Optics of short-pitch deformed-helix ferroelectric liquid crystals: Symmetries, exceptional points, and polarization-resolved angular patterns

In order to explore electric-field-induced transformations of polarization singularities in the polarization-resolved angular (conoscopic) patterns emerging after deformed-helix ferroelectric liquid crystal (DHFLC) cells with subwavelength helix pitch, we combine the transfer matrix formalism with the results for the effective dielectric tensor of biaxial FLCs evaluated using an improved technique of averaging over distorted helical structures. Within the framework of the transfer matrix method, we deduce a number of symmetry relations and show that the symmetry axis of $L$ lines (curves of linear polarization) is directed along the major in-plane optical axis which rotates under the action of the electric field. When the angle between this axis and the polarization plane of incident linearly polarized light is above its critical value, the $C$ points (points of circular polarization) appear in the form of symmetrically arranged chains of densely packed star-monstar pairs. We also emphasize the role of phase singularities of a different kind and discuss the enhanced electro-optic response of DHFLCs near the exceptional point where the condition of zero-field isotropy is fulfilled.

Figure 1

Geometry of (a) smectic cone and (b) planar aligned FLC cell with uniform lying helix.

Figure 2

Four-wave geometry with two incoming (incident) waves ${\mathbf{E}}_{\mathrm{in}}^{(+)}$ and ${\mathbf{E}}_{\mathrm{in}}^{(-)}$, impinging onto the entrance $\left(z=0\right)$ and exit $\left(z=D\right)$ faces, respectively.

Figure 3

Transmittance of light passing through crossed polarizers $T_{xy}$ as a function of the applied electric field at the wavelength $\lambda =650$ nm for the DHFLC cell of thickness $D=130$ $\mu \mathrm{m}$ filled with the FLC mixture FLC-576A [14]. Parameters of the mixture are $n_{\perp }=\sqrt{{\epsilon }_{\perp }}=1.5$ $\left(n_{\parallel }=\sqrt{{\epsilon }_1}=1.72\right)$ is the ordinary (extraordinary) refractive index and $\theta ={32}^{\circ }$ is the tilt angle. The experimental points are marked by squares. Dashed and solid lines represent the theoretical curves computed using the linear approximation [14] and the improved method of averaging with $P_s/{\chi }_E\approx 3.4$ V/$\mu \mathrm{m}$, respectively.

Figure 4

Coupling coefficients as a function of the smectic tilt angle $\theta$ at $r_1={(1.72/1.5)}^2\approx 1.32$ for (a) a uniaxially anisotropic FLC with $r_2=1$ and (b) a biaxially anisotropic FLC with $r_2=1.1$.

Figure 5

Zero-field dielectric constants as a function of the smectic tilt angle $\theta$ at $r_1={(1.72/1.5)}^2\approx 1.32$ for (a) a uniaxially anisotropic FLC with $r_2=1$ and (b) a biaxially anisotropic FLC with $r_2=1.1$.

Figure 6

Principal values of the effective dielectric tensor as a function of the electric field parameter at $r_1\approx 1.32$ and $r_2=1$ for different values of the smectic tilt angles.

Figure 7

Principal axis azimuthal angle versus the electric field parameter at $r_1\approx 1.32$ for (a) a uniaxially anisotropic FLC with $r_2=1$ and (b) a biaxially anisotropic FLC with $r_2=1.1$.

Figure 8

Isotropization tilt angle versus the anisotropy ratio $r_1={\epsilon }_1/{\epsilon }_{\perp }$ at different values of the biaxiality ratio $r_2={\epsilon }_2/{\epsilon }_{\perp }$.

Figure 9

Electrically controlled birefringence $\Delta n_{\mathrm{eff}}\left(E\right)-\Delta n_{\mathrm{eff}}\left(0\right)$ $\left[\Delta n_{\mathrm{eff}}=n_{+}-n_{-}\right]$ versus the electric field parameter at $r_1=1.32$ and $r_2=1$.

Figure 10

Transmittance of light passing through crossed polarizers $T_{xy}$ as a function of the electric field parameter at (a) $\theta ={\theta }_{\mathrm{iso}}\approx 54.{11}^{\circ }$ and (b) $\theta =52.{11}^{\circ }$. Parameters are listed in the caption of Fig. 3.

Figure 11

Arrangement of the polarization ellipses around the $C$ points of three different types.

Figure 12

Polarization-resolved conoscopic patterns computed as polarization ellipse fields in the observation plane for the DHFLC cell filled with the FLC mixture FLC-576A (see Ref. [14] and the caption of Fig. 3). Two cases are shown: (a) ${\alpha }_E=0$ $(n_{+}\approx 1.65,n_{-}=n_z\approx 1.532$, and ${\psi }_d=0)$ and (b) ${\alpha }_E=0.3$ $(n_{+}\approx 1.66,n_{-}\approx 1.529,n_z\approx 1.527$, and ${\psi }_d=13$ deg). In both cases, the angle between the in-plane optical axis and the polarization plane of the incident light is fixed at ${\psi }_d-{\varphi }_p^{\left(\mathrm{inc}\right)}={20}^{\circ }$. $L$ lines are represented by thick black solid lines. Left-handed and right-handed polarization is, respectively, indicated by solid and open ellipses.

Figure 13

Polarization-resolved conoscopic patterns computed as polarization ellipse fields in the observation plane for the DHFLC cell at ${\psi }_d-{\varphi }_p^{\left(\mathrm{inc}\right)}=39.5^{\circ }$ (see the caption of Fig. 12). Two cases are shown: (a) ${\alpha }_E=0$ and (b) ${\alpha }_E=0.3$. $C$ points are indicated by red circles (stars with $I_C=-1/2)$ and blue diamonds (monstars with $I_C=1/2)$.

Figure 14

Polarization-resolved conoscopic patterns computed as polarization ellipse fields in the observation plane for the DHFLC cell at ${\psi }_d-{\varphi }_p^{\left(\mathrm{inc}\right)}={41}^{\circ }$ (see the caption of Fig. 12). Two cases are shown: (a) ${\alpha }_E=0$ and (b) ${\alpha }_E=0.3$. $C$ points with $I_C=-1/2$ and $+1/2$ are indicated by red circles (stars) and blue diamonds (monstars), respectively. Blue thin solid lines represent the approximate $S_2$ nodal lines computed by solving Eq. (89).