Structures for a photodetector or terminator and methods of fabricating a structure for a photodetector or terminator. The structure includes a waveguide core having a longitudinal axis, a pad connected to the waveguide core, and a light-absorbing layer on the pad adjacent to the waveguide core. The light-absorbing layer includes an annular portion, a first taper, and a second taper laterally spaced from the first taper. The first taper and the second taper are positioned adjacent to the waveguide core.