Quality factor scaling of resonances related to bound states in the continuum

In real samples, bound states in the continuum (BICs) are manifested as resonances with a finite, although significantly high, quality ($Q$) factor. Control over the $Q$ factor through an asymmetry parameter (i.e., an intentional defect of the structure, allowing structural imperfections to be introduced on purpose) allows tailoring the coupling strength with the modes of the free space. In most systems, $Q$ has an inverse quadratic dependence on the asymmetry parameter. However, various applications require different scaling laws. For instance, sensors require a steeper dependence, whereas light generation needs a less steep one for robustness. Here, we consider a metasurface consisting of dielectric rods with air holes inside of them, obtaining several different scaling laws. Our analysis reveals that BIC has dominant and asymmetry-induced multipole terms. Depending on the radiation properties of the induced multipoles and their amplitude in the case of vanishing asymmetry, the exponent in the scaling law lies in the range from $\ensuremath{-}4$ to $\ensuremath{-}1.75$, including the common case of $\ensuremath{-}2$.