M.S. Petrovi , A.I. Strini , N.B . Aleksi and M.R. Beli 1 OR THE - PowerPoint PPT Presentation

Solitons in strongly nonlocal nematic liquid crystals M.S. Petrovi ć , A.I. Strini ć, N.B . Aleksić and M.R. Beli ć 1 OR THE TALE OF THE PAPER THAT REFUSES TO GET PUBLISHED Institute of Physics, P.O.Box 68, 11001 Belgrade, Serbia, 1 Texas A&M University at Qatar, P. O. Box 5825 Doha, Qatar LENCOS 2012, Sevilla Spain

IOP research group in NL optics and dynamics M. R. Beli ć M. Petrović A. Strinić N. Aleksic N. Petrović R. Jovanovi ć D. Timotijević D. Jović S. Prvanović

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Nematicons: Solitons in NLCs The Model: Essentially nonlocal    A Paraxial wave  D  D e     2 OPT 2 2 2 ik A k sin sin A 0 equation for light  x , y 0 0 z   2   A   2   D   e D e  D e    Director’s angle DC DC OPT 2 2 K E sin 2  of reorientation x y , 0   t 2              o Boundary conditions x L / 2 x L / 2 2 X. Hutsebaut, C. Cambournac and + initial conditions M. Haelterman, JOSA B 22 (2005) 1424-1431       0 θ 0 – Angle without light (the pre-tilt angle)

There exist the old and the new model The old model, by Assanto et al.  A  D  e     2 2 2 Fixed BCs: 2 ik A k (sin sin ( )) A 0  x , y 0 a rest z  in = p/2                   z , V V V exp z / z rest 0 in 0    0 = p/4 1   D   e e  2 K sin( 2 ) A   x , y 0 a t 4      0 The new model, by Haelterman et al.    A  D  D e     2 OPT 2 2 2 ik A k sin sin A 0  x , y 0 0 z Note the differences!   2   e A   2   D    D e  D e   DC DC OPT 0 K E sin 2  x , y   t 2 2   + Boundary conditions on  0 as an eigenvalue problem This is where the problem occurred: Old models die hard

Our results: The temporal evolution of the angle of reorientation Successive Over Relaxation (SOR) method       0   0 ˆ  The angle of reorientation θ depends on both light and voltage, where the angle θ 0 ( x,y,z ) accounts for the ˆ  molecular orientation induced by the static electric field only, while the quantity ( x,y,z,t ) corresponds to the optically induced molecular reorientation. The profile  0 is determined in the beginning, using boundary conditions and the successive over relaxation algorithm for the stationary case of Eq.

Beam profiles: Breathers Result: Spatial solitons breathe; the beam preserves its new shape but its characteristic width and maximum intensity breathe as it propagates. Intensity and reorientation profiles suggest that the problem is strongly nonlocal! Beam intensity (left) and molecular reorientation (right), in the middle of the crystal, as functions of the propagation distance, shown for the input beam intensity I = 8.6x10 10 V 2 /m 2 . For the intensity of soliton breathing, ( I = 8.6x10 10 V 2 /m 2 ), we show the intensity and the optically induced molecular reorientation in the middle of the crystal, I ( 0,0,z ) and ( 0,0,z ) and the corresponding FWHM of its transverse profiles, as functions of the propagation distance.

Soliton and Gaussian propagation by two different propagation methods Spatial solitons in highly nonlocal media with quadratic response possess Gaussian profiles. However, the fundamental soliton profile is not Gaussian. The soliton intensity profile compared to a Gaussian is shown in Fig. 3(a). To check the stability of fundamental solitons, we propagate them numerically; peak intensities as functions of the propagation distance are presented in Fig. 3(b). Also included in Fig. 3(b) is a case presenting the propagation of a Gaussian with similar parameters, but obtained using two different numerical methods. In both methods a split-step beam propagation procedure based on the fast Fourier transform (FFT) is used for the propagation of the optical field. In the first method the diffusion equation for the optically induced molecular reorientation is treated using the SOR method; this procedure is referred to as the FFT+SOR. In the second method the diffusion equation is treated using the splitstep procedure again – this is the FFT+FFT procedure. The methods provide similar results; the first method is more accurate. When one considers the propagation of a Gaussian beam using the two propagation methods, the results are close. In a highly nonlocal system, the potential is broad and parabolic, making it impossible for the narrow localized solution to radiate and relax to the fundamental soliton. It just keeps oscillating, forming a quasi-stable breathing soliton. Therefore, the FFT procedure FIG. 3. (a) Fundamental soliton intensity profile obtained should be discarded. Even the SOR soliton solution slightly by the eigenvalue method (black dots), fitted with a oscillates at lower accuracy; this, however, becomes Gaussian. Parameters: P = 10.6 mW, m = 3 . 84 L D -1 ; zero imperceptible as the accuracy is improved. In Fig. 3(b) we show BCs. (b) Soliton and Gaussian propagation using two a case where the oscillation of the amplitude is still perceptible. different propagation methods, FFT+SOR and FFT+FFT. This brings us to an important point. The soliton power P = 10.6 mW; Gaussian power P =10.6 The propagation of a Gaussian invariably leads to breathing mW for FFT+SOR, P =10.1 mW for FFT+FFT. beams, regardless of the method of integration.

Propagation of fundamental solitons – the influence of noise When the fundamental soliton is propagated through the medium in which a small amplitude noise added to Ɵ 0 , a breathing solution is obtained. We introduce noise by adding randomly distributed white noise to Ɵ 0 at each propagation step. Adding more noise leads to larger and more irregular oscillations, although for awhile one can discern a simple sinusoidal breather in the background, with the same period (Fig. 4). Such a situation is physically plausible: the existence of noise or random fluctuations in the director field of NLCs is a well established fact. The same induced oscillation phenomenon happens as well when a small intensity noise is added to the fundamental profile, but Ɵ 0 , kept unchanged. In a highly NN medium any additional energy from noise, no matter how small, cannot be radiated away and the solution has no way to relax to the fundamental soliton. Therefore, it keeps oscillating about the shape-invariant soliton, forming a stable breathing soliton. Since noise is unavoidable in any realistic set-up, be it experimental or numerical, this fact opens the question of physical observability of shape- invariant fundamental solitons in highly NN media. FIG. 4. Propagation of the fundamental soliton in a noisy medium. An amount of (a) 0.1%, (b) 0.5%, and (c) 1% randomly changing noise is added to 0. Blue sinusoidal fit is to guide eye. Parameters: m = 4 L D -1 , zero BCs.

Publish these results: Mission impossible  Four versions – four rejections  Impossible to pass by certain reviewers  Finally placed in the arXiv: Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist? arXiv: 1110.5053v1 [physics.optics] 2011  However, by cannibalizing: adding/subtracting thus far two papers published and the 3 rd one submitted  The first: Breathers in biased highly nonlocal uniaxial nematic liquid crystals, Phys. Scr. 85 (2012) 015403 (6pp)

The second: Solitons in highly nonlocal nematic liquid crystals: Variational approach PHYSICAL REVIEW A 85 , 033826 (2012)  We investigate numerically and theoretically solitons in highly nonlocal three dimensional nematic liquid crystals.  We calculate the fundamental soliton profiles using the modified Petviashvili’s method.  We apply the variational method to the widely accepted scalar model of beam propagation in uniaxial nematic liquid crystals and compare the results with numerical simulations.  To check the stability of such solutions, we propagate them in the presence of noise. We discover that the presence of any noise induces the fundamental solitons – the so-called nematicons – to breathe.  Our results naturally explain the difficulties in experimental observation of steady nematicons.