Abstract

Resonant modes in a lossy periodic structure sandwiched between two lossless homogeneous media form bands that depend on the Bloch wave vector continuously and have a complex frequency due to radiation and absorption losses. A complex bound state in the continuum (cBIC) is a special state with a zero radiation loss in such a band. Plane waves incident upon the periodic structure can be resonantly enhanced. An existing scaling law states that the field enhancement $\eta$, defined as the ratio between the amplitudes of the local field and the incident wave, is proportional to $\sqrt {{Q_{{\rm rad}}}} {Q_{{\rm dis}}}/(\!{Q_{{\rm rad}}} + {Q_{{\rm dis}}})$, where ${Q_{{\rm rad}}}$ and ${Q_{{\rm dis}}}$ are the quality factors associated with radiation and absorption losses, respectively. In this paper, we show that this scaling law is not always valid when there are multiple radiation channels. Using a perturbation method, we derive a rigorous formula for field enhancement, and show the scaling law is valid for specific incident fields, but not valid for optimal incident fields that maximize the field enhancement.

© 2022 Optica Publishing Group

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