The gradient index (GRIN) lens law for Gaussian beams is derived by matrix formalism. By use of this law, image and focal shifts are evaluated. The results are applied to Selfoc lenses to calculate positioning errors when GRIN connectors are inserted into optical communication systems.
J R Flores,
"Gradient Index Lens Law For Gaussian Illumination: Image And Focal Shifts," Optical Engineering 28(11), 281168 (1 November 1989). https://doi.org/10.1117/12.7977114