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The depth discrimination in confocal microscopy is based on the digital analysis of depth response signals obtained by each camera pixel during measurement. Various signal processing algorithms are used for this purpose. The accuracy of these algorithms is inter alia restricted by the axial symmetry of the signals. However, in practice response signals are rather asymmetrical especially in case of measurement objects with critical surface structures such as edges or steep flanks. We present a novel signal processing algorithm based on an exponential function with a cubic argument to handle asymmetrical and also symmetrical depth response signals. Results obtained by this algorithm are compared to those of commonly used signal processing algorithms. It turns out that the novel algorithm is more robust, more accurate and exhibits a repeatability of a similar order compared to other algorithms.

Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can be obtained.Optical microscopes are probably the most wide-spread optical instruments in science and technology. While in biological applications microscopes are mostly operated in transmission mode, reflection-type microscopes dominate in materials science. A further important field of application of reflection microscopy is the reconstruction of the three-dimensional (3D) surface topography of an object. This kind of 3D microscopy obtains 3D image stacks by axially scanning through the focus. While in a conventional bright-field microscope point scatterers are necessary to enable height discrimination, interference microscopy even works on specularly reflecting surfaces. In both cases, according to optical systems theory the physical behaviour of the microscope is fully represented by a 3D point spread function in the object space or, equivalently, by the corresponding 3D transfer function in the spatial frequency domain. In order to be consistent with microscopic imaging theory, integrating the 3D transfer function along the axial spatial frequency is required to result in the well-known modulation transfer function of a diffraction limited optical imaging system. Since to the best of our knowledge no formula for the calculation of the 3D transfer function of a microscope operated in reflection mode has been published, we derive a mostly analytical formula, which is perfectly consistent with the modulation transfer function of a diffraction limited system. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can be obtained. Utilizing the rotational symmetry of the 3D transfer function we present a computationally efficient formula for the numerical calculation of the 3D point spread function.

Laser distance sensors are a widespread, fast and contactless approach for distance and surface topography measurements. Main characteristics of those sensors are given by resolution, measurement speed and sensor geometry. With decreasing sensor size, the alignment of the optical components in sensor setup becomes more challenging. The depth response of optical profilers is analyzed to obtain characteristic parameters and, thus, to value the alignment and the transfer behavior of those sensors. We present a novel miniaturized sensor setup comprising of confocal and interferometric confocal signals within one sensor in order to compare both principles simply by obscuring the reference arm by an absorber. Further, we introduce a theoretical signal modeling in order to analyze influences such as spatial coherence, Gaussian beam characteristics and tilted reflectors on depth response signals. In addition to this, we show that the coherent superposition significantly reduces the axial resolution due to the confocal effect in interferometric signals compared to simple confocal signals in measurement and simulation results. Finally, an appropriate fit function is presented, in order to figure out characteristic sensor parameters from the obtained depth response signal. In this context, a good agreement to simulated and measured signals is achieved.

Optical interference microscopes are widespread in topography and roughness measurement on the micro- and nanoscale. In spite of a wide range of scientific and industrial applications, systematic d...

Three-dimensional transfer functions (3D TFs) are generally assumed to fully describe the transfer behavior of optical topography measuring instruments such as coherence scanning interferometers in the spatial frequency domain. Therefore, 3D TFs are supposed to be independent of the surface under investigation resulting in a clear separation of surface properties and transfer characteristics. In this paper, we show that the 3D TF of an interference microscope differs depending on whether the object is specularly reflecting or consists of point scatterers. In addition to the 3D TF of a point scatterer, we will derive an analytical expression for the 3D TF corresponding to specular surfaces and demonstrate this as being most relevant in practical applications of coherence scanning interferometry (CSI). We additionally study the effects of temporal coherence and disclose that in conventional CSI temporal coherence effects dominate. However, narrowband light sources are advantageous if high spatial frequency components of weak phase objects are to be resolved, whereas, for low-frequency phase objects of higher amplitude, the temporal coherence is less affecting. Finally, we present an approach that explains the different transfer characteristics of coherence peak and phase detection in CSI signal analysis.

Although optical 3D topography measurement instruments are widespread, measured profiles suffer from systematic deviations occurring due to the wave characteristics of light. These deviations can be analyzed by numerical simulations. We present a 3D modeling of the image formation of confocal microscopes. For this, the light-surface interaction is simulated using two different rigorous methods, the finite element method and the rigorous coupled-wave analysis. The image formation in the confocal microscope is simulated using a Fourier optics approach. The model provides high accuracy and advantages with respect to the computational effort as a full 3D model is applied to 2D structures and the lateral scanning process of the confocal microscope is considered without repeating the time consuming rigorous simulation of the scattering process. The accuracy of the model is proved considering different deterministic surface structures, which usually cause strong systematic deviations in measurement results. Further, the influences of apodization and a finite pinhole size are demonstrated.

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.

We present an extended vectorial Kirchhoff model of coherence scanning interferometry including several vector rotations occurring in the imagining and scattering process as well as polarization dependent reflection coefficients. For validation simulated results are compared to those of the conventional scalar Kirchhoff model and a rigorous finite element modeling.

Enhancing the lateral resolution limit in optical microscopy and interferometry is of great interest in recent research. In order to laterally resolve structures including feature dimensions below the resolution limit, microspheres applied in the optical near-field of the specimen are shown to locally improve the resolution of the imaging system. Experimental and simulated results following this approach obtained by a high NA Linnik interferometer are analyzed in this contribution. For further understanding of the transfer characteristics, measured interference data are compared with FEM (finite element method) based simulations with respect to the polarization dependency of the relevant image information.

If high numerical apertures are used in coherence scanning interferometry, an extension of the interference signal’s spectral distribution to lower frequencies can be observed. Depending on the slope of the measured surface interference signal contributions belonging to higher frequencies will vanish. In addition, the high spatial frequency information of a measured surface structure will contribute to the low frequency components of the spectrum of the measured interference signals. These effects can be explained by analyzing both the measuring object as well as the transfer characteristics of the interference microscope in the 3D spatial frequency domain. In this study we analyze the mentioned effects based on Kirchhoff’s diffraction theory in the spatial frequency domain introducing the double foil model. The model explains why the choice of the wavelength, which is used for signal analysis, shows a substantial impact on the reconstructed topography. As a consequence, careful analysis of the 3D transfer function based on the Ewald sphere model enables a better understanding of the measuring process, the lateral resolution capabilities, and the improvement of the measurement results by choosing adequate signal processing parameters.