Engineered functional surfaces often feature varying slopes on macro- and micro-scales. When surfaces are mirror-like, the highest surface slope that can be measured by a far-field 3D imaging optical surface measuring instrument isthe arcsine of the numerical aperture (NA) of the objective lens, i.e. the acceptance angle of the lens. However, progress in instrument design has allowed for measurement of non-specular surfaces with slopes steeper than this “traditional” NA limit. Nonetheless, there is currently a lack of understanding about the instrument response to surfaces with steep slopes beyond this limit. It is unclear over what surface spatial frequencies we can expect to accurately report fine surface-feature details. Here we present results demonstrating the capability of a commercial coherence scanning interferometer for measuring surface topography of a roughened flat and a blazed grating with tilt angles greater than the NA slope limit. We show that the surface form, i.e. the tilted plane, can be measured correctly. But, while surface texture information that can appear useful is also obtained, tilting significantly influences the measurement accuracy of micro-scale texture, and for asymmetric gratings, can depend on the tilting direction. A simplified surface scattering model suggests that the loss of scattered power captured by the instrument and a low signal-to-noise ratio causes the reduction of measurement accuracy. However, a rigorous three-dimensional instrument model is needed for a full understanding; we will develop this in our future work.
Coherence scanning interferometry (CSI), a type of interference microscopy, has found broad applications in the advanced manufacturing industry, providing high-accuracy surface topography measurement. Enhancement of the metrological capability of CSI for complex surfaces, such as those featuring high slopes and spatial frequencies and high aspect-ratio structures, requires advances in modeling of CSI. However, current linear CSI models relying on approximate surface scattering models cannot accurately predict the instrument response for surfaces with complex geometries that cause multiple scattering. A boundary elements method is used as a rigorous scattering model to calculate the scattered field at a distant boundary. Then, the CSI signal is calculated by considering the holographic recording and reconstruction of the scattered field. Through this approach, the optical response of a CSI system can be predicted for almost any arbitrary surface geometry.
Surfaces featuring complex topographies, such as high slope angles, large curvatures and high aspect-ratio structures on both macro- and micro-scales, present significant challenges to optical measuring instruments. Here we demonstrate a method to characterise and correct the three-dimensional surface transfer function (3D STF) of a coherence scanning interferometer (CSI). Slope-dependent errors present in the original measurements are reduced after phase inversion of the 3D STF, and the final results agree with traceable contact stylus measurements within the 30 nm reproducibility of the stylus measurements. This method enables in-situ compensation for errors related to aberrations, defocus and diffraction.
Coherence scanning interferometry (CSI) is a well-established technique for measuring surface topography based on the coherence envelope and phase of interference fringes. The most commonly used surface reconstruction methods, i.e. frequency domain analysis, the envelope detection method, and the correlogram correlation method, obtain the phase of the measured field for each pixel and, from this obtain the surface height, by assuming the two are directly proportional. For surfaces with minor deviations from a plane, it is straightforward to show that the scattered field’s phase is a linear function of surface height. An alternative approach known as the “foil model” gives more generally the scattered field as the result of a linear filtering process operating on a “foil” representation of the surface. This model assumes that the surface slowly varies on the optical scale and that there is no multiple scattering. However, for surfaces that are rough at the optical scale or have coherent features (e.g. vee-grooves), the effect of multiple scattering cannot be neglected and remains a problem for reconstruction methods. Linear reconstruction methods cannot provide accurate surface topographies for complex surfaces, since for such surfaces, the measurement process of CSI is fundamentally non-linear. To develop an advanced reconstruction method for CSI, an accurate model of the imaging process is required. In this paper, a boundary elements method is used as a rigorous scattering model to calculate the scattered field at a distant boundary. Then, the CSI signal is calculated by considering the image formation as back-propagation of the scattered field, combined with the reflected reference field. Through this approach, the optical response of a CSI system can be predicted rigorously for almost any arbitrary surface geometry. Future work will include a comprehensive experimental verification of this model, and development of the non-linear surface reconstruction algorithm.
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