Semi-analytical approach to supermode spatial solitons formation in nematic liquid crystals

Pawel S. Jung; Wieslaw Krolikowski; Urszula A. Laudyn; Miroslaw A. Karpierz; Marek Trippenbach

doi:10.1364/OE.25.023893

1. Introduction

Due to their strong nonlinear response to light, nematic liquid crystals (NLC) represent very efficient nonlinear medium supporting strong localization of light and formation of bright solitons [1–3]. Their nonlinearity arises from light-induced reorientation of elongated molecules of NLC which tend to orient along the polarization direction of incoming beam. This leads to increase of effective refractive index and subsequently self-focusing and spatial localization. As a result of the long range interaction of molecules the nonlinearity is also nonlocal, in a sense that the refractive index change in a specific spatial location is determined by the light intensity distribution in the neighbourhood of this location. Nonlocality arrests collapse of finite size beams and supports formation of stable structures including bound states of solitons [4–10]. Typically these solitons exhibit single peak beams propagating without diffraction. However, we have recently shown that when reorientational response of liquid crystals is accompanied by defocusing thermal nonlinearity their interaction may lead to the formation of two-peak fundamental solitons, the so called super-solitons [11].

In this work we will introduce a simple phenomenological model capturing the complex nonlinear response of nematic liquid crystals comprising both, reorientational focusing and thermal defocusing, and find conditions for the formation of two-peak solitons. Our analytical results are confirmed by numerical simulations of the soliton formation in the full model of competing nonlinearities of nematic liquid crystals.

2. Theory

We consider propagation of a one-dimensional optical beam in the nematic liquid crystal cell comprising molecules located between closely placed parallel glass plates located in the y-z plane (see Fig. 1, typical distance between plates is tens of micrometers). We assume that the internal surfaces of both plates are conditioned (for instance, by rubbing) to ensure that the molecules are anchored and aligned at an angle θ = θ₀, with respect to the z-axis. Hence our system behaves like an uniaxial optical medium with a refractive index:

n (θ, T) = \frac{n_{o} (T) n_{e} (T)}{\sqrt{n_{o}^{2} (T) + Δ ϵ \cos^{2} θ}}

for the y-polarized (i.e., extraordinary polarized) light. Here n_o and n_e are ordinary and extraordinary refractive indices,

Δ ϵ (T) = n_{e}^{2} (T) - n_{o}^{2} (T)

and T represents temperature since, in general, all these quantities are temperature dependent. The electric field of the optical beam propagating in the cell along the z-axis modifies locally the orientation of molecules, leading to the intensity-dependent index change for extraordinary polarized light. Assuming constant amplitude along the y direction for planar configuration of NLC, the evolution of the amplitude of electric field E(x, z) of the beam is described by:

2 i k_{o} n (θ_{0}, T) \frac{\partial E}{\partial z} = \frac{\partial^{2} E}{\partial x^{2}} + k_{o}^{2} (n^{2} (θ, T) - n^{2} (θ_{0}, T)) E,

where k₀ = 2π/λ₀.

Fig. 1 Planar configuration of the nematic liquid crystal cell considered in this work. Beam propagation direction is represented by red arrow.

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The index n(θ) depends on the local orientation of molecules, which follows the direction of electric field of the beam according to the following diffusion-type relation [12]:

\frac{\partial^{2} θ}{\partial x^{2}} - \frac{Δ ε (T) ε_{o}}{2 K (T)} \sin 2 θ | E |^{2} = 0 .

Here K is an effective elastic constant [1, 12]. Note that Eq. (2) and Eq. (3) indicate that, the light-induced reorientational index change is spatially nonlocal and it is always positive as the molecules tend to align along the direction of electric field. This leads to self-focusing of extraordinary polarized optical beam and formation of bright solitons, so-called nematicons [2]. In the following we will assume that the propagation of light in the liquid crystal is accompanied by a weak absorption which causes heating of the crystals 17]. This process is governed by the heat equation

κ \frac{\partial^{2} T}{\partial x^{2}} + \frac{c ε_{0} α}{2} | E |^{2} = 0 .

where κ - thermal conductivity, α - absorption coefficient and c is speed of light.

In order to describe beam propagation in nematic liquid crystals we need to solve numerically three coupled equations: (2), (3) and (4), presented above. However, it appears that by inspecting temperature dependence of parameters of typical liquid crystals one can significantly simplify the nonlinear model thus enabling semi-analytical description of soliton effects. We discuss these simplifications below and confirm their validity by direct comparison with full model predictions. For the sake of concreteness we will focus here on two examples of NLC’s; one with relatively high birefringence, known as “6CHBT” and the second with very weak birefringence, called “1110” [18]. In both cases we consider samples of the same width d = 50 μm. To model the temperature dependence of refractive indices and elastic constants of NLC we employ an empirical polynomial formulae, which accurately represents thermal response of both types of liquid crystals in its nematic phase in the temperature range 20 – 40°C [13, 14].

Our first approximation is to neglect temperature dependence in term Δεε_o/K in Eq. (3). This is justified in Fig. 2, where we depict temperature dependence for Δε and K for both NLC’s. While temperature dependence of the elastic constant K is quite prominent, it is well compensated in the expression Δεε_o/K, as we can see from the Fig. 2(a). As a result of this approximation the reorientation angle depends solely on the intensity of the light beam (|E|²)

Fig. 2 (a) Ordinary n_o (solid line) and extraordinary n_e (dashed line) indices versus temperature T for two types of NLC’s. Red corresponds to “6CHBT” crystal and black to “1110”. (b) Analogous plot presenting temperature dependence of elastic constant K (solid line) and Δε/K (dashed line) [18]. (c) Quality of our approximation for 6CHBT and (d) 1110 described by $\frac{n^{2} (θ, T) - n^{2} (θ_{0}, T)}{Γ (T) Θ (θ, θ_{0})}$ vs θ and T for θ_o = 0^o, where the numerator is calculated from the exact model of NLC.

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The second approximation involves dependence of refractive index on reorientational angle, Eq. (1). Since the term Δϵ cos² θ in the denominator of the equation Eq. (1) is always much less than $n_{0}^{2}$ one can expand nonlinear polarization in Eq. (1) obtaining (see [11]):

n^{2} (θ, T) - n^{2} (θ_{0}, T) \approx (n_{e}^{2} (T) - n_{o}^{2} (T)) (\cos^{2} θ - \cos^{2} θ_{0}) = Γ (T) Θ (θ, θ_{0}),

reducing the light-induced refractive index difference to a simple product of auxiliary functions Θ and Γ, which depend separately on the temperature and the reorientation angle θ. In Fig. 2 (b), we present the temperature dependence of n₀ and n_e for two types of NLC “1110” and “6CHBT”. The quality of the above approximations is presented in Fig. 2(c)–2(d) where we plot the quotient

\frac{n^{2} (θ, T) - n^{2} (θ_{0}, T)}{Γ (T) Θ (θ, θ_{0})}

as a function of reorientation angle θ, and temperature T, for 1110″ and “6CHBT” liquid crystals. Note that the numerator is calculated using exact formula (Eq. (1), Eq. (3)), thus the closer this ratio is to unity the better the approximation. As a result of the above approximations one can write the propagation equation in the form

2 i k_{o} n (θ_{0}) \frac{\partial E}{\partial z} = \frac{\partial^{2} E}{\partial x^{2}} + k_{o}^{2} Γ (T) Θ (θ, θ_{0}) E,

We have shown earlier that the above model supports two types of solitons [11]. For lower power these are just standard single peak solitons. However, for power sufficiently high, the interplay between focusing and thermally induced defocusing nonlinearities leads to the formation of two-peak fundamental spatial solitons, the so called, supermode spatial solitons. These solitons are similar to well known supermodes of the linear and nonlinear waveguide arrays [15, 16]. In order to get better insight into the mechanism of super-soliton formation, let us consider a simple (dimensionless) model of nonlinear beam propagation in the form

i \frac{\partial ψ}{\partial ζ} = \frac{1}{2} \frac{\partial^{2} ψ}{\partial ξ^{2}} + N (| ψ |^{2}) ψ,

with the nonlinear response N in form of a product N (|ψ|²) = α (|ψ|²) β (|ψ|²). Below both α and β are represented by phenomenological functions of nonlocal nonlinearity

α (| ψ |^{2}) = \int R_{1} (ξ^{'} - ξ) | ψ |^{2} d ξ, β (| ψ |^{2}) = 1 - γ \int R_{2} (ξ^{'} - ξ) | ψ |^{2} d ξ,

where γ is a relative strength of the defocusing nonlinearity and

R_{1, 2} = \frac{1}{\sqrt{π} σ_{1, 2}} \exp (- \frac{ξ^{2}}{σ_{1, 2}^{2}})

are the nonlocal response function (with unit norm

\int_{- \infty}^{+ \infty} R_{1, 2} (ξ) d ξ = 1

and width σ_1,2). In this simple approxiamtion α and β depict focusing and defocusing effects respectively, in order to mimic the role of Θ and Γ functions introduced earlier (Eq. (6)).

In general, the extent of nonlocality of the focusing and defocusing nonlinearities can be quite different. However, it appears that the most interesting situation occurs when they become comparable. For simplicity, we will assume here that they are exactly equal.

In Fig. 3(a) we illustrate numerically calculated focusing and defocusing contributions to nonlinearity change for a single as well as for a two peak soliton. Figure 3(b) depicts the sequence of stationary soliton beam profiles (together with the profile of nonlinearity N (|ψ|²)) calculated numerically, from Eq. (7) for increasing values of the initial intensity. Our results clearly demonstrate gradual transformation of a single peak soliton into two peak supermode soliton, exactly as found earlier for the exact model of nonlinearity of liquid crystal [11]. We see that with increasing intensity effective nonlinear response function (α (|ψ|²) β (|ψ|²)) flattens around its maximum and eventually develops shallow dip in the center. At the same time soliton profile undergoes splitting and transforms into two peak structure. We would like to stress that for any power these graphs depict fundamental spatial solitons which remain stable throughout the whole transformation process.

Fig. 3 (a) An example of nonlinear response of α (black line) and β (red line) for spatial intensity (|ψ|²) profile (blue dashes line). (b) Soliton solutions for multiplicative model with competing focusing and defocusing nonlocal nonlinearities as a function of beam power (system parameters σ = 40 and γ = 0.1). Spatial intensity |ψ|² (solid black line) and nonlinear response α (|ψ|²) β (|ψ|²) (solid red line) profiles are shown [total power (b) P=350 (c) P=380 (d) P=390 (e) P=420].

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Our simple model enables us to find the threshold condition for the beam splitting and emergence of supermode soliton. Since the nonlinear response becomes flat at the threshold this indicates that second derivative of the nonlinearity should be zero at the soliton peak, i.e. ${\frac{d^{2} N (| ψ |^{2})}{d ξ^{2}} |}_{ξ = 0} \geq 0$ , Let us assume, for simplicity, that the soliton profile below and at the threshold is represented by Gaussian function, $ψ = A \exp (- \frac{x^{2}}{w^{2}})$ . Then the threshold condition yields: $\frac{A^{2} w^{2} (2 A^{2} γ - σ \sqrt{\frac{1}{σ^{2}} + \frac{2}{w^{2}}})}{{(2 σ^{2} + w^{2})}^{2}} \geq 0$ . Next, using expression for the soliton power $P = \sqrt{\frac{π}{2}} A^{2} w$ we find for the threshold power

P \geq \frac{\sqrt{π} \sqrt{0.5 w^{2} + σ^{2}}}{2 γ} .

Futher, if one assumes that the degree of nonlocality is high such that σ ≫ w, we find extremely simple relation for the threshold power

P_{t h} = \frac{σ \sqrt{π}}{2 γ}

. We verified the formula Eq. (9) using numerical simulations of the model Eq. (7), for various degrees of nonlocality and various beam intensity. The results are summarised in Fig. 4(a) where we observe very good agreement between our simple analytical predictions and numerical results. On the left axis we plot the threshold power vs σ for two different strengths of defocusing (γ). There is practically no difference between our predictions and numerical results. To show better the relation between analytical and exact results we plot in the same graph (right axis) the ratio

κ = P_{t h}^{(n u m)} / P_{t h}

, which shows that the difference in the analytical and exact results is more pronounced for larger σ. This can be explained by the fact that in the highly nonlocal regime the condition σ ≫ w cannot be actually fulfilled as the soliton width increases with σ, making our approximation inadequate. It is worth mentioning that our additional simulations (not shown) show that after the threshold power for beam splitting is exceeded the separation between two formed peaks grows linearly with input power

Fig. 4 (a) Illustrating the threshold power P_th (left axis) of supermode soliton formation as a function of the degree of nonlocality σ, for different values of the relative strength of defocusing, γ = 0.1 (red lines) and γ = 0.2 (blue lines). The right axis refers to the ratio $κ = P_{t h}^{(n u m)} / P_{t h}$ . (b) Comparison of the threshold power P_th calculated using three different models: Eq. (2) (exact), Eq. (6) (simplified) and Eq. (7) (phenomenological). In the phenomenological model (Eq. (7)) we used: σ = n_o (θ_o) k_od, γ = a/θ_o + b and d = 50μm for the width of the liquid crystal cell. Here a = 0.29(0.09), b = 0.09(0.04) for “1110” (“6CHBT”) crystal.

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In Fig. 4(b) we demonstrate the dependence of the threshold power, P_th for the supermode soliton formation, as a function of initial molecular reorientation angle θ_o for three nonlinear models discussed here, described by Eq. (2), Eq. (6) and Eq. (7). For the phenomenological model (Eq. (7)) we approximated the degree of nonlocality σ, by the width of the LC cell, while the nonlinear parameter γ was fitted to obtain good accuracy with other models (Eq. (7)), in order to properly describes formation of supermode-solitons for a wide range of initial conditions.

3. Conclusion

In conclusion, we discussed formation of supermode spatial solitons in liquid crystals with competing reorientational and thermal nonlinearities. We introduced simple multiplicative model of nonlinearity which adequately represents the actual complex nonlocal nonlinear response of liquid crystal and at the same time allows for simple semi-analytical analysis of the soliton formation process. We showed that semi-analytically found threshold power for the formation of super-solitons is in excellent agreement with numerical calculations.

Funding

Qatar National Research Fund (NPRP8-246-1-060) and National Science Centre grant agreement: UMO-2016/22/M/ST2/00261 and National Centre for Research and Development grant agreement LIDER/018/309/L-5/13/NCBR/2014.

References and links

1. G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13 (2003). [CrossRef]

2. G. Assanto, ed. Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals (Wiley, 2012). [CrossRef]

3. M. Warenghem, J. F. Blach, and J. F. Henninot, “Thermo-nematicon: an unnatural coexistence of solitons in liquid crystals?” J. Opt. Soc. Am. B 25, 1882 (2008). [CrossRef]

4. O. Bang, W. Krolikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002). [CrossRef]

5. X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 333, 211 (2004). [CrossRef]

6. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003). [CrossRef] [PubMed]

7. U. A. Laudyn, P. S. Jung, M.A. Karpierz, and G. Assanto, “Quasi two-dimensional astigmatic solitons in soft chiral metastructures,” Sci. Rep. 6, 22923 (2016). [CrossRef] [PubMed]

8. U. A. Laudyn, P. S. Jung, M. A. Karpierz, and G. Assanto, “Power-induced evolution and increased dimensionality of nonlinear modes in reorientational soft matter,” Opt. Lett. 39(22), 6399–6402 (2014). [CrossRef] [PubMed]

9. U. A. Laudyn, M. Kwasny, A. Piccardi, M. A. Karpierz, R. Dabrowski, O. Chojnowska, A. Alberucci, and G. Assanto, “Nonlinear competition in nematicon propagation,” Opt. Lett. 40(22), 5235–5238 (2015). [CrossRef] [PubMed]

10. M. Warenghem, J. F. Blach, and J. F. Henninot, “Thermo-nematicon: an unnatural coexistence of solitons in liquid crystals,” J. Opt. Soc. Am. B 11, 1882 (2008). [CrossRef]

11. P. S. Jung, W. Krolikowski, U. A. Laudyn, M. Trippenbach, and M. A. Karpierz, “Supermode spatial optical solitons in liquid crystals with competing nonlinearities,” Phys. Rev. A 95, 023820 (2017). [CrossRef]

12. I. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 2007), Vol. 64. [CrossRef]

13. R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mecomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241 (1985). [CrossRef]

14. J. Li, S. Gauza, and S. T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 20 (2004).

15. A.A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” IEEE J. Lightwave Techn. LT-3, 1135 (1985). [CrossRef]

16. M. Matuszewski, B.A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double channel potential,” Phys. Rev. A 75, 063621 (2007). [CrossRef]

17. F. Derrien, J.F. Henninot, M. Warenghem, and G. Abbate, “A thermal (2D+1) spatial optical soliton in a dye doped liquid crystal,” J. Optics A 2, 332 (2000). [CrossRef]

18. A. Alberucci, U. A. Laudyn, A. Piccardi, M. Kwasny, B. Klus, M.A. Karpierz, and G. Assanto, “Nonlinear continuous-wave optical propagation in nematic liquid crystals: Interplay between reorientational and thermal effects,”’ Phys. Rev. E 96, 012703 (2017). [CrossRef]

Semi-analytical approach to supermode spatial solitons formation in nematic liquid crystals

Abstract

1. Introduction

2. Theory

3. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (9)

Optics Express