3D modeling of coherence scanning interferometry on 2D surfaces using FEM

Effect of boundary conditions in modeling of microsphere-assisted imaging

Exploring the performance of label-free imaging relies heavily on adequate physical models and accurate numerical simulations. A particularly challenging situation is imaging through contact microspheres, which have demonstrated resolution values exceeding the diffraction limit. Here an ab initio modeling of microsphere-assisted imaging is reported and its results are analyzed. The key part of modeling is solving the light scattering problem, which requires handling a rather large computational domain and broad angle illumination made up of multiple mutually incoherent plane waves. To account for plane wave incidence, two simulation approaches are developed that differ only by boundary conditions-quasiperiodic and absorbing. The algorithms to find images in both approaches are discussed and the simulation results are compared for free space and microsphere-assisted imaging. It is shown that while the super-resolution in microsphere-assisted imaging can be demonstrated using both approaches, the latter allows a large reduction in the computational resources. This significantly extends the capability of the simulations, enabling a rigorous exploration of novel imaging regimes.

Ab initio simulation of imaging of wavelength-sized objects and estimation of resolution

Image characterization in microscopy, in particular, the estimation of its resolution, requires detailed knowledge of its relation to the object. For objects with sizes comparable to or smaller than the operating wavelength, such a relation can be obtained only by considering electromagnetic scattering described by the Maxwell equations. Here we follow precisely the steps involved in the image formation in microscopy with broad angle illumination-starting from the Maxwell equations to find the scattered far fields for each plane wave, projecting them into a sensor array, and finally assembling the incoherent image by adding all coherent contributions. We consider a classical object-a narrow slit in an absorbing screen, which is taken as a very thin chromium film deposited on a glass substrate. The inapplicability of the Kirchhoff approximation for such a slit is addressed, and the calculated image is subsequently analyzed to evaluate its intrinsic resolution using a point spread function. The difference in image intensities defined using the Poynting vector and the electric field intensity is also discussed.

Dark field imaging of high aspect ratio structures - a simple model

A simplified model for dark-field optical imaging of three-dimensional high aspect ratio micro- and nano- structures is proposed, to reduce the time taken to simulate object fields with in-plane scattering between different parts of the object. Primary scattering is found by assuming that illumination of Manhattan geometries generates a set of spherical edge waves, following the incremental theory of diffraction. Secondary scattering is found by assuming that primary scattering is re-scattered from nearby features. Diffraction coefficients are simplified, and the number of illuminating beams is limited to those generating waves that enter the objective lens. Images obtained using TE and TM polarizations are compared, and results are benchmarked against a vectorial finite element model. Applications lie in simulating optical inspection of structures containing vertically etched features including MEMS and NEMS.

Simulation-driven machine learning approach for high-speed correction of slope-dependent error in coherence scanning interferometry

Slope-dependent error often occurs in the coherence scanning interferometry (CSI) measurement of functional engineering surfaces with complex geometries. Previous studies have shown that these errors can be corrected through the characterization and phase inversion of the instrument's three-dimensional (3D) surface transfer function. However, since CSI instrument is usually not completely shift-invariant, the 3D surface transfer function characterization and correction must be repeated for different regions of the full field of view, resulting in a long computational process and a reduction of measurement efficiency. In this work, we introduce a machine learning approach based on a deep neural network that is trainable for slope-dependent error correction in CSI. Our method leverages a deep neural network to directly learn errors characteristics from simulated surface measurements provided by a previously validated physics-based virtual CSI method. The experimental results demonstrate that the trained network is capable of correcting the surface height map with 1024 × 1024 sampling points within 0.1 seconds, covering a 178 µm field of view. The accuracy is comparable to the previous phase inversion approach while the new method is two orders of magnitude faster under the same computational condition.

3D full-wave multi-scattering forward solver for coherent microscopes

A rigorous forward model solver for conventional coherent microscope is presented. The forward model is derived from Maxwell's equations and models the wave behaviour of light matter interaction. Vectorial waves and multiple-scattering effect are considered in this model. Scattered field can be calculated with given distribution of the refractive index of the biological sample. Bright field images can be obtained by combining the scattered field and reflected illumination, and experimental validation is included. Insights into the utility of the full-wave multi-scattering (FWMS) solver and comparison with the conventional Born approximation based solver are presented. The model is also generalizable to the other forms of label-free coherent microscopes, such as quantitative phase microscope and dark-field microscope.

Dielectric microspheres enhance microscopy resolution mainly due to increasing the effective numerical aperture

Microsphere-assisted microscopy utilizing a microsphere in immediate proximity of the specimen boosts the imaging resolution mainly as a result of an increase in the effective numerical aperture of the system.

Polarization dependency of the 3D transfer behavior in microsphere enhanced interferometry

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.

Surface measuring coherence scanning interferometry beyond the specular reflection limit

The capability of optical surface topography measurement methods for measurement of steep and tilted surfaces is investigated through modelling of a coherence scanning interferometer. Of particular interest is the effect on the interference signal and measured topography when tilting the object at angles larger than the numerical aperture slope limit (i.e. the specular reflection limit) of the instrument. Here we use theoretical modelling to predict the results across a range of tilt angles for a blazed diffraction grating. The theoretically predicted interference patterns and surface height measurements are then verified directly with experimental measurements. Results illustrate the capabilities, limitations and modelling methods for interferometers to measure beyond the specular reflection limit.

Matched coordinates for the analysis of 1D gratings

The Fourier modal method (FMM) is certainly one of the most popular and general methods for the modeling of diffraction gratings. However, for non-lamellar gratings it is associated with a staircase approximation of the profile, leading to poor convergence rate for metallic gratings in TM polarization. One way to overcome this weakness of the FMM is the use of the fast Fourier factorization (FFF) first derived for the differential method. That approach relies on the definition of normal and tangential vectors to the profile. Instead, we introduce a coordinate system that matches laterally the profile and solve the covariant Maxwell's equations in the new coordinate system, hence the name matched coordinate method (MCM). Comparison of efficiencies computed with MCM with other data from the literature validates the method.