Time reversal constraint limits unidirectional photon emission in slow-light photonic crystals

Photonic crystal waveguides are known to support C-points - point-like polarisation singularities with local chirality. Such points can couple with dipole-like emitters to produce highly directional emission, from which spin-photon entanglers can be built. Much is made of the promise of using slow-light modes to enhance this light-matter coupling. Here we explore the transition from travelling to standing waves for two different photonic crystal waveguide designs. We find that time-reversal symmetry and the reciprocal nature of light places constraints on using C-points in the slow-light regime. We observe two distinctly different mechanisms through which this condition is satisfied in the two waveguides. In the waveguide designs we consider, a modest group-velocity of $v_g \approx c/10$ is found to be the optimum for slow-light coupling to the C-points.