# Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Frank Free Energy

#### 2.2. Optical Imaging of Localized Topological Structures

## 3. Results

#### 3.1. Simulated Structures

#### 3.2. Analytical Design of Localized Topological Structures

#### 3.3. Approximated Structures

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**POM textures and director distributions for solitonic CLC structures S1, TS2, and S3. (

**a**) White light POM textures in crossed polarizers computed at ${n}_{e}=1.76$ and ${n}_{o}=1.53$ [44]. Orientation of the axes for (

**a**–

**c**) and the white scale bar for all figures are shown in the inset at the left and the right of the texture for S1, respectively. (

**b**) White light POM textures in crossed polarizers computed for the case of low birefringence with ${n}_{e}=1.55$ and ${n}_{o}=1.47$ [44]. (

**c**) Director field distributions in the $xy$ plane (the middle cross-section of the cell at $z=0$). The color palette in the inset at the right of the director distribution for S1 describes x and y projections of the director used in (

**c**,

**d**). This color wheel represents the northern hemisphere of director orientations where ${n}_{z}$ is non-negative. In the southern hemisphere, the colors become darker as ${n}_{z}$ decreases approaching the southern pole with ${n}_{z}=-1$. (

**d**) Director field distributions in the $xz$ plane with the origin placed at the center of the sample. Orientation of the axes is shown in the inset at the left of the distribution for S1.

**Figure 2.**Director fields and topology of S3 (looped CF1 finger), TS2 (transient structure), and S1 (toron). (

**a**,

**b**) Orientational structures in the $xy$ and $xz$ planes, respectively. (

**c**) Isosurfaces of the z projection of the director computed at ${n}_{z}\in \{\pm 0.83,\pm 0.5,\pm 0.17\}$. For S3, a pair of ${\lambda}^{+1/2}$ (${\lambda}^{-1/2}$) disclination loops is marked by red (blue) circles. For TS2 and S1, cyan (magenta) circles indicate two hyperbolic hedgehogs ($+1$ disclination ring).

**Figure 3.**Director field distributions of (1) tilted helix (see Equation (6)) and (2) cholesteric finger CF1 (see Equation (7)) computed for the case of straight cholesteric finger. (

**a**) Vertical cross-section of the director field. (

**b**) Isosurfaces of the z-projection of the director. Blue and red points indicate disclination lines.

**Figure 4.**Approximated and simulated POM textures and director field distributions for localized topological structures S3, TS2, and S1. (

**a**,

**c**) Results for approximated (designed) structures using analytical ansatz (12) (see text for the list of parameters). (

**b**,

**d**) Results for the simulated structures from the lattice model. (

**a**,

**b**) Green light POM images in crossed polarizers at the 532 nm wavelength for ${n}_{e}=1.76$ and ${n}_{o}=1.53$. Orientation of the axes and the white scale bar for (

**a**,

**b**) are shown in the inset at the left and the right of the texture for S1, respectively. (

**c**,

**d**) Isolines for the z projection of the director in the $xy$ and the $xz$ planes. Orientation of the axes for (

**c**,

**d**) is shown in the insets at the left of the distributions for S1. Chiral arms are marked as Region III, whereas Regions I and II indicate outer and inner bright rings, respectively.

**Figure 5.**POM images and director field distributions of structure S1 approximated using different ansatz functions in Equation (18): (1) $\mathbf{N}=\mathbf{S}$; (2) $\mathbf{N}=\mathbf{S}+{m}_{+}\left(\mathsf{\Phi}\right)\widehat{\mathbf{z}}$; (3) $\mathbf{N}=\mathbf{S}+{m}_{\mathrm{cone}}^{(+)}\widehat{\mathbf{z}}$; (4) $\mathbf{N}=\mathbf{S}+{\mathbf{S}}^{\prime}+({m}_{\mathrm{surf}}+{m}_{\mathrm{cone}}^{(+)})\widehat{\mathbf{z}}$. (

**a**) White light POM textures seen in crossed polarizers at ${n}_{e}=1.76$ and ${n}_{o}=1.53$. (

**b**) Director field distributions in the $xy$ plane (the origin is at the center of the cell). (

**c**) Director field distributions in the $xz$ plane (the origin is at the center of the cell). (

**d**) Isosurfaces of the z projection of the director computed at ${n}_{z}\in \{\pm 0.83,\pm 0.5,\pm 0.17\}$. The helix pitch is $P=15$$\mathsf{\mu}$m and $tan\mathsf{\Phi}=0.7$, where $\mathsf{\Phi}$ is the helix tilt angle.

**Figure 6.**White-light POM images in crossed polarizers for the structure S3 approximated using ansatz (16) with varying pitch P and helix tilt angle $\mathsf{\Phi}$. (

**a**) $P=12$$\mathsf{\mu}$m and $tan\mathsf{\Phi}=0.7$ (see Figure 2); (

**b**) $P=12$$\mathsf{\mu}$m and $tan\mathsf{\Phi}=1.4$; (

**c**) $P=6$$\mathsf{\mu}$m and $tan\mathsf{\Phi}=0.7$; (

**d**) $P=6$$\mathsf{\mu}$m and $tan\mathsf{\Phi}=1.4$; Other parameters are: ${n}_{e}=1.76$, ${n}_{o}=1.53$, ${\rho}_{0}=5$$\mathsf{\mu}$m and ${\rho}_{1}=23$$\mathsf{\mu}$m.

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**MDPI and ACS Style**

Lobanov, I.; Aksenova, E.; Orlova, T.; Darmoroz, D.; Uzdin, V.; Kiselev, A.D.
Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals. *Symmetry* **2022**, *14*, 2476.
https://doi.org/10.3390/sym14122476

**AMA Style**

Lobanov I, Aksenova E, Orlova T, Darmoroz D, Uzdin V, Kiselev AD.
Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals. *Symmetry*. 2022; 14(12):2476.
https://doi.org/10.3390/sym14122476

**Chicago/Turabian Style**

Lobanov, Igor, Elena Aksenova, Tetiana Orlova, Darina Darmoroz, Valery Uzdin, and Alexei D. Kiselev.
2022. "Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals" *Symmetry* 14, no. 12: 2476.
https://doi.org/10.3390/sym14122476