  We identify a family of unusual slow-light modes occurring in lossy
multi-mode grating waveguides, for which either the forward or backward mode
components, or both, become degenerate. In the fully-degenerate case, by
varying the wave amplitudes in a uniform input waveguide, one can modulate
between coherent perfect absorption (zero reflection) and perfect reflection.
The perfectly-absorbed wave has anomalously short absorption length, scaling as
the inverse 1/3 power of the absorptivity.
