Geometrical properties of random particles and the extraction of photons from electroluminescent diodes

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Abstract
A confined random-particle flux (such as photons in an electroluminescent diode, integrating sphere, or room, music or other phonons in an auditorium, gas molecules in an imperfect vacuum chamber, neutrons in a reactor, etc.) is analyzed, and simple expressions are shown to exist for the mean path length, transit time, surface transmissivity, projected area, absorption rate, etc., that arise in its description. The results are applied to a theoretical and experimental analysis of the coupling of photons from their source within an indirect-band-gap electroluminescent diode (or within a sufficiently transparent heterostructure direct-band-gap diode) into the ambient or into a multimode optical fiber. We find that about two-thirds of the internally generated photons will reach air from an encapsulated rough-sawed red-emitting GaP(Zn,O) electroluminescent diode of reasonably high purity and contact reflectivity, whereas in a comparable green-emitting GaP(N) diode only about one-fourth of the photons escape while three-fourths are absorbed within the diode bulk and at its contacts.