An illumination apparatus provides an output with high spatial uniformity of luminance. The illumination apparatus includes a cavity that has reflective surfaces and an output area. At least one light source is disposed in the cavity, with the light source including a point source (such as an LED) and an optical diverter having a flared (e.g., cuspated) reflecting surface. Light from the point source is distributed by the optical diverter towards the reflective surfaces within the cavity. Before exiting the device, light passes through an optical conditioning element that is positioned over the output area of the cavity. The optical conditioning element includes at least a diffuser, such as a translucent film or plastic sheet, and preferably also includes one or more prism sheets such as a brightness-enhancing film.