Density of photon states in dye-doped chiral nematic liquid crystal cells in the presence of losses and gain

We calculate the density of photon states (DOS) of the normal modes in dye-doped chiral nematic liquid crystal (LC) cells in the presence of various loss mechanisms. Losses and gain are incorporated into the transmission characteristics through the introduction of a small imaginary part in the dielectric constant perpendicular and along the director, for which we assume no frequency dispersion. Theoretical results are presented on the DOS in the region of the photonic band gap for a range of values of the loss coefficient and different values of the optical anisotropy. The obtained values of the DOS at the photonic band gap edges predict a reversal of the dominant modes in the structure. Our results are found to be in good agreement with the experimentally obtained excitation thresholds in chiral nematic LC lasers. The behavior of the DOS is also discussed for amplifying LC cells providing additional insight to the lasing mechanism of these structures.

Figure 1

Theoretical results of the normalized DOS of the eigenwave $E_1$ (${\rho }_1/{\rho }_{\mathrm{iso}}$) for a chiral nematic cell as a function of reduced wavelength for different values of the loss coefficient and the optical anisotropy. For all plots, the cell thickness was fixed at $L=30p.$ (a) $n=1.581$ relative dielectric anisotropy at optical frequencies, $\alpha =0.1$ loss coefficient $\gamma =2{\gamma }_{\parallel }=0.0002$, (b) $n=1.581,\alpha =0.1$ loss coefficient $\gamma =2{\gamma }_{\parallel }=0.002$, (c) $n=1.541$, $\alpha =0.0526$ and loss coefficient $\gamma =2{\gamma }_{\parallel }=0.0002$ and (d) $n=1.541$, $\alpha =0.0526$ loss coefficient $\gamma =2{\gamma }_{\parallel }=0.002$.

Figure 2

The asymmetry in the DOS at the first edge modes either side of the photonic band gap of a chiral nematic LC. A plot of the difference of the maximum DOS between the short and long wavelength edges (${\rho }_{\mathrm{SWE}}-{\rho }_{\mathrm{LWE}}$) of the eigenwave $E_1$ that is normalized to isotropic DOS as a function of the loss coefficient. In this case $n=1.581$, the optical anisotropy is $\alpha =0.1$, and the cell thickness is $L=30p$.

Figure 3

Theoretical plots of the maximum normalized DOS (${\rho }_{1\mathrm{Max}}/{\rho }_{\mathrm{iso}}$) of the eigenwave $E_1$ at the two wavelength edges either side of the stop band as a function of the chiral nematic LC. The cell thickness is $L=Np$. (a) For $n=1.581$, $\alpha =0.1$ and $\gamma =2{\gamma }_{\parallel }=0.0002$, (b) for $n=1.581$, $\alpha =0.1$ and $\gamma =2{\gamma }_{\parallel }=0.006$, (c) for $n=1.603,\alpha =0.1245$ and $\gamma =2{\gamma }_{\parallel }=0.0002$, and (d) for $n=1.603$, $\alpha =0.1245$ and $\gamma =2{\gamma }_{\parallel }=0.006$. The arrows mark the number of pitches where the peak DOS is attained for each wavelength edge. The short wavelength and long wavelength edges are shown as solid and dashed lines, respectively.

Figure 4

(a) Experimental results of the dependence of the laser excitation threshold on the cell thickness for a low optical anisotropy ($\alpha =0.13$) (closed squares) and a high optical anisotropy ($\alpha =0.15$) (open circles) LC host. (b) Theoretical plots of the DOS for the long wavelength band edge for two different chiral nematic LCs for the same optical anisotropies as those used in the experiment. The loss coefficient in this case is $\gamma =0.0084$ for $n=1.608$ and $n=1.627,$ respectively.

Figure 5

The behavior of the DOS in the presence of gain. Normalized DOS of the eigenwave $E_1$ for a chiral nematic LC cell with thickness $L=30p$. $n=1.523$, $\alpha =0.03$ and (a) $\gamma =2{\gamma }_{\parallel }=-0.0304$, (b) $\gamma =2{\gamma }_{\parallel }=-0.0320$, (c) $\gamma =2{\gamma }_{\parallel }=-0.0540,$ (d) $n=1.541$, $\alpha =0.0526$ and $\gamma =2{\gamma }_{\parallel }=-0.0134$.