Modeling Aspects in Optical Metrology IX

The coherence function offers new possibilities for optical metrology not available with conventional wave field sensing. Its measurement involves spatio-temporal sampling of wave fields modulated by the object under investigation. While the evaluation of the coherence function is more elaborate than conventional approaches, an information theoretical treatment shows that it also delivers more information about the object under investigation. In order to achieve an efficient information extraction from the coherence function, advanced approaches involving compressed sensing are required to obtain an optical metrology technique that is simultaneously precise, robust and fast as well as suited for complex measurement situations.

The development of non-destructive optical metrology methods that provide spatially resolved information on defects and inhomogeneities is crucial for multiple industries that rely on high quality semiconductor materials and devices. In this work, we present how a compressed sensing approach can benefit different optical metrology techniques and the principles of how to adopt and implement a compressed sensing optical system in practice for semiconductor metrology. As examples, we demonstrate a compressed sensing current mapping technique for semiconductor devices with megapixel resolution, and a compressed sensing spectral photoluminescence imaging system.

Nanoscale IR imaging is an emerging tool for the characterization of micro- and nanostructures. However, their quantitative characterization requires hyperspectral imaging, where at each point in space a complete spectrum is recorded. As hyperspectral nano-IR imaging is based on the combination of the optical near-field with an AFM, it is inherently recorded serially. This severely limits its applicability due to the long acquisition times involved and accompanied stability issues. In addition, industrial applications are limited due to these issues. In this work we implement a subsampling strategy [1] using a commercial nano-FTIR system to significantly reduce the measurement time in hyperspectral imaging measurements by compressing the measurements combined with a low-rank matrix reconstruction. We apply this scheme to materials from the field of power electronics, where the ongoing development of wide-bandgap compound semiconductors is limited by material defects which IR imaging is sensitive to. [1] Metzner, S., Kästner, B., Marschall, M., Wübbeler, G., Wundrack, S., Bakin, A., Hoehl, A., Rühl, E., & Elster, C. (2022). Assessment of Subsampling Schemes for Compressive Nano-FTIR Imaging. IEEE Transactions on Instrumentation and Measurement, 71, 1–8. https://doi.org/10.1109/TIM.2022.3204072

Continued improvement of production-scalable characterisation methods is necessary to support the growth of semiconductor industries. In this work we present the application of compressed sensing for photoluminescence imaging in the temporal and spectral domains. The application can be enabled by using a digital micromirror device to programmatically control the spatial information of the excitation or detection source, allowing the use of single-point detectors for imaging applications, with benefits in terms of reduced measurement time and dark noise. We present the methodology for successful compressed sensing acquisition and reconstruction of spectral and temporal photoluminescence signals, developed through computational modelling work.

Parametric sensitivity analysis (SA) is as an essential prerequisite step in optical metrology data modeling to quantify the relative importance of optical model parameters and to identify those with little influence in order to simplify a model. The Morris method, one of the variants of a global sensitivity analysis technique, varies m model parameters along multiple “trajectories” in the m-dimensional parameter space. However, it is reputed that the method relies heavily on a way how these trajectories are spread over the space to cover properly the entire realistic ranges of all model parameters. The present study investigates how various sampling strategies, i.e., the ways to select “optimized trajectories” in the parameter space, influence various sensitivity indices in spectroscopic ellipsometry data modeling.

Mueller ellipsometry is one of the most sensitive techniques for accurate measurements of sub-wavelength structures and is an attractive upgrade compared to simple intensity-based scatterometry. In conventional Mueller ellipsometry, the first Mueller matrix element is unknown and the remaining 15 Mueller elements are normalized to the first element. Here we show that knowledge of the first Mueller element, which can be acquired from a scatterometry measurement, greatly enhances the measurement accuracy. Comparing results obtained by the normalized and non-normalized method for different silicon line gratings including sub-wavelength pitches, we demonstrate the advantage of including the first Mueller element.

Nanowire structures can be used for energy harvesting from renewable sources. These pillar structures with lateral dimensions in the nanometre range offer several advantages due to their small physical size and their large surface-to-volume ratio. The required nanodimensional characterization can be achieved efficiently by optical methods such as Mueller matrix ellipsometry. Here the measured Mueller matrices need to be analysed with numerical simulations to reconstruct the structural parameters of interest. We present the measurement method and measurement results on silicon nanowire test structures as well as the simulations on exemplary structures and a discussion of the results.

Model-based Fourier scatterometry is a powerful non-destructive measuring technique widely used for retrieving features on wafers[1]. It serves as an attractive option for quality control and process monitoring in semiconductor manufacturing, especially for measuring even the smallest features on semiconductor devices. However, the increasing complexity of 3D nanoscale device structures presents significant challenges for scatterometry. Consequently, it becomes necessary to integrate different methods for measuring systems, modeling, and data analysis. To address these challenges, the integration of machine learning algorithms and their ecosystems offers promising enhancements in measurement accuracy and speed. In this report, we analyze two approaches for grating reconstruction: the traditional iterative method, which uses rigorous coupled wave analysis (RCWA) method as the forward simulation tool, and the deep learning method. Our analysis demonstrates that under proper constrains with low-resolution images in the Fourier plane, the Levenberg-Marquardt algorithm can yield satisfactory reconstruction results with a few iterations for four or five grating parameters. We also construct a convolutional network for image processing with Resnet 18 as the main component[3]. With relatively small size of image in the Fourier plane, grating parameters with high precision are able to be reconstructed with noise present. Nevertheless, a substantial dataset needs to be prepared in advance. We also perform comparative analysis concerning the input datasets, the Fourier plane image and the Fourier-plane Mueller Matrices, which converges differently due to their sensitivity to different structural parameters. As an outlook, we propose combination strategy of the two methods to effectively create a more powerful modeling system for practical applications. [1] Maria Laura Gödecke, Karsten Frenner and Wolfgang Osten, “Model-based characterisation of complex periodic nanostructures by white-light Mueller-matrix Fourier scatterometry”, Light: Advanced Manufacturing 2, 18 (2021) [2] H. Gavin, “The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems”, Mathematics, (2013). https://people.duke.edu/~hpgavin/ExperimentalSystems/lm.pdf [3] https://arxiv.org/pdf/1512.03385.pdf

Optical metrology faces significant challenges as functional devices continue to shrink in size due to new patterning processes for semiconductor chips. Consequently, there is a growing interest in modeling optical systems to achieve more accurate measurements and to compare measurements from different optical instruments, such as confocal microscopes, white light interference microscopes, and focus-varied microscopes. Previous models have employed either a thin layer approximation or 2D periodic structures to simulate light scattering. However, to accurately simulate more complex structures and compare them with experimental data, there is a need for a physically accurate modeling and simulation tool that can handle large-scale nonperiodic 3D surfaces. To address this need, we have developed a simulation tool called SpeckleSim, which utilizes the boundary element method[1]. By incorporating a multi-level fast multiple method, we are able to calculate light scattering from 3D nanostructures within a reasonable timeframe. In this report, we adapt the method to a confocal microscopy model and investigate the extent to which it can reproduce surface profiles for various types of structures. The obtained results will be compared with experimental measurements and the results from other rigorous simulation tools such as rigorous coupled wave analysis (RCWA) and finite element method (FEM) from our project partners. [1] Liwei Fu, Max Daiber-Huppert, Karsten Frenner, Wolfgang Osten, “Simulation of realistic speckle fields by using surface integral equation and multi-level fast multipole method”, Optics and Lasers in Engineering 162, 107438 (2023)

Imaging coordinate measuring machines (CMMs) are widely used because of their ability to perform non-contact and high-precision coordinate measurement. Imaging CMMs measures not only the coordinate values but also forms of complex structures. However, the accuracy evaluation of the complex form measurement is not sufficient. ISO10360-7 defines the method for evaluating the accuracy of imaging CMMs. In this ISO, the length measurement errors are to be evaluated using a calibrated line scale, and the probing errors are to be evaluated using a calibrated roundness standard. The evaluation of probing errors is important for the accuracy evaluation of complex form measurement. Therefore, we started development of a calibration system for the photomask test circles, which serves as reference roundness standards for imaging CMMs. We constructed a rotary-table-based roundness measuring system. This system consists of an optical microscope and a high-precision rotary table, which equipped with a self-calibrating rotary encoder (SelfA) developed at NMIJ, instead of orthogonal linear stage, to eliminate the influence of geometric errors. To evaluate the validity of the developed method, the roundness of a circular mask with a nominal diameter of 0.22 mm was measured using a multistep method. As a result, a measurement result of 39 nm was obtained with respect to a reference value of 0.02 μm.

The detection of gravitational waves requires a strain sensitivity at unprecedented precision. The planned space observatory LISA overcomes this extreme challenge by heterodyne laser interferometry at picometer-precision based on the exploitation of carrier phase measurements between spacecraft separated by millions of kilometers. In addition, data transmission and absolute ranging, necessary to mitigate effects of laser frequency fluctuations in post-processing, are achieved with direct-sequence spread spectrum signals. The foreseen receivers shall typically operate in a sequential phase-locked loop and delay-locked loop configuration for consecutive phase and distance measurement. Recent analysis observed code tracking delay variations, identified as ranging bias variations, as a result of this sequential arrangement. Hereafter, we present an analytical analysis of these ranging bias variations. Comparisons to numerical simulations reveal the compelling influence of the cross-correlation of the chip sequences on the ranging bias variations for a fixed modulation scheme and thus affirm the necessity of numerical analysis. In addition, a generic model for the quantisation error of a digital delay-locked loop is introduced that may be used for analysis and design of digital code tracking loops in various applications. Finally, comparison to a numerical simulation reveals that at small ranging bias variations, the code tracking error is fully described by the quantisation error, while at high ranging bias variations, this effect is negligible and the code tracking error is dominated by ranging bias variations.

Digital photoelasticity demands strategies to unwrap the stresses maps based on color images. Thus, this article is a new three-wavelength phase-shifting method. Three LEDs and some digital image procedures were used to generate a synthetic chromatic-corrected of bright field polarized image, and its respective inverted image. A phase map wrapped is extracted using the images mentioned above and with some trigonometric relations proposed by Ekman and Nurse. A standard algorithm was applied to unwrap the phase. The results show that the proposed method improves the maximum detected order, reduces phase distortions and is useful for dynamic cases.

Elastic lenses have been used in various optical systems, such as cameras, microscopes and vision systems, to name a few. A recently reported technique consists of making the lens's optical design, subsequently manufacturing an aluminium mould with the optical parameters of the design obtained, and finally injecting the polymer mixture into it to generate the lens. Where the lens surfaces take the shape of the mould surfaces, it is necessary to verify the finish of the mould surfaces so that the manufactured lenses meet the design requirements. In this work, the null screen technique is presented to evaluate the finish of these surfaces. An analysis of the results obtained and their conclusions is offered.

Devices with a pair of rotational Risley prisms are one of the most common and fastest 2D laser scanners. The present work builds on the novel, graphical method we have developed [Proc. of the Romanian Acad. Series A 19, 2018; Applied Sciences 11, 2021; Symmetry 15, 2023] to simulate and study scan patterns of such devices. Our method has the advantage to generate exact patterns, in contrast to approximate methods. Also, it is easy to use, in contrast to analytical methods. This graphical approach allows for fulfilling the aim of this work: to utilize ghost rays (which are usually avoided in optical systems) in order to increase the fill factor (FF) of scan patterns. Thus, secondary scan patterns are produced and studied. A radiometric threshold is set for such patterns – to maintain the emerging flux from the prisms above a certain minimum level. With this condition, three useful secondary ray trajectories can be identified through the system. One of the four possible configurations of this type of scanner is considered in this study. Dimensions and characteristics of the secondary scan patterns are pointed out, in comparison to the “main” scan pattern, generated when pure refraction is considered throughout the prisms. Simulations, as well as experiments are carried out.

The OptiX Ray Tracing Engine by NVIDIA is proposed as a powerful tool for scientific optical 3D modeling tasks. The OptiX Engine offers a highly customizable ray tracing framework, onboard GPU support for parallel computing, as well as access to optimized ray tracing algorithms. The capabilities of this approach are demonstrated by modeling a Focus Variation instrument and performing virtual measurements of arbitrary and standardized measurement samples. The performance and accuracy of these simulations are evaluated and achieved results are compared with measurement results acquired by a respective physical measuring device.

We designed a channel-hole approximating network in the electromagnetic aspect (CHANEL). Based on the characteristic that the horizontal cutting plane is topologically consistent along the vertical axis of the channel-hole, the most dominant part of V-NAND, CHANEL directly predicts the scattering matrix of each layer from its structural and optical parameters. In this paper, we demonstrate that CHANEL outperforms the traditional CPU-based RCWA implementations in terms of time and that the approximation is sufficiently sophisticated to replace the RCWA operation when simulating the diffraction by V-NAND.

A new non-integral optical scatterometry technique has been introduced to circumvent issues with traditional methods in the critical dimension (CD) characterization of micro and nano-structures in semiconductor inspections. This method uses the high spatial coherence of the laser source, and an adjustable numerical aperture (NA) for effective beam shaping, enabling precise measurement of high-aspect-ratio structures. It incorporates a model-based approach with a virtual optical system and the Finite-Difference Time-Domain (FDTD) method for multiple CD characterizations, improving measurement precision. Early tests indicate a minimal average bias of 1.74% from calibrated references and standard deviations within 7 nm.

Parameter reconstruction problems appear frequently in optical metrology. Here, one attempts to explain a set of K experimental measurements by fitting to them a parameterized forward model of the measurement process. We present a Bayesian target vector optimization scheme that can be used to perform this fit. It has been shown to be capable of outperforming established methods such as Levenberg-Marquardt, and can after a successful fit enable very efficient and accurate determination of the distribution of the reconstructed model parameters using Markov chain Monte Carlo sampling.

Inspecting the structure of the through silicon via (TSV) with high aspect ratio is important because they are used for 3D IC stacking. In reflectometry, simulation of near field data for TSV hole arrays is used to investigate reflection spectrum for TSV with different geometry parameters such as depth and top critical dimension. We investigate simulation results of electromagnetic field data for different TSV array using the finite-difference time-domain (FDTD) method. Near field simulation data are stored as n by n complex matrices, where n represent the number of simulation region grid points. The matrices are large in dimension, and it is necessary to compress a huge data set by looking for the dominant singular value terms to keep the information as much as possible. We find that the singular value terms shrink fast in the first few terms. It is shown that after using singular value decomposition to compress near field data, the far field reflectivity spectrum is still close to the accurate results. We propose to use data after singular value decomposition for data analysis to investigate the TSV parameters mapping to the near field data.

As the semiconductor industry moves forward towards advanced packaging, through silicon vias (TSVs) with a top critical dimension (CD) down to 1µm and aspect ratio 1:15 or higher are being projected. This poses a major challenge to optical metrology as the dimensions of the lattice get close to the wavelength of the visible light. White-light interferometry is commonly used to measure the depth of the vias, but suffers from decreasing signal-to-noise ratio depending on aspect ratio and diameter. To understand the limitations and develop new approaches for sensing these structures, modelling has proved to be a valuable tool. In this paper, we present results from electromagnetic wave propagation analysis compared to experimental interferometry data from high aspect ratio samples. Effects of via shape, spacing and dimension as well as simulation parameters on the resulting spectra are shown. Simulation can thus be used to predict for which type of vias a successful measurement can be expected.

In optical critical dimension metrology, experimental findings have shown that the line edge is systematically underestimated compared to SEM or AFM measurements. While these methods respond to the volume density, optical methods are sensitive to the permittivity. Due to line edge roughness there is a systematical deviation between these parameters. We discuss an analytic upper limit estimation for the contribution of LER. For low index gratings (resist, glass) the contribution is about 1 nm, for high index gratings the contribution may be as high as 5 nm rendering this crucial for sub-nanometer metrology.

Modelling the scattering of focused, coherent light by nano-scale structures is oftentimes used to reconstruct or infer geometrical or material properties of structures under investigation. This includes both scatterers with and without periodic spatial arrangements. Methods like the coherent Fourier scatterometry exploits the special structure of the far field in periodically structured samples to probe sub-wavelength gratings with a focused beam covering several grating periods. While the scattering of focused beam by a spatially isolated scatterer is standard modelling task for state-of-the art electromagnetic solver such as the finite element method, the case of periodically structured targets is more involved. We will present a coherent illumination model for scattering of focused beams such as Gaussian- and Bessel- beams by periodic structures such as line gratings. The model allows to model optical aberrations in optical systems used for both the illumination and detection of the scattered fields. We compare the model with strategies implemented on large-scale supercells and inverse Floquet-transform strategies to superimpose both near- and far fields coherently.

Fourier ptychographic microscopy (FPM) is a novel computational imaging method that enables high-resolution imaging while maintaining a wide field-of-view. This is achieved by synthesizing frequency information of scattered light in various illuminations in both transmission and reflection modes. FPM for the characterization of nanoscale semiconductor devices requires high numerical aperture reflective mode illumination and a short-wavelength source. A recent demonstration of a reflective deep ultraviolet FPM system showed a minimum resolvable linewidth of 80 nm and a contrast that was enhanced by a factor up to six, indicating FPM’s potential for nanoscale semiconductor device metrology. To push the resolution limit of an imaging system to achieve deep-subwavelength imaging, reflective FPM can be combined with structured illumination microscopy for imaging organic samples, recently called pattern-illuminated FPM. In this presentation we integrate our previously implemented FPM approach with structured illumination in order to explore the possibilities of semiconductor metrology with super resolution microscopy by means of an extensive simulation study. Since the proper implementation of this approach in an actual semiconductor fab requires multiple measuring steps, we also show how to optimize the number of FPM images taken. These results can help to significantly reduce the time demands of this new measuring approach.

Microsphere and -cylinder-assisted microscopy (MAM) has grown to an intensively studied optical far-field measurement technique over the last decade that overcomes the fundamental lateral resolution limit of a given microscope. However, the physical effects leading to resolution enhancement are still frequently debated. In addition, various configurations of MAMs operating in transmission as well as reflection mode are examined and results generalized. In a recently published study (Pahl et al., LAM 3, 2022), we present a rigorous simulation model of microcylinder-assisted interference microscopy. In this contribution, the model is extended to conventional MAM and used to compare resolution enhancement for reflection and transmission mode.

Plasmonic lenses are metastructures that use the excitation of surface plasmon polaritons in metallic nanoslits to focus light to particularly small focal spots at arbitrary distances. This facilitates possibilities for improving nano-optical methods, for example in ellipsometry. We developed two- and three-dimensional plasmonic lenses with a new inverted design that complies the fabrication process. However, plasmonic lenses show chromatic aberrations. In this contribution, we explore different approaches and limitations to expand the inverted plasmonic lens design to achromatic applications. We use numerical simulations based on the Finite Element Method to investigate in different lens geometries.

We present numerical computation of reflected and transmitted near- and far- fields resulting from the interaction of 2D cylindrical shaped particles with photonic structures illuminated with planar and Gaussian incident fields. The interaction between the cylindrical/spherical particle and the photonic structure is generally too complex to be handled analytically, so we will use the semi-analytical Fourier Modal Method (FMM) to calculate the near- and far-fields. The Gaussian field is written as a sum of plane waves with varying amplitudes. We present a very general method for obtaining the plane wave amplitudes by combining the angular spectrum theory, Parseval's theorem and Shannon's sampling theorem. We demonstrate the use of this method in our rigorous calculation of reflected and transmitted near- and far-fields of single cylindrical/spherical particle located in the vicinity of a periodic photonic structure or a planar silicon surface. The examples demonstrate that the method may be applied to investigation of photonic jets with a photonic structure, the determination of the point spread function in microscopy and particle counting.

Tomographic phase microscopy in flow cytometry allows the label-free imaging of single cells in 3D through the volumetric distribution of their refractive index. Differently from the conventional tomographic imaging systems, in which fixed samples are imaged from controlled and well-known directions and the tomographic reconstruction is achieved by well-established techniques, here an effective and robust computational processing pipeline is essential to allow reliable tomographic data. Here we show how to model both 3D pose and the shape of flowing cells to achieve this goal. Moreover, a new strategy for encoding tomographic data, based on the 3D Zernike polynomials, is reported.

Performance characteristics for interferometers that measure surface topography include the ability to resolve closely spaced surface features, referred to as topographic spatial resolution. Within well-defined limits, scalar diffraction theory and classical Fourier optics provide a software model for prediction of the resolution and spatial frequency response for interference phase-based measurements of surface topography. Analytical solutions and adaptive sampling allow for rapid simulation of both the nominal linear transfer function and an estimate of intrinsic residual nonlinearities.

In this contribution, we present a technique for the determination of optical aberrations, which is based on measurements of the point spread function and a Bayesian optimization of rigorous simulations. The measuring system is a UV-microscope in a reflected light configuration with a 200x magnification, unpolarized light, and an illumination and imaging NA of 0.44 and 0.55, respectively. The PSF is measured by imaging a small quadratic chrome dot (side length ≈ 180 nm) on a glass substrate. We investigate the impact of different adjustment states, different dot locations and different optical microscopes.

Dimensional optical microscopy allows for the rapid inspection of devices at the cost of limited accuracy. Introducing a model-based approach that includes diffraction effects allows for increased accuracies. The model needs to be efficient and accurate to evaluate the measurements in an acceptable time frame. We present an overview of the illumination model and different incidence-pupil sampling techniques. Furthermore, we will demonstrate strategies for efficiently calculating the near-field scattering response from structures using the finite element method. Using these aspects, we demonstrate a significant increase in the accuracy of dimensional estimates for a range of structures.

Within the framework of the European TracOptic project we developed the UFO (Universal Fourier Optics) model, which simulates virtual CSI measurements of surface topography treating the surface as a two-dimensional phase object. The scattered light field is multiplied by a two-dimensional transfer function in the Fourier domain prior to surface reconstruction. Parameters affecting the final results are the central wavelength and the spectral bandwidth of light as well as the numerical aperture of the objective lens and the chosen evaluation wavelength. We introduce the UFO model and discuss its capability to predict systematic measurement errors.

I will present a photonic sensor that can be used for remote sensing of many biomedical parameters simultaneously and continuously. The technology is based upon illuminating a surface with a laser and then using an imaging camera to perform temporal and spatial tracking of secondary speckle patterns in order to have nano metric accurate estimation of the movement of the back reflecting surface. The capability of sensing those movements in nano-metric precision allows connecting the movement with remote bio-sensing and with medical diagnosis capabilities. The proposed technology was already applied for remote and continuous estimation of vital bio-signs (such as heart beats, respiration, blood pulse pressure and intra ocular pressure), for molecular sensing of chemicals in the blood stream (such as for estimation of alcohol, glucose and lactate concentrations in blood stream, blood coagulation and oximetry) as well as for sensing of hemodynamic characteristics such as blood flow related to brain activity. The sensor can be used for early diagnosis of diseases such as otitis, melanoma and breast cancer and lately it was tested in large scale clinical trials and provided highly efficient medical diagnosis capabilities for cardiopulmonary diseases. The capability of the sensor was also tested and verified in providing remote high-quality characterization of brain activity.

Laser scanning is employed in a variety of modalities, including the most common raster scanning, as well as spiral, Lissajous, or Risley prism-based (the later a type of rhodonea) scanning. The first two can have the advantage of linearity, while the last two are faster, although they are highly non-linear. The present study performs an exploration of such aspects. Thus, we compare the characteristics of the above-mentioned modalities, with an emphasis on parameters such as field of view (FOV), linearity, resolution, and fill factor (FF). Simulations are carried out regarding different types of scanning patterns, as well as some experimental validations. Also, results from some of our recent studies on galvanometer- [Appl Math Modelling 67, 2019, https://doi.org/10.1016/j.apm.2018.11.001], polygon mirror- [Appl Sci 12, 2022, https://doi.org/10.3390/app12115592], and Risley prisms-based scanning [Appl Sci 11, 2021, https://doi.org/10.3390/app11188451] are utilized. Some of the specific performances of the considered scanning modalities are discussed, highlighting advantages and drawbacks.

Optical inspection systems allow faster detection of defects on semiconductor wafers than scanning electron microscopy (SEM) inspection systems. However, optical detection becomes more challenging as the structure feature size shrinks below the optical diffraction limit with the advancement of technology nodes in semiconductor manufacturing. To overcome this challenge and achieve optimal performance, the optical system must be tailored to the specific characteristics of the wafer sample which requires knowledge of the underlying microscopic and macroscopic optical phenomena. In this work, we proposed a multiphysics simulation workflow to model the microscopic light interaction with the wafer sample using Ansys Lumerical FDTD and the macroscopic optics of the inspection system using Ansys Zemax OpticStudio. The optimum optical system design with maximum defect signal strength could be achieved through defect image analysis. Together, FDTD and OpticStudio facilitate the design of complex optical inspection systems and reduce the cycle time for creating inspection recipes in the development of advanced technology nodes in semiconductor manufacturing.

We show that the fluence rate in a non-scattering disc under Lambertian illumination shows a plateau near the disc’s center, and, starting at a critical radius determined by the refractive index, decays towards the boundary. Furthermore, the influence of scattering is investigated by means of a Monte Carlo simulation of light propagation. It is found that a non-vanishing very small scattering coefficient leads to the abrupt transition to a homogeneous fluence rate. New analytic solutions of the radiative transfer equation for radiance and fluence rate of unscattered and singly scattered light show good agreement with their counterparts obtained via Monte Carlo simulation.

SE (Spectroscopic Elipsometer) has the advantages of non-destructive, high-speed, and non-contact, but it has high-precision and excellent characteristics that can be detected before the monoatomic layer is completely grown, so its demand is increasing as a key measurement equipment in the semiconductor device manufacturing process. SE has excellent measurement uncertainty according to the basic principle, but since the measurement uncertainty can be easily affected by even a minor change in the measurement device, careful attention is required to the measurement quality control of the SE. For SE measurement quality control, certified reference materials of silicon dioxide thin layer thickness are mainly used. Recently, semiconductor device manufacturing technologies have been promoted, such as complicating semiconductor device structures, introducing atomic layer-level processes, and introducing new materials. Therefore, in order to improve the measurement quality control level of SE for semiconductor processing, research is being conducted worldwide to develop a higher quality standard material for certification of thin film layer thickness. As the first research step of this study, an uncertainty evaluation method for the measurand of various types of rotating-element spectroscopic ellipsometers with excellent real-time measurement performance was developed [1]. Here, the estimation of the measurement uncertainty of the ellipsometric transfer quantities (ETQ, e.g. Psi and Delta) has the advantage that it can be calculated directly from the observed ETQ without knowing the optical properties of a given sample. As a second research step, a method was developed to estimate the uncertainty of the values of unknown optical properties of a sample obtained from the observed ETQ spectra using the nonlinear least squares method [2]. To this end, implicit function theory has been introduced to develop a method for estimating measurement uncertainty for unknown optical property values of samples [3]. Finally, the result of applying the uncertainty evaluation of the thickness of the thin film layer and the refractive index spectrum of the silicon oxide thin film sample obtained from SE is introduced. < Funding> This research was partially supported by the Korea Agency for Technology and Standards (KATS) (23201015) and by the Commercialization Promotion Agency for R&D Outcomes (COMPA) (23102009) with funded from the Ministry of Science and ICT (MSIT). < References> [1] Y. J. Cho, W. Chegal, J. P. Lee, and H. M. Cho, Opt. Express 24 26215 (2016). [2] Y. J. Cho and W Chegal, Opt. Express 29 394428 (2021). [3] J. A. Fessler, IEEE Trans. Image Processing 5, 493 (1996).

This work investigates image formation in a confocal laser scanning microscope (CLSM) for multi-cylinder phantoms. The overall dimensions of the 3D-printed phantoms are about 200 x 200 x 200 µm3, with individual cylinder radii ranging from 5 to 10 µm. By varying the refractive index difference and other parameters of the measuring system, such as the size of the pinhole or the numerical aperture, the experimental results are compared with theoretical calculations using in-house developed Monte Carlo software. The key features of the experimental CLSM images, including the shape of the individual cylindrical scatterers and the changing penetration depth, can be reproduced and understood from the simulation.

Further improving the axial resolution is paramount for three-dimensional optical imaging systems. Vortex beams are being widely applied in 3D microscopy techniques. Here we theoretically investigate the ultimate resolution limits using Laguerre-Gauss (LG) beams. Various kinds of superpositions can nowadays be easily prepared by spatial light modulators (SLM). It has been keenly shown that LG beams' superpositions possess more information than pure LG beams yet do not saturate the ultimate limit with a simple intensity scan. More sophisticated detection schemes based on quantum super-resolution protocols are investigated here to retrieve the discarded information.

For the development of integrated circuits, the accompanying metrology inside the fabrication process is essential. Non-imaging metrology of nanostructure has to be quick and non-destructive. The multilayers are crucial components of today's microprocessor nanostructures and reflective coatings. Coherent Fourier scatterometry (CFS), which is currently employed as a method for determining certain parameters of nanostructures and isolated particle detection, has not been investigated in the context of multilayer characterization. Retrieving the thickness of many wavelength-thick films using a coherent visible-range source at a full-complex-field measurement is the specific application where CFS might be advantageous. Furthermore, due to polishing in the realistic multilayers, the anticipated optical performance suffers from stochastic changes relating to surface roughness. Few non-imaging metrology methods take into consideration these statistic variances and thus are of interest for this study. Operating in the visible regime, CFS can become a viable candidate to provide cover layer reconstruction in the presence of surface roughness that has a correlation length bigger than the characteristic spot size i.e., in the range of microns. We present forward model results of multilayer structure as measured with visible range CFS modality. The influence of surface roughness is taken into account and the simulation results are discussed. Simulations of micron-sized layers of dielectric on silicon substrate suggest an influence on the far field intensity that motivates a future extended study on experimental multiple wavelength thick cover layer reconstruction in the presence of roughness. European Metrology Programme for Innovation and Research (20IND04 ATMOC)

Nowadays, optical systems commonly use either aspheric or free-form surfaces to improve their performance; however, to ensure that the manufactured surfaces become successful in concordance with to the nominal design, some geometrical parameters such as radius of curvature, conic constant, aspheric coefficients, etc., must be measured, including the surface shape under test. In this work, we propose a simple method to evaluate the optical quality of a plano-convex aspheric lens, where the convex face is modeled as an aspheric or free-form surface. We design a non-uniform pattern on the plane face of the plano-convex lens, to obtain a uniform pattern on a predefined detection plane by using the law of refraction in vector form. Additionally, implementing numerical simulations, we calculate the synthetic images produced through a predefined optical surface that we will use as if they were obtained from an experimental test. Finally, we apply an iterative method to retrieve the shape of the surface by using the normal vector field to demonstrate the feasibility of our proposal.

The standardized material measures of ISO 25178-70 are categorized as either profile or areal surface geometry with regard to their application. This categorization limits many types of material measures to be only applicable to certain types of measuring instruments. To enable comparability and uncertainty estimation for many different types of measuring instruments, we examine the adaption of the either profile or areal assigned standardized material measures of ISO 25178-70 to multiple types of measuring instruments. For this enhancement the structures can be either imaged in different lateral directions, or changed to circular geometries that allow a sampling in different directions. The revised geometries are designed, manufactured, and measured to practically demonstrate the possibilities for a multifunctional calibration of different measuring instrument categories and to illustrate the effect of directionality on the results. Many types of material measures which are assigned to the calibration of profile measuring instruments can be enhanced to also allow a specific calibration of areal surface topography measuring instruments and the additional axes that is introduced hereby.

Although stitching is frequently applied in optical surface metrology, so far no method has been proposed to evaluate the uncertainty of a stitching result. Since the accuracy of a stitching result depends on many different interdependent influence factors, in this contribution a method to determine the uncertainty of a stitching result based on a Monte Carlo simulation is proposed.

One important cause for limited traceability in optical metrology is the presence of systematic measurement errors caused by the interaction of the sensor and the measured object. These effects are complex and influenced by many factors, hence, they may differ significantly even among similar measurement systems. This also implies, that it is usually necessary to model the whole measurement chain including the relevant characteristics of the measured surface. We are currently developing a model of a chromatic confocal point sensor dedicated to simulate object-dependent systematic measurement errors and estimating task-specific measurement uncertainties. The simulations already cover all relevant fundamental aspects of the system, some important details are currently being developed. We recently introduced realistic reflection characteristics based on methods originating in physically based rendering. We show how to phenomenologically describe the light-object-interaction using bidirectional reflectance distribution functions and how the principle of Monte Carlo Ray Tracing can be adopted for this use case. We can already show the general influence of surface curvature and slope and can qualitatively predict systematic effects. However, simulations using the current model still show clear deviations from measurement results. While some effects are caused by non-ideal characteristics of the real system, others are likely caused by the approximations within our model. Therefore, further investigations and model developments are pursued.

A reliable tool for simulations of confocal microscopes shall be developed to enable improved model-based dimensional metrology. To simulate measurements on rough surfaces the BEM simulation tool SpeckleSim, provided by the ITO of the University of Stuttgart, is combined with a Fourier optics based image formation. SpeckleSim, which calculates the light-structure interaction by solving Maxwells equations, is compared with the well-known FEM based solver JCMsuite. As an example, a rectangular shaped line is used as an object. Due to different boundary conditions the results show expected small deviations, which require further investigations. First results and the general concept will be presented.

Coherence scanning interferometry (CSI) is a widely used optical method for surface topography measurement of industrial and biomedical surfaces. Approximate methods use physics-based approaches to model a CSI instrument with minimal computational effort. A crucial aspect of CSI modelling is defining the transfer function for the imaging properties of the instrument, to predict the interference fringes created by the instrument. Approximate methods, such as elementary Fourier optics, universal Fourier optics and foil models, use scalar diffraction theory and the imaging properties of the optical system to model surface topography measurement. In this paper, the surface topographies obtained by these three methods are compared.