Angle spread function degradation by surface roughness in imaging optics

Abstract

A formula is derived for the total width of the angle spread function in an imaging optical system, the performance of which is degraded by surface roughness. The derivation is done in the framework of the scalar theory of diffraction, but it does not rely on the small roughness approximation. The contribution of scattering from surface roughness to the total width of the angle spread function is found to depend only on the variance of the roughness slope. It is also shown that the common rule of obtaining the total width of the angle spread function as a square sum of the pure scattering component and the pure specular component is actually incorrect and does not fully account for the contribution of the specular component.

Roughness scattering in X-ray grazing-incidence telescopes

“…The relation between the half-energy width in grazing-incidence telescopes and the roughness variance was studied in (Spiga 2007) by applying the Debye-Waller expression for the total integrated scattering (Spiller 1994) in 1-dimensional geometry. In the framework of the scalar theory of diffraction (Harvey 1995b;Goodman 1968;Gaskill 1978;Born & Wolf 1980), an exact derivation of the rms width of the angle spread function from the corresponding optical transfer function (Goodman 1968;Gaskill 1978;Born & Wolf 1980;Smith 1963;Harvey et al 1988) in 2-dimensional geometry was reported in (Zocchi 2009), in which the contribution of scattering to the rms width of the angle spread function was shown to depend on only the variance of the roughness slope. However, even in these two papers (Spiga 2007;Zocchi 2009), the cylindrical symmetry of an X-ray telescope was not taken into account.…”