  On a lossless periodic dielectric structure sandwiched between two
homogeneous media, bound states in the continuum (BICs) with real frequencies
and real Bloch wavevectors may exist, and they decay exponentially in the
surrounding homogeneous media and do not couple with propagating plane waves
with the same frequencies and wavevectors. The BICs are of significant current
interest, because they give rise to high-$Q$ resonances when the structure or
the Bloch wavevector is slightly perturbed. In this paper, the effect of a
small material loss on the BICs is analyzed by a perturbation method and
illustrated by numerical results. It is shown that bound states with complex
frequencies near the continuum appear, but they behave differently depending on
whether the BIC is symmetry-protected or not. The Bloch wavevector of a bound
state with a complex frequency can be real if the original BIC is
symmetry-protected, and it is usually complex if the original BIC is not
symmetry-protected. Our study improves the theoretical understanding on BICs
and provides useful insight for their practical applications.
