Propagation of light in gradient index media with symmetrical iso-indicial (constant refractive index) contours

Light propagation in non-homogeneous media with varying refractive index is challenging to analyse, even with knowledge of the gradient index (GRIN) structure n(x,y,z). Differential equations can be used to describe the path light rays take when travelling through a medium. Solutions to a differential equation are used to calculate the focal length of a GRIN lens with focus on quadratic GRIN profiles. Ray-tracing in GRIN media is notoriously difficult, and usually not possible without numerical methods, however for several types of GRIN lenses, the analytical solution exists. Examples of spherical GRIN lenses can be found in nature such as the lens in an octopus eye. A formula is proposed here to calculate the focal length of such a lens which will be compared to previous studies and models created on ray-tracing software, ZEMAX.