title: Focus variation microscope: linear theory and surface tilt sensitivity.
abstract: In a recent publication [3rd International Conference on Surface Metrology, Annecy, France, 2012, p. 1] it was shown that surface roughness measurements made using a focus variation microscope (FVM) are influenced by surface tilt. The effect appears to be most significant when the surface has microscale roughness (Ra≈50  nm) that is sufficient to provide a diffusely scattered signal that is comparable in magnitude to the specular component. This paper explores, from first principles, image formation using the focus variation method. With the assumption of incoherent scattering, it is shown that the process is linear and the 3D point spread characteristics and transfer characteristics of the instrument are well defined. It is argued that for the case of microscale roughness and through the objective illumination, the assumption of incoherence cannot be justified and more rigorous analysis is required. Using a foil model of surface scattering, the images that are recorded by a FVM have been calculated. It is shown that for the case of through-the-objective illumination at small tilt angles, the signal quality is degraded in a systematic manner. This is attributed to the mixing of specular and diffusely reflected components and leads to an asymmetry in the k-space representation of the output signals. It is shown that by using extra-aperture illumination or tilt angles greater than the acceptance angle of aperture (such that the specular component is lost), the incoherent assumption can be justified once again. The work highlights the importance of using ring-light illumination and/or polarizing optics, which are often available as options on commercial instruments, as a means to mitigate or prevent these effects.
authors.forename: Nikolay
authors.initials: N
authors.lastname: Nikolaev
authors.forename: Jon
authors.initials: J
authors.lastname: Petzing
authors.forename: Jeremy
authors.initials: J
authors.lastname: Coupland