I show how to simulate the scattering generated by a scratch on the surface of high-quality optics and their elements. This is accomplished by first describing how the present cosmetic scratch standards tend to be used in the optics industry. Second, I derive from first principles, using the scalar model for electromagnetic radiation, the first-order scattering coefficients for the far-field radiation due to a particular scratch pattern. There are approximations made to get these coefficients. The results allow construction of a set of secondary scratch standards. These are a pattern of rectangular grooves that can be made precisely reproducible during the manufacturing phase. Appropriate selection from this set can provide the same range of scattering power and character as is present in the current scratch standards, which are not easily reproducible. Because the method for construction of these new secondary standards is nonrandom, to guarantee the reproducible construction between these standards it is necessary to restrict the observation range 5–10° from the direct beam.
Eric G. Johnson, Jr., "Simulating the scratch standards for optical surfaces: theory," Appl. Opt. 22, 4056-4068 (1983)