Focused beam propagation in cholesteric liquid crystals

In this paper, we develop a numerical technique that allows us to study the propagation of tightly focused scalar or vector beams through a stratified birefringent material. The technique is based on the Berreman 4 × 4 transfer matrix method and modal decomposition using the Fourier transform. Instead of the usual Berreman vector, describing the electric and magnetic field components, we introduce a novel concept and represent the field vector components using a generalized bi-directional Jones vector. The Jones vector formulation simplifies the creation of focused beams and allows studying the beams' polarization state, intensity, and propagation direction using the usual 2 × 2 Jones matrix formalism. We develop a numerical model for an ideal high numerical aperture lens transform, and we use the technique to focus a tilted beam and propagate the beam through the cholesteric liquid crystal. Compared to plane waves at normal incidence, the transmission spectra of focused beams start to differ when the beam's Rayleigh length becomes comparable to the cholesteric thickness. The bandgap of the cholesteric liquid crystal remains unchanged, but we observe a reduced transmission in the side lobes near the edge of the bandgap. The effect is significantly enhanced when propagation is at oblique incidence with respect to the helical axis and for tightly focused beams, where the resonances near the bandgap disappear.