# Design of a three freeform mirror aplanat

Freeform optical surfaces have been in much demand recently due to improved techniques in their manufacturability and design methodology, and the degrees of freedom it gives the designers. Specifically in the case of off-axis mirror systems, freeform surfaces can considerably reduce the number of surfaces and compensate for some of the higher order aberrations as well, which improves the overall system performance. In this paper, we explore the design of freeform surfaces to obtain full aplanatic mirror systems, i.e., free of spherical aberration and circular coma of all orders. It is well know that such a system must be stigmatic and satisfy the Abbe sine condition. This problem is well known (Schwarzschild, 1905) to be solvable with two aspheric when the system has rotational symmetry. Here we prove that a rigorous solution to the general non-symmetric problem needs at least three free form surfaces, which are solutions of a system of partial differential equations. The examples considered have one plane of symmetry, where a consistent 2D solution is used as boundary condition for the 3D problem. We have used the x-y polynomial representations for all the surfaces used, and the iterative algorithm formulated for solving the above mentioned partial differential equations has shown very fast convergence.