High power white light LEDs are distributed within a thin reflective cavity. The cavity depth may be less than 3 cm and, in one embodiment, is about 1 cm. A light output surface of the cavity is a flat reflector with many small openings. A small plastic lens is positioned over each opening for causing the light emitted from each opening to form a cone of light between approximately 50-75 degrees. Alternatively, each hole may be shaped to be a truncated cone to control the dispersion. The light emitted by the LEDs is mixed in the cavity by reflecting off all six reflective walls of the cavity. The light will ultimately escape through the many holes, forming a relatively uniform pattern of light on a surface to be illuminated by the luminaire.