  We create a one-dimensional photonic crystal with strong polarization
dependence and tunable by an applied electric field. We accomplish this in a
planar microcavity by embedding a cholesteric liquid crystal (LC), which
spontaneously forms a uniform lying helix (ULH). The applied voltage controls
the orientation of the LC molecules and, consequently, the strength of a
polarization-dependent periodic potential. It leads to opening or closing of
photonic band gaps in the dispersion of the massive photons in the microcavity.
In addition, when the ULH structure possesses a molecular tilt, it induces a
spin-orbit coupling between the lattice bands of different parity. This
interband spin-orbit coupling (ISOC) is analogous to optical activity and can
be treated as a synthetic non-Abelian gauge potential. Finally, we show that
doping the LC with dyes allows us to achieve lasing that inherits all the
above-mentioned tunable properties of LC microcavity, including dual and
circularly-polarized lasing.
