KEYWORDS: Perovskite, Solar cells, Color, Photovoltaics, Optical filters, Reflectivity, Design and modelling, Thin films, Thin film solar cells, Standards development
Color aesthetics in photovoltaic modules are essential, especially in design-sensitive applications like building integrated photovoltaics (BIPV). Distributed Bragg reflector-based color filters can modify the appearance of silicon solar cells. This study extends the aesthetic evaluation to emerging perovskite solar cells, typically gray-brown, by integrating them with a MorphoColor® color filter. We present simulated and measured angular resolved reflectance measurements and evaluate the color appearance from varied viewing angles. The used simulation environment is based on wave optics and raytracing. Next, we study the impact of individual layer parameters on the color appearance and the maximum achievable short circuit current density in the perovskite solar cell. Findings show that changes in color filter-perovskite interfacing layers influence the saturation and hue of the color impression as well as the angular color stability. Finally, we present initial concepts of optimizing the specific layer stack, demonstrating its potential to tailor a customized color design.
KEYWORDS: Photovoltaics, Solar energy, Solar cells, Multijunction solar cells, Energy efficiency, Standards development, Silver, Silicon solar cells, Silicon, Renewable energy
We present a critical review on optical research in photovoltaics guided by the question: Which topics do we regard as most relevant to accelerate the large-scale implementation of photovoltaics? The following areas were identified: 1) The development of high-performance earth-abundant solar cell materials and the reduction of indium and silver in the device. 2) Color concepts for an appealing optical appearance of solar modules as photovoltaic modules enter urban environments as well as agricultural areas on a large scale. 3) Optical optimization of multijunction solar cells based on perovskite, III-V semiconductors and silicon to overcome the single-junction efficiency limit. 4) Accurate energy yield predictions considering the full complex illumination conditions particularly regarding bifacial and multijunction solar cells. 5) Advanced concepts with strong momentum such as radiative cooling and spectral conversion.
Luminescent coupling is a characteristic of multi-junction solar cells which has often been neglected in models of their performance. The effect describes the absorption of light emitted from a higher band gap semiconductor by a lower band gap semiconductor. In this way, light which might have been lost can be utilized for current generation. We present a framework for modeling this effect in both planar layer stacks and devices with periodic nanostructuring. As a case study, we evaluate how luminescent coupling is affected by the inclusion of nanostructuring in a perovskite silicon tandem solar cell. We find that nanostructuring, while reducing the reflection loss for tandem solar cells also reduces the luminescent coupling, allowing more light to be emitted to the surroundings, when compared to planar devices. This highlights the need to include modeling of this effect into optimization schemes in order to find the trade-off between these two effects.
The efficient design of metasurfaces presents a challenging optimization problem due to the relatively large number of meta-atoms and their mutual coupling. In this work, we present two novel multi-output surrogate models used in the context of the Bayesian optimization of a beam splitter. We show how learning the vectorial quantities forming the final objective can lead to more accurate results and significant speed-ups when compared to classical optimization of scalar objectives. Furthermore, we discuss how to incorporate gradient information with respect to design parameters to further accelerate the optimization.
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 in optical scatterometry. This comprises both periodic and aperiodic nano-structures. Coherent Fourier scatterometry with focused light exploits the diffraction pattern formed by the nano-structures in Fourier plane. While the scattering of a focused beam by a spatially isolated scatterer is a standard modelling task for state-of-the art electromagnetic solvers based, e.g., on the finite element method, the case of periodically structured samples is more involved. In particular when the focused light covers several grating periods of as it is commonly the case. We will present a coherent illumination model for scattering of focused beams such as Gaussian- and Besselbeams by periodic structures such as line gratings. The model allows to take into account optical wavefront 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 super-cells and inverse Floquet-transform strategies to superimpose both near- and far fields coherently.
KEYWORDS: Mathematical optimization, Optical metrology, Monte Carlo methods, Education and training, Process modeling, Sand, Global Positioning System, Computer programming languages
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.
We investigate the calculation of resonance modes of a VCSEL with a Riesz projection eigenvalue solver. The eigenvalue solver is based on the principle of contour integration where for the solution of scattering problems physical right sides are used. Here, it is investigated how numerical parameters impact the performance of the method, where we focus on the computation of the fundamental VCSEL mode.
Spectroscopic ellipsometry is a sensitive and optical model-supported quantitative tool to monitor interfaces. In this work, solid-liquid interfaces are studied using the Kretschmann-Raether configuration for biosensing applications. The interface layers support two purposes simultaneously: (i) chemical suitability for the adsorption of molecules to be detected and (ii) the optical enhancement of the signal to increase the sensitivity. Ellipsometry is not only used as a sensor but also as a quantitative measurement tool to study and understand the interface phenomena, and to develop the sensing layers for the largest possible optical sensitivity. Plasmonic and structured layers are of primary importance in terms of optical sensitivity. Layers structured both in lateral and vertical directions have been studied. Optical models based on both the transfer matrix and the finite element method were developed and used for the structural analysis where the material and geometrical derivatives are used in the inverse fitting process of the model data to the measurement. Structures utilizing plasmonic, diffraction, multilayer field enhancement, and other methods were analyzed as possible candidates for the improvement of the optical performance of the cell. Combinatorial and periodic plasmonic surface structures were developed to enhance the sensitivity of in-situ ellipsometry at solid-liquid interfaces utilizing the Kretschmann-Raether (KR) geometry. AgxAl1−x layers with variable compositions and Au layers with changing periods and critical dimensions were investigated to improve the performance of sensors based on the KR arrangement.
Manual optimization of experimental parameters can quickly become too complex and time-consuming if more than a few correlated parameters need to be adjusted. We discuss automating this process using Bayesian optimization. This machine learning-based method is particularly suitable because it can handle noisy measurements, performs a global search and requires relatively few experimental runs. We discuss the efficient, scalable implementation of Bayesian optimization, present practical applications for tuning experimental parameters, and compare it with other local and global heuristic methods to show its application range.
In many industrial sectors, dimensional microscopy enables non-destructive and rapid inspection of manufacturing processes. However, wave-optical effects and imaging errors of the optical system limit the accuracy. With modelbased approaches it is possible to measure the physical position of edges and corners with submicron uncertainty. This requires an accurate model for phase aberrations of the optical system. We present a method to model and quantify those phase aberrations by an efficient inverse measurement.
The deterministic integration of quantum emitters into on-chip photonic elements is crucial for the implementation of scalable on-chip quantum circuits. Here, we report on the deterministic integration of single quantum dots (QD) into tapered multimode interference beam splitters using in-situ electron beam lithography (EBL). We demonstrate the functionality of the deterministic QD-waveguide structures by µPL spectroscopy and by studying the photon cross-correlation between the two MMI output ports. The latter confirms single-photon emission and on-chip splitting associated with g(2)(0) << 0.5. Moreover, the deterministic integration of QDs enables the demonstration and controlled study of chiral light-matter effects and directional emission in QD-WGs, and the realization of low-loss heterogenous QD-WG systems with excellent quantum optical properties.
Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell’s equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations Χ2 . On the other hand, curve fitting algorithms, such as the Levenberg-Marquardt method, take the deviations between all model outputs and corresponding measurement values into account which enables a fast local convergence. In this paper we present a Bayesian Target Vector Optimization scheme which combines these two approaches. We compare the performance of the presented method against a standard Levenberg-Marquardt-like algorithm, a conventional Bayesian optimization scheme, and the L-BFGS-B and Nelder-Mead simplex algorithms. As a stand-in for problems from nano metrology, we employ a non-linear least-square problem from the NIST Standard Reference Database. We find that the presented method generally uses fewer calls of the model function than any of the competing schemes to achieve similar reconstruction performance.
This Conference Presentation, “Numerical optimization of resonant photonic devices,” was recorded at SPIE Optical Metrology 2019, held in Munich, Germany.
For the reliable fabrication of the current and next generation of nanostructures it is essential to be able to determine their material composition and dimensional parameters. Using the grazing incidence X-ray fluoresence technique, which is taking advantage of the X-ray standing wave field effect, nanostructures can be investigated with a high sensitivity with respect to the structural and elemental composition. This is demonstrated using lamellar gratings made of Si3N4. Rigorous field simulations obtained from a Maxwell solver based on the finite element method allow to determine the spatial distribution of elemental species and the geometrical shape with sub-nm resolution. The increasing complexity of nanostructures and demanded sensitivity for small changes quickly turn the curse of dimensionality for numerical simulation into a problem which can no longer be solved rationally even with massive parallelisation. New optimization schemes, e.g. machine learning, are required to satisfy the metrological requirements. We present reconstruction results obtained with a Bayesian optimization approach to reduce the computational effort.
Optical scatterometry is a method to measure the size and shape of periodic micro- or nanostructures on surfaces. For this purpose the geometry parameters of the structures are obtained by reproducing experimental measurement results through numerical simulations. We compare the performance of Bayesian optimization to different local minimization algorithms for this numerical optimization problem. Bayesian optimization uses Gaussian-process regression to find promising parameter values. We examine how pre-computed simulation results can be used to train the Gaussian process and to accelerate the optimization.
Modern optical systems are subject to very restrictive performance, size and cost requirements. Especially in portable systems size often is the most important factor, which necessitates elaborate designs to achieve the desired specifications. However, current designs already operate very close to the physical limits and further progress is difficult to achieve by changing only the complexity of the design. Another way of improving the performance is to tailor the optical properties of materials specifically to the application at hand. A class of novel, customizable materials that enables the tailoring of the optical properties, and promises to overcome many of the intrinsic disadvantages of polymers, are nanocomposites. However, despite considerable past research efforts, these types of materials are largely underutilized in optical systems. To shed light into this issue we, in this paper, discuss how nanocomposites can be modeled using effective medium theories. In the second part, we then investigate the fundamental requirements that have to be fulfilled to make nanocomposites suitable for optical applications, and show that it is indeed possible to fabricate such a material using existing methods. Furthermore, we show how nanocomposites can be used to tailor the refractive index and dispersion properties towards specific applications..
We present optical simulations for a tandem solar cell consisting of a nanostructured thin-film perovskite top cell and a silicon heterojunction (SHJ) wafer bottom cell. The absorption and related current density are calculated using the rigorous simulations in the form of the finite element method for the nanostructured perovskite cell and a semi-empirical method for the SHJ cell. In order to reach the optimal value for the perovskite layer thickness we employ Newton’s method using derivatives obtained directly from the rigorous simulation. Using this we obtain an optimal layer thickness using typically one iteration step and eliminate the need for a parameter scan.
We compare the results for different sinusoidal nanotextures applied to different layers in the multilayer thin-film perovskite top cell. The nanotextures lead to a gain in absorption and power conversion efficiency in comparison to an optimized planar reference. We also present experimental results towards a realisation of the proposed structure. These results give valuable insight for the emerging field of high efficiency perovskite/SHJ tandem solar cells.
For optimizing specific functionalities of optical components which include structures on a micrometer or nanometer scale, typically high-dimensional optimization problems have to be solved. We use Gaussian process regression to this aim. Gaussian processes can be viewed as machine-learning algorithms where results from evaluations at specific points in the parameter space (training data) are used to predict values and their uncertainty in the full parameter space. The forward-problem (evaluation at a given point in parameter space) requires to rigorously solve Maxwell’s equations, i.e. to compute light propagation in a specific setup. We use our finite-element method (FEM) implementation JCMsuite to this aim. The general framework of FEM allows to employ adaptive numerical resolution and accurate geometry modelling for arbitrary shapes. We discuss application of Bayesian optimization for the inverse problem in parameter retrieval from scatterometric data.
Recently, we studied the effect of hexagonal sinusoidal textures on the reflective properties of perovskite-silicon tandem solar cells using the finite element method (FEM). We saw that such nanotextures, applied to the perovskite top cell, can strongly increase the current density utilization from 91% for the optimized planar reference to 98% for the best nanotextured device (period 500 nm and peak-to-valley height 500 nm), where 100% refers to the Tiedje-Yablonovitch limit.* In this manuscript we elaborate on some numerical details of that work: we validate an assumption based on the Tiedje-Yablonovitch limit, we present a convergence study for simulations with the finite-element method, and we compare different configurations for sinusoidal nanotextures.
We report on the investigation of an approach for modelling light transmission through systems consisting of several jointed optical fibres, in which the analytical modelling of the waveguides was replaced by Finite Element Modelling (FEM) simulations. To validate this approach we first performed FEM analysis of standard fibres and used this to evaluate the coupling efficiency between two singlemode fibres under different conditions. The results of these simulations were successfully compared with those obtained using classical analytical approaches, by demonstrating a maximum loss deviation of about 0.4 %. Further, we performed other more complex simulations that we compared again to the analytical models. FEM simulations allow addressing any type of guiding structure, without limitations on the complexity of the geometrical waveguide cross section and involved materials. We propose as example of application the modelling of the light transmitted through a system made of a hollow core photonic crystal fibre spliced between two singlemode standard optical fibres, and qualitatively compare the results of the simulation with experimental results.
Online measurement of diameters and concentricities of optical fibre layers, and the coating layer in particular, is one of the challenges in fibre manufacturing. Currently available instruments can measure concentricity and diameter of layers offline, and are not suitable for precise monitoring or control of the manufacturing process in real time. In this work, we use two laser beams, positioned orthogonally to illuminate the fibre from two sides, and calculate deviations from the expected geometry by analysing the scattering pattern. To measure the diffraction pattern we use two 8K linear array detectors, with the scattered light incident directly on the sensors. Each detector is capturing approximately 90° angular range directly behind the fibre. The two measurement channels are positioned at different heights. The scattered pattern is modelled mathematically with finite-element and Fourier-modal methods, with various diameter and concentricity deviations. The sensitivities of the changes in the scattering pattern are identified in respect to these deviations. Since calculations are computationally intensive, the sensitivities are pre-calculated in advance, and the realtime measurement is based on pattern recognition. The symmetry of the pattern is used to differentiate between diameter and concentricity variations. We performed online measurements with the prototype instrument in production conditions, and show that this method is sensitive enough to measure deviations of under 1 µm in diameter and concentricity of the coating layer.
KEYWORDS: Vertical cavity surface emitting lasers, Oxides, Finite element methods, Computer simulations, Near field optics, Maxwell's equations, 3D modeling
Open resonant optical devices such as an oxide-confined vertical cavity surface emitting laser (VCSEL) can be characterized by a quasi-normal mode (QNM) expansion. In contrast to eigenmodes of a closed system, QNMs exhibit an exponential divergence in the exterior of the device and are no longer normalizable. This behavior renders the mathematical treatment and physical understanding very challenging. As an alternative we investigate the constant-flux mode (CFM) expansion which avoids the divergence in the exterior domain. Besides numerical studies, we present results for different oxide aperture sizes and positions inside the investigated VCSEL. Here, we apply CFMs to describe the impact on the resonance wavelength and on the mode profile.
In this work, we perform numerical studies of two photonic crystal membrane microcavities, a short line-defect L5 cavity with relatively low quality (Q) factor and a longer L9 cavity with high Q. We compute the cavity Q factor and the resonance wavelength λ of the fundamental M1 mode in the two structures using five state-of- the-art computational methods. We study the convergence and the associated numerical uncertainty of Q and λ with respect to the relevant computational parameters for each method. Convergence is not obtained for all the methods, indicating that some are more suitable than others for analyzing photonic crystal line defect cavities.
KEYWORDS: Perovskite, Tandem solar cells, Solar cells, Silicon, Absorption, Nanophotonics, Finite element methods, Computer architecture, Fourier transforms, Crystals
Currently, perovskite–silicon tandem solar cells are one of the most investigated concepts for overcoming the theoretical limit for the power conversion efficiency of silicon solar cells. For monolithic tandem solar cells, the available light must be distributed equally between the two subcells, which is known as current matching. For a planar device design, a global optimization of the layer thicknesses in the perovskite top cell allows current matching to be reached and reflective losses of the solar cell to be minimized at the same time. However, even after this optimization, the reflection and parasitic absorption losses add up to 7 mA / cm2. In this contribution, we use numerical simulations to study how well hexagonal sinusoidal nanotextures in the perovskite top-cell can reduce the reflective losses of the combined tandem device. We investigate three configurations. The current density utilization can be increased from 91% for the optimized planar reference to 98% for the best nanotextured device (period 500 nm and peak-to-valley height 500 nm), where 100% refers to the Tiedje–Yablonovitch limit. In a first attempt to experimentally realize such nanophotonically structured perovskite solar cells for monolithic tandems, we investigate the morphology of perovskite layers deposited onto sinusoidally structured substrates.
Sub-wavelength patterned structures like cavity resonator integrated grating filters are relevant in various optical applications. Due to their highly resonant nature and due to their relatively large size it can be demanding to perform accurate optical simulations for such devices. In this contribution we investigate performance of hp finite-element based methods for this simulation problem. We demonstrate numerical convergence of the obtained solutions up to very high accuracy levels. We compare performance of our methods to results from the literature and investigate the influence of physical parameters on device performance, enhancing the Q-factor and reducing the horizontal size of the device.
Light management is a key issue for highly efficient liquid-phase crystallized silicon (LPC-Si) thin-film solar cells and can be achieved with periodic nanotextures. They are fabricated with nanoimprint lithography and situated between the glass superstrate and the silicon absorber. To combine excellent optical performance and LPC-Si material quality leading to open circuit voltages exceeding 640 mV, the nanotextures must be smooth. Optical simulations of these solar cells can be performed with the finite element method (FEM). Accurately simulating the optics of such layer stacks requires not only to consider the nanotextured glass-silicon interface, but also to adequately account for the air-glass interface on top of this stack. When using rigorous Maxwell solvers like the finite element method (FEM), the air-glass interface has to be taken into account a posteriori, because the solar cells are prepared on thick glass superstrates, in which light is to be treated incoherently. In this contribution we discuss two different incoherent a posteriori corrections, which we test for nanotextures between glass and silicon. A comparison with experimental data reveals that a first-order correction can predict the measured reflectivity of the samples much better than an often-applied zeroth-order correction.
Numerical simulation of complex optical structures enables their optimization with respect to specific objectives. Often, optimization is done by multiple successive parameter scans, which are time consuming and computationally expensive. We employ here Bayesian optimization with Gaussian processes in order to automatize and speed up the optimization process. As a toy example, we demonstrate optimization of the shape of a free-form reflective meta surface such that it diffracts light into a specific diffraction order. For this example, we compare the performance of six different Bayesian optimization approaches with various acquisition functions and various kernels of the Gaussian process.
We present a Newton-like method to solve inverse problems and to quantify parameter uncertainties. We apply
the method to parameter reconstruction in optical scatterometry, where we take into account a priori information
and measurement uncertainties using a Bayesian approach. Further, we discuss the influence of numerical
accuracy on the reconstruction result.
KEYWORDS: Silicon, Thin film solar cells, Solar cells, Thin films, Antireflective coatings, Diffraction gratings, Numerical simulations, Finite element methods, Thin film solar cells, Refractive index, Interfaces, Polarization, Maxwell's equations, Absorption
Hexagonal sinusoidal nanotextures are well suited to couple light into silicon on glass at normal incidence, as we have shown in an earlier publication [K. Jäger et al., Opt. Express 24, A569 (2016)]. In this manuscript we discuss how these nanotextures perform under oblique incidence illumination. For this numerical study we use a rigorous solver for the Maxwell equations. We discuss nanotextures with periods between 350 nm and 730 nm and an aspect ratio of 0.5.
Maxwell solvers based on the hp-adaptive finite element method allow for accurate geometrical modeling and high numerical accuracy. These features are indispensable for the optimization of optical properties or reconstruction of parameters through inverse processes. High computational complexity prohibits the evaluation of the solution for many parameters. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem allowing to compute solutions for a parameter configuration orders of magnitude faster. The RBM allows to evaluate linear and nonlinear outputs of interest like Fourier transform or the enhancement of the electromagnetic field in milliseconds. We apply the RBM to compute light-scattering off two dimensional photonic crystal structures made of silicon and reconstruct geometrical parameters.
Optical chirality has been recently suggested to complement the physically relevant conserved quantities of the well-known Maxwell's equations. This time-even pseudoscalar is expected to provide further insight in polarization phenomena of electrodynamics such as spectroscopy of chiral molecules. Previously, the corresponding continuity equation was stated for homogeneous lossless media only. We extend the underlying theory to arbitrary setups and analyse piecewise-constant material distributions in particular. Our implementation in a Finite Element Method framework is applied to illustrative examples in order to introduce this novel tool for the analysis of time-harmonic simulations of nano-optical devices.
An efficient numerical method for computing angle-resolved light scattering off periodic arrays is presented. The method combines finite-element discretization with a Schur complement solver. A significant speed-up of the computations in comparison to standard finite-element method computations is observed.
Fields such as optical metrology and computational lithography require fast and efficient methods for solving the time-harmonic Maxwell's equation. Highly accurate geometrical modelling and numerical accuracy at low computational costs are a prerequisite for any simulation study of complex nano-structured photonic devices. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scattering at optical wavelengths of periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such as the width and height of the fin and gate can vary in a large range.
Oxide–confined apertures in vertical cavity surface emitting laser (VCSEL) can be engineered such that they promote
leakage of the transverse optical modes from the non– oxidized core region to the selectively oxidized periphery of the
device. The reason of the leakage is that the VCSEL modes in the core can be coupled to tilted modes in the periphery if
the orthogonality between the core mode and the modes at the periphery is broken by the oxidation–induced optical field
redistribution. Three–dimensional modeling of a practical VCSEL design reveals i) significantly stronger leakage losses
for high–order transverse modes than that of the fundamental one as high–order modes have a higher field intensity close
to the oxide layers and ii) narrow peaks in the far–field profile generated by the leaky component of the optical modes.
Experimental 850–nm GaAlAs leaky VCSELs produced in the modeled design demonstrate i) single–mode lasing with
the aperture diameters up to 5μm with side mode suppression ratio >20dB at the current density of 10kA/cm2; and ii)
narrow peaks tilted at 37 degrees with respect to the vertical axis in excellent agreement with the modeling data and
confirming the leaky nature of the modes and the proposed mechanism of mode selection. The results indicate that in–
plane coupling of VCSELs, VCSELs and p–i–n photodiodes, VCSEL and delay lines is possible allowing novel photonic
integrated circuits. We show that the approach enables design of oxide apertures, air–gap apertures, devices created by
impurity–induced intermixing or any combinations of such designs through quantitative evaluation of the leaky
emission.
Rigorous optical simulations of 3-dimensional nano-photonic structures are an important tool in the analysis and optimization of scattering properties of nano-photonic devices or parameter reconstruction. To construct geometrically accurate models of complex structured nano-photonic devices the finite element method (FEM) is ideally suited due to its flexibility in the geometrical modeling and superior convergence properties. Reduced order models such as the reduced basis method (RBM) allow to construct self-adaptive, error-controlled, very low dimensional approximations for input-output relationships which can be evaluated orders of magnitude faster than the full model. This is advantageous in applications requiring the solution of Maxwell's equations for multiple parameters or a single parameter but in real time. We present a reduced basis method for 3D Maxwell's equations based on the finite element method which allows variations of geometric as well as material and frequency parameters. We demonstrate accuracy and efficiency of the method for a light scattering problem exhibiting a resonance in the electric field.
The finite-element method is a preferred numerical method when electromagnetic fields at high accuracy are to be computed in nano-optics design. Here, we demonstrate a finite-element method using hp-adaptivity on tetrahedral meshes for computation of electromagnetic fields in a device with rough textures. The method allows for efficient computations on meshes with strong variations in element sizes. This enables to use precise geometry resolution of the rough textures. Convergence to highly accurate results is observed.
High quality scatterometry standard samples have been developed to improve the tool matching between different scatterometry methods and tools as well as with high resolution microscopic methods such as scanning electron microscopy or atomic force microscopy and to support traceable and absolute scatterometric critical dimension metrology in lithographic nanomanufacturing. First samples based on one dimensional Si or on Si3N4 grating targets have been manufactured and characterized for this purpose. The etched gratings have periods down to 50 nm and contain areas of reduced density to enable AFM measurements for comparison. Each sample contains additionally at least one large area scatterometry target suitable for grazing incidence small angle X-ray scattering. We present the current design and the characterization of structure details and the grating quality based on AFM, optical, EUV and X-Ray scatterometry as well as spectroscopic ellipsometry measurements. The final traceable calibration of these standards is currently performed by applying and combining different scatterometric as well as imaging calibration methods. We present first calibration results and discuss the final design and the aimed specifications of the standard samples to face the tough requirements for future technology nodes in lithography.
KEYWORDS: Vertical cavity surface emitting lasers, Oxides, Near field optics, Near field, Refractive index, 3D modeling, Waveguides, Reflectivity, Resistance, Semiconductors
Oxide–confined vertical cavity surface emitting lasers (VCSEL) are inherently leaky structures, despite the fact that the oxidized periphery region surrounding the all–semiconductor core has a lower refractive index. The reason is that the VCSEL modes in the non–oxidized core region can be coupled to tilted modes in the selectively oxidized periphery as the orthogonality between the core mode and the modes at the periphery is broken by the oxidation–induced optical field redistribution. Engineered VCSEL designs show that the overlap between the VCSEL mode of the core and the tilted mode in the periphery can reach >30% resulting in significant leakage. Three–dimensional modeling confirms that the leakage losses are much stronger for high order transverse modes which have a higher field intensity close to the oxidized region. Single mode lasing in the fundamental mode can thus proceed up to large aperture diameters. A 850–nm GaAlAs leaky VCSEL based on this concept is designed, modeled and fabricated, showing single–mode lasing with aperture diameters up to 5 μm. Side mode suppression ratio >20dB is realized at the current density of 10kA/cm2 in devices with the series resistance of 90 Ω.
Methods for solving Maxwell’s equations are integral part of optical metrology and computational lithography setups. Applications require accurate geometrical resolution, high numerical accuracy and/or low computation times. We present a finite-element based electromagnetic field solver relying on unstructured 3D meshes and adaptive hp-refinement. We apply the method for simulating light scattering off arrays of high aspect-ratio nano-posts and FinFETs.
We address demands and challenges for GaAs–based Vertical–Cavity Surface–Emitting Lasers (VCSEL) in data communication. High speed modulation (~50Gb/s) at a high reliability can be realized with a proper VCSEL design providing a high differential gain. In cases where extreme temperatures are required electrooptic modulation in duo– cavity VCSELs can be applied as the modulation speed and the differential gain are decoupled. Single mode operation of VCSELs is necessary to counteract the chromatic dispersion of glass fibers and extend distances to above 1 km while using standard multimode fibers. Oxide layer engineering or using of photonic crystals can be applied. Parallel error–free 25Gb/s transmission over OM3 and OM4 multimode fiber (~0.5 and 1 km, respectively) is realized in large aperture oxide–engineered VCSEL arrays. Passive cavity VCSELs with gain medium placed in the bottom DBR and the upper part made of dielectric materials a complete temperature insensitivity of the emission wavelength can be realized. Engineering of the oxide aperture region enables near field vertical cavity lasers. Such devices can operate in a high– order transverse mode with an effective mode angle beyond the angle of the total internal reflection at the semiconductor–air interface. Near filed coupling to optical fibers and waveguides becomes possible in this case.
Non-imaging techniques like X-ray scattering are supposed to play an important role in the further development of CD metrology for the semiconductor industry. GISAXS provides directly assessable information on structure roughness and long-range periodic perturbations. The disadvantage of the method is the large footprint of the X-ray beam on the sample due to the extremely shallow angle of incidence. This can be overcome by using wavelengths in the extreme ultraviolet (EUV) spectral range which allow for much steeper angles of incidence but preserve the large range of momentum transfer that can be observed. At the Physikalisch-Technische Bundesanstalt (PTB), the available photon energy range extends from 50 eV up to 10 keV at two adjacent beamlines. PTB commissioned a new versatile Ellipso-Scatterometer which is capable of measuring 6” square substrates in a clean, hydrocarbon-free environment with full flexibility regarding the direction of the incident light polarization. The reconstruction of line profiles using a geometrical model with six free parameters, a finite element method (FEM) Maxwell solver and least-squares optimization yielded consistent results for EUV and deep ultraviolet (DUV) scatterometry. For EUV photomasks, the actinic wavelength EUV scatterometry yields particular advantages. A significant polarization dependence of the diffraction intensities for 0th and +1st orders in the geometry with the grating lines perpendicular to the plane of reflection is observed and the 0th order intensity shows sufficient sensitivity to the line width such that a CD-resolution below 0.1 nm is within reach. In this contribution we present scatterometry data for line gratings using GISAXS, and EUV and DUV scatterometry and consistent reconstruction results of the line geometry for EUV and DUV scatterometry.
Non-imaging techniques like X-ray scattering are supposed to play an important role in the further development of CD
metrology for the semiconductor industry. Grazing Incidence Small Angle X-ray Scattering (GISAXS) provides directly
assessable information on structure roughness and long-range periodic perturbations. The disadvantage of the method is
the large footprint of the X-ray beam on the sample due to the extremely shallow angle of incidence. This can be
overcome by using wavelengths in the extreme ultraviolet (EUV) spectral range, EUV small angle scattering (EUVSAS),
which allows for much steeper angles of incidence but preserves the range of momentum transfer that can be
observed. Generally, the potentially higher momentum transfer at shorter wavelengths is counterbalanced by decreasing
diffraction efficiency. This results in a practical limit of about 10 nm pitch for which it is possible to observe at least the
± 1st diffraction orders with reasonable efficiency. At the Physikalisch-Technische Bundesanstalt (PTB), the available
photon energy range extends from 50 eV up to 10 keV at two adjacent beamlines. PTB commissioned a new versatile
Ellipso-Scatterometer which is capable of measuring 6" square substrates in a clean, hydrocarbon-free environment with
full flexibility regarding the direction of the incident light polarization.
The reconstruction of line profiles using a geometrical model with six free parameters, based on a finite element method
(FEM) Maxwell solver and a particle swarm based least-squares optimization yielded consistent results for EUV-SAS
and GISAXS. In this contribution we present scatterometry data for line gratings and consistent reconstruction results of
the line geometry for EUV-SAS and GISAXS.
A numerical investigation of a two dimensional integrated fiber grating coupler capable of exciting several LP fiber modes in both TE and TM polarization is presented. Simulation results and an assessment of the numerical complexity of the 3D, fully vectorial finite element model of the device are shown.
KEYWORDS: 3D modeling, Solar cells, Error analysis, MATLAB, Finite element methods, Chemical elements, Nanophotonics, Metrology, Scattering, Device simulation
Rigorous optical simulations are an important tool in optimizing scattering properties of nano-photonic devices and are used, for example, in solar cell optimization. The finite element method (FEM) yields rigorous, timeharmonic, high accuracy solutions of the full 3D vectorial Maxwell's equations1 and furthermore allows for great flexibility and accuracy in the geometrical modeling of these often complex shaped 3D nano-structures. A major drawback of frequency domain methods is the limitation of single frequency evaluations. For example the accurate computation of the short circuit current density of an amorphous silicon/micro-crystalline multi-junction thin film solar cell may require the solution of Maxwell's equations for over a hundred different wavelengths if an equidistant sampling strategy is employed. Also in optical metrology, wavelength scans are frequently used to reconstruct unknown geometrical and material properties of optical systems numerically from measured
scatterometric data. In our contribution we present several adaptive numerical integration and sampling routines and study their efficiency in the context of the determination of generation rate profiles of solar cells. We show that these strategies lead to a reduction in the computational effort without loss of accuracy. We discuss the employment of tangential information in a Hermite interpolation scheme to achieve similar accuracy on coarser grids. We explore the usability of these strategies for scatterometry and solar cell simulations.
KEYWORDS: Finite element methods, Electromagnetism, Computer simulations, Metrology, Stanford Linear Collider, Silver, Near field, Sensors, Numerical analysis, Chemical elements
A smart light trapping scheme is essential to tap the full potential of polycrystalline silicon (poly-Si) thin-film solar cells. Periodic nanophotonic structures are of particular interest as they allow to substantially surpass the Lambertian limit from ray optics in selected spectral ranges. We use nanoimprint-lithography for the periodic patterning of sol-gel coated glass substrates, ensuring a cost-effective, large-area production of thin-film solar cell devices. Periodic crystalline silicon nanoarchitectures are prepared on these textured substrates by high-rate silicon film evaporation, solid phase crystallization and chemical etching. Poly-Si microhole arrays in square lattice geometry with an effective thickness of about 2μm and with comparatively large pitch (2 μm) exhibit a large absorption enhancement (A900nm = 52%) compared to a planar film (A900nm ~ 7%). For the optimization of light trapping in the desired spectral region, the geometry of the nanophotonic structures with varying pitch from 600 nm to 800 nm is tailored and investigated for the cases of poly-Si nanopillar arrays of hexagonal lattice geometry, exhibiting an increase in absorption in comparison to planar film attributed to nanophotonic wave optic effects. These structures inspire the design of prospective applications such as highly-efficient nanostructured poly-Si thin-film solar cells and large-area photonic crystals.
Hollow-core photonic bandgap fibers guide light using diffraction rather than total internal reflection as is the case with
normal single- mode communications fibers. The fibers consist of a hollow capillary (~19 micrometers in diameter)
surrounded by capillary (~4 micrometers in diameter) arranged in a honey-comb like structure. The honey-comb
structure scatters light in the core such that light within the bandgap wavelengths cannot escape from the core. However,
the bandgap properties greatly depend on the accuracy with which the microstructures can be controlled during the
fabrication process. For measuring the geometrical properties of hollow core photonic crystal fibers with a honeycomb
cladding structure we use an angular scatterometric setup. For analyzing the experimentally obtained data we rigorously compute the scattering signal by solving Maxwell's equations with finite-element methods. This contribution focuses on the numerical analysis of the problem. A convergence analysis demonstrates that we reach highly accurate solutions. Our results show very good qualitative agreement between experimental and numerical results. We furthermore demonstrate concepts for accurately monitoring dimensional parameters in the fiber manufacturing process.
Extreme W scatterometry using radiation in the extreme ultraviolet photon energy range, with wavelengths around 13.5 nm, provides direct information on the performance of EUV optical components, e.g. EUV pho tomasks, in their working wavelength regime. Scatterometry with horizontal diffraction geometry, parallel to the grating lines (conical), and vertical scattering geometry, perpendicular to the lines (in-plane), was performed on EUV lithography mask test structures. Numerical FEM based simulations, using a rigorous Maxwell solver, compare both experimental set-ups with focus on the sensitivity of the diffraction intensities particularly with respect to the side wall angle.
At PTB we investigate the prospects of scatterometric methods for quantitative dimensional metrology of periodic
micro- and nanostructures. Commonly applied approximations and simplifications may lead to contributions to the
measurement uncertainty or even to systematic measurement errors. Here we present a short overview about the main
effects connected with these simplifications. In particular we present numerical investigations of the influence of a finite
beam size on the scatterometry results. The results indicate, that an impact of the finite beam size becomes significant
only for tightly focused beams with a beam waist radius wo smaller than 10 μm.
A method for automatic computation of parameter derivatives of numerically computed light scattering signals is demonstrated. The finite-element based method is validated in a numerical convergence study, and it is applied to investigate the sensitivity of a scatterometric setup with respect to geometrical parameters of the scattering target. The method can significantly improve numerical performance of design optimization, parameter reconstruction, sensitivity analysis, and other applications.
KEYWORDS: Monte Carlo methods, Absorption, Interfaces, Solar cells, Chemical elements, 3D modeling, Computer simulations, Transparent conductors, Finite element methods, Prisms
Light trapping techniques are one of the key research areas in thin film silicon photovoltaics. Since the 1980s randomly rough textured front transparent oxides (TCOs) have been the methods of choice as light trapping strategies for thin-film devices. Light-trapping efficiency can be optimized by means of optical simulations of nano-structured solar cells. We present a FEM based simulator for 3D rigorous optical modeling of amorphous silicon / microcrystalline silicon tandem thin-film solar cells with randomly textured layer interfaces. We focus strongly on an error analysis study for the presented simulator to demonstrate the numerical convergence of the method and investigate grid and finite element degree refinement strategies in order to obtain reliable simulation results.
This work addresses a versatile modeling of complex photonic integrated circuits (PICs). We introduce a co-simulation solution for combining the efficient modeling capabilities of a circuit-level simulator, based on analytical models of PIC sub-elements and frequency-dependent scattering matrix (S-matrix) description, and an accurate electromagnetic field simulator that implements the finite element method (FEM) for solving photonic structures with complicated geometries. This is exemplified with the model of a coupled-resonator induced transparency (CRIT), where resonator elements are first modeled in the field simulator. Afterwards, the whole structure is created at a circuit level and statistical analysis of tolerances is investigated.
An overview on recent applications of the finite-element method Maxwell-solver JCMsuite to simulation tasks in nanooptics is given. Numerical achievements in the fields of optical metamaterials, plasmonics, photonic crystal fibers, light emitting devices, solar cells, optical lithography, optical metrology, integrated optics, and photonic crystals are summarized.
Nanostructures, like periodic arrays of scatters or low-index gratings, are used to improve the light outcoupling from organic light-emitting diodes (OLED). In order to optimize geometrical and material properties of such structures, simulations of the outcoupling process are very helpful. The finite element method is best suited for an accurate discretization of the geometry and the singular-like field profile within the structured layer and the emitting layer. However, a finite element simulation of the overall OLED stack is often beyond available computer resources. The main focus of this paper is the simulation of a single dipole source embedded into a twofold infinitely periodic OLED structure. To overcome the numerical burden we apply the Floquet transform, so that the computational domain reduces to the unit cell. The relevant outcoupling data are than gained by inverse Flouqet transforming. This step requires a careful numerical treatment as reported in this paper.
Supported by the European Commission and EURAMET, a consortium of 10 participants from national metrology
institutes, universities and companies has recently started a joint research project with the aim of overcoming current
challenges in optical scatterometry for traceable linewidth metrology and to establish scatterometry as a traceable and
absolute metrological method for dimensional measurements. This requires a thorough investigation of the influence of all significant sample, tool and data analysis parameters, which affect the scatterometric measurement results. For this purpose and to improve the tool matching between scatterometers, CD-SEMs and CD-AFMs, experimental and
modelling methods will be enhanced. The different scatterometry methods will be compared with each other and with
specially adapted atomic force microscopy (AFM) and scanning electron microscopy (SEM) measurement systems.
Additionally novel methods for sophisticated data analysis will be developed and investigated to reach significant
reductions of the measurement uncertainties in critical dimension (CD) metrology. To transfer traceability to industrial
applications of scatterometry an important step and one final goal of this project is the realisation of different waferbased
reference standard materials for calibration of scatterometers. The approaches to reach these goals and first design
considerations and preliminary specification of the scatterometry standards are presented and discussed.
Photolithography simulations are widely used to predict, to analyze and to design imaging
processes in scanners used for IC manufacture. The success of these efforts is strongly dependent
on their ability to accurately capture the key drivers responsible for the image formation. Much
effort has been devoted to understanding the impacts of illuminator and projection lens models on
the accuracy of the lithography simulations [1-3]. However, of equal significance is the role of
the mask models and their interactions with the illuminator models.
We present a finite element method (FEM) solver for computation of optical resonance modes in VCSELs. We
perform a convergence study and demonstrate that high accuracies for 3D setups can be attained on standard
computers. We also demonstrate simulations of thero-optical effects in VCSELs.
Numerical simulations are an important tool for the design of opto-electronical components and devices. In
order to obtain realistic results, a multitude of physical effects and theories have to be included, e.g., Maxwell's
equations for lasing mode computations, heat transfer in active devices, and electronic transport. In our contribution
we perform coupled electro-thermal simulations of high power diode lasers. We analyze the temperature
dependence of the mode profile and far field characteristics. Our results will be compared to experimental
measurements of broad area lasers and will quantitatively describe the effect of thermal blooming.
For nanostructures such as semiconductor quantum dot emitters or biological systems like light harvesting complexes
(photosynthesis) the coupling between individual constituents leads to the formation of delocalized exciton
states. Coherent two dimensional spectroscopy is a versatile tool to investigate the structure of the excitonic
states, whereas nanoplasmonics allows to localize optical fields on a nanoscale: We combine these two methods
in a theoretical study and propose new experiments, such as the two dimensional spectra containing spatial
resolution via localized fields. Using post processing of different spectra with localized fields, we can enhance
certain spectroscopic features in standard coherent spectroscopy, e.g. by suppressing unwanted resonances.
EUV scatterometry is performed on 3D patterns on EUV lithography masks. Numerical simulations of the
experimental setup are performed using a rigorous Maxwell solver. Mask geometry is determined by minimizing
the difference between experimental results and numerical results for varied geometrical input parameters for
the simulations.
Optical metrology by scatterometry usually bases on the comparison of experimental and modeled light field
data. When solving inverse scatterometric problems, often not only a single simulation has to be carried out, but
multiple electromagnetic field solutions have to be computed for varying material and geometrical parameters
of the system under consideration. Then, high computational times for a single forward solution can make the
complete simulation task infeasible. Table based parameter reconstruction on the other hand has the disadvantage
of long offline computational times for creation of the library. Also an increasing number of variable parameters
can not be handled efficiently.
In this contribution we introduce the reduced basis method for creation of highly accurate reduced order
models of parametrized electromagnetic scattering problems. We apply our method to a real-world EUV metrology
application and show speed up factors of about 3000 in reconstruction time. Instead of several minutes or
hours EUV mask parameters can now be obtained in seconds, i.e., in real-time.
Comparison to direct microscopical measurements of the reconstructed geometry demonstrate the good performance
and maturity of our method.
Finite element methods (FEM) for the rigorous electromagnetic solution of Maxwell's equations are known to be
very accurate. They possess a high convergence rate for the determination of near field and far field quantities
of scattering and diffraction processes of light with structures having feature sizes in the range of the light
wavelength. We are using FEM software for 3D scatterometric diffraction calculations allowing the application
of a brilliant and extremely fast solution method: the reduced basis method (RBM). The RBM constructs a
reduced model of the scattering problem from precalculated snapshot solutions, guided self-adaptively by an error
estimator. Using RBM, we achieve an efficiency accuracy of about 10-4 compared to the direct problem with
only 35 precalculated snapshots being the reduced basis dimension. This speeds up the calculation of diffraction
amplitudes by a factor of about 1000 compared to the conventional solution of Maxwell's equations by FEM.
This allows us to reconstruct the three geometrical parameters of our phase grating from "measured" scattering
data in a 3D parameter manifold online in a minute having the full FEM accuracy available. Additionally, also
a sensitivity analysis or the choice of robust measuring strategies, for example, can be done online in a few
minutes.
Simulations of light scattering off an extreme ultraviolet lithography mask with a 2D-periodic absorber pattern
are presented. In a detailed convergence study it is shown that accurate results can be attained for relatively
large 3D computational domains and in the presence of sidewall-angles and corner-roundings.
KEYWORDS: Monte Carlo methods, Silicon, Absorption, Transparent conductors, Interfaces, Solar cells, Glasses, Chemical elements, Silicon films, Optical simulations
We apply a hybrid finite element / transfer matrix solver to calculate generation rate spectra of thin film
silicon solar cells with textured interfaces. Our focus lies on interfaces with statistical rough textures. Due to
limited computational domain size the treatment of such textures requires a Monte Carlo sampling of texture
representations to obtain a statistical average of integral target quantities. This contribution discusses our choice
of synthetic rough interface generation, the Monte Carlo sampling and the need for an incorporation of the cell's
substrate into optical simulation when illumination of the cell happens through the substrate. We present results
of the numerical characterization and generation rates for a single junction cell layout.
We present algorithmic details and applications of the reduced basis method as efficient Maxwell solver to
nanophotonic applications including examples from mask optimization in photolithography and parameter retrieval
in inverse problems, e.g., in optical metrology. The reduced basis method is a currently studied approach
to the multiple solution of problems depending on a number of geometrical, material and source parameters.
Such problems occur frequently in optimization tasks where parameters have to be adjusted in order to minimize
some error functionals or in production environments where deviations from ideal structures have to be
controlled.
Image modeling and simulation are critical to extending the limits of leading edge lithography technologies used
for IC making. Simultaneous source mask optimization (SMO) has become an important objective in the field of
computational lithography. SMO is considered essential to extending immersion lithography beyond the 45nm
node. However, SMO is computationally extremely challenging and time-consuming. The key challenges are due
to run time vs. accuracy tradeoffs of the imaging models used for the computational lithography.
We present a new technique to be incorporated in the SMO flow. This new approach is based on the reduced
basis method (RBM) applied to the simulation of light transmission through the lithography masks. It provides a
rigorous approximation to the exact lithographical problem, based on fully vectorial Maxwell's equations. Using
the reduced basis method, the optimization process is divided into an offline and an online steps. In the offline
step, a RBM model with variable geometrical parameters is built self-adaptively and using a Finite Element
(FEM) based solver. In the online step, the RBM model can be solved very fast for arbitrary illumination
and geometrical parameters, such as dimensions of OPC features, line widths, etc. This approach dramatically
reduces computational costs of the optimization procedure while providing accuracy superior to the approaches
involving simplified mask models. RBM furthermore provides rigorous error estimators, which assure the quality
and reliability of the reduced basis solutions.
We apply the reduced basis method to a 3D SMO example. We quantify performance, computational costs
and accuracy of our method.
Optical resonances in 1D photonic crystal microcavities are investigated numerically using finite-element light
scattering and eigenmode solvers. The results are validated by comparison to experimental and theoretical
findings from the literature. The influence of nanometer-scale geometry variations on the resonator performance
is studied. Limiting factors to ultra-high Q-factor performance are identified.
Optical properties of circular grating resonators in a silicon-on-insulator system are investigated numerically.
These structures are of interest as building-blocks of integrated photonic devices. We use a time-harmonic 3D
finite-element solver for studying transmission of waveguide modes through the system. We compare numerical
results to experimental findings from the literature.
Optical properties of hybrid plasmonic waveguides and of low-Q cavities, formed by waveguides of finite length
are investigated numerically. These structures are of interest as building-blocks of plasmon lasers. We use
a time-harmonic finite-element package including a propagation-mode solver, a resonance-mode solver and a
scattering solver for studying various properties of the system. Numerical convergence of all used methods is
demonstrated.
A bottleneck for computational lithography and optical metrology are long computational times for near field
simulations. For design, optimization, and inverse scatterometry usually the same basic layout has to be simulated
multiple times for different values of geometrical parameters.
The reduced basis method allows to split up the solution process of a parameterized model into an expensive
offline and a cheap online part. After constructing the reduced basis offline, the reduced model can be solved
online very fast in the order of seconds or below. Error estimators assure the reliability of the reduced basis
solution and are used for self adaptive construction of the reduced system.
We explain the idea of reduced basis and use the finite element solver JCMsuite constructing the reduced
basis system. We present a 3D optimization application from optical proximity correction (OPC).
Plasmonic nano antennas are highly attractive at optical frequencies due to their strong resonances - even when their size
is smaller than the wavelength - and because of their potential of extreme field enhancement. Such antennas may be
applied for sensing of biological nano particles as well as for single molecule detection. Because of considerable material
losses and strong dispersion of metals at optical frequencies, the numerical analysis of plasmonic antennas is very
demanding. An additional difficulty is caused when very narrow gaps between nano particles are utilized for increasing
the field enhancement. In this paper we discuss the main difficulties of time domain solvers, namely FDTD and FVTD
and we compare various frequency domain solvers, namely the commercial FEM packages JCMsuite, Comsol, HFSS,
and Microwave Studio with the semi-analytic MMP code that may be used as a reference due to its fast convergence and
high accuracy. The current version of this paper has had a correction made to it at the request of the author. Please see the linked Errata for further details.
Light transmission through a 2D-periodic array of small rectangular apertures in a film of highly conductive
material is simulated using a finite-element method. It is demonstrated that well converged results are obtained
using higher-order finite-elements. The influence of the array periodicity and of corner roundings on transmission
properties is investigated.
Light transmission through circular subwavelength apertures in metallic films with surrounding nanostructures
is investigated numerically. Numerical results are obtained with a frequency-domain finite-element method.
Convergence of the obtained observables to very low levels of numerical error is demonstrated. Very good
agreement to experimental results from the literature is reached, and the utility of the method is demonstrated
in the investigation of the influence of geometrical parameters on enhanced transmission through the apertures.
We have developed an interface which allows to perform rigorous electromagnetic field (EMF) simulations with
the simulator JCMsuite and subsequent aerial imaging and resist simulations with the simulator Dr.LiTHO.With
the combined tools we investigate the convergence of near-field and far-field results for different DUV masks.
We also benchmark results obtained with the waveguide-method EMF solver included in Dr.LiTHO and with
the finite-element-method EMF solver JCMsuite. We demonstrate results on convergence for dense and isolated
hole arrays, for masks including diagonal structures, and for a large 3D mask pattern of lateral size 10 microns
by 10 microns.
Extreme ultraviolet (EUV) lithography is seen as a main candidate for production of future generation computer
technology. Due to the short wavelength of EUV light (≈ 13 nm) novel reflective masks have to be used in the
production process. A prerequisite to meet the high quality requirements for these EUV masks is a simple and
accurate method for absorber pattern profile characterization.
In our previous work we demonstrated that the Finite Element Method (FEM) is very well suited for the simulation
of EUV scatterometry and can be used to reconstruct EUV mask profiles from experimental scatterometric
data.
In this contribution we apply an indirect metrology method to periodic EUV line masks with different critical
dimensions (140 nm and 540 nm) over a large range of duty cycles (1:2, ... , 1:20). We quantitatively compare
the reconstructed absorber pattern parameters to values obtained from direct AFM and CD-SEM measurements.
We analyze the reliability of the reconstruction for the given experimental data. For the CD of the absorber
lines, the comparison shows agreement of the order of 1nm.
Furthermore we discuss special numerical techniques like domain decomposition algorithms and high order
finite elements and their importance for fast and accurate solution of the inverse problem.
Scatterometry, the analysis of light diffracted from a periodic structure, is a versatile metrology for characterizing
periodic structures, regarding critical dimension (CD) and other profile properties. For extreme ultraviolet (EUV) masks,
only EUV radiation provides direct information on the mask performance comparable to the operating regime in an EUV
lithography tool. With respect to the small feature dimensions on EUV masks, the short wavelength of EUV is also
advantageous since it increases the sensitivity for small structural details. Measurements using PTB's EUV reflectometer
at the storage ring BESSY II showed that it is feasible to derive information on the absorber line profile in periodic areas
of lines and spaces by means of rigorous numerical modeling with the finite element method (FEM). A prototype EUV
mask with fields of nominally identical lines was used for the measurements. In this contribution we correlate the
scatterometry data to CD-SEM and surface nano probe measurements of the line profiles as provided by the mask
supplier. We discuss status of the determination of CD and side-wall geometry by scatterometry using rigorous FEM
calculations of EUV diffraction and directions for further investigations.
Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical
metrology and design of nanostructured optical components. A vectorial finite element method (FEM) is a good
choice for an accurate modeling of complicated geometrical features. However, from a numerical point of view
solving the arising system of linear equations is very demanding even for medium sized 3D domains. In numerics,
a domain decomposition method is a commonly used strategy to overcome this problem. Within this approach
the overall computational domain is split up into smaller domains and interface conditions are used to assure
continuity of the electromagnetic field. Unfortunately, standard implementations of the domain decomposition
method as developed for electrostatic problems are not appropriate for wave propagation problems. In an earlier
paper we therefore proposed a domain decomposition method adapted to electromagnetic field wave propagation
problems. In this paper we apply this method to 3D mask simulation.
We discuss realization, properties and performance of the adaptive finite element approach to the design of optical
waveguides. Central issues like the construction of higher-order vectorial finite elements, local error estimation,
automatic and adaptive grid refinement, transparent boundary conditions and fast linear system solution by
domain decomposition techniques will be discussed.
We present rigorous 3D EMF simulations of isolated features on photomasks using a newly developed finite-element
method. We report on the current status of the finite-element solver JCMsuite, incorporating higher-order
edge elements, adaptive refinement methods, and fast solution algorithms. We demonstrate that rigorous
and accurate results on light scattering off isolated features can be achived at relatively low computational cost,
compared to the standard approach of simulations on large-pitch, periodic computational domains.
Scatterometry, the analysis of light diffracted from a periodic structure, is a versatile metrology for characterizing
periodic structures, regarding critical dimension (CD) and other profile properties. For extreme ultraviolet (EUV) masks,
only EUV radiation provides direct information on the mask performance comparable to the operating regime in an EUV
lithography tool. With respect to the small feature dimensions on EUV masks, the short wavelength of EUV is also
advantageous since it provides more diffraction orders as compared to UV. First measurements using PTB's EUV
reflectometer at the storage ring BESSY II showed that it is feasible to derive information on the line profile in periodic
areas of lines and spaces by means of rigorous numerical modeling. A prototype EUV mask with a matrix of test fields
each divided into subfields containing among others test fields with lines & spaces was used for the measurements. In this
contribution we summarize our present results in determining line profile parameters using scatterometry and
reflectometry to provide the input data for the determination of CD and side-wall geometry using rigorous calculations of
EUV diffraction. Particularly, we present a first investigation on the influence of line edge roughness and CD uniformity
by correlating in-plane scatterometry data for the discrete diffraction orders corresponding to the pitch of the structure to
out-of-plane measurements of diffusely scattered light induced by line edge roughness and CD uniformity. We
demonstrate the influence of diffuse scattering on the determination of CD and side-wall geometry using only the
discrete in-plane diffraction orders. To this aim we perform finite element (FEM) simulations on 2D computational domains.
Extreme ultraviolet (EUV) lithography is seen as the main candidate for production of next generation computer
technology. Due to the short wavelength of EUV light (≈ 13 nm) novel reflective masks have to be used in the
production process. The high quality requirements for these EUV masks make it necessary to measure and
characterize their pattern profile.
Here we present numerical simulations of EUV masks with the finite element method (FEM) which allow the
reconstruction of geometrical mask parameters like critical dimension (CD), sidewall angles, layer thicknesses
from experimental scatterometry data.1 Special numerical techniques like domain decomposition algorithms and
high order finite elements become very important to obtain accurate numerical results in small computational
time.
We analyze the sensitivity of scatterometry with respect to the geometrical line profile. We demonstrate the
determination of line profiles from experimental scatterometry data and compare our values to direct microscopic
measurements using CD-SEM and AFM.
Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical
metrology and optics design of nanostructured components. As has been shown in previous benchmarks some of
the presently used methods suffer from low convergence rates and/or low accuracy of the results and exhibit very
long computation times1, 2 which makes application to extended 2D layout patterns impractical. We address 3D
simulation tasks by using a finite-element solver which has been shown to be superior to competing methods by
several orders of magnitude in accuracy and computational time for typical microlithography simulations.2 We
report on the current status of the solver, incorporating higher order edge elements, adaptive refinement methods,
and fast solution algorithms. Further, we investigate the performance of the solver in the 3D simulation project
of light diffraction off an alternating phase-shift contact-hole mask.
Adaptive finite elements are the method of choice for accurate simulations of optical components. However as
shown recently by Bienstman et al. many finite element mode solvers fail to compute the propagation constant's
imaginary part of a leaky waveguide with sufficient accuracy. In this paper we show that with a special goal
oriented error estimator for capturing radiation losses this problem is overcome.
Hollow-core holey fibers are promising candidates for low-loss guidance of light in various applications, e.g., for the use in laser guide star adaptive optics systems in optical astronomy. We present an accurate and fast method for the computation of light modes in arbitrarily shaped waveguides. Maxwell's equations are discretized using vectorial finite elements (FEM). We discuss how we utilize concepts like adaptive grid refinement, higher-order finite elements, and transparent boundary conditions for the computation of leaky modes in photonic crystal fibers. Further, we investigate the convergence behavior of our methods. We employ our FEM solver to design hollow-core photonic crystal fibers (HCPCF) whose cores are formed from 19 omitted cladding unit cells. We optimize the fiber geometry for minimal attenuation using multidimensional optimization taking into account radiation loss (leaky modes).
We present rigorous simulations of EUV masks with technological imperfections like side-wall angles and corner roundings. We perform an optimization of two different geometrical parameters in order to fit the numerical results to results obtained from experimental scatterometry measurements. For the numerical simulations we use an adaptive finite element approach on irregular meshes. This gives us the opportunity to model geometrical structures accurately. Moreover we comment on the use of domain decomposition techniques for EUV mask simulations. Geometric mask parameters have a great influence on the diffraction pattern. We show that using accurate simulation tools it is possible to deduce the relevant geometrical parameters of EUV masks from scatterometry measurements. This work results from a collaboration between AMTC (mask fabrication), Physikalisch-Technische Bundesanstalt (scatterometry) and ZIB/JCMwave (numerical simulation).
We perform 3D lithography simulations by using a finite-element
solver.
To proof applicability to real 3D problems we investigate
DUV light propagation through a structure of size 9μm x 4μm x 65nm.
On this relatively large computational domain we
perform rigorous computations (No Hopkins) taking into account
a grid of 11 x 21 source points with two polarization directions
each.
We obtain well converged results with an accuracy of the
diffraction orders of about 1%.
The results compare well to experimental aerial imaging results.
We further investigate the convergence of 3D solutions towards
quasi-exact results obtained with different methods.
Rigorous numerical simulations of Maxwell's equations are extremely helpful in the understanding of physical effects in nano-optics and are essential for the design of nano-optical devices. We have developed a finite-element (FEM) package for the solution of eigenvalue and scattering problems resulting from Maxwell's equations. We have implemented higher order vectorial elements, adaptive mesh refinement, transparent boundary conditions based on the Pole condition, and fast algorithms. In this contribution we discuss the application of the FEM solvers to recent experiments in nano-photonics.
Miniaturized optical resonators with spatial dimensions of the order of the wavelength of the trapped light offer prospects for a variety of new applications like quantum processing or construction of meta-materials. Light propagation in these structures is modelled by Maxwell's equations. For a deeper numerical analysis one may compute the scattered field when the structure is illuminated or one may compute the resonances of the structure. We therefore address in this paper the electromagnetic scattering problem as well as the computation of resonances in an open system. For the simulation effcient and reliable numerical methods are required which cope with the infinite domain. We use transparent boundary conditions based on the Perfectly Matched Layer Method (PML) combined with a novel adaptive strategy to determine optimal discretization parameters like the thickness of the sponge layer or the mesh width. Further a novel iterative solver for time-harmonic Maxwell's equations is presented.
An extremely fast time-harmonic finite element solver developed for the transmission analysis of photonic crystals was applied to mask simulation problems. The applicability was proven by examining a set of typical problems and by a benchmarking against two established methods (FDTD and a differential method) and an analytical example. The new finite element approach was up to 100 times faster than the competing approaches for moderate target accuracies, and it was the only method which allowed to reach high target accuracies.
KEYWORDS: Magnetism, Split ring resonators, Geometrical optics, Near infrared, Light scattering, Chemical elements, Metamaterials, Finite element methods, Polarization, Gold
It seems to be feasible in the near future to exploit the properties of left-handed metamaterials in the telecom or even in the optical regime. Recently, split ring-resonators (SRR's) have been realized experimentally in the near infrared (NIR) and optical regime.1, 2 In this contribution we numerically investigate light propagation through an array of metallic SRR's in the NIR and optical regime and compare our results to experimental results. We find numerical solutions to the time-harmonic Maxwell's equations by using advanced finite-element-methods (FEM). The geometry of the problem is discretized with unstructured tetrahedral meshes. Higher order, vectorial elements (edge elements) are used as ansatz functions. Transparent boundary conditions (a modified PML method3) and periodic boundary conditions4 are implemented, which allow to treat light scattering problems off periodic structures.
This simulation tool enables us to obtain transmission and reflection spectra of plane waves which are incident onto the SRR array under arbitrary angles of incidence, with arbitrary polarization, and with arbitrary wavelength-dependencies of the permittivity tensor. We compare the computed spectra to experimental results and investigate resonances of the system.
We present a finite-element simulation tool for calculating
light fields in 3D nano-optical devices. This allows to solve challenging problems on a standard personal computer. We present solutions to eigenvalue problems, like Bloch-type eigenvalues in photonic crystals and photonic crystal waveguides, and to scattering problems, like the transmission through finite photonic crystals.
The discretization is based on unstructured tetrahedral grids with
an adaptive grid refinement controlled and steered by an error-estimator. As ansatz functions we use higher order, vectorial elements (Nedelec, edge elements). For a fast convergence of the solution we make use of advanced multi-grid algorithms adapted for the vectorial Maxwell's equations.
We present our simulation tool JCMmode for calculating propagating modes of an optical waveguide. As ansatz functions we use higher order, vectorial elements (Nedelec elements, edge elements). Further we construct transparent boundary conditions to deal with leaky modes even for problems with inhomogeneous exterior domains as for integrated hollow core Arrow waveguides. We have implemented an error estimator which steers the adaptive mesh refinement. This allows the precise computation of singularities near the metal's corner of a Plasmon-Polariton waveguide even for irregular shaped metal films on a standard personal computer.
The "HiPhoCs" program, a cluster of projects supported by the German
Ministry of Education and Research, is focused on the proof of
applicability of planar photonic crystals based on slab waveguides
within telecom transmission lines and optical networks. Results of
the "HiPhoCs" program on modeling and etching aiming at the optimization of device oriented structures in the III-V and SOI material systems are reported as well as their application to key components such as PhC-based WDM-filters, dispersion compensators,
integrated lasers and their coupling to the optical fiber infrastructure.
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