Understanding the effects of groove structures on the MTF

We develop a theory for understanding and modeling the effects of mid-spatial frequency (MSF) structures on the modulation transfer function (MTF) of an imaging system. We show that the MTF is related through Fourier transformation to what we call the pupil-difference probability density (PDPD) function. This relation is illustrated for several one-dimensional periodic groove patterns. We also consider two-dimensional pupils, particularly those presenting straight and circular concentric periodic surface errors, similar to those resulting from diamond turning and milling processes of freeform optical manufacturing, and develop simple approximations for the MTF based on the PDPD. Our approach provides an analytic way to understand the effects of tool profiles, as well as other freeform manufacturing parameters.