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As the requirements concerning lateral resolution, robustness and reliability continuously increase, appropri- ate modeling of coherence scanning interferometry (CSI) as an important method prerequisite of virtual CSI instruments gains in importance. We recently published the so-called double foil model that is based on the three-dimensional (3D) spatial frequency representation of interference signals. The model is consistent with Kirchhoff’s diffraction theory applied to surface reflection and scattering. Scattered light contributions belong- ing to certain plane wave components of incident light are superimposed incoherently. For an instrument of given numerical aperture the maximum lateral resolution provided by the diffraction limit and the capability of measuring steep surface slopes are closely related to the evaluation wavelength, i.e. the wavelength, at which the interference phase is analyzed. In this contribution we extend the model in order to describe the complete measuring process including the depth scan. Our approach introduces 3D representations of both, the surface under investigation as well as the reference mirror as thin foils in cartesian coordinates. Interference is shown to occur after Fourier transformation with respect to the axial coordinate z in the hybrid xyqz coordinate system, where the surface under investigation is treated as a phase object. Consequently, an axial shift of the measurement object or the reference mirror results in different phase shifts of the monochromatic interference patterns depending on the angle of incidence and the scattering angle. Our study combines theoretical considerations and simulations with exemplary experimental results. Conclusions are drawn with respect to signal filtering and analysis aiming at high topography fidelity of CSI systems.

Coherence scanning interferometry is one of the most frequently used techniques for optical profiling due to its outstanding axial resolution. However, optical profilers suffer from systematic deviations caused by their transfer characteristics and diffraction effects occurring by means of light-surface interaction with measurement objects. In order to predict these deviations and to get better insight into the physical effects leading to their appearance, analytical and rigorous numerical models are applied. Usually, rigorous models provide higher accuracy whereas analytical models require less computational effort since the light-surface interaction is considered by a phase object approximation. We present a full vectorial three-dimensional modeling of coherence scanning interferometry based on the phase object approximation. Further, we compare three different common approaches using the phase object approximation, usually called Richards Wolf model, Foilmodel and Kirchhoff model. The comparison is validated with respect to rigorously simulated and measured results shown elsewhere.

Model based approaches in dimensional metrology have great potential in terms of better accuracy. In some cases they may even help to overcome classical resolution criteria. A famous example is optical scatterometry for measuring critical dimensions on semiconductor chips in the tenth-nanometer range. Basically, these techniques rely on the solution of the inverse problem, i.e. retrieve the measured profile from a signal, e.g. a spectrum in optical scatterometry. Here, the appropriateness and accuracy of the model is of great importance to achieve the goals of quantitative metrology. On the other hand, the implemented models shall enable fast turnaround cycles for practical applications. Thus, long computation times and huge memory consumption can hardly be accepted. We investigate and compare different scenarios of model complexity and rigorousness related to application in two likewise well-established optical profiling techniques, laser focus scanning (LFS) and coherence scanning interferometry (CSI). Especially, two electromagnetic diffraction methods are considered, Finite Element Method (FEM) and Modal Methods (MM). In general, rigorous methods are rather expensive and time consuming compared to methods based on analytical approximations. Particularly, full 3D models require huge efforts and computing resources. Thus, some alternatives shall be discussed in this paper such as reduction of dimensionality and various other methods for acceleration such as symmetry usage and for MM. Moreover, the different approaches are compared and conclusions are drawn with respect to their practical applicability in both, CSI as well as LFS.