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Resonant modes in a lossy periodic structure sandwiched between two losslesshomogeneous media form bands that depend on the Bloch wavevector continuouslyand have a complex frequency due to radiation and absorption losses. A complexbound state in the continuum (cBIC) is a special state with a zero radiationloss in such a band. Plane waves incident upon the periodic structure inducelocal fields that are resonantly enhanced. In this paper, we derive a rigorousformula for field enhancement, and analyze its dependence on the frequency,wavevector and amplitude of the incident wave. For resonances with multipleradiation channels, we determine the incident wave that maximizes the fieldenhancement, and find conditions under which the field enhancement can berelated to the radiation and dissipation quality factors. We also show thatwith respect to the Bloch wavevector, the largest field enhancement is obtainedapproximately when the radiation and dissipation quality factors are equal. Ourstudy clarifies the various factors related to field enhancement, and providesa useful guideline for applications where a strong local field is important.

Resonant field enhancement in lossy periodic structures supporting complex bound states in the continuum