An inverse and analytic lens design method
  Traditional lens design is a numerical and forward process based on ray tracing and aberration theory. This method has limitations because the initial configuration of the lens has to be specified and the aberrations of the lenses have to considered. This paper is an initial attempt to investigate an analytic and inverse lens design method, called Lagrange, to overcome these barriers. Lagrange method tries to build differential equations in terms of the system parameters and the system input and output (object and image). The generalized Snell's law in three dimensional space and the normal of a surface in fundamental differential geometry are applied. Based on the Lagrange method equations for a single surface system are derived which can perfectly image a point object. 
Yang Lu and Vasudevan Lakshminarayanan
physics.optics