A backlight for a liquid crystal display, the backlight comprising a light emitting device comprising: a single side-emitting, non-lasing light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) having a generally square shape, comprising: a first semiconductor layer of a first type; a second semiconductor layer of a second type; and an active layer between the first semiconductor layer and the second semiconductor layer, the active layer having a major surface; a first electrode in contact with the first semiconductor layer; a second electrode in contact with the second semiconductor layer, the first electrode and the second electrode being on a first side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) for making a direct flip-chip connection to electrodes on a submount; optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) formed as layers over the first semiconductor layer on a second side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) opposite said first side, the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) having sides substantially perpendicular to the major surface of the active layer, wherein a majority of light generated in the active layer is emitted through said sides of the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77); and a reflector (34, 92) over the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) such that substantially all light impinging on the reflector (34, 92) is redirected back into the light-emitting diode and emitted from all sides of the light-emitting diode that are substantially perpendicular to the major surface of the active layer; the backlight further comprising a transparent waveguide having side walls coated with a reflective material, the transparent waveguide being molded to have a square hole, in which said side-emitting, non-lasing light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) is mounted, such that the sides of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) are at approximately 45 degrees relative to the internally reflective surfaces of the waveguide and no side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) is parallel to a reflective surface of the waveguide, for coupling side emission from the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) into the waveguide. a single side-emitting, non-lasing light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) having a generally square shape, comprising: a first semiconductor layer of a first type; a second semiconductor layer of a second type; and an active layer between the first semiconductor layer and the second semiconductor layer, a first semiconductor layer of a first type; a second semiconductor layer of a second type; and an active layer between the first semiconductor layer and the second semiconductor layer, the active layer having a major surface; a first electrode in contact with the first semiconductor layer; a second electrode in contact with the second semiconductor layer, the first electrode and the second electrode being on a first side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) for making a direct flip-chip connection to electrodes on a submount; optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) formed as layers over the first semiconductor layer on a second side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) opposite said first side, the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) having sides substantially perpendicular to the major surface of the active layer, wherein a majority of light generated in the active layer is emitted through said sides of the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77); and a reflector (34, 92) over the optical elements (30, 32, 36, 42, 44, 46, 51, 52, 53, 56, 62, 65, 66, 69, 77) such that substantially all light impinging on the reflector (34, 92) is redirected back into the light-emitting diode and emitted from all sides of the light-emitting diode that are substantially perpendicular to the major surface of the active layer; the backlight further comprising a transparent waveguide having side walls coated with a reflective material, the transparent waveguide being molded to have a square hole, in which said side-emitting, non-lasing light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) is mounted, such that the sides of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) are at approximately 45 degrees relative to the internally reflective surfaces of the waveguide and no side of the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) is parallel to a reflective surface of the waveguide, for coupling side emission from the light emitting diode (10, 40, 50, 60, 64, 68, 82, 90, 98) into the waveguide.