The paper is concerned with well-posedness of TE and TM polarizations of time-harmonic electromagnetic scattering by perfectly conducting periodic surfaces and periodically arrayed obstacles with local perturbations. The classical Rayleigh Expansion radiation condition does not always lead to well-posedness of the Helmholtz equation even in unperturbed periodic structures. We propose two equivalent radiation conditions to characterize the radiating behavior of time-harmonic wave fields incited by a source term in an open waveguide under impenetrable boundary conditions. With these open waveguide radiation conditions, uniqueness and existence of time-harmonic scattering by incoming point source waves, plane waves and surface waves from locally perturbed periodic structures are established under either the Dirichlet or Neumann boundary condition. A Dirichlet-to-Neumann operator without using the Green's function is constructed for proving well-posedness of perturbed scattering problems.
Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for extreme ultraviolet lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. The inverse problem can be formulated as a nonlinear operator equation in Euclidean space. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gauß–Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period in the range of 420 nm–840 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.
We investigate the impact of line-edge and line-width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The two-dimensional (2D) rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method providing a numerical solution of the 2D Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using simulated data with LER/LWR perturbed efficiencies for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.
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