A light emitting device includes a light emitting diode (LED); a transparent optic having a refractive index noptic; and a phosphor layer spaced apart from the LED and positioned between the LED and the transparent optic. The phosphor layer has an effective refractive index nphosphor, where a gap between the LED and the phosphor layer has a refractive index ngap that is less than nphosphor. The transparent optic has an inner convex surface in contact with the phosphor layer. The inner convex surface has an inner radius of curvature r; and an outer convex surface facing away from the phosphor layer and being a surface through which the light emitting device emits light into a medium adjacent the outer convex surface. The medium has a refractive index nmedium. The outer convex surface has an outer radius of curvature R, such that r/R is equal to nmedium/noptic.