Structure and grating efficiency of thin cells filled by a twist-bend nematic liquid crystal

A twist-bend nematic $\left(N_{\mathrm{TB}}\right)$ liquid crystalline phase spontaneously forms modulated structures on a microscale level when confined in thin planar cells. Preliminary studies showed that these cells can be used as polarization gratings. Here we present a theoretical description of the formation of a two-dimensionally modulated structure. By considering the $N_{\mathrm{TB}}$ phase as a pseudolayer medium, a threshold condition for the onset of a modulated structure is calculated for weak and strong boundary conditions in the case of initially bookshelf or pretilt alignment of pseudolayers. Based on the modeled structure we determine spatial variation of the optic axis and calculate properties of the transmitted diffracted light. Results of the beam propagation method (BPM) and transfer matrix method are compared and it is shown that a more complex BPM gives better agreement with experimental results, meaning that even in thin cells the diffraction of light inside the grating should not be neglected.

Figure 1

(a) The pseudolayer that was initially at $y_0$ is displaced by $u\left(x,z\right)$; $\stackrel{\rightharpoonup }{v}$ is the pseudolayer normal. (b) By undulation, the pseudolayer thickness is reduced from $d_s$ to $d_0\left(T\right)$. (c) A 2D pseudolayer undulation with the period of modulation $D$ along the $x$ direction.

Figure 2

Pretilt of pseudolayers. (a) The period of modulation $\tilde{D}$ is practically the same for weak and strong anchoring. Amplitude of modulation ${\tilde{u}}_0$ as a function of pretilt $\gamma$; the dot-dashed (blue), solid (green), and dotted (red) lines represent $\gamma =0.04,0.07,0.1$, respectively: (b) strong and (c) weak anchoring. Parameter values: $\tilde{B}=5\times {10}^5$, $\tilde{\mathrm{K}}=1$.

Figure 3

Definition of angles $\alpha$ and $\beta$; solid (red) arrow along the $z$ axis and dotted (blue) arrow represent the wave vector direction and optic axis, respectively.

Figure 4

(a) Polarization of light is defined by the azimuth angle $\psi$ and ellipticity $e$. (b) Polarization can be presented on the Poincaré sphere by a point the coordinates of which are given by the normalized values of the Stokes parameters ($s_1,s_2,s_3$), where $s_1=cos\left(2\psi \right)cos\left(2e\right)$ (solid arrow), $s_2=sin\left(2\psi \right)cos\left(2e\right)$ (dash-dotted arrow), and $s_3=sin\left(2e\right)$ (dashed arrow).

Figure 5

Temperature ($T$) dependence of birefringence $n_e-n_o$ (experimentally measured), $q_{0s}$, $\tilde{D}$, and ${\tilde{u}}_0$, the latter two calculated in the case of strong anchoring with pretilt at the values of $q_{0s}$ denoted by blue filled circles in the left graph. (a), (b) CB7CB with $\gamma =0.04$; (c), (d) CB6OCB with $\gamma =0.1$. Parameter values: $\tilde{B}=5\times {10}^5$, $\tilde{K}=1$. In (a), (c) we give also experimentally measured ratio $q_{0s}^{\left(\mathrm{expt}\right)}=q/q_{\min }$ (red filled squares), obtained by the resonant soft x-ray scattering [31, 58], where $q_{\min }=2\pi /d_s$ and $d_s$ is the pitch measured at the phase transition temperature. The error bars on two points are added to give the impression of the experimental error (approximately 10%).

Figure 6

Experimental and theoretical temperature dependence of the polarization of the $+2q_0$ (orange, light gray) and $-2q_0$ (blue, dark) diffraction peak for the LCP incident light for CB7CB (a–d) and CB6OCB (e–h). CB7CB: (a) experimental values, (b) preliminary model [48], (c) TMM, and (d) BPM. CB6OCB: (e) experimental values, (f) preliminary model [49], (g) TMM, and (h) BPM. The arrow heads represent the direction of temperature increase. Parameter values used in calculation are given in the text. The incident light is LCP.

Figure 7

Experimental and theoretically predicted ratios of intensities between the zeroth ($I_0$), first ($I_1$), and second ($I_2$) order diffraction peaks. (a), (b) CB7CB; (c), (d) CB6OCB. Black filled squares represent the experimental data and orange filled circles and blue filled diamonds show the theoretically calculated ratios from BPM and TMM, respectively.