  Vector and vortex laser beams are desired in many applications and are
usually created by manipulating the laser output or by inserting optical
components in the laser cavity. Distinctly, inserting liquid crystals into the
laser cavity allows for extensive control over the emitted light due to their
high susceptibility to external fields and birefringent nature. In this work we
demonstrate diverse optical modes for lasing as enabled and stablised by
topological birefringent soft matter structures using numerical modelling. We
show diverse structuring of light -- with different 3D intensity and
polarization profiles -- as realised by topological soft matter structures in
radial nematic droplet, in 2D nematic cavities of different geometry and
including topological defects with different charges and winding numbers, in
arbitrary varying birefringence fields with topological defects and in
pixelated birefringent profiles. We use custom written FDFD code to calculate
emergent electromagnetic eigenmodes. Control over lasing is of a particular
interest aiming towards the creation of general intensity, polarization and
topologically shaped laser beams.
