Jan Pomplun

Adaptive finite element method for simulation of optical nano structures

We discuss realization, properties and performance of the adaptive finite element approach to the design of nano-photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components into the unbounded and possibly heterogeneous exterior. We apply the finite element method to the optimization of the design of a hollow core photonic crystal fiber. Thereby we look at the convergence of the method and discuss automatic and adaptive grid refinement and the performance of higher order elements.

Accelerated A Posteriori Error Estimation for the Reduced Basis Method with Application to 3D Electromagnetic Scattering Problems

We propose a new method for fast estimation of error bounds for outputs of interest in the reduced basis context, efficiently applicable to real world 3D problems. Geometric parameterizations of complicated 2D, or even simple 3D, structures easily leads to affine expansions consisting of a high number of terms (

The influence of line edge roughness and CD uniformity on EUV scatterometry for CD characterization of EUV masks

\propto100-1000

Finite Element Simulation of Radiation Losses in Photonic Crystal Fibers

). Application of state-of-the-art techniques for computation of error bounds becomes practically impossible. As a way out we propose a new error estimator, inspired by the subdomain residuum method, which leads to substantial savings (orders of magnitude) regarding online and offline computational times and memory consumption. We apply certified reduced basis techniques with the newly developed error estimator to 3D electromagnetic scattering problems on unbounded domains. A numerical example from computational lithography demonstrates the good performance and effectivity of the proposed estimator.

Evaluation of EUV scatterometry for CD characterization of EUV masks using rigorous FEM-simulation

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 PTBs 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.

Metrology of EUV masks by EUV-scatterometry and finite element analysis

In our work we focus on the accurate computation of light propagation in finite size photonic crystal structures with the finite element method (FEM). We discuss how we utilize numerical concepts like high-order finite elements, transparent boundary conditions and goal-oriented error estimators for adaptive grid refinement in order to compute radiation leakage in photonic crystal fibers and waveguides. Due to the fast convergence of our method we can use it e.g. to optimize the design of photonic crystal structures with respect to geometrical parameters, to minimize radiation losses and to compute attenutation spectra for different geometries. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite element simulation of the optical modes of semiconductor lasers

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 PTBs 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 FEM simulation of EUV masks: influence of shape and material parameters

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.

Fast simulation method for parameter reconstruction in optical metrology

In the present paper we investigate optical near fields in semiconductor lasers. We perform finite element simulations for two different laser types, namely a super large optical waveguide (SLOW) laser, which is an edge emitter, and a vertical cavity surface emitting laser (VCSEL). We give the mathematical formulation of the different eigenvalue problems that arise for our examples and explain their numerical solution with the finite element method (FEM). Thereby, we also comment on the usage of transparent boundary conditions, which have to be applied to respect the exterior environment, e.g., the very large substrate and surrounding air. For the SLOW laser we compare the computed near fields to experimental data for different design parameters of the device. For the VCSEL example a comparison to simplified 1D mode calculations is carried out.

Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks

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).