Fresnel coefficients at an interface with a lamellar composite material

Different effective-medium theories (EMT's) are used to describe the high-frequency and optical properties of composite materials. However, these theories reveal not only differences in the evaluation of the effective permeability and permittivity, but also in their definitions. Rytov gave definitions of the effective permeability and permittivity that are clearly incompatible with the extended Bruggeman definitions when the skin effect occurs. An analysis of the exactly solvable case of a lamellar composite is performed using both approaches. Since most experimental determinations of the permeability and permittivity of composites rely on reflection-transmission measurements, it is of foremost importance to determine which definitions of the permeability and permittivity should be used to express the Fresnel coefficients under the conventional farm. For that purpose, we derive the reflection and transmission coefficients at an interface between a composite material and the air, without any effective-medium hypothesis for the composite. This derivation is performed on a periodic composite containing conducting inclusions separated by a dielectric plane. We point out that in the interface region, evanescent modes are present and cannot be described by an effective-medium approach. We infer the proper definitions of the permeability and the permittivity of a composite from the expression of the Fresnel coefficients and from the expression of the refractive index of the propagative mode. We show that the extended Bruggeman definitions are basically correct, but that small correction terms due to the modes at the interface should be taken into account in some cases. A numerical example is given to show these interface effects. An experimental result is also presented. It illustrates that the permeability determined from reflection-transmission measurement disagrees with the definitions given by Rytov but agrees with our definitions.