This contribution discusses lateral resolution capabilities of CSI instruments based on the three-dimensional optical transfer function. The 3D transfer function combines the lateral spatial frequency transfer of a measured surface topography with axial spatial frequency contributions corresponding to the fringe frequencies of CSI signals. A broad bandwidth of fringe frequencies contributing to CSI signals can be achieved by both, low temporal coherence of the light source employed, or high numerical apertures, which reduce the longitudinal spatial coherence while enhancing the lateral resolution capabilities of an instrument. We first analyze theoretical aspects of lateral resolution in CSI using the Universal Fourier Optics (UFO) model and then validate our theoretical findings by experimental results using custom-built CSI systems of high NA. In addition, we introduce a CSI microscope equipped with immersion objectives to further improve the lateral resolution. Results based on separated phase irregularities demonstrate that scattering centers can be resolved even if their distance is less than the minimum distance predicted by the Abbe limit.
KEYWORDS: Confocal microscopy, Modeling, Light sources and illumination, Microscopes, 3D modeling, Equipment, Optical surfaces, Electric fields, Objectives, 3D metrology
We present a unified electromagnetic modeling of coherence scanning interferometry, confocal microscopy, and focus variation microscopy as the most common techniques for surface topography inspection with micro- and nanometer resolution. The model aims at analyzing the instrument response and predicting systematic deviations. Since the main focus lies on the modeling of the microscopes, the light–surface interaction is considered, based on the Kirchhoff approximation extended to vectorial imaging theory. However, it can be replaced by rigorous methods without changing the microscope model. We demonstrate that all of the measuring instruments mentioned above can be modeled using the same theory with some adaption to the respective instrument. For validation, simulated results are confirmed by comparison with measurement results.
Various attempts have been discussed to overcome the lateral resolution limit and thus to enlarge the fields of application of optical interference microscopy. Microsphere-assisted microscopy and interferometry have proven that the imaging of structures well below Abbe’s resolution limit through near-field assistance is possible if microspheres are placed on the measured surface and utilized as near-field assisting imaging elements. The enhancement of the numerical aperture (NA) by the microspheres as well as photonic nanojets was identified to explain the resolution enhancement, but also whispering gallery modes and evanescent waves are assumed to have an influence. Up to now, to the best of our knowledge, there is no complete understanding of the underlying mechanisms and no model enabling to examine ideal imaging parameters. This contribution is intended to clarify how much the lateral resolution of an already highly resolving Linnik interferometer equipped with 100 × NA 0.9 objective lenses can be further improved by microspheres. Our simulation model developed so far is based on rigorous near-field calculations combined with the diffraction-limited illumination and imaging process in an interference microscope. Here, we extend the model with respect to microsphere-assisted interference microscopy providing a rigorous simulation of the scattered electric field directly above the sphere. Simulation and experimental results will be compared in the three-dimensional spatial frequency domain and discussed in context with ray-tracing computations to achieve an in-depth understanding of the underlying mechanism of resolution enhancement by the microsphere.
Various attempts have been discussed to overcome the lateral resolution limit and thus to enlarge the fields of application of optical interference microscopy. Microsphere assisted microscopy and interferometry have proven that the imaging of structures well below Abbe’s resolution limit through near-field assistance is possible if microspheres are placed on the measured surface and utilized as near-field assisting imaging elements. The enhancement of the numerical aperture by the microspheres as well as photonic nanojets were identified to explain the resolution enhancement, but also whispering gallery modes and evanescent waves are assumed to have an influence. Up to now, to the best of our knowledge there is no complete understanding of the underlying mechanisms and no model enabling to examine ideal imaging parameters. This contribution is intended to clarify, how much the lateral resolution of an already highly resolving Linnik interferometer equipped with 100x, NA 0.9 objective lenses can be further improved by microspheres. Our simulation model is based on rigorous near-field calculations combined with the diffraction limited illumination and imaging process in an interference microscope. Here, we extend the model with respect to microsphere assisted interference microscopy providing a rigorous simulation of the scattered electric field directly above the sphere. Simulation and experimental results will be compared in the 3D spatial frequency domain and discussed in context with ray-tracing computations in order to achieve an in-depth understanding of the underlying mechanism of resolution enhancement by the mircosphere.
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.
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.
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