  The size of the bandgap in a photonic crystal ring is typically intuitively
considered to monotonically grow as the modulation amplitude of the grating
increases, causing increasingly large frequency splittings between the
'dielectric' and 'air' bands. In contrast, here we report that as the
modulation amplitude in a photonic crystal ring increases, the bandgap does not
simply increase monotonically. Instead, after the initial increase, the bandgap
closes and then reopens again with the dielectric band and the air bands
flipped in energy. The air and dielectric band edges are degenerate at the
bandgap closing point. We demonstrate this behavior experimentally in silicon
nitride photonic crystal microrings, where we show that the bandgap is closed
to within the linewidth of the optical cavity mode, whose quality factor
remains unperturbed with a value $\approx$ 1$\times$10$^6$ (i.e., linewidth of
2 pm). Moreover, through finite-element simulations, we show that such bandgap
closing and band flipping phenomena exist in a variety of photonic crystal
rings with varying units cell geometries and cladding layers. At the bandgap
closing point, the two standing wave modes with a degenerate frequency are
particularly promising for single-frequency lasing applications. Along this
line, we propose a compact self-injection locking scheme that integrates many
core functionalities in one photonic crystal ring. Additionally, the
single-frequency lasing might be applicable to DFB lasers to increase their
manufacturing yield.
