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Linear and Nonlinear Light Beam Propagation in Liquid-Crystal Waveguide

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Nonlinear Phenomena in Complex Systems, vol. 20, no. 4 (2017), pp. 319 - 326

Linear and Nonlinear Light Beam Propagation inLiquid-Crystal Waveguide Arrays

O. S. Kabanova, E. A. Melnikova, O. G. Romanov, and A. L. Tolstik∗Faculty of Physics, Belarusian State University,4 Nezalezhnasti Ave., 220030 Minsk, BELARUS

(Received 02 August, 2017)

Methods of spatial switching of laser beams in a system of coupled optical liquid-crystal waveguides with an electrically controlled depth of refractive index modulation areexperimentally realized. Tuning of the parameters of the waveguide system was carried outusing the electrooptic effect and the nonlinear-optical response of the liquid-crystal medium.The proposed theoretical model allows us to describe the processes of low-frequency andoptical modulation of the dielectric properties of a liquid crystal, calculate the profilesof optical waveguides induced in the system, and interpret the basic laws governing thepropagation of light beams in a system of coupled waveguides.

PACS numbers: 42.79.Kr; 61.30.Gd; 42.82.EtKeywords: liquid crystals, optical anisotropy, photonic structures, discrete diffraction, waveguiding

1. Introduction

Optical liquid crystal (LC) systems withspatial modulation of the refractive indexrepresent a promising technological platformfor creation of modern photonic devices withenhanced functional characteristics. In the lastdecade, the features of the manifestation oflinear and nonlinear effects in discrete opticalsystems have been actively investigated [1–7]. The design of discrete systems basedon functional materials with special opticalproperties allows to significantly improve theperformance characteristics of the photonicdevices manufactured. In particular, nematicliquid crystals (NLC) are successfully used tocreate discrete systems with tunable opticalparameters.

An abnormally high optical anisotropy ofNLC, controlled under the action of low voltages(of the order of units of volts)[8, 9], makes itpossible to realize discrete LC structures with atunable refractive index modulation depth. Also,NLCs proved its importance for nonlinear optics:the study of the interaction of NLC with light

∗E-mail: tolstik@bsu.by

led to the discovery of giant optical nonlinearity,8-9 orders of magnitude higher than the Kerrnonlinearity of ordinary liquids [10]. Under theinfluence of polarized light, at comparatively lowlevels of laser radiation intensity (∼ 1 kW/cm2),optical reorientation of the direction of the NLCdirector (light-induced Fredericks transition) isperformed [11–13], which allows the developmentof a new class of photonic devices.

The pronounced electro-optical response ofthe NLC makes it possible to transform aplanar NLC layer into a one-dimensional array ofwaveguides. Thus, in the presence of an externalspatially modulated electric field within the NLClayer, a discrete structure with a controlledrefractive index contrast depth is formed. Forradiation polarized as an extraordinary wave,the guiding properties are realized when itpropagates in the system of electrically inducedNLC waveguides [14].

In this paper we present a method forcreating a system of coupled optical NLCwaveguides with an electrically controlled depthof modulation (contrast) of the refractive index.Numerical simulation methods have been usedto calculate linear and nonlinear propagation ofa Gaussian light beam with TM polarization

319

320 O. S. Kabanova, E. A. Melnikova, O. G. Romanov, and A. L. Tolstik

in the developed waveguide NLC system. Thespatial control of radiation in a system ofcoupled NLC waveguides based on the electro-optical and nonlinear-optical response has beenexperimentally realized.

2. Liquid-crystal cell with wave-guide array

To create a spatial modulation of therefractive index within the NLC layer, a planarcell of the sandwich type was used, whichcontained an opaque, comb-shaped electricallyconductive layer on the lower substrate. Theswitching of a NLC cell from a planar waveguideto a one-dimensional system of identical weaklycoupled optical waveguides was carried out bymeans of an external low-frequency electricvoltage. Fig. 1a, b shows the schematic diagramof a NLC cell and illustrates the principle ofthe formation of a system of electrically inducedoptical waveguides (period of the structure Λ =40 µm).

As the core of the planar waveguide, a layerof a NLC material with a positive dielectricanisotropy (ε∥ > ε⊥) was used, located betweentwo plane-parallel glass plates (substrates). Thethickness of the NLC layer was determined by thegap between the substrates and was d = 100 µm.The refractive indices of the NLC material usedin the work for laser radiation with wavelengthλ = 532 nm were: ne = 1.70 for an extraordinarywave and no = 1.52 for an ordinary wave. Theinitial planar orientation of the NLC director (Fig.1a) along the z axis (i.e., along the propagationdirection of the light wave) on the surface of theupper and lower substrates was realized using thephotostimulated rubbing alignment technology[15].

To observe propagation of light beams inthe NLC layer in the yz plane, glass was usedas the upper substrate, uniformly coated witha transparent electrically conductive layer ofindium-tin oxide (ITO) 50 nm thick. Deposition ofa comb-shaped chromium electrically conductive

layer on the lower substrate of the LC cell wascarried out by laser lithography. The ratio ofthe width of the electrically conductive chromiumbands to the gap between them is equal tounity. The geometry of the selected texture of theelectroconductive layer allows the formation of aspatially modulated voltage across the thicknessof the NLC layer (i.e., in the x−axis direction).Under the action of an external voltage, thedirector of the LC molecules in the xz planeis reoriented, that leads to formation of amodulation of the refractive index in the NLClayer along the x− and y− directions. Thus, whenan external electric field is connected to a cell, thewaveguide distribution of the refractive index forthe TM polarization mode (

−→E ↑↑ x) is formed as a

result of the Fredericks transition within the NLClayer, as it is shown in Fig. 1b. The electricallyinduced system of optical waveguides consists ofidentical parallel channels that are weakly coupledalong the y-direction. The coupling coefficient ofNLC waveguides is determined by the magnitudeof the control voltage [1]. The optical parametersof the represented system of coupled waveguidesare rearranged on the basis of the electroopticalresponse and/or the nonlinear optical response ofthe NLC medium.

The formation in the NLC layer of theperiodic lattice of the refractive index, which isa waveguide system, was verified experimentallyby using a polarization microscope that studiesobjects in transmitted light by placing the cellin crossed polarizers. In Fig. 1c, polarizationmicrograph of a NLC cell with an electricallyinduced system of optical waveguides is presented.

3. Theoretical model andnumerical results

Theoretical models of propagation of laserbeams in LC structures, including waveguidesand spatially periodic structures with modulationof the refractive index of material, developed todate, can be divided into three main groups:1) the phenomenological model of the discrete

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Linear and Nonlinear Light Beam Propagation in Liquid-Crystal Waveguide Arrays 321

FIG. 1. (Color online) Schematic diagram of a NLCcell (view from the end) (a); the principle of theformation of a system of NLC waveguides (b); amicrograph of a NLC lattice (top view) in polarizedlight (crossed polarizers) (c).

nonlinear Schrodinger equation for describingpropagation of laser radiation in a system ofcoupled waveguides [16]; 2) the model based onthe solution of nonlinear wave equation in theparaxial approximation (nonlinear Schrodingerequation) for the complex light field amplitude,that takes into account the possible nonlineardependence of the dielectric constant on the lightfield intensity and/or periodic spatial modulationof the medium parameters [17]; 3) a directnumerical solution of the Maxwell equations forthe electric and magnetic field strengths with agiven spatial distribution of the dielectric functionof the medium and its dependence on the intensityof the light field and external electric or magneticfields [18].

In this paper we use an approach based onthe analysis and solution of the nonlinear wave

equation in the paraxial approximation for thecomplex amplitude of the light wave field (thenonlinear Schrodinger equation), which in generalform is written as follows [17]:

2ik∂E

∂z+∆⊥E =

ω20

c2∆ε

(∣∣∣E∣∣∣2, r) E. (1)

Here, the function ∆ε

(∣∣∣E∣∣∣2, r) takes into

account the possible nonlinear dependence ofthe permittivity on the intensity of the lightfield and/or the periodic spatial modulation ofthe parameters of the medium. Allowance forthe anisotropy effects characteristic of liquid-crystalline media makes it necessary to considera system of wave equations of the form (1) fordifferent polarization modes.

In further analysis, we assume that a focusedlaser beam having an initial Gaussian intensityprofile is directed to the NLC cell from the endz = 0. In accordance with the experimentalconditions, vertical polarization of the radiation(along the x axis) is assumed, the modulation ofthe dielectric permittivity of the medium is givenin the form ∆ε = ∆ε (x, y)+εnl (I), where the firstterm is related to the spatially inhomogeneousreorientation of NLC molecules in a low-frequencyexternal electric field E0, and the second termdescribes the orientational action of the electricfield of a light wave E. Under these conditions, thenormalized wave equation for the x-component ofthe electric field E can be written in the form:

∂Ex

∂z= i∆⊥Ex + i

kLD∆ε

εlEx (2)

where εl ≡ ε⊥ is the linear dielectric constant, k isthe wavenumber, the spatial variables x and y arenormalized to the half-width of the Gaussian lightbeam r0, and the variable z is normalized to thediffraction length: z = 2LDz, LD = kr20 =

2πr20n0

λ0.

To find the stationary distribution of theorientation of NLC molecules in a low-frequencyand/or optical electric field, we find the minimumfor the bulk density of free energy F = FD + FE ,

Nonlinear Phenomena in Complex Systems Vol. 20, no. 4, 2017

322 O. S. Kabanova, E. A. Melnikova, O. G. Romanov, and A. L. Tolstik

where the first term FD describes the energy ofelementary deformations of the bulk NLC, andthe second term FE represents the interactionenergy of NLC molecules and electric fields. In theone-constant approximation, in which the elasticFranck constants for each of the deformationtypes (splay, twist, and bend) are assumed to bethe same K1 = K2 = K3 = K, the formula for FD

has the form: FD = 12K

[(∇ · n

)2+

(∇ × n

)2],

and the interaction energy of NLC molecules andthe external electric field is expressed by the

formula FE = −∆ε2

(n · E

)2. The minimum of free

energy is found from the expression [19]:

∂

∂x

(∂F

∂θx

)+

∂

∂y

(∂F

∂θy

)+

∂

∂z

(∂F

∂θz

)−∂F

∂θ= 0 (3)

where θx, θy, θz are the derivatives of the angleof rotation of the director of the LC layer θ alongthe corresponding coordinate axes.

In accordance with the NLC cell diagramshown in Figure 1, the initial orientation of LCmolecules is planar along the z axis. Since theexternal electric field is modulated in the x, yplane and is a constant along the z axis, we obtainthe explicit dependence of the orientation angleof the LC director and the field itself on only twocoordinates θ (x, y) and E0 (x, y). In the proposedcoordinate system, the director of the nematicLC-layer n has only x− and z−components (nx =sin θ, ny = 0, nz = cos θ), and the vector of theexternal electric field has the components E0x andE0y. Using these assumptions, one can obtain anequation for the tilt angle of the NLC directorwith respect to the initial planar orientation:

K

[∂2θ

∂x2+

∂2θ

∂y2

]+

∆εE20x

2sin 2θ = 0. (4)

The intensity of the low-frequency electricfield between the electrodes E0 can be calculatedfrom the Maxwell equation ∇ · D = 0, taking

into account the expression D = εLF⊥E0 +(εLF∥ − εLF⊥

) (n · E0

)n:

∂

∂x

[(εLF⊥ +

(εLF∥ − εLF⊥

)sin2θ

)E0x

]+εLF⊥

∂E0y

∂y= 0 (5)

Taking into account that E0 = −∇φ0, we get:

∂

∂x

[(εLF⊥ +

(εLF∥ − εLF⊥

)sin2θ

) ∂φ0

∂x

]+εLF⊥

∂2φ0

∂y2= 0. (6)

Thus, the distribution of the NLC director n,which is completely determined by the tilt angleθ, can be found from the solution of equations(4), (6). In this case, the spatial distribution ofthe refractive index for a light wave propagatingalong the z axis and polarized along the xaxis is determined by the formula: neff (θ) =

n⊥n∥√n2∥cos

2θ+n2⊥sin2θ

. As follows from this expression,

as the tilt angle of the NLC director changes, therefractive index for the polarized wave vertically(along the x axis) changes from the value n⊥ ≡ no

with the initial planar orientation of LC moleculesto n∥ ≡ ne with the angle of reorientation θ = π

2 .Thus, conditions for a waveguide propagation arerealized for a vertically polarized wave.

Numerical modeling of the process of thelight beam propagation has been carried out forthe following cases: 1) linear propagation of anarrow low-power laser beam in an LC layer withan electrically induced array of waveguides; 2)nonlinear self-focusing of a wide laser beam ina system of weakly modulated waveguides; 3)propagation of a wide laser beam in a waveguidearray with simultaneous electrically and opticallyinduced modulation of the refractive index ofthe material. The latter variant corresponds tothe conditions of the experimental study. Thesimulation results are shown in Figure 2.

Figure 2b shows the intensity profiles of thelight beam with the initial half-width r0 = 4 µm,and the wavelength λ0 = 1 µm propagating

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Linear and Nonlinear Light Beam Propagation in Liquid-Crystal Waveguide Arrays 323

in the waveguide array with the period Λ =4 µm and depth of modulation ∆εmax = 0.3.As can be seen from the calculation, under theseconditions at a depth of penetration of the orderof 100 − 400 µm there is a clearly pronouncedtransfer of the energy of the light beam from thecentral waveguide to the neighboring ones, whichis a typical picture for observing the process ofdiscrete diffraction.

Experimental study of the propagation oflaser beams was carried out under conditionsof focusing radiation with a wavelength λ0 =0.532 nm on the input of the NLC layer into aspot with diameter 40 µm and for a period ofthe waveguide structure Λ = 40 µm, so numericalsimulation was performed for these parametersto study the effect of the modulation depth ofthe waveguide structure and the nonlinear opticalresponse of NLC.

In the case of weak (∆εmax ≪ 1) spatialmodulation of the permittivity of the NLC layer,the main contribution to the change in the spatialstructure of the light beam is determined bythe effect of self-focusing (Figure 2, c), whichunder the indicated conditions manifests itselfat penetration depths z ∼ 2000 µm. For smallangles of reorientation of LC molecules (θ ≪ π

2 ),the nonlinear term of the permittivity can bedescribed in the framework of Kerr-nonlinearitymodel [20]: εnl ≈ 2n0∆n = 2n0n2I. Thesaturation of the reorientation effect (θ ∼ π

2 ) canbe taken into account in the framework of thesaturation nonlinearity model, in which ∆n =(n∥ − n⊥

)n2I

1+n2I.

The combined effect of electrically inducedspatial modulation of the dielectric permittivity ofthe NLC layer and optically induced nonlinearityin the propagation of a laser beam with aninitial radius r0 = 40 µm in a waveguide arraywith a period Λ = 40 µm leads to formationof a complex multimode radiation structure(Figure 2d), a characteristic feature of which isthe manifestation of the discrete diffraction atpenetration depths z ∼ 400− 500 µm.

FIG. 2. (Color online) The structure of the waveguidearray (a). (b–d) Intensity profiles of the light beamI(y) inside NLC layer.

Nonlinear Phenomena in Complex Systems Vol. 20, no. 4, 2017

324 O. S. Kabanova, E. A. Melnikova, O. G. Romanov, and A. L. Tolstik

4. Experimental results

The experiments were performed using thesecond harmonic of the Nd: YAG laser (λ =532 nm). To analyze the spatial distribution ofthe light field in a discrete NLC structure, weused the standard method of fixing the scatteringpattern of laser radiation on inhomogeneitiesin the orientation of the director of a liquidcrystal [12]. The registration of the scatteringpattern of the light beam in the yz plane wascarried out using a high-resolution photosensitivecamera coupled with an objective lens. With thehelp of the multifunctional measuring complex"UNIPRO" , an alternating voltage of 0–10 Vamplitude was applied to the electrodes of theNLC cell with a repetition rate of the pulsesν = 1 kHz.

To study the nonlinear propagation of alight beam in a system of coupled optical NLCwaveguides, linearly polarized laser radiation(λ = 532 nm,

−→E ∥x, P = 5 mW) was focused by

objective lens to a spot with a diameter 40 µm anddirected to the end of the NLC cell in the regionof the comb-shaped electrode. To reduce thethreshold for the nonlinear orientational effect, weused the method of applying a voltage to the LCcell near the threshold of the Fredericks transition(Uthr = 1.1 V) [20]. Figure 3a, shows theexperimental scattering patterns of a light beamin a system of coupled NLC waveguides underdifferent control voltages, and the correspondingdependencies of the intensity distribution profilesfor the propagation length z = 2 mm are shownin Figure 3b.

At a voltage U = 1.0 V, in the regionof the textured electrode, the pretilt angle ofthe director of the NLC molecules in the xzplane is created, which leads to a decrease in thethreshold value of the intensity of the light beamproviding the photoinduced Fredericks transition[21]. Since the maximum optical power is reachedat the center of the beam cross section, withina given region the light field self-channeling dueto the nonlinear orientation effect and acquiresthe form of an optical spatial soliton (nematicon)

FIG. 3. (Color online) Scattering patterns of laserradiation in a system of electrically induced NLCwaveguides under different control voltages (a).Dependencies of the distribution profiles of theintensity I(y) of the light beam at a depth z = 2 mm(dashed line) (b).

with the energy concentrated within one NLCchannel. At a voltage on the cell U = 2.0 V, anexternal electric field plays the predominant role

Нелинейные явления в сложных системах Т. 20, № 4, 2017

Linear and Nonlinear Light Beam Propagation in Liquid-Crystal Waveguide Arrays 325

in reorientation of the NLC director, as a resultof which the effective refractive index in the coreregion of the NLC waveguide increases.

With further increase of the voltage on thecell (U ≫ 2.0 V) saturation of the reorientationprocess of NLC molecules occurs in the regionof the comb-shaped electrode, which resultsin smoothing the modulation of the refractiveindex within the discrete system. Indeed, atexcessively high voltages, the refractive indexincreases not only in the core region, but also inthe region of the cladding of NLC waveguides,which is due to the nonlocal response of the NLCmedium, as well as the output of the electricfield lines beyond the electrodes. The electricallyinduced change in the coupling coefficient of NLCwaveguides, responsible for the process of transfer(redistribution) of light energy between adjacentchannels, is accompanied by the manifestationof discrete diffraction of the light beam in asystem of coupled NLC waveguides. In accordancewith Figure 3, as a result of discrete diffraction,a redistribution of the energy of light radiationoccurs between adjacent NLC waveguides. Thus,

at a voltage of U = 6.0V , the light energy fromthe region of one NLC waveguide is redistributedbetween ten adjacent channels of the discretesystem. As a result, an external voltage can beused to redistribute the energy of the light fieldbetween a given number of waveguide channelswithin a discrete NLC system.

5. Conclusion

The method of forming a discrete NLCstructure with electrically controllable depth ofrefractive index modulation for spatial control oflight fields is proposed. The features of linear andnonlinear propagation of light beams in a systemof coupled optical NLC waveguides are studiedexperimentally. It is shown that, depending onthe power of the light beam and the magnitudeof the control voltage, the mode of waveguidepropagation of light is realized, as well as theregime of the discrete diffraction of the light beamin a system of coupled NLC waveguides.

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