  A photonic-crystal slab can support bound states in the continuum (BICs)
which have infinite lifetimes but embedded into the continuous spectrum of
optical modes in free space. The formation of BICs requires a total internal
reflection (TIR) condition at both interfaces between the slab and free space.
Here, we show that the TIR of Bloch waves can be directly obtained based on the
generalized Fresnel equations proposed. If each of these Bloch waves picks up a
phase with integer multiples of 2pi for traveling a round trip, light can be
perfectly guided in the slab, namely, forming a BIC. A BIC solver with low
computational complexity and fast convergence speed is developed, which can
also work efficiently at high frequencies beyond the diffraction limit where
multiple radiation channels exist. Two examples of multi-channel BICs are
shown, and their topological nature in momentum space is also revealed. Both
can be attributed to the coincidence of the topological charges of far-field
radiations from different radiation channels. The concept of the generalized
TIR and the TIR-based BIC solver developed offer highly effective approaches
for explorations of BICs which could have many potential applications in
guided-wave optics and enhanced light-matter interactions.
