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

Institute of Physics, P.O.Box 68, 11001 Belgrade, Serbia,

1Texas A&M University at Qatar, P. O. Box 5825 Doha, Qatar

LENCOS 2012, Sevilla Spain

Solitons in strongly nonlocal nematic liquid crystals

M.S. Petrović, A.I. Strinić, N.B. Aleksić and M.R. Belić1

OR THE TALE OF THE PAPER THAT REFUSES TO GET PUBLISHED

IOP research group in NL optics and dynamics

• M. R. Belić
• M. Petrović
• A. Strinić
• D. Jović
• S. Prvanović
• R. Jovanović
• N. Petrović
• D. Timotijević
• N. Aleksic

A typical research group hierarchy…

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 

sin sin 2

2 2 2 ,

  D  D    A k A z A ik

OPT y x

  e

• X. Hutsebaut, C. Cambournac and
• M. Haelterman, JOSA B 22 (2005) 1424-1431

 

2 2 ,

2 2 sin 2 2

DC DC OPT x y

A K E t    e e e      D  D  D       

   

• L

x L x 2 2 / 2 /       

Paraxial wave equation for light Director’s angle

• f reorientation

Boundary conditions

θ0 – Angle without light (the pre-tilt angle)

     

Nematicons: Solitons in NLCs The Model: Essentially nonlocal

+ initial conditions

There exist the old and the new model

)) ( sin (sin z 2

rest 2 2 2 ,

   D    A k A A ik

a y x

  e

         

z z V V V z

in

/ exp ,

rest

       

 

sin sin 2

2 2 2 ,

  D  D    A k A z A ik

OPT y x

  e

2 ,

) 2 sin( 4 1 t A K

a y x

 e e     D   

 

 e e e    2 sin 2 2

2 2 ,

        D  D  D    A E K t

OPT DC DC y x

The old model, by Assanto et al. The new model, by Haelterman et al.

     

in = p/2 0 = p/4

Fixed BCs:

+ Boundary conditions on 0 as an eigenvalue problem This is where the problem occurred: Old models die hard Note the differences!

 0

     

Successive Over Relaxation (SOR) method

The angle of reorientation θ depends on both light and voltage, where the angle θ0(x,y,z) accounts for the 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.

Our results: The temporal evolution of the angle of reorientation

ˆ  ˆ 

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 = 8.6x1010 V2/m2. For the intensity of soliton breathing, (I = 8.6x1010 V2/m2), 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.

Result: Spatial solitons breathe; the beam preserves its new shape but its characteristic width and maximum intensity breathe as it propagates.

Beam profiles: Breathers

• FIG. 3. (a) Fundamental soliton intensity profile obtained

by the eigenvalue method (black dots), fitted with a

• Gaussian. Parameters: P = 10.6 mW, m = 3.84LD
• 1; zero
• BCs. (b) Soliton and Gaussian propagation using two

different propagation methods, FFT+SOR and FFT+FFT. The soliton power P = 10.6 mW; Gaussian power P=10.6 mW for FFT+SOR, P=10.1 mW for FFT+FFT.

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

• n the fast Fourier transform (FFT) is used for the propagation of

the optical field. In the first method the diffusion equation for the

• ptically 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 should be discarded. Even the SOR soliton solution slightly

• scillates

at lower accuracy; this, however, becomes imperceptible as the accuracy is improved. In Fig. 3(b) we show a case where the oscillation of the amplitude is still perceptible. This brings us to an important point. The propagation of a Gaussian invariably leads to breathing beams, regardless of the method of integration.

• 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 = 4LD

• 1, zero BCs.

When the fundamental soliton is propagated through the medium in which a small amplitude noise added to Ɵ0, a breathing solution is obtained. We introduce noise by adding randomly distributed white noise to Ɵ0 at each propagation

• step. Adding more noise leads to larger and more irregular
• scillations, 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

• f NLCs is a well established fact. The same induced
• scillation phenomenon happens as well when a small

intensity noise is added to the fundamental profile, but Ɵ0, 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

• pens the question of physical observability of shape-

invariant fundamental solitons in highly NN media.

Propagation of fundamental solitons – the influence of noise

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 3rd 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

• f 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

• f steady nematicons.

BACK TO THE MODEL

The optical beam polarized along the x axis propagates in the z direction, while the NLC molecules can rotate in the x-z plane only;  (x, y, z) -the total orientation of the molecules with respect to the z axis. 0 -the orientation induced by the static electric field only.

• the optically induced molecular reorientation.
• FIG. 1: Liquid-crystal cell

considered. The system of equations of interest consists of the NL Schrȍdinger-like paraxial wave equation for the propagation of the optical field E, and the diffusion-like equation for the orientation angle. After the rescaling x/x0 → x, y/x0 → y, z/LD → z: : 1 2

THE EIGENVALUE PROCEDURE

The solitonic solutions can be found from the system of Eqs. (1,2) by using the modified Petviashvili’s iteration method. Equation (1) allows the existence of a fundamental soliton solution in the form

E = a(x,y)eimz m -the propagation constant.

Then the real-valued amplitude function a(x, y), after the separation of linear and nonlinear terms onto different sides of the equation, satisfies the following relation:

−△a + (2μ + P)a = Q, P = a sin2(0), Q = a sin2()a

We perform Fourier transformation of that equation, to find: 4 3 Including convergence coefficients:

• FIG. 2: Possible solitonic solutions for m=1LD

−1.

Two cases of intensity profiles and reorientation angles are presented for zero and mixed BCs. A family of fundamental solitons is found, depending on the BCs applied. Examples of such fundamental solitonic solutions are presented in Fig. 2, where the optical field intensity is denoted as I = |E|2. Shape and power of spatial shape-invariant solutions depend on the BCs applied to the

• ptically

induced molecular reorientation angle . Zero BCs present Dirichlet BCs. Periodic BCs correspond to the mixed BCs (along the y direction Dirichlet BC and along the x axis Neumann BC). The solution with the periodic BCs is more appropriate to the geometry and symmetry of the problem. It is also more acceptable on physical grounds, since it requires less beam power for the same value of the propagation

• constant. An approximation used in many

publications Ɵ0 =p /4 leads to the solution with zero BCs.

Zero BCs : on all boundaries Periodic (Dirichlet) BCs:

ˆ  

/ ) 2 / , ( ˆ / ) 2 / , ( ˆ        y D x y D x

C C

  ) , 2 / ( ˆ ) , 2 / ( ˆ    y D y D

C C

 

The fundamental solitons

• FIG. 3: Fundamental soliton intensity profiles (left)

and the corresponding profiles of the optically induced molecular reorientation (right), in the case

• f zero BCs (top) and periodic BCs (bottom).

For each set of reasonable physical parameters one can find a family of fundamental solitonic solutions, depending

• n the BCs applied, with

different propagation constants and beam powers. On physical grounds, the solutions with periodic BCs should be preferred!

The two families of fundamental solitonic solutions

VARIATIONAL CALCULUS

There are no known exact analytical solitonic solutions of the nematicon equations. A powerful approximate technique for studying this problem is based on the variational approach to the governing Eqs. (1,2). We start from the model equations in the lowest-order approximation for the fields: (5) (6)

) 2 sin(  a a 

Abbreviations:

) 2 sin( 2    

1 ˆ  

Assume: The Lagrangian density for the system of Eqs. (5,6) Gaussian trial function for the electric field

(7) (8)

r - the distance from the z axis in cylindrical coordinates, A - amplitude, R - the width of the beam, C - its curvature, ϕ is the peak-intensity phase. R ≪ DC

The trial function for the reorientation angle

(9)

the nonlocal contributions the local contribution

B and D - the amplitudes, W - width of the nonlocality, d = DC / 2, and Ei - the exponential integral function

VARIATIONAL CALCULUS: DETAILS

Substituting the trial functions Eqs. (8,9) into the Lagrangian Eq. (7) and integrating over r, we obtain the averaged Lagrangian (still in the limit R ≪ d):

(10)

• the prime denotes the derivative with respect to z, γ - Euler’s constant,

δ = max{β, (R/d) ln(d/R)}, P = πA2R2 - power of the spatial soliton

We require the minimization of the averaged Lagrangian by the variational parameters R(z), C(z), ϕ(z),B(z), W(z) and D(z).

three ODEs between them three algebraic relations

(11) (12) (13) (14) (15) (16)

From Eqs. (11,12) we obtain

(17)

In the stationary state dR/dz = dC/dz = 0,  the fixed point of the system of

• Eqs. (11,12,13,14,15,16) is:

and also W0 = W(R0), B0 = B(R0), and D0 = D(R0).

(18) (19)

the propagation constant μ = (dϕ/dz)0

(20)

The period of small oscillations around the fixed point (T) is calculated from Eq. (17). We assume a small perturbation

• f the beam width

R R R ~

0 



(17) (21) (22)

The period of oscillations

• FIG. 4: Fundamental soliton power, width, and

peak intensity as functions of the propagation constant, for zero BCs. Solid lines represent results of variational approach, dots represent numerical solitonic solutions.

The most important quantities of variational calculation are presented in Figs. 4 and 5. The fundamental soliton power, width, and peak intensity, as functions

• f

the propagation constant, are shown in Fig. 4,

• respectively. We note very good agreement

between the results of variational approach and the numerical solitonic solutions,

• btained by integrating directly the system
• f Eqs. (1,2).

One can discern from Fig. 4 that the

• btained variational solution is stable,

according to the Vakhitov-Kolokolov stability criterion [24]. According to this criterion, the solitary wave should be stable as long as dP/dμ > 0.

The variational calculus results

• FIG. 5: Period of small oscillations

T as a function of the propagation

• constant. Solid line represents

results of variational approach, dots represent results obtained in numerical simulations.

The period of small oscillations T as a function of the propagation constant is represented in Fig. 5 (solid line). To check this result, we propagate a fundamental soliton in the case of small perturbations, and find that it oscillates regularly with a period in good agreement with that

• btained by the variational approach.

Numerical procedure applied to the propagation equation is the splitstep beam propagation method based on the fast Fourier transform (FFT); the diffusion-type equation for the

• ptically

induced molecular reorientation is treated using the SOR method.

Shape-invariant solitons in nematic liquid crystals: The influence of noise

 We investigate the influence of noise on the shape-invariant solitons in nematic liquid crystals.  We check the stability of such solutions by propagating them for long

• distances. We displayed their stability in propagation, utilizing two

different propagation methods.  We discovered that adding a small amount of white noise to the medium causes the fundamental solitons to breathe, rendering them practically unobservable.  In addition, by increasing the amount of noise, the quasi-stable propagation distance of breathers is reduced, rendering them absolutely unstable.

• FIG. 6: (a) Propagation of the non-ideal

soliton solution calculated with accuracy of 10−6. (b), (c) Soliton peak intensity evolution in a noisy medium. The level of amplitude noise is indicated in each figure. Parameters: m=3LD

−1, P = 8.6 mW, T = 7.5LD; zero BCs.

The influence of small perturbations on the propagation of fundamental solitons

We investigate the influence of small perturbations

• n the propagation of fundamental soliton solutions

by monitoring the peak intensity as a function of the propagation distance (Figs. 6 and 7). A non-ideal fundamental soliton oscillates regularly during propagation, as in Fig. 6(a), where the numerical accuracy of the beam profile calculation is defined as the relative error between the last two iterations in the Petviashvili’s iterative procedure. To achieve more proper input soliton shape, we needed better numerical resolution in our numerics: as the accuracy of the eigenfunction profile improved, the amplitude of its oscillation diminished. In all realistic media, noise is unavoidable. Thermal fluctuations of the director field are inherent to the nematic phase. Even if one launches a perfect fundamental soliton, some noise in the medium is bound to influence its propagation. We introduce noise in our simulations by adding randomly distributed white noise to the q0 at each propagation

• step. Propagation of an ideal (double-precision)

fundamental soliton solution in a noisy medium is presented in Figs.6(b) and (c), where the red curve is a sinusoidal fit to the perturbed soliton. It is seen that the perturbed soliton propagates similar to a

• breather. For all three cases in Fig. 6, the period
• f oscillations is 7.5LD; Eq. (22) gives T = 6.9LD.

The results presented are obtained using the fundamental solitons with zero BCs.

• FIG. 7: Peak intensity as a function of the

propagation distance in the case of 0.5% noise added to the pre-tilt angle q0 (dots), fitted with a sine function (red line). For zero BCs (top) the period of oscillation is T = 5.3LD; for periodic BCs (bottom) the period is T = 5.7LD. The propagation constant for input solitons is m = 5LD

−1.

The influence of small perturbations on the propagation of fundamental solitons We investigate the influence of small perturbations on the propagation of fundamental soliton solutions by monitoring the peak intensity as a function of the propagation distance . It is seen that the perturbed soliton propagates similar to a breather. The results presented are obtained using the fundamental solitons with zero BCs and for the case of periodic BCs, Fig. 7. Our variational approach is valid for the case of zero BCs, and for the case presented in Fig. 7, Eq. (22) gives T = 4.4LD. Extending the propagation distance, the perturbed solitons cease to propa- gate stably generally beyond 100 LD.

Submitted for publication: Shape-invariant solitons in nematic liquid crystals: The influence of noise

LESSON LEARNED

Be persistent – do not give up Be resilient: come up with new ideas Be smart: avoid certain reviewer mafia