Using symmetry bandgaps to create a line of bound states in the continuum in 3D photonic crystals

In many types of optical systems, it is of paramount importance to create states with large quality factors to enhance light-matter interactions, for example to facilitate lasing [1] or harmonic generation [2] . Over the last decade, bound states in the continuum (BICs) [3] , which are states with infinite quality factors despite the availability of a radiative continuum at the same frequency, have emerged as an important design principle for such systems. Previously, BICs have been realized in optical systems by 1) protecting the BIC through symmetry, 2) creating it through modal interference, or 3) finding one accidentally when the number of constraints on a state's radiative emission rate is less than or equal to the number of tunable parameters in the system [3] . Of these three methods, symmetry-protection offers the most reliable route to creating BICs, as both using modal interference and finding accidental BICs present certain difficulties in system design. However, despite the desirability of symmetry-protected BICs, there has been no systematic study of the requirements necessary to find such symmetry protection in planar systems, such as photonic crystal slabs or metasurfaces.