Algebraic approach to finding the number of parameters required to obtain a bound state in the continuum

Photonic bound states in the continuum (BICs) are nonradiating eigenmodes of structures with open scattering channels. Most often, BICs are studied in highly symmetric structures with one or two open scattering channels. In this simplest case, the so-called symmetry-protected BICs can be found by tuning a single parameter, which is the light frequency. Another kind of BIC---accidental BICs---can be obtained by tuning two parameters, e.g., the frequency and a wave-vector component. For more complex structures lacking certain symmetries or having many open scattering channels, more than two parameters might be required. In the present work, we propose an algebraic approach for computing the number of parameters required to obtain a BIC by expressing it through the dimension of the solution set of certain algebraic equations. Computing this dimension allows us to relate the required number of parameters to the number of open scattering channels without solving Maxwell's equations. We show that different relations take place when the scattering matrices describing the system are symmetric or asymmetric. The obtained theoretical results are confirmed by the results of rigorous electromagnetic simulations.