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We perform a statistical analysis of the optical solitary wave propagation in an ultra-slow stochastic non-local focusing Kerr medium such as liquid crystals. Our experimental results show that the localized beam trajectory presents a dynamical random walk whose beam position versus the propagation distance z depicts two different kind of evolutions A power law is found for the beam position standard deviation during the first stage of propagation. It obeys approximately z3/2 up to ten times the power threshold for solitary wave generation.
© 2016 Optical Society of America
1. Introduction
The concept of noise induced transition either in spatially homogeneous [1] or in extended [2] systems plays an important role in the dynamics far from equilibrium. In particlular, instabilities of optical soliton in nonlinear medium are drawing considerable attention both from fundamental as well as applied point of views [3–5]. It is also expected that the stability of the solitary wave (SW) increases with the nonlocality-induced finite correlation length of the noise [6]. More recently, the influence of the correlation length of noise on soliton destruction for long propagating distance was adressed [7]. Longitudinal effects were also studied in liquid crystals, such as filamentation and undulation [8, 9]. On the other hand, in systems where the nonlinearity response is very much slower than that of the field propagating time, a dynamics appears since now we observe the behavior at the time scale of the nonlinear medium [10] and no more its averaged value at the time scale of the optical field. This situation prevents the adiabatic elimination of one of the variables and turns any analytics into a hard task. It is also completely unexplored experimentally.
In this paper, we investigate experimentally the role of the ultra-slow response of the nonlinearity on the propagation of solitary waves in focusing stochastic media such as liquid crystals (LC). We show that during a first stage of propagation, the standard deviation of the solitary wave location versus the propagation distance z follows the approximate power law z3/2. On the other hand, for further propagating distance, no clear evidence of power laws can be given. Numerical simulations agree with the experimental observations. Note that, stationary regimes have been preditced wen the nonlinearity is ultra-fast nonlinearity [6, 11].
2. Experimental dynamical random walk of the solitary wave
We consider the well-known experimental setup of light propagation in nematic liquid crystals [12, 13]. The experimental setup is sketched in Fig. 1(a). The nonlinear medium consists of a 75 µm thick planar anchored nematic liquid crystal cell filled with E7. The cell glass plates are ITO coated to allow for the application of a voltage. The applied one is 1.15Vrms at 5 KHz to maximize the input beam intensity domain leading to SW. The input beam is provided by a Nd3+:YVO4 laser emitting at 532nm. A microscope objective 20X is used to inject an approximately 12 µm radius gaussian beam within the cell. The beam propagation is recorded collecting the scattered light from the top of the cell as sketched in Fig. 1(a).
Fig. 1 (a) Sketch of the experimental setup. MO: Microscope Objective Lens 20X. (b) Experimental recordings of the beam propagation for: (I) P = 0.5mW, (II) P = 10.5mW, (III) P = 30.5mW. (c): Evolution of the solitary wave radius ω versus the injected power for different positions in the LC cell, namely z = 230µm (circles) and z = 330µm (triangles); inset: typical intensity profile with its corresponding fit using an amended exponential dependence exp(−|x/ω|β) [14], P = 14.5mW, z = 330 µm and β = 1.11.
A typical longitudinal evolution of the beam propagation versus injected beam power, that was already observed by several authors [13], is shown in Fig. 1(b). At low power, the beam diffracts as shown in Fig. 1(b-I). Increasing the injected power, a solitary wave takes place over some hundreds of micrometers. The SW transverse shape evidences a spatial dependence that evolves with z [14]. An example of such a profile is depicted on the inset of Fig. 1(c). We thus use an amended exponential dependence exp(−|x/ω|β) to fit our profiles. Plotting the longitudinal averaged solitary wave width [Fig. 1(c)] allows to define a pseudo reference power (here, Pref ≈5 mW) above which the beam is almost collimated, that is, a solitary wave (SW) state is achieved. The SW trajectory reveals temporal fluctuations. The coming detailed analysis of the SW trajectory will show that whatever the injected power, this is the true. Indeed, the trajectory depicts transverse undulations that continuously and permanently erratically wander with time. This temporal evolution is a main difference with studies on solitary wave propagation through ultra-fast nonlinearities [6, 7, 11, 15, 16] where only stationary regimes are predicted. This latter absence of dynamics is not surprising since the studied timescale is that of the optical field which is very much slower than the nonlinear response time, thus averaging any dynamics of the medium nonlinearity. On the contrary, here, the observed timescale is that of the nonlinearity which is very much slower than that of the optical field, thus allowing to directly observe the dynamics of the medium. To characterize this dynamics, we perform a statistical analysis of the optical solitary wave propagation through its beam position standard deviation . Since we can not modify the noise intensity level we will study versus the propagating coordinate z.
3. Numerical simulations
Our system is modeled by a coupled set of equations, one for the light propagation and a second one for the medium response [13]. In the paraxial approximation, the propagation of the optical field E through a nonlinear medium with nonlinear refractive index n is described by
where the optical wavenumber k0 = 2π/λ, n0 is the medium linear index, α accounts for losses and χ parameterizes the nonlinearity (positive for focusing case). For non local, ultra-slow, noisy Kerr media, such as aligned nematic liquid crystal slices, the refractive index n is well discribed by [17, 18] with σ and σz are the transverse and longitudinal diffusion lengths, respectively [19], τ is the relaxation time and ζ is the additive noise. In liquid crystals, noise comes from the molecular orientationnal fluctuations, is white, Gaussian and delta correlated in space and time. Its intensity is given by ε. Time dependence for refractive index in liquid crystal was already considered e.g. in [20] for soliton formation. The temporal dependence of n in Eq. (2) is crucial since it transcribes the dynamics at the timescale of the nonlinear medium. Indeed, τ ≈ 2s is many order of magnitude larger than the time for the optical field to propagate over the few millimeters of the cell, that is, ≈ 10−11s. Removing the temporal dynamics and the longitudinal laplacian in Eq. (2), we recover the model used for ultra-fast nonlinearities such as in [11]. Eq. (1) is solved using the beam propagation procedure based on a split-step Fourier method and Eq. (2) is integrated using the stochastic algorithm described in [21]. Figure 2(a) depicts the beam propagation for increasing input power P without scattering losses (α = 0) for parameters corresponding to the experimental ones. That is, a transverse non-local response of the nonlinearity whose half width is σ ≈ 21µm and the input beam half diameter is w ≈ 8µm. This beam radius value is smaller than that of the recorded experimental one. Indeed, the measured radius with help of CCD cameras is the one of the diffused light which is larger than that of the optical beam [22]. The solitary wave evolution versus P [Fig. 2(a)] mimics the experimental one [Fig. 1(b)]. Trajectory undulation and random walk behavior is obtained above a critical Pref. The transverse location X of the solitary structure is obtained calculating its barycenter. This location is estimated for each snapshot at each distance of propagation. The evolution of the SW standard deviation versus z is sketched on Fig. 2(b) in Log-Log scale. It is clear that a power law can be associated with a first stage of propagation ( ). In addition, the longitudinal extension of the first regime only slightly reduces versus P [Fig. 2(b)] indicating that its dynamics is almost independent from the injected power P up to here 6 times Pref. On the contrary, during the further stage of propagation no clear linear profile is found and the shape of continuously change with the injected power P as can be seen on Fig. 2(b). Thus, the SW experiences two dynamical regimes during its propagation. A linear fit allows to show that the early propagation follows a z1.45 law [upper Fig. 2(c)]. Note that scattering losses α do not affect the first regime since the γ coefficient of the power law remains almost unchanged. Indeed, its value is γ = 1.49 [lower Fig. 2(c)]. We now take advantage of these results/insights to analyze the experimental recordings.Fig. 2 (a) Numerical simulations of the beam propagation without losses for: (I) P = 0.025Pref, (II) P = 1.2Pref and (III) P = 2Pref. (b): Evolution of the standard deviation (dots) of the SW position versus z for increasing values of P (Pref to 6Pref) together with corresponding linear fits (dashed lines) within the range z = 100µm to 400µm (vertical dashed line) without losses. (c): Evolutions of the power law exponent γ versus input beam power P without (top) and with (bottom) losses, dashed lines correspond to fits giving respectively γ=1.45±0.14 and γ=1.49±0.24. σ = 21µm, σz = 25µm, w = 8µm, n0 = 1.5269, α = 600, ε = 2.5.105.
4. Exploitation of experimental results and discussions
The experimental standard deviation of the SW beam is achieved recording movies of 1000 pictures over 250 seconds (LC response time is τ ≈ 2s). The evolution of the standard deviation versus propagation distance z in Log-Log scales is depicted in Fig. 3(a) for increasing P. Two disctinct evolutions of clearly appear for the range for . A linear fit associated with the linear part of the slope ( ) allows to plot the evolution of the power law coefficient γ versus injected beam power P. Results are depicted on Fig. 3(b). They evidence that during the first stage of propagation, γ is independent from the injected beam power P [Fig. 3(b)]. The value of γ is 1.57 ±0.15 for our experimental non-local ratio σ/w ≈ 2 and an applied voltage equals to 1.15 Vrms. As for numerical simulations, for further propagating distances, no power law can be evidenced. The longitudinal extension of the first regime is limited to . We want to notice that the values of power law coefficient γ very slightly vary from an experimental beam injection to another one. As well, the longitudinal extension of the domain where the z1.57 power law is observed can change from a sample to another one, depending on the provider. However, the two dynamical regimes and their features are always kept. Thus, the statistical analysis of the experimental recordings evidences two successive dynamical regimes for the solitary wave dynamics when propagating in an ultra-slow stochastic non-local focusing Kerr medium.
Fig. 3 (a) Evolution of (dots) of the SW position versus z for increasing values of P. The dashed gray lines correspond to linear fits between z = 100µm and z = 400µm (vertical dashed lines). (b) Evolution of the power law coefficient γ versus input beam power P. The dashed line gives γ=1.57 ±0.15. (c) Evolution of the temporal averaged SW scattering intensity integral < ISI >t over x versus z for P = 6Pref in semilog scale (the integral is normalized to its maximum value).
Two regimes of propagation for SW in presence of noise and nonlocality are predicted in [6, 7] but for nonlinearity whose response time is very much faster that the optical field evolution. In both articles, the effects of the emission of radiation and subsequent loss of power are crucial for the SW dynamics. Thus, in [6], a first regime is mentioned to where the SW is preserved while in a second stage it as affected by the emission of radiation. In [7], the two regimes are referred to a breathing of the SW at short propagation distance and to its dispersal at longer distances, almost without radiation losses. They both defines distances over which the SW keeps its shape. Even if our experimental statistical analysis clearly provides evidence of two dynamical regimes, we can not assess that they are similar to ones discussed in either [6] or [7]. Indeed, the injected beam in theoretical works is the deterministic SW solution whereas it is a Gaussian profile in the experiments. Likewise, our nonlinearity response time is ultra slow unlike to the ones in [6, 7].
The plot of the temporal averaged SW scattering intensity integral < ISI >t over x versus z in semi-log scale evidences two exponential decaying regions [Fig. 3(c)]. They coincide with the two dynamical regime longitudinal extensions [Fig. 3(a)]. The slope is almost the same, indicating that losses are due to Rayleigh scattering [Fig. 3(c)]. Since we do not observe the emission of radiation but rather a spreading of the beam for long z, we cannot therefore impute the transition between the two regimes to radiation emission. A bump separates these two regions (@ z = 400µm) that looks like breathing in the SW intensity. The origin of these two regimes require further investigation.
Small discrepancies between numerical simulations and experimental recordings, such as the very early stage of propagation [Fig. 3(a) and Fig. 2(b)], are attributed to the adaptation of the input beam due to the specific LC molecules entrance alignment as well as to the noise level intensity ε chosen in numerical simulations. Even if we do not see on experimental pictures the emission of radiation losses but a spreading of the beam for long z, the first dynamical regime corresponds to the first decreasing part of the SW integral intensity as can be seen on Fig. 3(c) whereas for further propagating distance this amplitude considerably decreases again leading to the probable destruction of the SW. Note that the SW intensity integral evolution damped oscillations [Fig. 3(c)] whose longitudinal period remains unchanged versus P are attributed to the anisotropy of the medium [23].
5. Conclusions
We have studied experimentally the propagation of a solitary wave in an ultra-slow stochastic non-local Kerr medium such as liquid crystal cells. We find that the localized structure is dynamic with a temporally randomly fluctuating trajectory to contrast with the stationary regime obtained for the corresponding ultra-fast nonlinearity counterpart. A statistical analysis of the trajectory fluctuations gives an indicator that evidences the existence of two successive dynamical regimes for the propagation of the SW.
Acknowledgments
We thank J.F. Blach for providing the LC cell and C. Conti for fruitfull discussions. M.T. received support from the Fonds National de la Recherche Scientifique (Belgium). This research was partially supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 photonics@be and by Ministry of Higher Education and Research as well as by the Agence Nationale de la Recherche through the LABEX CEMPI project (ANR-11-LABX-0007).
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