Electromagnetic Metasurfaces. Christophe Caloz

Electromagnetic Metasurfaces

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, bold-script upper E left-parenthesis bold r comma t right-parenthesis and to the polarization density as bold-script upper E Subscript loc Baseline left-parenthesis bold r comma t right-parenthesis equals bold-script upper E left-parenthesis bold r comma t right-parenthesis plus bold-script upper P left-parenthesis bold r comma t right-parenthesis slash left-parenthesis 3 epsilon 0 right-parenthesis [140], which reduces (2.20) into

(2.21)

where omega 0 equals StartRoot omega Subscript normal r Superscript 2 Baseline minus upper N q Subscript normal e Superscript 2 Baseline slash left-parenthesis 3 m Subscript normal e Baseline epsilon 0 right-parenthesis EndRoot is the resonant frequency. In the harmonic regime, this equation becomes

(2.22) minus omega squared bold upper P plus j Baseline 2 omega normal upper Gamma bold upper P plus omega 0 squared bold upper P equals StartFraction upper N q Subscript normal e Superscript 2 Baseline Over m Subscript normal e Baseline EndFraction bold upper E comma

where bold upper P is the time-domain Fourier transform of bold-script upper P left-parenthesis bold r comma t right-parenthesis and bold upper E that of bold-script upper E left-parenthesis bold r comma t right-parenthesis . Finally substituting bold upper P equals epsilon 0 chi Subscript ee Baseline bold upper E , eliminating , and solving for yields the dispersive susceptibility

(2.23) chi Subscript ee Baseline left-parenthesis omega right-parenthesis equals StartFraction omega Subscript normal p Superscript 2 Baseline Over omega 0 squared minus omega squared plus j Baseline 2 omega normal upper Gamma EndFraction comma

where omega Subscript normal p Baseline equals StartRoot upper N q Subscript normal e Superscript 2 Baseline slash left-parenthesis epsilon 0 m Subscript normal e Baseline right-parenthesis EndRoot 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 omega 0 equals 0 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.

Graph depicts the dispersive response of the electric susceptibility of a resonant structure for the parameter ,!ωp/Γ=10.

Figure 2.2 Dispersive response of the electric susceptibility of a resonant structure for the parameter omega Subscript normal p Baseline slash normal upper Gamma equals 10 (Eq. (2.23)).

Table 2.1 Plasma frequency

(corresponding wavelength) and damping

for three common metals [116].

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Metal	Plasma frequency, (eV)	Damping, (eV)
Ag	9.013 (137.56 nm)	0.018