An illumination system (100; 300), comprising: a light source (106; 306), wherein said light source comprises at least one light-emitting diode that emits light, wherein said at least one light-emitting diode is further comprised of an emitting layer (108; 308) that emits said light and a reflecting layer (110; 310) that reflects said light, wherein the reflectivity of said reflecting layer (110; 310) is reflectivity Rs, wherein the total light-emitting area of said light source (106; 306) is area As and wherein said light emitted by said light source has a maximum intrinsic source radiance; a light-recycling envelope (102; 302), wherein said light-recycling envelope at least partially encloses said light source (106; 306), wherein said light-recycling envelope (102; 302) has reflectivity Re and wherein said light-recycling envelope reflects part of said light back to said reflecting layer (110; 310) of said at least one light-emitting diode; and at least one light output aperture (104; 304), wherein said at least one light output aperture is located in a surface of said light-recycling envelope (102; 302), wherein the total light output aperture area is area Ao, wherein said area Ao is less than said area As, wherein said light source (106; 306) and said light-recycling envelope (102; 302) direct at least a fraction of said light out of said light-recycling envelope (102; 302) through said at least one light output aperture (104; 304); wherein said reflectivity Rs is at least 50%; said reflectivity Re is at least 50%; said area Ao is less than or equal to 30% of said area As; and wherein said fraction of said light exits said at least one light output aperture as incoherent light having a maximum exiting radiance greater than said maximum intrinsic source radiance. a light source (106; 306), wherein said light source comprises at least one light-emitting diode that emits light, wherein said at least one light-emitting diode is further comprised of an emitting layer (108; 308) that emits said light and a reflecting layer (110; 310) that reflects said light, wherein the reflectivity of said reflecting layer (110; 310) is reflectivity Rs, wherein the total light-emitting area of said light source (106; 306) is area As and wherein said light emitted by said light source has a maximum intrinsic source radiance; a light-recycling envelope (102; 302), wherein said light-recycling envelope at least partially encloses said light source (106; 306), wherein said light-recycling envelope (102; 302) has reflectivity Re and wherein said light-recycling envelope reflects part of said light back to said reflecting layer (110; 310) of said at least one light-emitting diode; and at least one light output aperture (104; 304), wherein said at least one light output aperture is located in a surface of said light-recycling envelope (102; 302), wherein the total light output aperture area is area Ao, wherein said area Ao is less than said area As, wherein said light source (106; 306) and said light-recycling envelope (102; 302) direct at least a fraction of said light out of said light-recycling envelope (102; 302) through said at least one light output aperture (104; 304);