Light scattering by cholesteric skyrmions

We study the light scattering by localized quasiplanar excitations of a cholesteric liquid crystal known as spherulites. We quantitatively evaluate the cross section of the axis rotation for polarized light by taking into account the anisotropic optical properties of the medium and the peculiar shape of the excitations. Because of the complexity of the system under consideration, we exploit also a simplified analytical description of the spherulite and evaluate the scattering cross sections in the Born approximation. We compare the scattering data by the analytical and the numerical skyrmion solutions for several choices of the model paramenters. The effects of changing values of the driving external static electric (or magnetic) field is also considered. Possible applications of the phenomenon are envisaged.

Figure 1

Comparison for $\frac{E}{E_0}=1.02$ among the numerical solution of (2.8), the linearly piecewise approximation (2.12), and the analytic solution of the linearized equation.

Figure 2

Comparison for $\frac{E}{E_0}=1.5$ among the numerical solution of (2.8), the linearly piecewise approximation (2.12), and the analytic solution of the linearized equation.

Figure 3

Profiles $\theta \left(\rho \right)$ for $E/E_0=1.02, k_s=0.1$ (a), and $k_s=6$ (b). Different curves refer to different values of $\mid z\mid$. Dashed curves have to be referred to $z=0$ (the thin one) and to $\mid z\mid =\nu /2$ (the thicker one).

Figure 4

Profiles $\theta \left(\rho \right)$ for $E/E_0=1.5, k_s=0.1$ (a), and $k_s=6$ (b). Different curves refer to different values of $\mid z\mid$. Dashed curves have to be referred to $z=0$ (the thin one) and to $\mid z\mid =\nu /2$ (the thicker one). We note that the effect of a greater external electric field is to narrow the size of the vortices for fixed values of $k_s$.

Figure 5

Frontal and equatorial sections of the director field distribution for a spherulite in CLC. Here the parameters are the same as in Fig. 3, i.e., $\left({\displaystyle \frac{E}{E_0}}\right)=1.02, k_s=6, \nu =1.8$. Notice how the transversal radius of the spherulite decreases near the bounding plane surfaces.

Figure 6

The numerical values of ${\mathcal{I}}_m^{\varphi }$ as function of $0\le m\le 50$ for three different values of $k{\rho }_0=8,10,12$. They can be approximated by continuous functions in $m$, for instance, by linear combinations of Gaussian functions but still a clear pattern for a systematic approximation has to be developed. It is evident a linear dependence, with a slope $\sim 4$, on the number of significant terms with the size parameter $k{\rho }_0$. This is in agreement with the scattering features on localized central potentials.

Figure 7

The numerical evaluation of the conversion cross section (4.9), in arbitrary units, for the three different values of $k{\rho }_0$ used in Fig. 6.

Figure 8

The numerical evaluation of the conversion cross section (4.9), in arbitrary units, for ${\rho }_0$ fixed by (2.11) and the ratio $E/E_0=1.02, k{\rho }_0=10$, for the numerical solution (solid line) and the approximated solution (2.12) (dashed line).

Figure 9

The numerical evaluation of the conversion cross section (4.9), in arbitrary units, for ${\rho }_0$ fixed by (2.11) and the ratio $E/E_0=1.5, k{\rho }_0=10$, for the numerical solution (solid line) and the approximated solution (2.12) (dashed line).

Figure 10

The numerical values of $I_m^{\left(\rho \right)}$ as function of $0\le m\le 150$, for two different values of $E/E_0$ with fixed $k{\rho }_0$, computed through the use of the numerical solution (stars) and the approximated solution (2.12) (circles).

Figure 11

The numerical evaluation of the conversion cross section (4.22), in arbitrary units, for $E/E_0=1.02$ and $k{\rho }_0=10$. The numerical solution corresponds to the solid line and the approximated solution (2.12) to the dashed one.

Figure 12

The numerical evaluation of the conversion cross section (4.22), in arbitrary units, for $E/E_0=1.02$ and $k{\rho }_0=10$. The numerical solution corresponds to the solid line and the approximated solution (2.12) to the dashed one.

Figure 13

Comparison of the log-polar plots of the in-plane–out-plane conversion differential cross sections for two different values of the external electric field. In both cases the solid line curves refer to the numerical spherulite profile, while the dashed ones correspond to the piecewise linear approximation of it.

Figure 14

The density plot has been obtained computing the cross section (4.22) at different values of $0\le z\le \frac{1}{2}L/p, k{\rho }_0=10$ and $E/E_0=1.5$, without any modification in (4.22) but changing the spherulite radius accordingly to ${\rho }_{0}Z\left(z\right)$ in (2.13)and (2.14).