Dielectric function representation by fractional (non-integer) oscillators in the model-based optical metrology

The fractional oscillator model, which generalizes the classical linear harmonic oscillator in the framework of fractional calculus, is described. A concept of fractional integral/derivative, i.e., the integration and differentiation of arbitrary order, was known for a quite long time, but it is largely unfamiliar to the ellipsometry community. In this paper, an application of the fractional oscillator model with an external multiplicative noise for accurate parameterization of the dielectric response of materials is presented. Supported by several examples, we demonstrate that this approach can be used to explicitly represent the complex dielectric function of various materials in the context of model-based spectroscopic ellipsometry data analysis. In particular, this parametric formulation enables a smooth interpolation between various process stages in real-time ellipsometric applications using a minimal number of free parameters.