Effective dielectric tensor of deformed-helix ferroelectric liquid crystals with subwavelength pitch and large tilt angle

Short pitch deformed helix ferroelectric liquid crystals have numerous applications as active materials in displays, optical telemetry, and biomedical devices. In this paper, we derive convenient analytical formulas to calculate the effective dielectric tensor of these materials beyond the space average approximation. By comparison with exact numerical calculations, we show that our formulas are remarkably accurate in predicting optical properties in virtually all practical situations, including the important case of large tilt angles, where the space average approximation breaks down. We also present a comparison between the two complementary approaches of expanding the mesoscopic dielectric tensor versus the mesoscopic transfer matrix, by deriving an expression for the effective transfer matrix as an infinite expansion and explicitly calculating the corresponding effective dielectric tensor for the first time. Our results demonstrate that both methods give accurate predictions when two-photon scattering terms are taken into account.

Figure 1

Geometry of the problem. A planar dielectric grating of thickness $D$ is placed in an isotropic ambient medium with refractive index $n_m$. The dielectric tensor of the grating, $\mathbf{\epsilon }\left(x\right)$, varies periodically in a direction parallel to the slab's surface with pitch ${\lambda }_g$. This geometry describes deformed helix ferroelectric liquid crystals with homogeneous alignment.

Figure 2

Geometry of a DHFLC. In each smectic layer (delimited by dashed lines) the direction of the director is shown by an arrow. The resulting helical structure, with pitch ${\lambda }_g$, has its axis along $x$. The tilt angle, ${\theta }_t$, and the azimuthal angle, $\mathrm{\Phi }$, are also indicated. The electric field, $\mathbf{E}$, is perpendicular to the helix axis.

Figure 3

Birefringence as a function of the incident light's wavelength (${\lambda }_0$) for the exact numerical calculations (black crosses) and for the six analytical approximations, as indicated in the legend. A summary of the various approximations is also reported in Table 1 for convenience.

Figure 4

Rotation of the optic axes as a function of the incident light's wavelength ${\lambda }_0$. Symbols indicate the exact numerical calculations, while lines denote the six analytical approximations, as indicated in the legend.

Figure 5

Birefringence as a function of the tilt angle. Symbols indicate the exact numerical calculations, while lines denote the six analytical approximations, as indicated in the legend.

Figure 6

Rotation of the optic axes as a function of the tilt angle. Symbols indicate the exact numerical calculations, while lines denote the six analytical approximations, as indicated in the legend.

Figure 7

Normal incidence transmission at crossed polarizers calculated as a function of the electric field. Incident light is linearly polarized along the direction of the helix axis. Symbols indicate the exact numerical calculations, while lines denote the six analytical approximations, as indicated in the legend.

Figure 8

Transmission at crossed polarizers as a function of the incident angle. Incident light is linearly polarized along the direction of the helix axis. Symbols indicate the exact numerical calculations, while lines denote the six analytical approximations, as indicated in the legend.