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The coupling of a spherical-surfaced LED and a spherical-ended fiber is analyzed using a model in which a light source and an aperture stop are on the anterior and posterior focal planes, respectively. The launching efficiency is defined as the ratio of the power launched into a fiber to the optical power generated in the LED. The efficiencies for step- and graded-index fibers are derived. Maximum efficiency can be obtained when d0=dc, where d0 is the emitting diameter and dc is the critical diameter defined by the focal length and the critical angle of the fiber. When d0?dc, the ratio χ of improvement in the efficiency for a spherical-surfaced LED to that for a flat LED is approximately proportional to the square of the lateral magnification β, which is the image diameter divided by d0. χ?β2 is confirmed experimentally with couplings using spherical-surfaced and flat LED’s.
Hasegawa et al. (1980) studied this question.