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An analytical and general form factor for any polyhedron is derived on the basis of a projection method, in terms of the vertex coordinates and topology of the polyhedron. An integral over the polyhedron equals the sum of the signed integrals over a set of dissected tetrahedra by defining a sign function, and a general tetrahedral form factor is established by defining a projection method. All possible singularities present in the formula are discussed in detail. Using a MATLAB implementation, illustrative examples are discussed to verify the accuracy and generality of the method. The use of the scalar product operation and the sign function in this work allows a general and neat formula to be obtained for any polyhedron, including convex and concave polyhedra. The formulas and discussions presented here will be useful for the characterization of nanoparticles using small-angle scattering techniques.

The quality of the measured signature is influenced not only by the instrument’s precision but also by the selected measurement configuration. In optical scatterometry, the purpose of measurement configuration optimization (MCO) is to select an optimal or suboptimal combination of measurement conditions, such as the angles of incidence, azimuth, polarization and wavelength, to achieve higher measurement accuracy. This analysis not only requires an effective optimization strategy but is also time-consuming. In this work, we propose a general MCO method that incorporates error propagation theory and condition-number-based error estimation technique, by which the MCO problem can be formulated as an optimization problem for the condition number of the coefficient matrix in the linear estimation of parameter deviations. The method is demonstrated on a multi-wavelength Mueller matrix scatterometry measuring a Si grating. With the help of the neural-network-based surrogate model, the feasibility of the method is verified by making a comparison with Latin Hypercube sampling. Fitting results of the measured and calculated Mueller matrix spectra obtained at the selected optimal measurement configuration show a good agreement. The proposed method is promising to provide an alternate solution to globally evaluate the MCO problem in optical scatterometry and other measurement scenarios.