Bragg reflection band width and optical rotatory dispersion of cubic blue-phase liquid crystals

The Bragg reflection band width and optical rotatory dispersion of liquid crystalline cholesteric blue phases (BPs) I and II are compared by numerical simulations. Attention is paid to the wavelength regions for which the reflection bands with lowest photon energies appear, i.e., the $\left[110\right]$ direction for BP I and the $\left[100\right]$ direction for BP II. Finite difference time domain and $4\times 4$ matrix calculations performed on the theoretical director tensor distribution of BPs with the same material parameters show that BP II, which has simple cubic symmetry, has a wider photonic band gap than BP I, which has body centered cubic symmetry, possibly due to the fact that the density of the double-twist cylinders in BP II are twice that in BP I. The theoretical results on the Bragg reflection band width are supported by reflectance measurements performed on BPs I and II for light incident along the $\left[110\right]$ and $\left[100\right]$ directions, respectively.

Figure 1

Photonic band structure of BPs I and II.

Figure 2

Closeup of the photonic band structure near the reflection band with lowest energy in (a) BP I, (b) BP II, (c) the cholesteric phase, (d) the cholesteric phase with birefringence reduced to 22.8%, and (e) the cholesteric phase with birefringence reduced to 33.7%.

Figure 3

(a) Transmittance and (b) polarization spectra of BPs I and II along the $\left[101\right]$ and $\left[100\right]$ directions by the FDTD assuming a three-dimensional dielectric tensor distribution for the BPs (3D-FDTD) and Berreman's $4\times 4$ matrix formulation using the dielectric tensor distribution averaged on the plane perpendicular to the direction of polarization [23].

Figure 4

Electric field amplitudes of light transmitted through BPs I and II at wavelengths of (a) 550 nm, (b) 639 nm, and (c) 750 nm, averaged on the plane normal to the propagation direction. The maximum and minimum values within the plane are also plotted to show the plane wavelike nature of light propagation in the BP.

Figure 5

Simulated transmittance through ideal crossed polarizers for light at $\lambda =400\mathrm{nm}$ propagating along the $\langle 110\rangle$ and $\langle 100\rangle$ directions of BPs I and II, respectively. The horizontal axis shows the period of BPs I and II along the $\langle 110\rangle$ and $\langle 100\rangle$ directions, respectively, and refractive indices of $n_e=1.7,n_o=1.5$ are assumed with different values of $S_{max}$. The vertical axis has a logarithmic scale to emphasize the difference in cross-polarized transmittance.

Figure 6

(a) Polarized optical microscope image of a sample as the temperature is raised slowly to induce a transition from BP I to BP II. The length of the scale bar is $100\mu \mathrm{m}$, and the arrows indicated the direction of the polarizers. (b) Kossel diagrams observed for BPs I and II using a probe wavelength of 436 nm.

Figure 7

Polarized reflectance spectra of the sample as the temperature is raised slowly to induce a transition from BP I to BP II. $T_{\mathrm{BP}\mathrm{I}\to \mathrm{BP}\mathrm{II}}$ indicates the BP I to BP II transition temperature. Arrows are drawn as guides to show how the reflectance spectrum evolves.

Figure 8

Simulated polarized reflectance spectra calculated using the combination of numerical parameters summarized in Table 1 and the order parameter distributions for BPs I and II. OP in the annotation refers to “order parameter.” The dotted curves are experimental reflectance spectra, corresponding to temperatures $T_{\mathrm{iso}}-0.06{}^{\circ }\mathrm{C}$ and $T_{\mathrm{iso}}+0.09{}^{\circ }\mathrm{C}$ in Fig. 7.