interplay between totalinternal and Bragg reflection [16].

It has been shown that two types of Bloch oscillations can beexpected, depending on the input light beamwidth [10]. In this paperwe study Bloch oscillations obtained for a broad beam excitation withfull width at half maximum (FWHM) of the order of several waveguidewidths. Since the light beam in this case oscillates along the latticewith almost unchanged width, it might be possible to gradually shiftsignal without deterioration, which has already been indicated in Ref.

[10]. Our intention is to theoretically investigate possible application ofthis type of oscillations for input beam shifting into a desiredwaveguide or input beam splitting. It was shown that a linearlygrowing effective refractive index can be induced via the electro- orthermo-optical effect, bending, etc. [12–15,17]. Therefore, it might beuseful from the practical point of view to manage the light beampropagation by externally inducing the refractive index gradient sincethe period and the amplitude of oscillations depend on the externalfield gradient. The aim of this work is to propose a potential usage ofthe obtained results to control the light beam propagation along thePL. Continuing our research in the field of light propagation manage-ment in 1D complex PLs [18,19] we include the same systemparameters from easily experimentally realizable setup described indetails in [20,21]. Since we examine the PLs which can be fabricatedon photorefractive–photovoltaic lithium niobate substrate, we simu-late light propagation through these PLs by solving the paraxial time-independent Helmholtz equation with included nonlinear saturablemedia response [18,21] (Section 2). The model equation is numericallysolved by adopting the split-step Fourier method [18,22]. Obtainedconclusions can be directly extended to other nonlinear media withsaturable nonlinearity, for example SBN crystal media characterized byhigher nonlinearity and shorter response time than presented LiNbO3

media [23].

2. Mathematical model

The light propagation in 1D nonlinear PL is described by theparaxial time-independent Helmholtz equation [20]

idEdz

þ 12k0n0

∂2E∂x2

þk0n0n xð ÞE¼ 0; ð1Þ

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journal homepage: www.elsevier.com/locate/optcom

Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.09.0350030-4018/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author at: School of Electrical Engineering, University ofBelgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia.Tel.: þ381 113408632.

E-mail address: anar@vinca.rs (A. Radosavljević).

Optics Communications 335 (2015) 194–198

where E(x, z) is the component of the light electric field in thepropagation z-direction, x is a transverse direction along which theperiodic optical lattice is realized [20], k0¼2π⧸λ is the wavenumber, n0 is the refractive index of the substrate, and λ repre-sents the wavelength of light field. A stationary transverse latticepotential Δn(x) consists of term nl (x) defining the lattice, satur-able nonlinear term described by the function nnl (x) and refractiveindex gradient αx:

n xð Þ ¼ nl xð Þþnnl xð Þþαx¼ΔnG xð Þ�12n20rEpv

Ej j2Idþ Ej j2

þαx: ð2Þ

Parameter α denotes coefficient of linear refractive indexgradient. Lattice potential is represented with a periodic set ofGaussians

Gn xð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi4 ln 2πω2

g

se�4 x� xn

ωg

� �2

ln 2; ð3Þ

where wg is the waveguide width, and xn centre of the respectivewaveguides. 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 ξ¼k0 x, η¼k0 z anddimensionless wave amplitude ψ ¼ E=

ffiffiffiffiId

p, Eq. (1) can be rewritten

in dimensionless form:

i∂ψ ξ;η

� �∂η

þ 12n0

∂2ψ ξ;η� �∂ξ2

þn0Δn G ξ� �þαsξ

� �ψ ξ;η� �

�γψ ðξ;ηÞ�� ��2

1þ ψ ðξ;ηÞ�� ��2ψ ξ;η

� �¼ 0; ð4Þ

where γ ¼ 0:5n30rEpv and αs ¼ αk�1

0 .The parameter set describing the lattice system and input light

characteristics in our study is taken from the experimentallyrealizable setup presented in details in Refs. [20,21]. In order toanalyse the effect of induced refractive index gradient on the lightpropagation, we selected in the numerical simulation those valuesof parameters for which in the case of regular, periodic PL theinput light beam diffracts through the lattice, i.e. wave packetassociated to the light ballistically extends during the propagation:n0¼2.242, r¼30 pm/V, Epv¼7 kV/mm and γE0.001. The resultspresented in the following are obtained for N¼101 lattice ele-ments (waveguides), Δn¼0.00038 and width of a single wave-guide in periodic lattice wg0¼4 μm, except it is not stateddifferently. Without any loss of generality, the lattice is initiallyexcited at the lattice centre by a broad Gaussian beam with thewavelength λ¼514.5 nm and FWHM¼20 μm. We vary the valueof αS in our calculations.

3. Linear Bloch oscillations in the absence/presence ofnonlinearity or disorder

Bloch oscillations can be understood as one kind of light localiza-tion introduced by the presence of refractive index gradient. Thislocalization is identified as light oscillations within a limited space inlattice. Period and amplitude of these oscillations depend on theexternally induced refractive index gradient and it has been shownexperimentally and confirmed numerically that higher gradient leadsto oscillations within narrower space. Therefore, very high gradientconfines Bloch oscillations within a space of the order of just a fewwaveguides. The corresponding period of the Bloch oscillations can beestimated as zB�αS.

In this section we give a brief overview of light oscillationsbehaviour in our linear setup with refractive index gradient andtheir destruction in the presence of nonlinear and disorder effects.Firstly, by changing αS we observe the expected correspondence

between αS and period of oscillations [1–16]. The numericallyobtained oscillation period vs. αS is depicted in Fig. 1, as wellas the theoretically predicted oscillation period dependence onαS: z¼ 2 λ0=Λk0

� �w=αS where w is rescaled width of the input

beam with respect to the chosen numerical grids and Λ is the PLperiod.

Next instance was to include nonlinearity in this setup.Detailed studies of various dynamical regimes in the param-eter space of nonlinearity and potential gradient can be found in[24–26]. Light beam propagation through the gradient PL wassimulated for a several values of coefficient of linear refractiveindex gradient and different nonlinearity strengths for each valueof αS. All of our findings are qualitatively in accordance withtheoretical studies in [24–26], as well as with experimental resultspresented in [10]. Therefore, we illustrate the light oscillations inour setup only for two values of nonlinearity strength and achosen value of αS¼0.05. The presence of weak nonlinearityγE0.001, which is not high enough to cause the localization incorresponding regular nonlinear lattice, is manifested only in aslight smearing of the light oscillations in comparison to the casewithout nonlinearity (Fig. 2a). When nonlinearity is increased,oscillations become smeared after a certain propagation length inthe direction of refractive index gradient, independently of thenonlinearity strength which is depicted in Fig. 2b for a very strongnonlinearity γE0.01.

Additionally, we investigate the simultaneous action of disor-der and gradient refractive index on light propagation in thechosen setup, with the intention to see which effect, nonlinearityor disorder, has higher impact on light oscillations. Disorder wasintroduced in the way presented in our paper [18]. Disorderstrength was increased in the physically feasible limit. However,we did not observe any significant changes in light beam propaga-tion. Our presumption is that disorder induced by changingrandomly the widths of waveguides comprising the PL, cannotaffect light propagation for the chosen parameter set since thecharacteristic length of this process is not comparable withcorresponding one due to the refractive index gradient. In otherwords, the gradient is too high to be affected by local changes inthe waveguide widths, which are of the order of a few tenths ofmicron. Therefore, we sought another combination of parametervalues for which disorder can compete with linear change ofrefractive index gradient. Results are depicted in Fig. 3 for thefollowing parameters: disorder strength d¼0.8, waveguide widthsof 6 μm and distances between waveguides of 5 μm in thecorresponding periodic lattice, 301 waveguides, Δn¼0.005, andαS¼0.0001. Following the procedure in [18] we estimate that

Fig. 1. Bloch oscillation amplitude vs. αS obtained from simulations – black dashedline and theoretically predicted – red solid line. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of thisarticle.)

A. Radosavljević et al. / Optics Communications 335 (2015) 194–198 195

Anderson localization length for the chosen set of parameters is ofthe order of 200 μm, which corresponds to 18 waveguides. Asexpected, disorder causes smearing of oscillations with theirinclination towards attenuation [17,27,28].

4. Interrupted Bloch oscillations and possible applications

Here we investigate how the light propagation is affected if theexternal field, which induces refractive index gradient, suddenlygets turned off together with the periodic potential after a certainpropagation distance. This situation is not uncommon in experi-mental practice. During lattice inscription using high-power lasers[20], refractive index modulation can be missing at some part ofthe sample. In order to examine how the interruption point affectslight propagation, we break off Bloch oscillations at four differentpoints: before a half period of oscillations is over, right at the endof a half period, between the half and a full period, and at last,at the end of the full period (Fig. 4).

Calculations are performed for linear composite PL systemsformed of the gradient lattice (L1), comprised of 201 waveguides,followed by continual medium (L2) in the propagation direction,which corresponds to turning off the gradient and periodic potential.

The transverse length of the component lattices is the same in all thestudied cases, and the total system length in propagation direction iskept to be L1þL2¼15 mm. It is evident that oscillations are destroyedin the second part of the composite system in all of the studied cases.Similar idea has been exploited in experiment with Bose–Einsteincondensate [29]. Also, one could note that turning off the gradientcorresponds to the time-of-flight technique in the field of cold atoms[30]. We observe that by varying the point of interruption lightchanges its propagation direction in the part of the system withoutthe index gradient. Particularly interesting case is the light beamsplitting when oscillations are interrupted at the end of a half-period(Fig. 4b). Very simple analogy of this ramping of Bloch-oscillationsand interrupted oscillations of the linear harmonic oscillator orpendulum can be established. In the last by cutting tension spring,or unstretchable thread, the oscillating pellet freely continue to movewith the kinetic energy possessed at the moment of interruption.This opens a way to govern light propagation by properly adjustingthe position in the propagation direction of the lattice after whichthe refractive index change is suppressed. However, a strong disper-sion of the free beams should be addressed as a disadvantage ofsuch setup.

Finally, we investigate which effect could be used to effectivelyconfine light beam, acting against its momentum, after it has been

Fig. 2. Two-dimensional (2D) amplitude plots of Bloch oscillation when αS¼0.05, Δn¼0.00038 and the nonlinearity strength is (a) γE0.001 and (b) γE0.01. The amplitudevalues are scaled with the strength of the grey areas—darker area corresponds to higher amplitude.

Fig. 3. 2D amplitude plots of Bloch oscillation when αS¼0.0001, Δn¼0.005 and the disorder strength is (a) d¼0, (b) d¼0.8.

A. Radosavljević et al. / Optics Communications 335 (2015) 194–198196

sent to a desired direction. Systems comprising three parts werenumerically simulated with total propagation length L1þL2þL3¼20mm.We focused our attention on configuration exhibiting light splitting.Therefore, the first and the second part (L1þL2) of the system are thesame as those presented in Fig. 4b. The third part contains PL withdifferent effects included to test their light localization performances.

Controlling the length of the continual part of composite system (L2),one can determine which waveguides will be excited when the latticepotential is switched on again. Switching on the lattice potentialtogether with higher gradient refractive index in the third part (L3) cancause oscillations revival. If the gradient is high enough, period andamplitude of oscillations will be accordingly small leading to oscilla-tions confined to narrow space and causing apparent localization.

Fig. 4. 2D amplitude plots of interrupted Bloch oscillation when αS¼0.005, Δn¼0.001 and the nonlinearity strength is γ¼0, when the oscillations are interrupted afterlength L for (a) Lo1.5 periods (b) L¼1.5 periods, (c) 1.5 periods oLo2 periods and (d) L¼2 periods.

Fig. 5. 2D amplitude plots of interrupted Bloch oscillation when αS¼0.005, Δn¼0.001 and the nonlinearity strength is γ¼0, when the oscillations are interrupted afterlength L¼1.5 periods¼7.3 mm. After the interruption, at L¼9.17 mm, with intention to localize light we include (a) lattice potential with linear refractive index gradientcharacterized by αS¼0.09, higher than in the first part and (b) high nonlinearity γ¼0.05.

A. Radosavljević et al. / Optics Communications 335 (2015) 194–198 197

The drawback in this approach is that beam is not strongly localized toa small number of waveguides, which is shown in Fig. 5a. Another ideais to employ high nonlinearity in the third part, after the latticepotential has been established again, to localize beam in the desiredwaveguides. Fig. 5b depicts one example of the case of nonlinear thirdpart of composite system. Noticeably, high nonlinearity can localizebeams with “higher precision” in the intended waveguides at theexpense of energy loses which emerge at the transition point betweenthe continual medium (L2) and new lattice potential (L3) [31].

5. Conclusions

The phenomenon of Bloch oscillations in the experimentallyrealizable setup that includes the photorefractive lattice with saturablenonlinearity and additional refractive index gradient of waveguidesconstituting the lattice is investigated. In addition, we consider theeffect of the nonlinearity on the oscillations in this type of PL. Weobserve that nonlinearity affects the light propagation properties bysmearing the oscillations, which is in accordance with previous studiesof combined action of these phenomena. In our setting, the qualitativechange of the process of localization occurs only in the presence of avery strong nonlinearity, when the nonlinearity quickly destroysoscillations. On the other hand, the effect of simultaneously incorpo-rated disorder and linear change of refractive index induces lesspronounced changes in light behaviour, in comparison with thenonlinearity, since in this case refractive index gradient predominantlymanages light propagation in our setup.

We propose feasible composite photonic structures which may beused for performing different functions, such as beam splitting andswitching, in future components of all-optical networks. The proposedcomposite photonic structures comprise of gradient lattice followed bycontinual part and second lattice, respectively, in the propagationdirection. By interrupting lattice potential with linear refractive indexchange, we showed how oscillating beam can be sent to differentdirections or split in the continual part. Second lattice (gradient ornonlinear periodic) introduced after the continual part, confines andguides shifted or split beams in the proposed setup. Depending on thedemands for a specific application, whether it is a precise control ofexcited waveguides number or low energy loss in the third part ofdescribed composite system, one of the proposed approaches for lightconfinement might be useful. By properly adjusting the value ofgradient, which determines the period of Bloch oscillations in the firstpart of composite photonic structure, and the length of continual partwhich follows the interrupted potential it might be possible to controlthe function of the composite structure, as well as the positions ofexcited waveguides in the second lattice. Therefore, our study, besidesthe fundamental understanding of the investigated phenomena in thecomplex lattices, can be of interest in designing the optical circuits intelecommunication and sensing networks.

Acknowledgment

This work was supported by the Ministry of Education, Scienceand Technological Development (Republic of Serbia), ev. no. III45010.

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