Resonant modes in a lossy periodic structure sandwiched between two lossless homogeneous media form bands that depend on the Bloch wavevector 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 induce local fields that are resonantly enhanced. In this paper, we derive a rigorous formula for field enhancement, and analyze its dependence on the frequency, wavevector and amplitude of the incident wave. For resonances with multiple radiation channels, we determine the incident wave that maximizes the field enhancement, and find conditions under which the field enhancement can be related to the radiation and dissipation quality factors. We also show that with respect to the Bloch wavevector, the largest field enhancement is obtained approximately when the radiation and dissipation quality factors are equal. Our study clarifies the various factors related to field enhancement, and provides a useful guideline for applications where a strong local field is important.
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