Waveguide structures in which light confinement strength varies along the direction of light propagation are described. The waveguides include a core adapted to propagate light along a path defined by the core and a cladding material that at least partially surrounds the core. The core and the cladding material each have a refractive index profile in the direction of light propagation. One or more of the profiles or lateral core dimensions are varied along the direction of light propagation, thereby causing the degree of light confinement to vary in the direction of light propagation. With such structures it is possible to tailor the velocity of the light propagating through the core and the confinement of the light to the core for any given core cross section. The structures may be used, for example, in mode-matching an input or output of a waveguide and in fabricating compact directional couplers.