Optical 3D simulations in many-query and real-time contexts require new solution strategies. We study an adaptive, error controlled reduced basis method for solving parameterized time-harmonic optical scattering problems. Application fields are, among others, design and optimization problems of nano-optical devices as well as inverse problems for parameter reconstructions occurring e. g. in optical metrology. The reduced basis method presented here relies on a finite element modeling of the scattering problem with parameterization of materials, geometries and sources.

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.

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.

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.

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.

Optical 3D simulations in many-query and real time contexts require new solution strategies. We study an adaptive, error controlled reduced basis method for solving parametrized time-harmonic optical scattering problems. Application fields are, among others, design and optimization problems of nano-optical devices as well as inverse problems for parameter reconstructions, as they occur, e. g., in optical metrology. The reduced basis method presented here relies on a finite element modeling of the problem plus parametrization of materials, geometries and sources.

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.

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.

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.

Of keen interest to the IC industry are advanced computational lithography applications such as Optical Proximity Correction of IC layouts (OPC), scanner matching by optical proximity effect matching (OPEM), and Source Optimization (SO) and Source-Mask Optimization (SMO) used as advanced reticle enhancement techniques. The success of these tasks is strongly dependent on the integrity of the lithographic simulators used in computational lithography (CL) optimizers. Lithographic mask models used by these simulators are key drivers impacting the accuracy of the image predications, and as a consequence, determine the validity of these CL solutions. Much of the CL work involves Kirchhoff mask models, a.k.a. thin masks approximation, simplifying the treatment of the mask near-field images. On the other hand, imaging models for hyper-NA scanner require that the interactions of the illumination fields with the mask topography be rigorously accounted for, by numerically solving Maxwell's Equations. The simulators used to predict the image formation in the hyper-NA scanners must rigorously treat the masks topography and its interaction with the scanner illuminators. Such imaging models come at a high computational cost and pose challenging accuracy vs. compute time tradeoffs. Additional complication comes from the fact that the performance metrics used in computational lithography tasks show highly non-linear response to the optimization parameters. Finally, the number of patterns used for tasks such as OPC, OPEM, SO, or SMO range from tens to hundreds. These requirements determine the complexity and the workload of the lithography optimization tasks. The tools to build rigorous imaging optimizers based on first-principles governing imaging in scanners are available, but the quantifiable benefits they might provide are not very well understood. To quantify the performance of OPE matching solutions, we have compared the results of various imaging optimization trials obtained with Kirchhoff mask models to those obtained with rigorous models involving solutions of Maxwell's Equations. In both sets of trials, we used sets of large numbers of patterns, with specifications representative of CL tasks commonly encountered in hyper-NA imaging. In this report we present OPEM solutions based on various mask models and discuss the models' impact on hyper- NA scanner matching accuracy. We draw conclusions on the accuracy of results obtained with thin mask models vs. the topographic OPEM solutions. We present various examples representative of the scanner image matching for patterns representative of the current generation of IC designs.

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, time-harmonic, high accuracy solutions of the full 3D vectorial Maxwell's equations(1) 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.

We use a finite-element method to obtain highly converged results for a nano-optical light scattering setup with a non-periodic geometry.

Of keen interest to the IC industry are advanced computational lithography applications such as Optical Proximity Correction, OPC, Optical Proximity Effect matching, OPEM, and Source-Mask Optimization, SMO. Lithographic mask models used by these simulators and their interactions with scanner illuminator models are key drivers impacting the accuracy of the image predications of the computational lithography applications. To construct topographic mask model for hyper-NA scanner, the interactions of the fields with the mask topography have to be accounted for by numerically solving Maxwell’s equations. The simulators used to predict the image formation in the hyper-NA scanners have to rigorously treat the topographic masks and the interaction of the mask topography with the scanner illuminators. Such mask models come at a high computational cost and pose challenging accuracy vs. compute time tradeoffs. To address the high costs of the computational lithography for hyper-NA scanners, we have adopted Reduced Basis, RB, method to efficiently extract accurate, near field images from a modest sample of rigorous, Finite Element, FE, solutions of Maxwell’s equations for the topographic masks. The combination of RB and FE methods provides means to efficiently generate near filed images of the topographic masks illuminated at oblique angles representing complex illuminator designs. The RB method’s ability to provide reliable results from a small set of pre-computed, rigorous results provides potentially tremendous computational cost advantage. In this report we present RB/FE technique and discuss the accuracy vs. compute time tradeoffs of hyper-NA imaging models incorporating topographic mask images obtained with the RB/FE method. The examples we present are representative of the analysis of the optical proximity effects for the current generation of IC designs.

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.

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.