Photonic band structure of diamond colloidal crystals in a cholesteric liquid crystal

In this paper, we demonstrate the presence of a photonic band gap for a diamond lattice structure made of particles with normal anchoring inside a cholesteric liquid crystal. As is typical for liquid crystals (LCs), there is considerable contrast between the dielectric constant parallel ${\epsilon }_{\parallel }$ and perpendicular ${\epsilon }_{\perp }$ to the director, with ${\epsilon }_{\parallel }/{\epsilon }_{\perp }\sim 4$ here. It is shown that the size of the photonic band gap is directly related to the size of colloidal particles and the contrast between the dielectric constant in the particles and the extreme values of $\epsilon$ in the LC medium (one needs either $\epsilon$ in the particle much smaller than ${\epsilon }_{\perp } or$ much bigger than ${\epsilon }_{\parallel }$). No opening is seen in the band diagrams for small particles. For larger particles a partial gap opens when the particles are composed of very low dielectric material but never a complete gap. On the other hand, a complete gap starts to be revealed when the size of the colloidal particles is increased and when a high dielectric constant is used for filling inside the particles. The maximum size of the gap is observed when the particles are large enough so that their surfaces overlap.

Figure 1

High, $\epsilon =70$, contour of $\epsilon$ for the particles of radius $17\mathrm{\Delta }x(\sim 1.1\mu \mathrm{m}$). The blue contour shows the regions inside of which $\epsilon$ is isotropic and its value is set to 70. The liquid crystal, with anisotropic $\epsilon$, given by Eq. (7), fills the space outside this contour.

Figure 2

Irreducible Brillouin zone for the simple cubic lattice.

Figure 3

The diamond lattice made of colloids with radius $10\mathrm{\Delta }x$ with a (yellow) contour of the scalar order parameter ($14\%$ below the bulk value) showing the Saturn-ring-like defects surrounding each colloid.

Figure 4

The photonic band structures for particles of radius $10\mathrm{\Delta }x$ filled with (a) $\epsilon =1$ and (b) $\epsilon =70$.

Figure 5

The diamond lattice made of colloids with radius $17\mathrm{\Delta }x$ with a (yellow) contour of the scalar order parameter showing the disclination lines traveling along symmetry lines of the lattice.

Figure 6

The photonic band structures for particles of radius $17\mathrm{\Delta }x$ filled with (a) $\epsilon =1$, (b) $\epsilon =48$, and (c) $\epsilon =70$.

Figure 7

The diamond lattice made of colloids with radius $19\mathrm{\Delta }x$ with the contour plot of scalar order parameter showing the line defects surrounding each colloid. The front-center sphere is drawn transparently so that the circular distortion where the spheres meet is visible.

Figure 8

The photonic band structures for particles of radius $19\mathrm{\Delta }x$ filled with (a) $\epsilon =1$, (b) $\epsilon =2.31$, and (c) $\epsilon =70$.

Figure 9

The photonic band structures for particles of radius $20\mathrm{\Delta }x$ filled with (a) $\epsilon =1$ and (b) $\epsilon =70$.