Analytical Expressions for Power Coupling Coefficients Into Graded-Index Fibers With Generalized Beam Launch Conditions

In this paper, we derive the closed-form analytical expressions of the power coupling coefficients for a Laguerre-Gaussian beam coupled into a graded-index fiber with either central, offset, or tilted launch. We further generalize to an arbitrary launching condition with the radial offset and the three-dimension angular tilt simultaneously occurring. The analytical results are in good agreement with the numerical ones and can be applied to manufactured graded-index multimode fibers with an index exponent parameter from 1.8 to 2.2. Using the derived analytical expressions, the computation time shortens by at least a four-order of magnitude compared to conventional numerical methods.