Christophe Caloz - Electromagnetic Metasurfaces

( 2.23)

where is the plasma frequency. Equation ( 2.23) is referred to as the Lorentz model and its frequency dependence is plotted in Figure 2.2. Note that the susceptibility ( 2.23) is a complex quantity that naturally satisfies Kramers–Kronig relations ( Eqs. (2.13)and ( 2.14)).

An important particular case of the Lorentz model ( 2.23) is the Drude model, which applies to metals. In a metal, the electrons are free charge carriers, which implies that the restoring force ( 2.16) is zero, leading to in ( 2.23). The Drude model is particularly useful for modeling metals in the optical regime, where the frequency of light approaches the plasma frequency.

The plasma frequency and damping of aluminum, gold, and silver, which are metals frequently used in the optical regime, are reported in Table 2.1. As the frequency increases and approaches the plasma frequency, these metals behave more and more as lossy dielectrics. In contrast, in the microwave regime, metals behave as perfect electric conductors (PEC), with negligible dispersion. This is of particular importance for the practical realization of metallic-based metasurfaces, especially in the optical regime where the lossy nature of these metals becomes substantial.

To further illustrate the dispersive behavior of these metals, Figure 2.3plots the real and imaginary parts of their permittivity in the optical regime. At the longer wavelengths, they behave as expected, i.e. they exhibit a permittivity with large negative real and imaginary parts. However, as the wavelength tends toward the plasma frequency, their optical behavior changes: they become less opaque and thus let light penetrate deeper in them.

Figure 2.2 Dispersive response of the electric susceptibility of a resonant structure for the parameter ( Eq. (2.23)).

Table 2.1 Plasma frequency (corresponding wavelength) and damping for three common metals [116].

Metal	Plasma frequency, (eV)	Damping, (eV)
Ag	9.013 (137.56 nm)	0.018
Au	9.026 (137.36 nm)	0.0267
Al	14.75 (84.06 nm)	0.0818

In addition of being temporally dispersive, a medium may also be spatially dispersive. As temporal dispersion is a temporally nonlocal phenomenon, spatial dispersion is a spatially nonlocal phenomenon, whereby the response of the medium at the position depends on its excitation at position [9, 29, 52, 148]. This effect is important in metamaterials because it is fundamentally related to bianisotropy and artificial magnetism, as we shall now demonstrate.

Figure 2.3 Experimental dispersion curves of the permittivity of silver, gold, and aluminum [72, 102]. (a) Real part. (b) Imaginary part.

Note that an extensive treatment of the topic of spatial dispersion would be beyond the scope of this book. Here, we thus limit ourselves to a brief and simplified description of this phenomenon, while more advanced presentations may be found in [9, 29, 52, 148].

In order to show how spatial dispersion brings about bianisotropy and artificial magnetism in the constitutive relations, consider a medium with the conventional constitutive relations

( 2.24a)

( 2.24b)

where we have expressed the material polarization density, , in terms of the induced current density, (A/ ). Spatial dispersion is now introduced via the relationship between the induced current and the electric field [29],

( 2.25)

where the dyadic tensor represents the response of the medium and is a volume of integration, centered at . We next restrict our attention to the case of weak spatial dispersion, where . We can then simplify ( 2.25) by using the three-dimensional Taylor expansion of truncated to the second order [29],

( 2.26)

where the subscripts and run over , , and , which decomposes the second term in three terms and the third term in nine terms. Substituting ( 2.26) into ( 2.25) leads, after somewhat involved manipulations and simplifications [29], to