The spectral function of composites: the inverse problem

The dielectric function of a composite depends on the geometry of the composite and the dielectric functions of the constituent materials. In the Bergman–Milton spectral representation for a two-component composite, all of the relevant geometric information can be captured in a spectral function which is independent of the material properties. Extracting the spectral function from experimental values of the dielectric function would be a compact way of presenting a large body of data and highlight the role of geometry in determining the electrical properties of the composite. We show that known constraints on the spectral function make it possible to solve the inverse problem of determining the spectral function directly from experimental measurements of the reflectance if one of the components has a resonance and data are taken in the restrahlen band, where the real part of the dielectric function of the optically active material is negative. We demonstrate the method using numerical simulations of the reflectance of a model system with physically reasonable values for the dielectric functions of the two components. Our results show that the spectral function determined by this method is stable against the introduction of noise and agrees with that previously calculated directly for the same model geometry. We suggest that this technique will be useful when used with real experimental data.