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Vortex stabilization by means of spatial solitons in nonlocal media

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2016 J. Opt. 18 054006

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Vortex stabilization by means of spatialsolitons in nonlocal media

Yana Izdebskaya1,2, Wieslaw Krolikowski1,3, Noel F Smyth4 andGaetano Assanto5,6

1 Laser Physics Center, Research School of Physics and Engineering, The Australian National University,Canberra ACT 0200, Australia2Nonlinear Physics Center, Research School of Physics and Engineering, The Australian NationalUniversity, Canberra ACT 0200, Australia3 Texas A & M University at Qatar, Doha, Qatar4 School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK5NooEL— Nonlinear Optics and OptoElectronics Lab, University of Rome ‘Roma Tre’, I-00146 Rome,Italy6Optics Laboratory, Tampere University of Technology, FI-33101 Tampere, Finland

E-mail: yana.izdebskaya@anu.edu.au

Received 6 December 2015, revised 14 January 2016Accepted for publication 27 January 2016Published 14 March 2016

AbstractWe investigate how optical vortices, which tend to be azimuthally unstable in local nonlinearmaterials, can be stabilized by a copropagating coaxial spatial solitary wave in nonlocal,nonlinear media. We focus on the formation of nonlinear vortex-soliton vector beams inreorientational soft matter, namely nematic liquid crystals, and report on experimental results, aswell as numerical simulations.

Keywords: optical vortex, spatial solitons, self-action effects, liquid crystals

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

1. Introduction

Light beams carrying optical vortices have attracted growingattention during the past decades [1–4]. Such complex beamsare usually associated with a ring-like intensity structure witha zero value at the centre, where the electromagnetic fieldvanishes, with a phase distribution spiralling around it.Besides their rich and intriguing properties, including phasesingularities, internal energy circulation and the unique fea-tures of their linear and angular momentum distributions,optical vortices offer a wide range of prospective applicationsin such areas as micro-manipulation [5–9], optical encoding/processing of information [10–14], sensitivity and resolutionenhancement in optical measurements [15, 16]. Optical vortexbeams can be generated in different types of linear [1–4, 17]and nonlinear media [18]. However, they are usually prone tostrong dynamical instabilities in self-focusing nonlinearmedia local azimuthal modulations of the initial donut shapeand split it into fragments which fly away from the initialvortex ring [18].

Nevertheless, as was shown in a few theoretical papers, ifthe nonlinearity is accompanied by spatial nonlocality, so thatthe overall nonlinear perturbation in the medium extends farbeyond the beam waist in the transverse plane [19], the pro-pagation of an optical vortex can be stabilized. In particular,stable propagation of spatially localized vortices may becomepossible in highly nonlocal nonlinear media with a self-focusing response [20, 21]. This prediction was confirmed byexperimental results on the existence of stable vortex solitonswith unit topological charge in thermal nonlinear media [22].Nonlocal spatial solitons were also found to be able to stablyguide and route vortex beams across an interface or around adefect by counteracting the diffraction and instabilitiesenhanced by such refractive perturbations [23]. An intensevortex and a co-propagating spatial soliton are expected toform a stable vector soliton in a self-focusing nonlocal media[18], in a similar fashion to bright vector solitons with light ofdifferent colors [24, 25], spiralling solitons with angularmomentum [26–29] and multi-hump soliton structures con-sisting of two (or more) components which mutually self-trap

Journal of Optics

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[18, 30–32]. As was reported in recent theoretical and num-erical studies [33], such two-component vector vortex solitonsmay be stable in nematic liquid crystals (NLCs) due to thestrong reorientational nonlinearity of such soft matter [34, 35]and the stabilizing character of the resulting nonlinear, non-local potential generated by the superposition of both beamcomponents [33, 36]. Thus, an important challenge in non-linear singular optics remains to reveal the physicalmechanisms which will allow the experimental observation ofstable optical vortices in realistic nonlinear media.

In this paper, we observe experimentally and describetheoretically the formation of stable two-component vectorvortex solitons in NLCs. These vector solitons appear in theform of two-wavelength self-trapped beams with one ofthem components carrying a phase singularity and beingstabilized by its nonlocally enhanced interaction with theother transverse localized beam, a fundamental spatialsoliton. We also find that such vector beams can be gen-erated for certain ranges of the soliton excitation, that is theinput beam power.

2. Sample and experimental results

For our experiments we used a planar cell realized by poly-carbonate slides held parallel to one another and separated by110 mm across y in order to contain the 6CHBT NLC mixture.The cell structure is sketched in figures 1(a) and (b). Theplanar interfaces between the polycarbonate and the NCLs

provide molecular anchoring by means of mechanical rub-bing, ensuring that the elongated molecules are orientatedwith their main axes in the plane (x, z) of the slides, at anangle 40q p= with the input wavevector k along the z-direction. Two additional 150 mm thick glass slides at theinput and output interfaces seal the cell to prevent lens-likeeffects and avoid light depolarization. The maximum propa-gation length along the z-axis was 1.1mm from the input tothe output facets.

The experimental setup for the generation of two-component, two-color vector vortex solitons is shown infigure 1(c). One beam component (red) carries the extra-ordinarily polarized single-charge vortex beam generated by afork-type amplitude diffraction hologram (DH) using a cwlaser beam at wavelength 671 nm01l = and power Pr. Thesecond (green) component is an extraordinarily polarizedfundamental Gaussian beam of wavelength 532 nm02l =and power Pg. Figure 2 illustrates the input beams, withfigures 2(a) and (d) depicting the corresponding input inten-sity distributions, figures 2(b) and (e) the schematic phasefronts and figures 2(c) and (f) the intensity profiles of eachbeam: the green Gaussian beam (left) and the red singlecharged vortex (right). The two co-polarized beams wereinjected into the NLC with collinear Poynting vectors along zby a 10×microscope objective (MO1). The input waists werew 7 mv m» and w 4 mg m» for the vortex and fundamentalGaussian beams, respectively. With the half-wave plates

201l and 202l we controlled the polarization state of bothbeams. In order to study the singular phase structure of the

Figure 1. (a) Perspective and (b) top views of the planar cell; the ellipses indicate the oriented molecules; (c) experimental setup: L1,2 are thegreen ( 671 nm01l = ) and red ( 532 nm02l = ) cw lasers, 20l —half wave plates, BS—beam splitters, DH—vortex hologram, M—mirrors,MO1—10 ́microscope objective, MO2—20×microscope objective, NLC—sample, F—filter, CCD—charge-coupled device camera.

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J. Opt. 18 (2016) 054006 Y Izdebskaya et al

vector vortex beams we used a Mach–Zehnder arrangement(beam splitters BS1 and BS3, mirrors M1 and M2). The outputintensity after propagation was monitored by collecting thelight at the output using a 20×microscope objective (MO2)and a high-resolution CCD camera. We monitored the evol-ution at both wavelengths, but in order to prevent chromaticeffects and record the output images of the (red) vortex or

green (fundamental) beams separately, we used either red orgreen filters (F) placed in front of the camera.

We initially investigated the linear and nonlinear beha-vior of either component when each beam propagated in theabsence of the other. As visible in figures 3(a) and (c), for lowinput powers of both beams, P 0.9 mWg,r < , the self-focusingis too weak to overcome diffraction. On increasing the power

Figure 2. Each vector beam component, i.e. a Gaussian (left) or a single-charge vortex (right) propagating separately in the NLC cell. (a) and(d) Measured input intensity distributions; (b) and (e) schematic wavefront views and (c) and (f) normalized intensity profiles of each beamfrom the input images (a) and (d).

Figure 3.Measured output intensity distributions (a)–(d) and corresponding normalized intensity profiles (e)–(h) of Gaussian (a), (b), (e), and(f) and vortex (c), (d), (g), and (h) components propagating separately for various input powers Pg and Pr.

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of the green component to P 2.4 mWg > the fundamentalGaussian beam undergoes self-focusing and the transverseintensity distribution at the output visibly reduces, so that aspatial soliton is formed, also called a nematicon in thesematerials (see figure 3(b)) [35]. On the other hand, increasingthe input power of the beam which carries angular momentumto P 5 mWr > leads to its self-focusing; however, such abeam is affected by strong dynamic instabilities which tend toamplify local azimuthal modulations of the initial ring shapeand so split the vortex beam into two fragments (figure 3(d))[20]. Such a symmetry-breaking azimuthal instability isenhanced by the nonsymmetric configuration of the planarcell and the related anisotropy in the induced refractive indexprofile. This dynamics is consistent with recently reportedresults on the astigmatic deformation of a vortex beam in aplanar NLC cell and its transformation into a dipole-liketransverse distribution [29]. Figures 3(e)–(h) show the nor-malized x cross-sections of the intensity profiles in the plane(x, y) for two different powers of both components, asacquired from the output images in figures 3(a)–(d).

To prevent the symmetry-breaking azimuthal instabilityof the vortex beam in the planar NLC cell we studied theinteraction of two mutually incoherent two-color components—a vortex and a Gaussian beam—co-propagating in the cellas co-polarized coaxial wavepackets. Both components wereextraordinarily polarized (input E-field along the x-axis) andlaunched with the same Poynting direction along z. Figure 4shows the results for two series of experiments for differentinput powers of the red vortex beam, namely P 5 mWr =(figures 4(a)–(e)) and P 8 mWr = (figures 4(f)–(j)). Thesevalues of Pr were high enough to excite nonlinear effects andinstabilities in the absence of the fundamental Gaussian beam.In both sets of experiments we kept Pr constant while we

gradually changing the input power of the green beam Pg.Firstly, we monitored the power dependent dynamics of atwo-color vector soliton at P 5 mWr = , using the red band-pass filter in front of the CCD camera in order to block thegreen light. For low powers of the green nematicon,

P0 mW 3.8g [ ]< < , the initially ring-shaped (red) vortexbeam transformed into a dipole-like mode due to the sym-metry breaking azimuthal instability, which is similar to thedipole-like vector solitons observed in other systems [37, 38].A further increase of the input power, P3.9 mW 5.5g [ ]< < ,led to a dramatic reshaping of the intensity distribution of thevortex, see figure 4(b). Noteworthy, at higher powers

P5.6 mW 8.5g [ ]< < we observed a remarkable stabilizationof the intensity profile of the red vortex and the formation of avector vortex soliton which, in the red component, had anannular shape around a dark core, see figure 4(c). At higherexcitations, P 8.6 mWg > , the spatial dynamics was amplifiedand the vector beam developed temporally unstable intensitydistributions [39], as seen in figure 4(d).

We also observed an analogous behavior of the vectorbeam at the other values of Pr. Figures 4(f)–(i) presentexperimental results for a two-color vector vortex beampropagating at P 8 mWr = in the same NLC cell. As expec-ted, at relatively low input powers P0 mW 4.8g [ ]< < , thesymmetry-breaking mechanism of the vortex beam leads to itsazimuthal instability. However, at higher power excitations,

P4.9 mW 8g [ ]< < , the composite beam is converted into astable circularly symmetric vector vortex soliton. Finally, asexpected, for higher powers P 8.1 mWg > we observe atemporally unstable behavior. By comparing both cases forvarious values of the input power, i.e. P 5 mWr = andP 8 mWr = , we see that a stable vortex soliton can exist for a

Figure 4. Experimental results for the output intensity distribution (a)–(d) and (f)–(i) and the corresponding interferograms (e) and (j) of thevortex (red) component of the composite vector beam for various input powers of the fundamental (Gaussian) component Pg. Images wereacquired for two different input powers of the vortex (red) component P 5 mWr = (a)–(e) and P 8 mWr = (f–j).

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relatively broad power range which is determined by thegreen nematicon power and the mode mixing process.

Finally, we investigated the interferograms of the vectorvortex solitons by employing a Mach–Zehnder arrangement:a tilted broad Gaussian beam at an angle with the vector beaminterferes with it at the output (see figure 1(c)). The inter-ferograms in figures 4(e) and (j) show the phase singularitiesby the characteristic presence of fork dislocations. Therefore,we ascertain that the vortex character of the red input beampersists after the formation of a vector soliton, owing to itsinteraction with the nonlocal potential induced by the greennematicon. The highly nonlocal, nonlinear response of softmatter, specifically reorientational NLCs, dramaticallyenhances the incoherent coupling of two co-polarized wave-packets of different colors, leading to the stabilization of avortex soliton when the soliton power is large enough to trapthe nonlinear vortex.

3. Vector vortex-beam equations

To model the experiments presented above, let us consider thepropagation of two beams of polarized, monochromatic lightof wavenumbers k01 and k02 through a cell containingundoped NLCs. As in the experiments, the z-direction is takenas the propagation direction of the beam, with the (x, y)coordinates orthogonal to this. In the absence of the opticalwave-packets the nematic molecules lie at an angle 0q to the z-direction in the (x, z) plane. Furthermore, the molecules areconstrained to rotate in the (x, z) plane under the influence ofthe electric fields of the optical beams. Let us denote the extra(nonlinear) rotation of the nematic molecules due to theoptical beams by θ. It can be assumed that this extra rotationis small, 0∣ ∣q q . Then in the slowly varying, paraxialapproximation the equations for the electric field envelopesA1 and A2 of the beams and the optically induced rotation θare [24, 34, 35, 43]

k nA

zD A k n A

k nA

zD A k n A

K n A

n A

2i sin 2 0,

2i sin 2 0,

1

4sin 2

1

4sin 2 0, 1

a

a

a

a

01 011

12

1 012

12

0 1

02 022

22

2 022

22

0 2

20 1

20 1

2

0 22

0 22

( )

( )

( )∣ ∣

( )∣ ∣ ( )

d q q

d q q

q d q

d q

¶¶

+ + =

¶¶

+ + =

+

+ =

with the Laplacian ∇2 in the transverse (x, y) plane. Thequantities n01 and n02 are the background refractive indices ofthe medium and na1d and na2d are the optical anisotropies atthe two-wavelengths, respectively [35, 43]. In general,n n na

2 2 2d = - ^ , with nP and n⊥ being the refractive indicesfor electric fields parallel and perpendicular to the optic axis(director) of the NLC, respectively. In the present work thevalues n 1.67= and n 1.51=^ are used, as for the nematicmixture 6CHBT at room temperature [44, 45], which is thereorientational medium used in the experiments. The

parameters D1 and D2 are the diffraction coefficients at thetwo-wavelengths and K is the scalar elastic constant ofthe NLC.

The two-color nematic equations (1) can be simplified bysetting them in nondimensional form via the variable andcoordinate transformations

X Wx Y Wy A u A v Z z, , , , .2

1 2

( )a b g= = = = =

The non-dimensional two-color nematic equations are then

u

Zu ui

1

22 0, 32 ( )q¶

¶+ + =

v

ZD v A vi

1

22 0, 4v v2 ( )q

¶¶

+ + =

u A v2 2 0, 5v2 2 2∣ ∣ ∣ ∣ ( )n q + + =

where the Laplacian ∇2 is now in the transverse (X, Y) planeand the scale width W, scale amplitude β for the A2 mode andscale propagation length γ are respectively

WD

k n

k

k

n

k n

2

sin 2, ,

4

sin 2. 6

a

a

1

01 1 0

02

01

01

01 12

0

( )

( )( )

d qb a

gd q

= =

=

The scale amplitude α for the mode A1 is determined from thepower P1 and width wb of the input beam in the A1 mode. Fora Gaussian input beam

P cn w2 2

. 710

012

b2 ( ) pa=

The nondimensional nonlocal response ν of the soft mediumin the director equation (5) is given by

K

n W

8

sin 2. 8

a0 12 2 2

0( )( )

n

d a q=

Finally, the non-dimensional diffraction coefficient Dv and thecoupling coefficient Av in equation (4) for the beam v aregiven by

DD

D

k n

k nA

k n n

k n nand . 9v v

a

a

2

1

01 01

02 02

02 01 22

01 02 12

( )dd

= =

The parameter values used to non-dimensionalise thenematic equation (1) are taken from experimental data [46].The vortex was formed from a beam at 671 nm02l = and thesolitary wave at 532 nm01l = , with k 201 01p l= andk 202 02p l= . The anchoring angle was 40q p= in theplane (X, Z). The amplitude of the beams was scaled using (7)with the power P 4.9 mW1 = and width w 4 mb m= of thesolitary wave which was found to stabilize the vortex in theexperiments. Finally, the diffraction coefficients were taken tobe D D 11 2= = and the Frank constant K 1.2 10 N12= ´ - .

The input vortex and nematicon at Z=0 are taken tohave Gaussian profiles, as in the experiments, so that

u a e , 10u r wu2 2 ( )= -

v a re e , 11v r w iv2 2 ( )= j-

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J. Opt. 18 (2016) 054006 Y Izdebskaya et al

where r X Y2 2 2= + and j is the related polar angle.Figure 5 shows results for an input vortex of power

10.7 mW and radius of 2.35 mm at its maximum without a co-propagating nematicon. In non-dimensional variables,w 10v = and a 0.25v = . Figure 5(a) shows the vortex after ithas propagated a distance Z=300, i.e. the physical distancez 333 mm= . It can be seen that the vortex becomes unstabledue to the standard n=2 symmetry breaking instability andcollapses into two beams [20]. Figure 5(b) shows the evol-ution of the vortex in the (X, Z) plane (Y= 0). Clearly thevortex beam spreads apart as it becomes unstable, which ismost apparent at the upper end of its propagation range. Thisis due to it breaking up into two nematicons due to the azi-muthal instability.

Figure 6 shows the corresponding results when the vortexco-propagates with a nematicon of input power 0.85 mW andGaussian radius 3.33 mm , so that w 10u = and a 0.5u = . Theco-propagating nematicon stabilizes the vortex, in agreement

with the experiments. Figure 6(a) shows the vortex atZ=300, after propagating a physical distance of 333 mm ,and figure 6(c) the co-propagating nematicon after the samepropagation distance. Figure 6(b) for the evolution of thevortex shows that it oscillates in amplitude and width, butholds together and does not broaden as for the isolated vortexof figure 5(b). Note, the difference between the evolution ofthe unstable and stable vortices in the (X, Z) plane (Y= 0) aremost clearly seen around Z=300 (see figures 5(b) and 6(b)).The input powers for the vortex and nematicon are similar tothose in the experiments, for which the input vortex hadpower 8 mW and width at its maximum of 7 mm and thenematicon which stabilised the vortex had a power of 4.9 mWand a Gaussian width of 4 mm .

A vortex and a nematicon with the nominal powers andwidths as used in the experiments (i.e. not accounting forinput coupling and propagation losses) are far from steadybeams for the nematicon equation (1), so they undergo large

Figure 5.Vortex v∣ ∣ (a) in (X, Y) at Z=300 and (b) in (X, Z) without a co-propagating nematicon. The parameters for the beams are a 0.0u = ,a 0.25v = and w 10.0v = . The nonlocality figure is 219n = .

Figure 6. Vortex v∣ ∣ (a) in X Y,( ) at Z=300 and (b) in (X, Z) with a co-propagating nematicon. (c) Nematicon in (X,Y) at Z=300. Theparameters for the beams are a 0.5u = , w 10u = , a 0.25v = and w 10.0v = . The nonlocality is 219n = .

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amplitude and width oscillations. The nematicon may splitinto two filaments [18], which is typical solitary wave beha-viour for large power beams [47]; the vortex initially under-goes significant shape changes, as can be seen fromfigures 5(b) and 6(b), accompanied by the shedding of dif-fractive radiation. This is partly due to the Gaussian profile(11) not being the exact vortex solution of the nematicequations (3)–(5), so radiation shedding moves it towards thevortex solution.

4. Conclusions

In conclusion, we have shown experimentally and numeri-cally that two-component vector vortex solitons, for whichone of the components carries an optical vortex with a singletopological charge, can be stabilized using the nonlocalreorientational nonlinearity of NLCs. We have found that thecoupling with the fundamental soliton avoids astigmatictransformations of the input vortex into spiralling dipolestates that can occur in this anisotropic medium when a vortexcarrying beam propagates alone. Remarkably, such compositevector solitons are observed for comparable powers of the redand green light components, indicating a strong nonlinearcoupling between them. We expect that our results will fur-ther stimulate the generation of, till now, elusive types ofcomposite solitons, such as multi-pole or multi-ring com-plexes and their periodic dynamical transformations andoscillations. The great potential of such controllable stablerouting of vortex-carrying excitations and the informationencoded in their nontrivial phase distributions requires furtherdetailed studies.

Acknowledgments

YI and WK acknowledge the Australian Research Council forfinancial support. GA thanks the Academy of Finland forfinancial support through the Finland Distinguished Professorgrant no 282858. WK acknowledges support from the QatarNational Research Fund.

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1. Introduction2. Sample and experimental results3. Vector vortex-beam equations4. ConclusionsAcknowledgmentsReferences