Finding starting points analytically for optical system optimization

Understanding the structure of the design space in optical system optimization is difficult, because common human intuition fails when it encounters the challenge of high dimensionality, resulting from the many optimization parameters of lens systems. However, a deep mathematical idea, that critical points structure the properties of the space around them, is fruitful in lens design as well. Here we discuss simple systems, triplets with curvatures as variables, for which the design space is still simple enough to be studied in detail, but complex enough to be non-trivial. A one-to-one correspondence between the possible design shapes and the critical points resulting from a simplified model based on third-order spherical aberration within the framework of thin-lens theory could lead in the future to a new way to determine good starting points for subsequent local optimization.