Structures including a grating coupler and methods of fabricating a structure including a grating coupler. The structure includes structure includes a dielectric layer on a substrate, a first waveguide core positioned in a first level over the dielectric layer, and a second waveguide core positioned in a second level over the dielectric layer. The second level differs in elevation above the dielectric layer from the first level. The first waveguide core includes a tapered section. The structure further includes a grating coupler having a plurality of segments positioned in the second level adjacent to the second waveguide core. The segments of the grating coupler and the tapered section of the first waveguide core are positioned in an overlapping arrangement.