The power launching into multimode optical fibers with any profile of refractive index is analyzed by means of the geometrical optics. First two formulas of the numerical aperture are obtained, one including and the other excluding the contribution of leaky skew rays. Each one of these two numerical apertures has a certain domain of application, and the launching efficiencies from incoherent sources into optical fibers are calculated in both cases. Useful formulas and plots of launching efficiency for a few refractive index profiles are given. Finally the power losses due to source-fiber coupling errors (separation, displacement, or misalignment) are evaluated, and an interesting effect of virtual lengthening of the fiber is found.
P. Di Vita and R. Vannucci, "Multimode optical waveguides with graded refractive index: theory of power launching," Appl. Opt. 15, 2765-2772 (1976)