  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.
