Intensity, Brightness and Étendue of an Aperture Lamp

Phosphor molecules are excited by ultraviolet light emitted during the steady electrical discharge of the low-pressure gas inside the lamp, and subsequently de-excite via emission of visible light. Some of this light is absorbed and re-emitted by the phosphor. In this problem, assume that there are no losses in this absorption/re-emission. Compare the intensity, brightness and étendue of the light from an ordinary fluorescent lamp (with no reflector and phosphor over the full azimuth) with that of an aperture lamp, with small angular aperture Δφ, with the same power output in the visible light. You may assume that the phosphor surface emits radiation according to Lambert’s cosine law [2]. Note that the principle of this problem applies equally well to a lamp consisting of an array of light-emitting diodes (which also absorb and re-emit light with little loss). Show that the effect of the phosphor/reflector (optical insulator) is to increase the intensity/brightness/temperature of the light inside the lamp (and the light emitted by it) for a given input power. The phosphors are typically metal oxides with a PO4 radical, often with rare-earth elements. The first variant of an aperture lamp may be from 1936 [1]. See, for example, sec. 4.8 of [3]. The principle also applies to an incandescent lamp in which a reflector (mirrorlike or diffuse) directs some of the light back onto the filament. The small size of tungsten filaments makes this a small effect, which apparently went unnoticed until (2000) [4]). Subsequently it was noted that the principle (there called “light recycling”) applies to extended light sources [5], such as LEDs, and that these sources can also serve as diffuse reflectors as in the fluorescent aperture lamp. The effect of the “light recycling” by the phosphor/reflector is similar to that of insulation of the walls of a house in permitting a higher internal temperature for a given heat input.