Modeling and determination of 3D transfer functions of high-resolution 3D optical microscopes for surface topography measurement

• Applicant: Professor Dr.-Ing. Peter Lehmann
• Subject Area: Measurement Systems
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 510953418

Transfer functions characterize the properties of optical imaging systems in the spatial frequency domain. Depth-scanning 3D microscopes such as interference, confocal and focus variation microscopes record a series of microscopic images at different focus positions in order to reconstruct the depth information of a surface. Thus, the three-dimensional frequency representation of such an image series includes not only the transversal spatial frequency components of the images but also axial spatial frequency coefficients. The complete characterization of optical 3D microscopes in the spatial frequency domain not only leads to the axial and lateral resolution capabilities but also enables the simulation of intensity output signals and the reconstruction of surface topography. Therefore, 3D transfer functions build a substantial requirement for understanding physical mechanisms. In addition, determining the 3D transfer function of an actual measuring instrument enables the reduction of systematic measurement deviations, e. g. caused by optical aberrations, in practice. This is achieved by a filter applied in the 3D spatial frequency domain, providing a series of filtered images corresponding to the result of an ideal aberration-free system. Nevertheless, in the application of this concept scientific issues occur that demand for elaborate research, which will be addressed in this proposal. So far, the Kirchhoff or physical optics approximation is the theoretical basis on which 3D transfer functions are defined. This requires the local minimum radii of curvature of the surface’s micro-topography to be much greater than the wavelength of light. In addition, the reflectivity of the measured surface is assumed to be independent of the incidence angle and the polarization dependence of reflection and scattering processes are neglected. Recent methods to determine 3D transfer functions experimentally require the measurement of the scattered light intensity using reflective micro-spheres with diameters of 40-110 µm. This leads to low intensity values on the one and to systematic deviations of the measured transfer function compared to the transfer function that takes effect, if diffractive structures on a plane substrate are measured, on the other hand. Preparatory investigations show that 3D transfer functions can be obtained from a plane mirror under different tilt angles with respect to the optical axis. However, this method is time-consuming so that the suitability of available surface calibration standards needs to be investigated. Besides angle and polarization-dependent reflection coefficients also edge and single point scattering on a surface will be considered by appropriate rigorous FEM computations of scattered light fields. Based on 3D transfer function systematic measurement deviations can be understood and both, signal processing algorithms as well as hardware instrumentation will be improved.

• DFG Programme: Research Grants