The missing ingredient in effective-medium theories: Standard deviations

TL;DR

This paper highlights the importance of including standard deviations in effective-medium theories for composite materials, showing that averages alone can be insufficient due to variability depending on incident angles and polarization.

Contribution

It introduces a rigorous method to calculate standard deviations in reflection properties, revealing limitations of average-based effective-medium theories.

Findings

Standard deviations can be significant depending on incident conditions.

Average reflectivities match well with multilayer models.

Variability increases with differences in material parameters.

Abstract

Effective-medium theories for electromagnetic constitutive parameters of particulate composite materials are theories of averages. Standard deviations are absent because of the lack of rigorous theories. But ensemble averages and standard deviations can be calculated from a rigorous theory of reflection by planar multilayers. Average reflectivities at all angles of incidence and two orthogonal polarization states for a multilayer composed of two kinds of electrically thin layers agree well with reflectivities for a single layer with the same overall thickness and a volume-weighted average of the relative permittivities of these two components. But the relative standard deviation can be appreciable depending on the angle of incidence and the polarization state of the incident illumination, and increases with increasing difference between the constitutive parameters of the two layers. This suggests that average constitutive parameters obtained from effective-medium theories do not have uniform validity for all calculations in which they might be used.