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

^{1}

^{2}

^{*}

## 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

## References

- DeGennes, P.G.; Prost, J. The Physics of Liquid Crystals; Oxford Science: New York, NY, USA, 1993. [Google Scholar]
- Khoo, I.C. Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Braun, E.; Faucheux, L.P.; Libchaber, A. Strong self-focusing in nematic liquid crystals. Phys. Rev. A
**1993**, 48, 611–622. [Google Scholar] [CrossRef] [PubMed] - Peccianti, M.; Assanto, G. Nematicons. Phys. Rep.
**2012**, 516, 147–208. [Google Scholar] [CrossRef] - Bang, O.; Krolikowski, W.; Wyller, J.; Rasmussen, J.J. Collapse arrest and soliton stabilization in nonlocal nonlinear media. Phys. Rev. E
**2002**, 66, 046619. [Google Scholar] [CrossRef] [PubMed] - Conti, C.; Peccianti, M.; Assanto, G. Route to nonlocality and observation of accessible solitons. Phys. Rev. Lett.
**2003**, 91, 073901. [Google Scholar] [CrossRef] [PubMed] - Peccianti, M.; de Rossi, A.; Assanto, G.; Luca, A.D.; Umeton, C.; Khoo, I.C. Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells. Appl. Phys. Lett.
**2000**, 77, 7–9. [Google Scholar] [CrossRef] - Peccianti, M.; Assanto, G. Incoherent spatial solitary waves in nematic liquid crystals. Opt. Lett.
**2001**, 26, 1791. [Google Scholar] [CrossRef] [PubMed] - Alberucci, A.; Peccianti, M.; Assanto, G.; Dyadyusha, A.; Kaczmarek, M. Two-color vector solitons in nonlocal media. Phys. Rev. Lett.
**2006**, 97, 153903. [Google Scholar] [CrossRef] [PubMed] - Laudyn, U.; Kwasny, M.; Karpierz, M.; Assanto, G. Three-color vector nematicon. Photon. Lett. Pol.
**2017**, 9, 36–38. [Google Scholar] [CrossRef] - Beeckman, J.; Neyts, K.; Haelterman, M. Patterned electrode steering of nematicons. J. Opt. A Pure Appl. Opt.
**2006**, 8, 214. [Google Scholar] [CrossRef] - Piccardi, A.; Peccianti, M.; Assanto, G.; Dyadyusha, A.; Kaczmarek, M. Voltage-driven in-plane steering of nematicons. Appl. Phys. Lett.
**2009**, 94, 091106. [Google Scholar] [CrossRef][Green Version] - Barboza, R.; Alberucci, A.; Assanto, G. Large electro-optic beam steering with nematicons. Opt. Lett.
**2011**, 36, 2725–2727. [Google Scholar] [CrossRef] [PubMed] - Piccardi, A.; Alberucci, A.; Barboza, R.; Buchnev, O.; Kaczmarek, M.; Assanto, G. In-plane steering of nematicon waveguides across an electrically adjusted interface. Appl. Phys. Lett.
**2012**, 100, 251107. [Google Scholar] [CrossRef] - Izdebskaya, Y.V. Routing of spatial solitons by interaction with rod microelectrodes. Opt. Lett.
**2014**, 39, 1681–1684. [Google Scholar] [CrossRef] [PubMed] - Izdebskaya, Y.V.; Shvedov, V.; Assanto, G.; Krolikowski, W. Magnetic routing of light-induced waveguides. Nat. Commun.
**2017**, 8, 14452. [Google Scholar] [CrossRef] [PubMed][Green Version] - Peccianti, M.; Conti, C.; Assanto, G.; DeLuca, A.; Umeton, C. All optical switching and logic gating with spatial solitons in liquid crystals. Appl. Phys. Lett.
**2002**, 81, 3335. [Google Scholar] [CrossRef] - Pasquazi, A.; Alberucci, A.; Peccianti, M.; Assanto, G. Signal processing by opto-optical interactions between self-localized and free propagating beams in liquid crystals. Appl. Phys. Lett.
**2005**, 87, 261104. [Google Scholar] [CrossRef][Green Version] - Alberucci, A.; Piccardi, A.; Bortolozzo, U.; Residori, S.; Assanto, G. Nematicon all-optical control in liquid crystal light valves. Opt. Lett.
**2010**, 35, 390–392. [Google Scholar] [CrossRef] [PubMed] - Assanto, G.; Minzoni, A.A.; Smyth, N.F.; Worthy, A.L. Refraction of nonlinear beams by localized refractive index changes in nematic liquid crystals. Phys. Rev. A
**2010**, 82, 053843. [Google Scholar] [CrossRef] - Laudyn, U.A.; Karpierz, M.A. Nematicons deflection through interaction with disclination lines in chiral nematic liquid crystals. Appl. Phys. Lett.
**2013**, 103, 221104. [Google Scholar] [CrossRef] - Peccianti, M.; Dyadyusha, A.; Kaczmarek, M.; Assanto, G. Escaping solitons from a trapping potential. Phys. Rev. Lett.
**2008**, 101, 153902. [Google Scholar] [CrossRef] [PubMed] - Piccardi, A.; Alberucci, A.; Assanto, G. Soliton self-deflection via power-dependent walk-off. Appl. Phys. Lett.
**2010**, 96, 061105. [Google Scholar] [CrossRef] - Piccardi, A.; Alberucci, A.; Assanto, G. Self-turning self-confined light beams in guest–host media. Phys. Rev. Lett.
**2010**, 104, 213904. [Google Scholar] [CrossRef] [PubMed] - Karpierz, M.A. Spatial Solitons in Liquid Crystals in Soliton-Driven Photonics; Springer: New York, NY, USA, 2001. [Google Scholar]
- Assanto, G.; Peccianti, M. Spatial solitons in nematic liquid crystals. IEEE J. Quantum Electron.
**2003**, 39, 13–21. [Google Scholar] [CrossRef] - Assanto, G.; Karpierz, M. Nematicons: Self-localized beams in nematic liquid crystals. Liq. Cryst.
**2009**, 36, 1161. [Google Scholar] [CrossRef] - Piccardi, A.; Alberucci, A.; Bortolozzo, U.; Residori, S.; Assanto, G. Readdressable Interconnects with spatial soliton waveguides in liquid crystal light valves. IEEE Photon. Techn. Lett.
**2010**, 22, 694–696. [Google Scholar] [CrossRef] - Perumbilavil, S.; Piccardi, A.; Buchnev, O.; Kauranen, M.; Strangi, G.; Assanto, G. Soliton-assisted random lasing in optically-pumped liquid crystals. Appl. Phys. Lett.
**2016**, 109, 161105. [Google Scholar] [CrossRef][Green Version] - Perumbilavil, S.; Piccardi, A.; Barboza, R.; Buchnev, O.; Strangi, G.; Kauranen, M.; Assanto, G. Beaming random lasers with soliton control. Nat. Commun.
**2018**, 9, 3863. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.K.; Senyuk, B.; Lavrentovich, O.D. Molecular reorientation of a nematic liquid crystal by thermal expansion. Nat. Commun.
**2012**, 3, 1133. [Google Scholar] [CrossRef] [PubMed][Green Version] - Piccardi, A.; Alberucci, A.; Tabiryan, N.; Assanto, G. Dark nematicons. Opt. Lett.
**2011**, 36, 1356–1358. [Google Scholar] [CrossRef] [PubMed] - Derrien, F.; Henninot, J.F.; Warenghem, M.; Abbate, G. A thermal (2D+1) spatial optical soliton in a dye doped liquid crystal. J. Opt. A Pure Appl. Opt.
**2000**, 2, 332. [Google Scholar] [CrossRef] - Janossy, I.; Taghizadeh, M.; Mathew, J.; Smith, S. Thermally induced optical bistability in thin film devices. IEE J. Quantum Electron.
**1985**, 21, 1447–1452. [Google Scholar] [CrossRef] - Tsai, M.S.; Jiang, I.M.; Huang, C.Y.; Shih, C.C. Reorientational optical nonlinearity of nematic liquid-crystal cells near the nematic isotropic phase transition temperature. Opt. Lett.
**2003**, 28, 2357–2359. [Google Scholar] [CrossRef] [PubMed] - Warenghem, M.; Blach, J.; Henninot, J. Measuring and monitoring optically induced thermal or orientational non-locality in nematic liquid crystal. Mol. Cryst. Liq. Cryst.
**2006**, 454, 297–314. [Google Scholar] [CrossRef] - Assanto, G.; Svensson, B.; Kuchibhatla, D.; Gibson, U.J.; Seaton, C.T.; Stegeman, G.I. Prism coupling into ZnS waveguides: A classic example of a nonlinear coupler. Opt. Lett.
**1986**, 11, 644. [Google Scholar] [CrossRef] [PubMed] - Vitrant, G.; Reinisch, R.; Paumier, J.C.; Assanto, G.; Stegeman, G.I. Non-linear prism coupling with nonlocality. Opt. Lett.
**1989**, 14, 898. [Google Scholar] [CrossRef] [PubMed] - Rotschild, C.; Cohen, O.; Manela, O.; Segev, M.; Carmon, T. Solitons in nonlinear media with an infinite range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons. Phys. Rev. Lett.
**2005**, 95, 213904. [Google Scholar] [CrossRef] [PubMed] - Rothschild, C.; Alfassi, B.; Cohen, O.; Segev, M. Long-range interactions between optical solitons. Nat. Phys.
**2006**, 2, 769. [Google Scholar] [CrossRef] - Laudyn, U.; Piccardi, A.; Kwasny, M.; Karpierz, M.; Assanto, G. Thermo-optic soliton routing in nematic liquid crystals. Opt. Lett.
**2018**, 43, 2296–2299. [Google Scholar] [CrossRef] [PubMed] - Laudyn, U.A.; Kwasny, M.; Piccardi, A.; Karpierz, M.A.; Dabrowski, R.; Chojnowska, O.; Alberucci, A.; Assanto, G. Nonlinear competition in nematicon propagation. Opt. Lett.
**2015**, 40, 5235–5238. [Google Scholar] [CrossRef] [PubMed] - Alberucci, A.; Laudyn, U.; Piccardi, A.; Kwasny, M.; Klus, B.; Karpierz, M.; Assanto, G. Nonlinear continuous-wave optical propagation in nematic liquid crystals: Interplay between reorientational and thermal effects. Phys. Rev. E
**2017**, 96. [Google Scholar] [CrossRef] [PubMed][Green Version] - Warenghem, M.; Blach, J.; Henninot, J.F. Thermo-nematicon: An unnatural coexistence of solitons in liquid crystals? J. Opt. Soc. Am. B
**2008**, 25, 1882–1887. [Google Scholar] [CrossRef] - Li, J.; Wu, S.T. Extended Cauchy equations for the refractive indices of liquid crystals. J. Appl. Phys.
**2004**, 95, 896–901. [Google Scholar] [CrossRef] - Alberucci, A.; Piccardi, A.; Peccianti, M.; Kaczmarek, M.; Assanto, G. Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity. Phys. Rev. A
**2010**, 82, 023806. [Google Scholar] [CrossRef] - Jánossy, I.; Kósa, T. Influence of anthraquinone dyes on optical reorientation of nematic liquid crystals. Opt. Lett.
**1992**, 17, 1183–1185. [Google Scholar] [CrossRef] [PubMed] - Dąbrowski, R. New liquid crystalline materials for photonic applications. Mol. Cryst. Liq. Cryst.
**2004**, 421, 1–21. [Google Scholar] [CrossRef] - Baran, W.; Raszewski, Z.; Dabrowski, R.; Kedzierski, J.; Rutkowska, J. Some physical properties of mesogenic 4-(trans-4-n-Alkylcyclohexyl) isothiocyanatobenzenes. Mol. Cryst. Liq. Cryst.
**1985**, 123, 237–245. [Google Scholar] [CrossRef] - Schirmer, J.; Kohns, P.; Schmidt-kaler, T.; Muravski, A.A.; Yakovenko, S.Y.; Bezborodov, V.S.; Dabrowski, R.; Adomenas, P. Birefringence and refractive indices dispersion of different liquid crystalline structures. Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. A Mol. Cryst. Liq. Cryst.
**1997**, 307, 17–42. [Google Scholar] [CrossRef] - Conti, C.; Peccianti, M.; Assanto, G. Observation of optical spatial solitons in a highly nonlocal medium. Phys. Rev. Lett.
**2004**, 92, 113902. [Google Scholar] [CrossRef] [PubMed] - Alberucci, A.; Jisha, C.; Assanto, G. Breather solitons in highly nonlocal media. J. Opt.
**2016**, 18, 12. [Google Scholar] [CrossRef] - Klus, B.; Laudyn, U.; Karpierz, M.; Sahraoui, B. All-optical measurement of elastic constants in nematic liquid crystals. Opt. Express
**2014**, 22, 30257. [Google Scholar] [CrossRef] [PubMed] - Milanchian, K.; Abdi, E.; Tajalli, H.; Ahmadi, S.K.; Zakerhamidi, M. Nonlinear refractive index of some anthraquinone dyes in 1294-1b liquid crystal. Opt. Commun.
**2012**, 285, 761. [Google Scholar] [CrossRef]

**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 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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