Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures
- Gross, Hermann
- Rathsfeld, Andreas
2010 Mathematics Subject Classification
- 78A46 65N30 65K05
2008 Physics and Astronomy Classification Scheme
- Diffraction gratings, inverse problems, sensitivity analysis
In this work, we discuss some aspects of numerical algorithms for the determination of periodic surface structures (gratings) from light diffraction patterns. With decreasing structure details of lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method are applied to simple periodic line structures in order to evaluate the quality of the manufacturing process. Using scatterometry, geometrical parameters of periodic structures including period (pitch), side-wall angles, heights, top and bottom widths of trapezoid shaped bridges can be determined. The mathematical model for the scattering is based on the time-harmonic Maxwell's equations and reduces in case of grating structures to the Helmholtz equation. For the numerical simulation, e.g. finite element methods can be applied to solve the corresponding boundary value problems. More challenging is the inverse problem, where the grating geometry is to be reconstructed from the measured diffraction patterns. Restricting the class of gratings and the set of measurements, the inverse problem can be reformulated as a non-linear operator equation in Euclidean spaces. The operator maps the parameters describing the grating to special efficiencies of plane wave modes diffracted by the grating. We employ a Newton type iterative method to solve this operator equation. The reconstruction properties and the convergence of the numerical algorithm, however, is controlled by the local conditioning of the non-linear mapping, i.e. by the condition numbers of its Jacobian matrix. To improve the convergence of the iteration and the accuracy of the reconstruction, we determine optimal sets of efficiencies for the measurements by optimizing the condition numbers of the corresponding Jacobians. Numerical examples for a chrome-glass mask and for an inspecting light of wave length 632.8 nm confirm that an optimization of the measurement data results in better solutions.
- Waves Random Complex Media, 18 (2008) pp. 129--149.