Electromagnetic Metasurfaces. Christophe Caloz

t EndFraction bold-script upper P left-parenthesis bold r comma t right-parenthesis plus omega Subscript normal r Superscript 2 Baseline bold-script upper P left-parenthesis bold r comma t right-parenthesis equals StartFraction upper N q Subscript normal e Superscript 2 Baseline Over m Subscript normal e Baseline EndFraction bold-script upper E Subscript loc Baseline left-parenthesis bold r comma t right-parenthesis period"/>

For electrically small particles, we may use the Clausius–Mosotti expression, which relates the local field to the total electric, and to the polarization density as [140], which reduces (2.20) into

      (2.21)

      where is the resonant frequency. In the harmonic regime, this equation becomes

      (2.22)

      where is the time-domain Fourier transform of and that of . Finally substituting , eliminating , and solving for yields the dispersive susceptibility

(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 2.3 plots 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