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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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 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).
Numerical design inverse reconstruction and parameter estimation of optical systems usually involves the multiple
solution of an e.g. geometrically parameterized system. Long computational times however can rule out many
possible applications like inverse scatterometry.
The reduced basis method allows to split up the solution process of an e.g. geometrically parameterized
system into an expensive offline and a cheap online part. In the offline phase the reduced basis is computed selfadaptively
by solving the underlying model several times. During the real-time application the reduced system
is solved in the order of seconds even for 3D problems. Error estimators assure the reliability of the reduced
basis solutions.
In our contribution we explain general ideas of the reduced basis method and apply it to the simulation of
light scattering from 2D and 3D parameterized photo masks. We compare computational times and accuracy of
reduced basis and rigorous finite element simulations.
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.
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.
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.
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.
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.
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).
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