# Interplay of Thermo-Optic and Reorientational Responses in Nematicon Generation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**n**, and an order parameter defining the average direction and the variance of the molecular orientation, respectively. Macroscopically, nematic liquid crystals (NLC) exhibit anisotropic optical properties: when prepared on properly treated substrates, the order extends to long-range and they become uniaxial media with refractive indices ${n}_{\perp}$ and ${n}_{\left|\right|}$ for electric fields perpendicular and parallel to the molecular director

**n**, respectively, with the latter taking the role of the optic axis [2].

**n**and the wave-vector

**k**) the transverse distribution of refractive index resulting from reorientation is focusing or even confining [3], with the beam power determining the magnitude of the nonlinear perturbation. When reorientational self-focusing balances linear diffraction, the beam can propagate without energy spreading and maintain an invariant profile: a spatial optical soliton or nematicon is generated [4]. Due to the elastic intermolecular forces, the refractive index perturbation extends well beyond the beam size, i.e., NLC are strongly nonlocal; hence, two-dimensional solitons are stable as the catastrophic collapse typical of Kerr nonlinear materials is avoided [5,6]. The large reorientational response and the high nonlocality of NLC favor the formation of 2D+1 spatial solitons with continuous-wave bell-shaped beams at milliwatt powers and waists of a few micrometers [7], even in the limit of spatial incoherence [8,9,10]. Owing to the guiding properties of the light-induced index channels and the NLC sensitivity to stimuli, nematicons have been employed as waveguides for co-polarized signals of different wavelengths, to be switched and steered by way of electric or magnetic fields [11,12,13,14,15,16], extra self-confined beams or refractive index perturbations [17,18,19,20,21], even self-induced changes in director distribution [22,23,24]. In this framework, nematicons are candidates for novel generations of electro-optic and all-optical devices based on or controlled by self-induced waveguides [25,26,27,28], including cavity-less lasers [29,30].

## 2. Temperature Effects on Beam Propagation

**n**uniformly oriented in the plane $yz$ at an angle ${\theta}_{0}$ with the z axis, as sketched in Figure 1a. The optical excitation is a Gaussian beam launched with $\mathbf{k}$ along z. When polarized as an ordinary wave, the beam propagates along z and diffracts as in isotropic media, with refractive index ${n}_{o}={n}_{\perp}$, as shown in the top panel of Figure 1b; conversely, the beam in the extraordinary-wave polarization propagates in a ${\theta}_{0}$-dependent refractive index ${n}_{e}={\left({\mathrm{cos}}^{2}{\theta}_{0}/{n}_{\perp}^{2}+{\mathrm{sin}}^{2}{\theta}_{0}/{n}_{\left|\right|}^{2}\right)}^{-1/2}$, with energy flux (Poynting vector) angularly displaced by the walk-off $\delta ({\theta}_{0})=\frac{1}{{n}_{e}}\frac{\partial {n}_{e}}{\partial {\theta}_{0}}$ from the wave vector $\mathbf{k}$.

**n**at ${\theta}_{0}={60}^{\circ}$ with respect to the input beam wave-vector ($\mathbf{k}$ parallel to z).

## 3. Temperature-Dependent Nematicon Propagation

## 4. Discussion

## 5. Materials and Methods

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) NLC sample geometry. (

**b**) Typical linear and nonlinear propagation of light beams. Top: ordinary-wave diffracting beam. Center: extraordinary-wave beam at low power, undergoing diffraction. Bottom: extraordinary-wave beam at high power, undergoing self-confinement. Measured (

**c**) dielectric and (

**d**) elastic constant ${K}_{22}$ versus temperature in five NLC mixtures: 1110 (black squares), 903 (red circles), 6CHBT (blue triangles), E7 (brown hexagons), 2007 (green diamonds).

**Figure 2.**(

**a**–

**e**) Walk-off sensitivity to temperature for various ${\theta}_{0}$: black line ${\theta}_{0}={75}^{\circ}$, red line ${\theta}_{0}={60}^{\circ}$, blue line ${\theta}_{0}={45}^{\circ}$ for the NLC mixtures as labelled and (

**f**) walk-off variation versus temperature with respect to its reference value, in the five NLC mixtures: 1110 (black), 903 (red), 6CHBT (blue), E7 (brown), 2007 (green).

**Figure 3.**(

**Left**): beam trajectory vs. power. (

**Center**): beam trajectory vs. temperature. (

**Right**): comparison of calculated and measured walk-off changes with respect to its initial value. Rows (

**a**–

**e**) refer to the five mixtures 1110 (

**a**), 903 (

**b**), 6CHBT (

**c**), E7 (

**d**), and 2007 (

**e**), respectively.

**Figure 4.**(

**a**) Acquired images of $P=10$ mW nematicon propagation in the five NLC mixtures, as labelled. (

**b**) Calculated nonlinearity versus ${\theta}_{0}$ for each material. (

**c**–

**g**) Beam-width evolution versus propagation for various input powers (see legends), and for the five mixtures, as labelled. As power increases, the breathing period decreases. The relative values are coherent with the graphs in panel (

**b**): a larger nonlinear figure is associated to a shorter period.

**Figure 5.**(

**Left**): acquired images of beam evolution in the plane $yz$ at two temperatures in each mixture from (

**a**–

**e**), respectively, as labeled. (

**Center**): measured beam width versus propagation at various temperatures (see legends). (

**Right**): calculated parameter ${n}_{2}$ and measured breathing period after scaling to the maximum value.

**Figure 6.**Acquired images of beam evolution at two wavelengths: top, ${\lambda}_{i}=1064$ nm, bottom, ${\lambda}_{g}=532$ nm. (

**a**,

**b**) The near-infrared beam diffracts when polarized as an ordinary wave, irrespective of the power; (

**c**,

**d**) when extraordinarily polarized, the beam goes from diffraction at low power ($P<1$ mW) in (

**c**) to self-confinement for $P=4$ mW in (

**d**). (

**e**,

**f**) The green beam (within the absorption band of the DDNLC) in the ordinary polarization undergoes self-focusing when increasing power from ${P}_{g}=1$ mW (

**e**) to ${P}_{g}=6$ mW (

**f**); (

**g**,

**h**) when polarized as an extraordinary wave it self-defocuses.

**Figure 7.**Competing nonlinearities in DDNLC. (

**a**–

**c**) Acquired images of green beam evolution for various linear input polarizations and ${P}_{g}=6$ mW, in the presence of a ${P}_{i}=4$ mW NIR nematicon. (

**a**) In the ordinary polarization, the visible beam self-confines and does not sense the extraordinary-wave nematicon waveguide. (

**b**) When polarized at ${45}^{\circ}$ the ordinary component is less confined (owing to reduced power) while the extraordinary component gets confined within the nematicon, as in the case (

**c**) of both co-polarized extraordinary-wave beams. (

**d**) The ${P}_{g}=6$ mW extraordinary-wave green without NIR undergoes self-defocusing. (

**e**,

**f**) Acquired images of the NIR nematicon (${P}_{i}=4$ mW) interacting with a collinear co-polarized green beam for (

**e**) ${P}_{g}=1$ mW and (

**f**) ${P}_{g}=6$ mW. NIR beam (

**g**) trajectories and (

**h**) width evolutions when varying the green power from 0 to 6 mW (darker to lighter lines), respectively.

**Table 1.**Measured values of refractive indices and Frank elastic constant $K22$ for different temperatures.

1110 | 6CHBT | 903 | |||||||||

T [${}^{\circ}$] | ${n}_{\perp}$ | ${n}_{\left|\right|}$ | ${K}_{22}[pN]$ | T [${}^{\circ}$] | ${n}_{\perp}$ | ${n}_{\left|\right|}$ | ${K}_{22}[pN]$ | T [${}^{\circ}$] | ${n}_{\perp}$ | ${n}_{\left|\right|}$ | ${K}_{22}[pN]$ |

20 | 1.4517 | 1.4976 | 8.36 | 20 | 1.4967 | 1.6335 | 3.61 | 20 | 1.4696 | 1.5422 | 7.36 |

24 | 1.4506 | 1.4947 | 7.95 | 24 | 1.5021 | 1.6314 | 3.44 | 24 | 1.4681 | 1.5410 | 7.17 |

28 | 1.4496 | 1.4916 | 7.38 | 28 | 1.5046 | 1.6262 | 3.01 | 28 | 1.4662 | 1.5371 | 6.95 |

32 | 1.4488 | 1.4882 | 5.98 | 32 | 1.5097 | 1.6203 | 2.55 | 32 | 1.4649 | 1.5345 | 6.68 |

36 | 1.4482 | 1.4844 | 4.72 | 36 | 1.5135 | 1.6138 | 2.14 | 36 | 1.4626 | 1.5317 | 6.35 |

40 | 1.4480 | 1.4797 | 3.65 | 40 | 1.5161 | 1.5923 | 1.64 | 40 | 1.4609 | 1.5269 | 5.96 |

2007A | E7 | ||||||||||

T [${}^{\circ}$] | ${n}_{\perp}$ | ${n}_{\left|\right|}$ | ${K}_{22}[pN]$ | T [${}^{\circ}$] | ${n}_{\perp}$ | ${n}_{\left|\right|}$ | ${K}_{22}[pN]$ | ||||

20 | 1.5090 | 1.7773 | 15.01 | 20 | 1.5290 | 1.7314 | 4.39 | ||||

24 | 1.5091 | 1.7739 | 14.79 | 25 | 1.5340 | 1.7298 | 3.70 | ||||

28 | 1.5093 | 1.7704 | 14.50 | 30 | 1.5393 | 1.7276 | 3.33 | ||||

32 | 1.5096 | 1.7668 | 14.16 | 35 | 1.5450 | 1.7246 | 3.05 | ||||

36 | 1.5100 | 1.7630 | 13.78 | 40 | 1.5513 | 1.7203 | 2.85 | ||||

40 | 1.5105 | 1.7590 | 13.37 | 45 | 1.5586 | 1.7141 | 2.59 | ||||

50 | 1.5122 | 1.7580 | 12.25 | 50 | 1.5679 | 1.7039 | 2.33 | ||||

60 | 1.5149 | 1.7450 | 10.85 | 55 | 1.5856 | 1.6770 | 2.22 |

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**MDPI and ACS Style**

Laudyn, U.A.; Piccardi, A.; Kwasny, M.; Klus, B.; Karpierz, M.A.; Assanto, G.
Interplay of Thermo-Optic and Reorientational Responses in Nematicon Generation. *Materials* **2018**, *11*, 1837.
https://doi.org/10.3390/ma11101837

**AMA Style**

Laudyn UA, Piccardi A, Kwasny M, Klus B, Karpierz MA, Assanto G.
Interplay of Thermo-Optic and Reorientational Responses in Nematicon Generation. *Materials*. 2018; 11(10):1837.
https://doi.org/10.3390/ma11101837

**Chicago/Turabian Style**

Laudyn, Urszula A., Armando Piccardi, Michal Kwasny, Bartlomiej Klus, Miroslaw A. Karpierz, and Gaetano Assanto.
2018. "Interplay of Thermo-Optic and Reorientational Responses in Nematicon Generation" *Materials* 11, no. 10: 1837.
https://doi.org/10.3390/ma11101837