Structures for a directional coupler and methods of fabricating a structure for a directional coupler. A first section of a first waveguide core is laterally spaced from a second section of a second waveguide core. A coupling element is arranged either over or under the first section of the first waveguide core and the second section of the second waveguide core. The first and second waveguide cores are comprised of a material having a first refractive index, and the first coupling element is comprised of a material having a second refractive index that is different from the first refractive index. The first coupling element is surrounded by a side surface that overlaps with the first section of the first waveguide core and the second section of the second waveguide core.