Density of photonic states in cholesteric liquid crystals

Density of photonic states $\rho \left(\omega \right)$, group $v_g$, and phase $v_{\mathrm{ph}}$ velocity of light, and the dispersion relation between wave vector $k$, and frequency $\omega \left(k\right)$ were determined in a cholesteric photonic crystal. A highly sensitive method (measurement of rotation of the plane of polarization of light) was used to determine $\rho \left(\omega \right)$ in samples of different quality. In high-quality samples a drastic increase in $\rho \left(\omega \right)$ near the boundaries of the stop band and oscillations related to Pendellösung beatings are observed. In low-quality samples photonic properties are strongly modified. The maximal value of $\rho \left(\omega \right)$ is substantially smaller, and density of photonic states increases near the selective reflection band without oscillations in $\rho \left(\omega \right)$. Peculiarities of $\rho \left(\omega \right), v_g$, and $\omega \left(k\right)$ are discussed. Comparison of the experimental results with theory was performed.

Figure 1

Transmission spectrum in nonpolarized light (dashed curve) and rotation of the plane of polarization of light (points) for a perfect sample with aligning coating on the cell surfaces. Oscillations in transmission and rotation of the plane of polarization of light are related to Pendellösung beatings and demonstrate high quality of the sample. $T=23.5$ °C, sample thickness $L=5.8\mu \mathrm{m}$.

Figure 2

Transmission spectrum in nonpolarized light (dashed curve) and rotation of the plane of polarization of light (points) for an imperfect sample without aligning coating on the cell surfaces. Pendellösung oscillations in the spectra are absent. $T=26$ °C, sample thickness $L=7.75\mu \mathrm{m}$.

Figure 3

Normalized DOS of perfect (a) and imperfect (b) samples of cholesteric photonic crystal (solid curves) obtained from rotation of the plane of polarization of light in Figs. 1 and 2. The dashed line in (a) is the theoretical spectrum of DOS calculated using Eqs. (2), (4), and (5) with material parameters ($n_0^2=2.491, p=362\mathrm{nm}, \delta =0.0638$) corresponding to transmission spectra of Fig. 1. Oscillations in $\rho \left(\omega \right)$ [Fig. 3] are related to the finite size of the photonic crystal. Experimentally obtained $\rho \left(\omega \right)$ is in agreement with calculated DOS. Two vertical lines show the edges of the forbidden photonic band.

Figure 4

Plot of the dimensionless group velocity of light $v_g/c$ (solid curve) for the perfect sample of finite thickness. The extremes of the group velocity appear in the finite sample on either sides of the stop band. Dimensionless phase velocity of light $v_{\mathrm{ph}}/c$ for the sample of finite thickness (dashed curve). Two vertical lines show the edges of the stop band. $L=5.8\mu \mathrm{m}$.

Figure 5

Experimental dispersion of light $\omega \left(k\right)$ for the perfect sample of finite thickness (points). $\omega =2\pi c/\lambda$. Theoretical dispersion (solid lines). Two horizontal lines show the edges of the forbidden photonic band. The dashed line shows the dispersion for the medium without the photonic band. $L=5.8\mu \mathrm{m}$.