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The use of the Weierstrass sphere geometry in luminescence spectroscopy

Calculations are presented for the photon flux that may be focused into the entrance slit of a monochromator at the correct f-number from a sample shaped as a parallel slab and from a sample shaped as a Weierstrass sphere (a sphere truncated at the inner aplanatic radius). When various experimental factors are taken into account it is shown that, for materials with refractive indices between 3 and 4, the flux gain factor that may be achieved by using a Weierstrass sphere is around 10 for cathodoluminescence measurements and around 20 for photoluminescence measurements.