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Describing freeform surfaces with orthogonal functions

In optical design with freeform surfaces descriptions of the surfaces are needed that use only few parameters and are suitable for optimisation. Depending on the merit function – spot size or wavefront error – and the position of the surface in the system, different surface types can yield different optimisation performance. It has been demonstrated by G. Forbes that slope orthogonal polynomials are an advantageous freeform description. From literature on Gaussian moments it is known that this can be achieved using differences of Zernike polynomials, which are easy to compute and implement with recent algorithms. We will demonstrate the benefits of Zernike polynomials with optimisation examples. Furthermore we present an orthogonal surface representation on a rectangular aperture based on Chebyshev polynomials. This description is very convenient when the aperture has a very high aspect ratio, or when designing a system with a rectangular pupil.