nd Reading - School of Mathematics | School of Mathematicsnoel/jnopmkingussie...2 nd Reading December 12, 2014 16:11 WSPC/S0218-8635 145-JNOPM 1450045 L. W. Sciberras et al. optical

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December 12, 2014 16:11 WSPC/S0218-8635 145-JNOPM 1450045

Journal of Nonlinear Optical Physics & MaterialsVol. 23, No. 4 (2014) 1450045 (19 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218863514500453

Steering of optical solitary waves by coplanar low powerbeams in reorientational media

Luke W. Sciberras

Instituto de Investigaciones en Materiales,Universidad Nacional Autonoma de Mexico,

Apdo. Postal 70-360, Cd. Universitaria Mexico, 04510 D.F., [email protected]

Antonmaria A. Minzoni

Fenomenos Nonlineales y Mecanica (FENOMEC),

Department of Mathematics and Mechanics,Instituto de Investigacion en Matematicas Aplicadas y Sistemas,

Universidad Nacional Autonoma de Mexico, 01000 Mexico D.F., [email protected]

Noel F. Smyth

School of Mathematics and Maxwell Institute for Mathematical Sciences,University of Edinburgh, The King’s Buildings,Edinburgh, Scotland EH9 3FD, United Kingdom

[email protected]

Gaetano Assanto

NooEL Nonlinear Optics and OptoElectronics Lab,University ‘Roma Tre,’ Via della Vasca Navale 84, 00146 Rome, Italy

Optics Lab, Department of Physics, Tampere University of TechnologyFI 330101 Tampere, Finland

[email protected]

Received 15 July 2014

The interaction of solitary waves in a nematic liquid crystal (NLC) with a coplanar lowpower optical beam is investigated. The emphasis of the study is on the control of thesolitary wave trajectory by the low power beam and the transfer of momentum betweenthe beams. The results of numerical studies are confirmed by a theoretical analysis ofthis momentum transfer. The implications for all-optical signal control are discussed.

Keywords: Nonlinear optics; optical solitons; all-optical switching; liquid crystals.

1. Introduction

Nonlinear light propagation in nematic liquid crystals (NLCs) has attracted manyresearchers in recent years, both experimentalists and theorists.1–6 The “huge”

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L. W. Sciberras et al.

optical nonlinearity, as well as the nonlocality which extends the medium responsewell beyond the size of the beam, render NLCs an ideal material platform forthe excitation, propagation and management of optical solitary waves. The nonlo-cal response, in particular, prevents the collapse of two-dimensional bulk solitarywaves,4,7,8 termed “nematicons”, making them extremely robust to perturbationsand collisions.1,9 Experiments conducted on nematicons have shown that they couldbe employed in several all-optical devices, including circuits, reconfigurable inter-connects, logic and bistable gates, owing to the waveguide character of Kerr-likesolitary waves with respect to weaker copolarized signals10–13 and provided theresponse time is not a crucial issue.1,14–19 The formation of solitary waves in NLCrequires that the Freedericksz threshold is overcome3,10,20 and that beam spatialdispersion (diffraction) is balanced out by self-focusing.8 To avoid this threshold theorganic molecules of NLCs can be pre-tilted by applying an external low frequencyelectric field across the cell thickness1,10 or by anchoring the nematic moleculesat a finite angle (typically close to π/4 (see Ref. 21)) with respect to the beamwavevector.1,2,22

The path of nematicons can be altered and controlled by various means,including other solitary waves,9,23,24 external beams and refractive index perturba-tions,15,16,18,19,25–29 applied voltage(s)22,30,31 and input power.32,33 In this paper,we discuss the use of a low power beam launched in the same principal plane as thenematicon in order to modify the latter’s trajectory. The geometry under consider-ation, since the principal plane is defined by the optic axis (or molecular director)and the beam wavevector, corresponds to propagation either in the plane (x, z)orthogonal to the parallel interfaces of a planar cell or in the plane (y, z) paral-lel to them, depending on the initial orientation of the optic axis. In both casesthe extraordinarily polarized electric field lies in the same principal plane. Thecopolarized and coplanar control beam is assumed to share the wavelength and becoherent with the nematicon, i.e., to affect its path through both the nonlinearnonlocal induced index perturbation and interference effects.

2. Governing Equations

Two extraordinarily polarized coherent light beams are launched into a planar NLCcell, as sketched in Fig. 1. We take the z-axis as the propagation direction, the x-axisas the direction of the input beam polarisation, with y completing the coordinatesystem. A solitary wave results owing to the balance between linear diffraction andthe self-focusing of the beam via optical reorientation, i.e., the change in molecu-lar director orientation due to the Coulomb torque between the electric field vectorand the induced molecular dipoles. In positive uniaxial NLC the optic axis coincideswith the molecular director, i.e., the average orientation of the molecular main axesin space. To maximize the nonlinearity (reorientational response) and be able toobtain solitary waves at milliwatt power levels, the molecular director is assumedto be pre-oriented at an angle θ0 ∼ π/4 with respect to the beam wavevector, the

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Steering of optical solitary waves by coplanar low power beams in reorientational media

Nematicon

Low power beam

Y

X

Z

Fig. 1. Sketch of the planar NLC cell with principal plane (x, z). Two collinear copolarized(extraordinary wave) beams are launched in the NLC, one is a low power wider beam and theother forms a self-confined nematicon.

latter taken parallel to z in Fig. 1. The pre-orientation, obtained by either an appliedvoltage bias across the cell thickness or suitable anchoring of the molecules at theplanar (glass or plastic) interfaces, also eliminates the Freedericksz threshold3,10

and helps the medium to mimic a Kerr-like (saturable and nonlocal) nonlinearresponse. Neglecting birefringent walk-off (i.e., the angular offset between Poyntingand wavevectors), the nondimensional equations governing the propagation of non-linear, extraordinarily polarized light beams in the paraxial approximation are34–36

i∂E

∂z+

12∇2E + 2Eθ = 0, ν∇2θ − 2qθ = −2|E|2. (2.1)

Here the Laplacian ∇2 is in the (x, y) plane. The first of (2.1) is a nonlinearSchrodinger (NLS)-type equation governing the complex electric field envelope E

of the beam, with z being a “time-like” coordinate. The second equation of (2.1)is the director equation (an elliptic equation) which rules the perturbation θ of themolecular director angle from the rest angle θ0. The parameter ν relates to theintermolecular elasticity of the liquid crystal, with large values corresponding toa highly nonlocal response. In typical experiments ν = O(100).1,10,37 We take thecell to be infinitely extended as beam sizes are much smaller than any character-istic (thickness, width, length) dimension of a cell. Hereby, we focus on coherent,copolarized, copropagating beams of the same wavelength with their electric fieldvectors in the principal plane. Therefore, the excitation is chosen as a one colourbeam with two components, one representing the nematicon with a Gaussian, prop-agation invariant, profile and the other representing a low power Gaussian beam.There are no general nematicon solutions. Hence, the electric field of the opticalbeam will be assumed to have the form of the trial function

E = a1e−r2

1/w21ei(σ1+Vx1(x−ξ1)+Vy1(y−η1)) + a2e

−r22/w2

2ei(σ2+Vx2(x−ξ2)+Vy2(y−η2)),

(2.2)

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where r21,2 = (x − ξ1,2)2 + (y − η1,2)2 with the subscripts 1 and 2 referring to the

nematicon and low power beams, respectively.

3. Discussion and Results

To demonstrate the effects and the diversity of in-plane steering of a nematicon witha low power beam, various numerical experiments were conducted with parametersselected to be consistent with those of typical experiments.

3.1. Change in power via a change in amplitude for the lower

power beam

Two coplanar and copolarized beams are input into an NLC cell. One beam is anematicon and the other is a low power beam with a fixed width. The effect of achange in the power of the low power beam is explored by increasing its amplitude.Due to the nonlocality of the NLC, the low power beam alters the refractive indexwell beyond its peak intensity, adding to the refractive index change self-induced bythe nematicon. The refractive index change due to the presence of the low powerbeam skews the director’s response, i.e., the nematic molecules reorientate morewhere the low power beam is present. Figure 2 shows the skewness of the directorfield at (ξ2, η2) = (−20, 0) where the peak intensity of the low power beam is located.Hence, by changing the amplitude of the low power beam, a larger reorientationtakes place. The corresponding increase in refractive index attracts the nematicon,as shown in Fig. 3(a). Figure 3(a) shows the trajectories of the nematicon as itevolves in the NLC when the low power beam is initially at (ξ2, η2) = (−10, 0). Thetrajectory of the nematicon can be steered more by increasing the initial distancebetween the two beams. Figure 4(a) shows the nematicon trajectory when the lowpower beam is initially at (ξ2, η2) = (−20, 0). An increased movement towards the

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Fig. 2. (Color online) The low power beam modifies the refractive index distribution in theNLC. (a) Side-view and (b) top-down view of the same refractive index change due to the controlbeam. By reducing the maximum value of θ (in (b)) the change in the refractive index can beseen more clearly. The initial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0)and (Vx1, Vy1) = (0, 0). The initial values for the low power beam are a2 = 0.3, w2 = 15,(ξ2, η2) = (−10, 0) and (Vx2, Vy2) = (0, 0), with ν = 200 and q = 2.

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Fig. 3. (Color online) Nematicon trajectories for different amplitudes of the low power beam,for a2 = 0.0 (red solid line), a2 = 0.1 (green long dashed line), a2 = 0.15 (blue short dashedline), a2 = 0.2 (magenta dotted-line) and a2 = 0.25 (light blue dot-long dashed line) for (a) theposition of low power beam at (ξ2, η2) = (−10, 0) (region between two black dotted lines). Final(b) nematicon electric field modulus and (c) director field for each of the amplitudes. The initialvalues for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0). The initialvalues for the low power beam are w2 = 15 and (Vx2, Vy2) = (0, 0), with ν = 200 and q = 2.

low power beam can be seen on comparing the trajectory with that in Fig. 3(a). Themovement of the nematicon in both cases is towards the refractive index changecreated by the low power beam if the two are launched in-phase. As the amplitudeof the low power beam increases, an in-phase nematicon moves closer towards it.This nonlinear interaction is highly enhanced due to the mutual phase relationshipof the two coherent beams.

When the fields of the nematicon and the control (low power) beams are π out ofphase, the beams tend to repel in spite of the attraction mediated by the additionalrefractive index change created by the low power beam. When the latter overcomesthis repulsion, then the nematicon “jumps” into the “potential well” associatedwith the control beam.

Figures 5(a)–5(e) show the evolution of nematicon and low power beams, withthe exception of Fig. 5(a) which shows the beams repelling as expected. In all other

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Fig. 4. (Color online) Nematicon trajectories for different amplitudes of the low power beam, fora2 = 0.0 (red solid line), a2 = 0.1 (green long dashed line), a2 = 0.15 (blue short dashed line),a2 = 0.2 (magenta dotted-line) and a2 = 0.25 (light blue dot-long dashed line) for (a) the positionof the low power beam at (ξ2, η2) = (−20, 0) (region between two black dotted lines). Comparisonof the final (b) nematicon electric field modulus and (c) director field for each of the amplitudes.The initial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0).The initial values for the low power beam are w2 = 15 and (Vx2, Vy2) = (0, 0), with ν = 200and q = 2.

cases the beams exchange positions. Figure 6 highlights the positional exchangeof the nematicon, showing the abrupt jump occurring within the region betweenz = 20 to z = 30 for all the nematicons, except for the nematicon colaunchedwith the lowest power control beam (Fig. 5(a)), which continues along a quasi-straight trajectory. This jump behaviour occurs when the refractive index of theNLC is altered sufficiently, so that the repulsion initially felt by the two beams isno longer strong enough to maintain their separation. Hence, the director providesa means for the beams to exchange positions. Figure 7(a) shows the electric fieldat z = 100, with the nematicon and low power beam initially coincident. Thisfigure also highlights the diffraction of the low power beam. Figure 7(b) shows thedirector distribution at z = 100, with the low power beam still influencing therefractive index. Finally, Fig. 7(c) illustrates the phase distribution at z = 100,

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(a) (b)

(c) (d)

(e)

Fig. 5. (Color online) Evolution of a nematicon and low power beam for different initial ampli-tudes of the low power beam. (a) a2 = 0.1, (b) a2 = 0.15, (c) a2 = 0.2, (d) a2 = 0.25 and(e) a2 = 0.3. The initial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and(Vx1, Vy1) = (0, 0). The initial values for the low power beam are w2 = 15, (ξ2, η2) = (−10, 0)and (Vx2, Vy2) = (0, 0), with ν = 200 and q = 2.

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Fig. 6. (Color online) Comparison of the nematicon trajectory for different amplitudes of a lowpower beam. a2 = 0.1 (red solid line), a2 = 0.15 (green long dashed line), a2 = 0.2 (blue shortdashed line) and a2 = 0.25 (magenta dotted-line) and a2 = 0.3 (light blue dot-long dashed

line). The trajectory of the unimpeded low power beam is the region between two black dottedlines. The initial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) =(0, 0). The initial values for the low power beam are w2 = 15, (ξ2, η2) = (−10, 0) and (Vx2, Vy2) =(0, 0), with ν = 200 and q = 2.

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(a) (b)

(c)

Fig. 7. (Color online) The beam at z = 100 showing (a) the modulus of the electric field,(b) the director field and (c) the phase of the nematicon and low power beam. The initial valuesfor the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0). The initial valuesfor the low power beam are a2 = 0.3, w2 = 15, (ξ2, η2) = (−10, 0) and (Vx2, Vy2) = (0, 0), withν = 200 and q = 2.

showing the formation of concentric circles around the position of the nematicon.These concentric circles also occur in the phase plot for the propagation of only onenematicon. However, for the coupled nematicon and low power beam, in the regionaround x = 7, there is a bulge due to the presence of the low power beam whichnow has the form of diffractive radiation.

3.2. Refraction

To demonstrate nematicon refraction, two coplanar, copolarized beams are inputinto a NLC cell, with the low power beam fixed at (ξ2, η2) = (0, 0). The nematicon isgiven an initial angle and is input at the position (ξ1, η1) = (−10, 0). The low powerbeam creates a refractive index change and the nematicon undergoes refraction.Figure 8(a) depicts the evolution of the nematicon without the low power beam,while Fig. 8(b) depicts the evolution of the nematicon with the low power beamperturbing the NLC. At the propagation length z = 40 the interaction of the twobeams is strongest and refraction of the nematicon can be clearly observed.

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(a) (b)

Fig. 8. (Color online) The evolution of a nematicon (a) without the presence a low power beamand (b) with an induced refractive index change due to the presence of a low power beam. Theinitial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (−10, 0) and (Vx1, Vy1) = (0.2, 0).The initial values for the low power beam are a2 = 0.25, w2 = 15, (ξ2, η2) = (0, 0) and (Vx2, Vy2) =(0, 0), with ν = 200 and q = 2.

Figures 9(a)–9(e) show the interaction of a low power beam of various inputpowers with a nematicon launched at different angles. Figure 9(a) uses the smallestinput angle for the nematicon. Within the propagation length the nematicons donot bend below their paths in the absence of the low power beam. However, thenematicons are attracted to the low power beam, with the largest power controlbeam creating the strongest attraction of the nematicon, as evidenced by the largestbending. As the nematicon input angle is increased, the bending of its trajectoryincreases and the nematicon is attracted back towards the low power beam. Thus,the trajectory of the nematicon is bent, as shown in Figs. 9(c)–9(e), where thenematicon deviates from the path of the unperturbed nematicon. Steering of thenematicon can then be achieved by using an initial tilt and controlling the finalposition by the amplitude (power) of the low power beam.

3.3. Change in width of the low power beam

The effects of increasing the power of the control beam are explored by fixing itsamplitude and widening its profile. The tails of the low power beam extend furtherand affect a larger region of NLC than in the case of a fixed width and increasedamplitude (Sec. 3.1). However, due to the small amplitude of the control beam,the rotation induced in the NLC molecules is less than that of Sec. 3.1 as thepeak intensity is smaller. Figure 10 displays the director distribution with skew-ness towards the low power beam at (ξ2, η2) = (−20, 0). Comparing Fig. 10(b)with Fig. 3(b) a larger NLC region is affected by the wider low power beam (notethe scales of the respective plots). Figures 11(a) and 11(b) show the nematicontrajectory after copropagating with a low power beam of different widths andtwo different initial positions, (ξ2, η2) = (−10, 0) and (ξ2, η2) = (−20, 0), respec-tively. The trajectories of the nematicons are similar to those in Figs. 3(a) and4(a).

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Fig. 9. (Color online) Nematicon trajectories for different initial amplitudes of a low powerbeam between a2 = 0.0 (red solid line), a2 = 0.15 (green long dashed line), a2 = 0.2 (blue shortdashed line) and a2 = 0.25 (magenta dotted-line) for different launch angles of the nematicon (a)(Vx1, Vy1) = (0.05, 0), (b) (Vx1, Vy1) = (0.1, 0), (c) (Vx1, Vy1) = (0.15, 0), (d) (Vx1, Vy1) = (0.2, 0),and (e) (Vx1, Vy1) = (0.25, 0), for an infinite cell. The trajectory of the unimpeded low power beam(region between two black dotted lines). The initial values for the nematicon are a1 = 2.5, w1 = 3,(ξ1, η1) = (−10, 0). The initial values for the low power beam are w2 = 15, (ξ2, η2) = (0, 0) and(Vx2, Vy2) = (0, 0), with ν = 200, and q = 2.

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(a) (b)

Fig. 10. (Color online) Skewness of the director distribution with a low power beam presentat (ξ2, η2) = (−20, 0). (a) total director field and (b) a reduction of the z-axis to emphasizethe skewness in the field towards the position of the low power beam. The initial values for thenematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0). The initial valuesfor the low power beam are a2 = 0.2, w2 = 25 and (Vx2, Vy2) = (0, 0), with ν = 200 andq = 2.

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Fig. 11. (Color online) Nematicon trajectories for different widths for a low power beam witha fixed launch angle for w2 = 10 (red solid line), w2 = 15 (green long dashed line), w2 = 20(blue short dashed line) and w2 = 25 (magenta dotted-line) for (a) position of low power beamat (ξ2, η2) = (−10, 0) and (b) position of low power beam at (ξ2, η2) = (−20, 0). Trajectory ofthe unimpeded low power beam only for w2 = 15: region between two black dotted lines. Theinitial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0).The initial values for the low power beam are a2 = 0.2 and (Vx2, Vy2) = (0, 0), with ν = 200 andq = 2.

3.4. Tilted low power beam

Two coplanar, copolarized beams are launched into a NLC cell, with the low powerbeam given an initial launch angle. Two different cases are studied, in the first casethe low power beam has a fixed width of w2 = 15, while in the second case the widthof the low power beam is doubled to w2 = 30. Figures 12(a) and 12(b) show thenematicon trajectory when the control beam has initial angles (Vx2, Vy2) = (0.2, 0)and (Vx2, Vy2) = (0.4, 0), respectively, with initial position at (ξ2, η2) = (−10, 0).

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Fig. 12. (Color online) Nematicon trajectories for different amplitudes with a low power beamwith a fixed launch angle for a2 = 0.1 (red solid line), a2 = 0.15 (green long dashed line),a2 = 0.2 (blue short dashed line) and a2 = 0.25 (magenta dotted-line) for (a) (ξ2, η2) = (−10, 0),(Vx2, Vy2) = (0.2, 0), (b) (ξ2, η2) = (−10, 0), (Vx2, Vy2) = (0.4, 0), (c) (ξ2, η2) = (−20, 0),(Vx2, Vy2) = (0.2, 0) and (d) (ξ2, η2) = (−20, 0), (Vx2, Vy2) = (0.4, 0) for an infinite cell. Trajec-tory of the unimpeded low power beam only for w2 = 15: region between two black dotted lines.The initial values for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0).The initial values for the low power beam are w2 = 15, with ν = 200 and q = 2.

Before the low power beam crosses the path of the nematicon, the latter is attractedtowards the low power beam. As the low power beam passes through the nematicon,there is a shift in the position of the nematicon and it tends to follow the low powerbeam. However, the initial momentum of the nematicon continues to move it in theopposite direction. Figure 12(b) indicates (by the trajectory of the nematicon) thatthe larger initial angle of the low power beam affects the refractive index only nearthe nematicon, which results in the nematicon trajectories converging to the samepoint regardless of the initial power of the control beam. Figures 12(c) and 12(d)show the nematicon trajectory when the control beam has initial angles (Vx2, Vy2) =(0.2, 0) and (Vx2, Vy2) = (0.4, 0), respectively, with initial position at (ξ2, η2) =(−20, 0). Very little movement of the nematicon trajectory is observed, but thereis a slight movement of the nematicon towards the control beam at z = 200.

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Fig. 13. (Color online) Nematicon trajectories for different amplitudes with a low power beamwith a fixed launch angle for a2 = 0.1 (red solid line), a2 = 0.15 (green long dashed line), a2 = 0.2(blue short dashed line) and a2 = 0.25 (magenta dotted-line). (a) (ξ2, η2) = (−10, 0), (Vx2, Vy2) =(0.2, 0), (b) (ξ2, η2) = (−10, 0), (Vx2, Vy2) = (0.4, 0), (c) (ξ2, η2) = (−20, 0), (Vx2, Vy2) = (0.2, 0)and (d) (ξ2, η2) = (−20, 0), (Vx2, Vy2) = (0.4, 0) for an infinite cell. Trajectory of the unimpededlow power beam only for w2 = 30: region between two black dotted lines. The initial values forthe nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0). The initial values forthe low power beam are w2 = 30 with ν = 200, and q = 2.

Figures 13(a) and 13(b) show a different result from the previous figure. Thenematicon follows the low power beam after the interaction, rather than continuingtowards the initial position of the low power beam. The movement of the nemati-con’s trajectory is more significant when the initial angle of the low power beam issmall, and its power is large, as the refractive index of the NLC is affected moreby both. Shifting the initial position of the low power beam to (ξ2, η2) = (−20, 0)results in the nematicon trajectory moving more towards the low power beam, asshown in Fig. 13(c). Due to the larger width of the low power beam, the director isperturbed more and as, a result, attracts the nematicon. The wider low power beamacts like a plane wave compared with the narrower low power beam of Fig. 12. Thesame is true when the initial angle of the low power beam is (Vx2, Vy2) = (0.4, 0),

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as shown in Fig. 13(d), for which the trajectory of the nematicon has been bentdramatically from its original input position.

3.4.1. Asymptotic analysis

An estimate for the position of the deflected nematicon of Sec. 3.4 can be obtainedby using an averaged Lagrangian38 as follows. We assume the impinging wave is alinear wave which, to leading order, is a solution of the Schrodinger equation, thatis the first equation of (2.1) with the nonlinear term involving θ neglected. Thisgives the usual plane wave solution

Enw = a2ei(V2·X−λz), (3.1)

where λ satisfies the linear dispersion relation for a plane wave in the form

λ =12V2 ·V2 − a2

2. (3.2)

The vector V2 gives the direction of the incident beam. We now assume that thetotal electric field can be taken in the form

Enem = a1e−r2/w2

1ei(V1·(X−ζ(z))+σz) + Q, (3.3)

where X = (x, y), ζ = (ξ1, η1), V1 = (Vx1, Vy1) and r = |X − ζ|. The nematiconparameters a1, w1 and σ are taken to be constants and to satisfy the approximateamplitude-width relation. Also, the amplitude a2 and the wavevector V2 are takento be constants, along with λ satisfying the linear dispersion relation, Eq. (3.2).The only z dependence is in the position and velocity of the nematicon.

The Lagrangian of the nematicon equations (2.1) is∫ zf

0

∫ ∞

−∞

∫ ∞

−∞

[i(EzE

∗ − E∗zE) − |∇E|2 + 2θE

]dx dy dz, (3.4)

where zf is the final propagation length, E = Enem + Enw and the director angle θ

satisfies

ν∇2θ − 2qθ = −2|a1e−r2/w2

1ei(V1·(X−ζ(z)+σ) + Q|2. (3.5)

We now obtain an approximation to the refractive index change induced by theintroduction of the linear beam to the nematicon. This is obtained by keeping onlythe linear terms in a1 on the right-hand side of Eq. (3.5). This gives for the totaldirector θ = θnem + θnw

ν∇2θnem − 2qθnem = −2a21e

−2r2/w21 , (3.6)

ν∇2θnw − 2qθnw = −4 Rea1a2e−|X−ζ(z)|2/w2

1

× cos (V2 ·X − λz − σz + V1 · (X − ζ(z))) . (3.7)

The solution for θnem is just the usual term sustaining the nematicon which givesthe usual contribution to the nematicon Lagrangian which is independent of ζ(z).39

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We approximate θnw using the delta function for the nematicon trial function toobtain

θnw = 4a1a2G(X − ζ(z)) cos(V2 · ζ(z) − λz − σz)∫ ∞

−∞

∫ ∞

−∞e−|X−ζ(z)|2/w2

1 dX,

(3.8)

where G(X,X′) is the Green’s function for the director equation, the second of(2.1).

This last expression is substituted in the Lagrangian (3.4) to obtain for thecontribution of the director to the Lagrangian

Lnw =∫ ∞

−∞

∫ ∞

−∞|E(X)|2 dX

∫ ∞

−∞

∫ ∞

−∞G(X,X′)|E(X′)|2 dX′

= 4a1a2 cos(V2 · ζ(z) − λz − σz)

×∫ ∞

−∞

∫ ∞

−∞e−|X−ζ(z)|2/w2

1 dX∫ ∞

−∞

∫ ∞

−∞G(X,X′)|E(X′)|2 dX′. (3.9)

The second integral is now approximated by taking |E|2 to be the nematicon field.It is then independent of ζ(z). The final Lagrangian is therefore

L =∫ zf

0

[Mζ · V1 − M

2|V1|2 − 4πa3

1a2w41 cos(V2 · ζ(z) − λz − σz)

]dz, (3.10)

where M =∫ ∞−∞

∫ ∞−∞ |E|2 dX and we have omitted the z independent terms.

Taking variations of the Lagrangian (3.10) gives the equations of motion for thecentre of the beam

ζ = V1, (3.11)

V1 =4πa3

1a2w41

MV2 sin(V2 · ζ(z) − λz − σz). (3.12)

These equations of motion have a simple solution which gives the nematicon tra-jectory. This solution must match the phase of the linear wave and the nematicon,which gives

V2 · ζ(z) = (λ + σ)z. (3.13)

Thus

ζ(z) =V2

|V2|( |V2|

2− a2

2

|V2| +σ

|V2|)

z (3.14)

is the required solution for the nematicon trajectory.To compare the modulation solution with numerical solutions, we set a2 = 0.25

and V2 = (0.4, 0) and the nematicon phase σ is estimated from the numericalevolution of the free nematicon. For the amplitudes and widths in this case wehave a1 = 2.5 and w1 = 3. The frequency of oscillation of the nematicon in thefield of the beam is estimated from the period in Fig. 14, which is a plot of thenumerical amplitude of the nematicon interactions with the low power beam. This

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1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 50 100 150 200

|E|

z

Fig. 14. (Color online) Amplitude for a nematicon and low power beam interaction. The initialvalues for the nematicon are a1 = 2.5, w1 = 3, (ξ1, η1) = (0, 0) and (Vx1, Vy1) = (0, 0). The initialvalues for the low power beam are a2 = 0.25, w2 = 30, (ξ2, η2) = (−20, 0) and (Vx2, Vy2) = (0.4, 0),with ν = 200, and q = 2.

gives the estimate σ = 2π/45, which is smaller than the value σ = π/10 for a freenematicon.39 With these values we have for the vertical component of the nematicondisplacement d(z) (the only one present in this case)

d(z) = 0.35(z − z0), (3.15)

where z0 is the point where the nematicon starts to move. From Fig. 13(d) weobtain for z0 = 100 the displacement of d(200) = 35, which compares well with thedisplacement calculated from the full numerical solution, which is d(z) = 30 (lightblue dot-long dashed line in Fig. 13(d)).

The reduction in σ can be explained qualitatively using the modulation equationfor σ, which is now modified by the interaction between the nematicon and the planewave. This term decreases the frequency. However, to complete the study it will benecessary to include radiation effects which will further slow the nematicon. Thisis currently under investigation.

4. Conclusion

A NLC has been demonstrated to show numerous different effects when used asa medium to propagate nematicons.1,2 Numerous different techniques for beamsteering have been applied experimentally and explored theoretically.36 However,in-plane steering by a low power beam is not one of them. In this article severalmethods have been employed to explore the potential for steering a nematicon withanother beam in the form of a low power beam. It was found that fixing eitherthe amplitude or width of the low power beam results in the deformation of thedirector which attracts the nematicon, thus altering its trajectory. The nematicon’strajectory can also be steered using a refractive index change induced by a lowpower beam, with the inputted nematicon tilted by an angle. The propagation of

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two out of phase beams lead to the interesting result of the nematicon swappingpositions with the low power beam, which is diffracted away afterwards. This effecthas the potential to be used in all-optical communications. Tilting the initial angleof the low power beam influences the nematicon trajectory which is dependent uponthe width of the low power beam. Finally, increasing the width of the low powerbeam from w2 = 15 to w2 = 30 created a beam which was similar to a plane waveand “transferred” momentum to the nematicon. The use of low power beams tosteer and control nematicons offers a new avenue to explore experimentally andtheoretically.

Acknowledgments

LWS acknowledges support from the postdoctoral fellowship program DGAPA-UNAM, Mexico. GA and NFS were supported by the Royal Society of Londonunder grant IE111560. GA acknowledges the Academy of Finland for his FinnishDistinguished Professor project, grant 282858.

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