1 January 1995 Laser diode to single-mode fiber coupling using an "out-of-plane misalignment" model
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An analytical model has been developed to account for coupling losses between two Gaussian beams with misalignments in different planes. The verification of the model is implemented via the coupling of laser diode light into an optical fiber. A nonzero end-face angle on the fiber tip allows for simple adjustment of misalignment planarity and can be interpreted as independent of traditional longitudinal, transverse, and tilt offsets. The model is intended for applications where variations in coupling efficiency must be accurately evaluated and where the potential for out-of-plane misalignments must be considered. Experimental data are presented for the coupling of light between a laser diode module and several single-mode fibers with different end-face angles. The predicted change in coupling efficiency agrees with theory. However, the absolute coupling efficiency is only 0.71 of that predicted by theory, as is common for initial launch. Inasmuch as the theory presented is for any two Gaussian beams, the absolute coupling efficiency is expected to be more accurate for fiber-to-fiber applications.
Darrin P. Clement, Ulf L. Oesterberg, "Laser diode to single-mode fiber coupling using an "out-of-plane misalignment" model," Optical Engineering 34(1), (1 January 1995). https://doi.org/10.1117/12.188309

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