Diffusion studies of anamorphic GRIN lenses

• Diffusion study of cations in fabrication of elliptic cross section gradient index (GRIN) lens by ion exchange is presented.
• The element is a kind of lens of unusual symmetry unlike planer or radial or axial GRIN and having different lens power along two orthogonal directions called ‘Anamorphic GRIN lens’, useful in transforming inherently elliptic beam from laser diode into spherical one.
• The expression of ion concentration in terms of Mathieu functions and its plots are analysed first time.

The present paper reports the diffusion study of cylindrical GRIN rod with elliptical cross section, developed by ion exchange process. The diffusion equation takes the form of Mathieu equations when transform into elliptic coordinate system and the solutions are derived in terms of angular and radial Mathieu functions. Computations of eigenvalues and expansion coefficients as well as angular and radial Mathieu functions are made which shows good agreement with the existing results. Simpler expression for ionic concentration is derived using asymptotic formulae of the functions which are used for final computation of ionic concentration of diffusing cations in elliptic GRIN. The plot of change in concentration versus diffusion depth along different directions approximately correlates with the results obtained by an earlier experimental study.