Electromagnetic Metasurfaces. Christophe Caloz

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      2

Electromagnetic Properties of Materials

      In order to effectively model, synthesize, and analyze a metasurface, we need to understand how it interacts with electromagnetic waves. From a general perspective, understanding the interactions of a given structure with an electromagnetic wave requires two fundamental prerequisites: (i) a description of the structure in terms of electromagnetic material parameters and (ii) the availability of appropriate boundary conditions. The purpose of this chapter is to address the first prerequisite, while the second one will be dealt with in Chapter 3.

      The concepts discussed in this chapter are presented for the general perspective of volume materials, and hence deal with 3D material parameters. However, they also essentially apply to metasurfaces, which are modeled by 2D material parameters throughout the book, as will be established in Chapter 3.

      This chapter presents a general description of the medium parameters and constitutive relations of materials. It also provides a detailed discussion of the physical properties that are inherently related to these parameters. Specifically, Section 2.1 introduces the conventional bianisotropic constitutive relations. Section 2.2 describes the temporal response of matter (temporal dispersion) and the mechanism responsible for resonances. It also provides the fundamental relationship between causality, and the real and imaginary parts of material parameters through the Kramers–Kronig relations. Section 2.3 presents the spatially dispersive nature of matter (spatial dispersion), which helps understanding the fundamental origin of bianisotropy. Sections 2.4 and 2.5 derive the Lorentz reciprocity theorem and the Poynting theorem, respectively. Based on the Poynting theorem, Section 2.6 then deduces energy conservation relations for lossless-gainless systems both in terms of susceptibilities and scattering parameters. Finally, Section 2.7 classifies bianisotropic media according to their fundamental material properties.

2.1 Bianisotropic Constitutive Relations

The electromagnetic response of a linear medium may be expressed in the frequency domain, using the MKS system of units, as

(2.1a)

      (2.1b)

      where (C/) and (Wb/) are, respectively, the electric displacement vector and the magnetic flux density vector, which depend on the applied electric field vector (V/m) and on the magnetic field vector (A/m) as well as on the material polarizations via the electric polarization density vector (C/) and the magnetic polarization density vector (A/m). The constants F/m and H/m are the vacuum permittivity and permeability, respectively.

The polarization densities in (2.1) are typically expressed in terms of quantities which are either microscopic, the polarizabilities, or macroscopic, the susceptibilities. Although both the microscopic and macroscopic descriptions are applicable for modeling metasurfaces, we will mostly use the macroscopic model throughout the book, because it more conveniently describes metasurfaces as homogeneous media. For a bianisotropic metasurface, the polarization densities in (2.1) read

(2.2a)

      (2.2b)

where m/s and are the speed of light in a vacuum and the vacuum impedance, respectively, and , , , and are the electric, magnetic, magnetic-to-electric, and electric-to-magnetic susceptibility tensors, respectively. Note that these susceptibilities are all unitless. Substituting (2.2) into (2.1) yields the bianisotropic constitutive relations

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