The application of Liouville’s theorem from statistical mechanics to a distribution of rays emanating from a light source provides a method of visualizing and calculating the maximum efficiency achievable when couping the radiation to a receiver. It is shown that the maximum coupling efficiency depends only on measurable spatial and angular boundary conditions of the source and receiver together with knowledge of the source radiance distribution. Knowledge of the maximum coupling efficiency permits evaluation of any optical system which might be used to effect coupling. A specific example is given for coupling radiation from a light emitting diode to a cylindrical, multimode, optical waveguide.
Marshall C. Hudson, "Calculation of the Maximum Optical Coupling Efficiency into Multimode Optical Waveguides," Appl. Opt. 13, 1029-1033 (1974)