Geometrical modelling of scanning probe microscopes and characterization of errors

Scanning probe microscopes (SPMs) allow quantitative evaluation of surface topography with ultra-high resolution, as a result of accurate actuation combined with the sharpness of tips. SPMs measure sequentially, by scanning surfaces in a raster fashion: topography maps commonly consist of data sets ideally reported in an orthonormal rectilinear Cartesian coordinate system. However, due to scanning errors and measurement distortions, the measurement process is far from the ideal Cartesian condition. The paper addresses geometrical modelling of the scanning system dynamics, presenting a mathematical model which describes the surface metric x-, y- and z- coordinates as a function of the measured x'-, y'- and z'-coordinates respectively. The complete mathematical model provides a relevant contribution to characterization and calibration, and ultimately to traceability, of SPMs, when applied for quantitative characterization.