Light transmission from a twisted nematic liquid crystal: Accurate methods to measure the azimuthal anchoring energy

In this paper, we analyze the light transmission from a twisted nematic liquid crystal (NLC) and we propose two accurate and very direct optical methods to measure the azimuthal anchoring energy. In both of them, a monochromatic beam of wavelength $\lambda$ with a polarization vector that rotates at an angular frequency $\omega$ impinges on a twisted nematic liquid crystal. The intensity of the transmitted beam is modulated at angular frequency $2\omega$ with a phase shift $\beta ,$ which is related to the surface azimuthal director angle ${\phi }_1$ at the investigated interface. It is shown that there exists a special geometry where the simple adiabatic relation ${\phi }_1=\beta /2\mathrm{}$ is satisfied up to second order in the small perturbative parameter $\alpha =\lambda /\left(2\mathrm{}\pi \Delta n\xi \right),$ where $\Delta n$ is the anisotropy of the refractive indices of the NLC and $\xi$ is the twist distortion length. Furthermore, the small residual higher order correction terms can be greatly reduced by choosing a proper geometry for the experiment. With this choice, the range of validity of the adiabatic theory is greatly extended. The perturbative theoretical results are fully confirmed by numerical calculations. The azimuthal anchoring energy coefficient can be obtained by measuring phase shift $\beta$ versus the intensity of an applied magnetic field. These methods greatly improve the accuracy of the previous transmitted light techniques and also provide accurate measurements of strong azimuthal anchoring energies.