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.

Aspheric and freeform surfaces are becoming more common as optical designs become more sophisticated and new generations of fabrication tools reduce cost. Unlike spherical surfaces, these surfaces are fabricated with processes that leave a signature or "structure" that is primarily in the mid-spatial frequency region. Tolerancing aspheric and freeform surfaces requires attention to both surface form and structured mid-spatial frequency fabrication errors. These structured surface errors are shown to create image artifacts such as ghosts, and ripples in the MTF profile. Spatial frequencies beyond "form" errors are often ignored or are modeled with statistical descriptors, which do not account for structured errors.This work explores and develops the theory to describe these errors without statistical assumptions. The analytic source of these artifacts in the image Point Spread Function and the Modulation Transfer Function are compared with computational models. The magnitudes of the image artifacts arising from structured surface errors are shown to be non-linear with surface height. It is also shown that multiple structured surface frequencies mix to create sum and difference diffraction orders that are not present in statistical models.An algorithm is developed that enables an optical designer to determine the important spatial frequencies and magnitudes of allowable errors given an MTF performance budget.