Zernike polynomials for mid-spatial frequency representation on optical surfaces

摘要

Mid-spatial frequency structure on freeform optical elements induces small-angle scatter and affects performance. Fabrication techniques involved in making freeform surfaces leave tooling marks on the surface due to the sub-aperture nature of the fabrication process. In recent years, there has been a growing need for specification and characterization of the mid-spatial frequencies for freeform surfaces. There are a range of methods to consider for representing the mid-spatial frequency content: the power spectral density (PSD), the structure function (SF) and a polynomial basis representation such as Zernike and Forbes Q-polynomials, as examples. In this paper, we investigate a Zernike polynomial representation for quantifying the mid-spatial frequency content in height maps. We will show fit coefficients of synthesized and real data sets to Zernike polynomials from low orders to very large orders. We also illustrate how this polynomial representation captures certain characteristics of the mid-spatial frequency error. The results are analyzed and compared with Forbes gradient orthogonal polynomials. Finally, limits of Zernike polynomials for representing mid-spatial frequency content of the surface are discussed.