Light-emitting diode spherical packages: an equation for the light   transmission efficiency
  Virtually all light-emitting diodes (LEDs) are encapsulated with a transparent epoxy or silicone-gel. In this paper we analyze the optical efficiency of spherical encapsulants. We develop a quasi-radiometric equation for the light transmission efficiency, which incorporates some ideas of Monte-Carlo ray tracing into the context of radiometry. The approach includes the extended source nature of the LED chip, and the chip radiance distribution. The equation is an explicit function of the size and the refractive index of the package, and also of several chip parameters such as shape, size, radiance, and location inside the package. To illustrate the use of this equation, we analyze several packaging configurations of practical interest; for example, a hemispherical dome with multiple chips, a flat encapsulation as a special case of the spherical package, and approximate calculations of an encapsulant with a photonic crystal LED or with a photonic quasi crystal LED. These calculations are compared with Monte-Carlo ray-tracing, giving almost identical values. 
Ivan Moreno, David Bermudez, and Maximino Avendano-Alejo
physics.optics