A method and apparatus is described that use an index-of-refraction profile having a significant central dip in refractive index (or another tailored index profile) within the core of a gain fiber or a gain waveguide on a substrate. The benefits of this central dip (more power with a given mode structure) are apparent when an input beam is akin to that of a Gaussian mode. In some embodiments, the invention provides a fiber or a substrate waveguide having an index profile with a central dip, but wherein the device has no doping. Some embodiments use a central dip surrounded by a higher-index ring in the index of refraction of the core of the fiber, while other embodiments use a trench between an intermediate-index central core portion and the ring, or use a plurality of rings and/or trenches. Some embodiments use an absorber in at least one core ring.