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, I choose to 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, ellipical, 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 value of the merit function as we scan say the radius of curvature of the mirror as follows:

we see:

Alternatively, if we want to look at the brightness of the central pixel directly, set up the Universal Plot to look at the NSDD operand directly, as follows:

to get this:

These graphs show why optimizing pure non-sequential systems is harder than sequential systems. In pure non-sequential mode, we detect rays in pixels, and a small change of a parameter may not result in any change of merit function, because the rays still land on the same pixels. For this reason, damped-least-squares by itself may not produce significant system improvements. The Global Search and Hammer optimizers, as they take larger steps, are often more suitable for this kind of optimization.