Free-form surfaces are usually described as multiple low-order polynomials such as splines or Bezier curves. They are often used to describe forms such as turbine blades, car bodies and boat hulls.

In optical system design, however it is often helpful to retain the concept of an underlying conicoid section, and to have free-form deviations from that section. The reason for this will be demonstrated soon. For this reason, we will use the Extended Polynomial Surface object. This surface is described by an equation of the form:



The first term is the standard conic asphere beloved of optical design, and is used for designing spherical, elliptical, parabolic, hyperboloid, etc. mirrors. The second term represents a series of increasingly high-order polynomial deformations from this surface. The polynomials are a power series in x and y. The first term is x, then y, then x*x, x*y, y*y, etc. There are 2 terms of order 1, 3 terms of order 2, 4 terms of order 3, etc. The maximum order is 20, which makes a maximum of 230 polynomial aspheric coefficients. The position values x and y are divided by a normalization radius so the polynomial coefficients are dimensionless.

In this design the maximum order of the polynomial is limited to 20 terms, so the highest freeform deviation goes as x0y5 and x5y0. This is neither necessary nor a recommendation: it was just a choice made during the design process.

Now if we use the Universal Plot to show the central pixel intensity as we scan say the radius of curvature of the mirror as follows:



we see:



This graph illustrates both the difficulty in optimizing NS systems and the need for a well defined merit function. If we look at the overall merit function value versus base radius, we can see why centroid and spot radius are better optimization targets.





Now that our merit function properly defines our design criteria, we will compare the results of optimization with DLS and OD using both local and global optimization algorithms.