A light emitting device is produced using a plurality of light emitting diodes within a light mixing cavity formed by surrounding sidewalls. The sidewalls may be integrally formed as part of a surrounding heat sink or alternatively may be an insert into a cavity within a heat sink. The reflective sidewalls may be coated with a diffusing material and/or covered with one or more phosphors. Multiple phosphors are located at different locations of the cavity, e.g., on the sidewalls, a window covering the output port, or on a reflector attached to the bottom of the cavity. The light emitting diodes may be positioned rotationally symmetrically around the optical axis on a board.