The structure function as a metric for roughness and figure

Abstract

As optical designs become more sophisticated and incorporate aspheric and free form surfaces, the need to specify limits on mid-spatial frequency manufacturing errors becomes more critical, particularly as we better understand the effects of these errors on image quality. While there already exist methods based on Fourier analysis to specify these errors in most commercial interferometry software, the method of calculation and the power spectral density (PSD) results remain obscure to many in the optical design and manufacturing field. We suggest that the structure functions (SF) contains the same information as in the Fourier based PSD but in a way that is much more transparent to analysis, interpretation and application as a specification. The units of measure are more familiar and the concept behind the analysis is simpler to understand. Further, the information contained in the structure function (or PSD) allows a complete specification of an optical surface from the finest measurable detail of roughness to the overall figure. We discuss the origin of the structure function in the field of astronomy to describe the effects of air turbulence on image quality, the simple mathematical definition of the structure function and its easy means of calculation and how its results should be scaled depending on the location of the optical surface in a system from pupil to image plane. Finally, we give an example of how to write a specification of an optical surface using the structure function.