Lorentz model of a metamaterial. In Section 11.2.1, we developed a model for the relative...

Lorentz model of a metamaterial. In Section 11.2.1, we developed a model for the relative permittivity by considering the dynamics of a bound electron in the presence of an external electric field. The formula for єr given by (11.28) is called the Lorentz model. Using (11.5), we can rewrite (11.28a) as

where we added the subscript e to associate the characteristic frequencies with the bound electron model. A similar model may be used to approximate the permeability of artificially constructed metamaterials:28

(a) First consider the lossless case, where κe = κm = 0. Using ωpe = 2π × 8.0× 109 rad-s−1, ωpm = 2π × 7.0 × 109 rad-s−1, ω0e = 2π × 2.8 × 109 rad-s−1, and ω0m = 2π × 2.5 × 109 rad-s−1: (i) Plot the real and imaginary components of єr and μr in the frequency range 4 f11 GHz. (ii) Plot the real and imaginary components of the index of refraction n in the same frequency range. Use the convention that Re{β} 0 for backward propagation. (iii) In the frequency range 4 f 11 GHz, which frequencies allow for wave propagation? What is the frequency range of backward wave propagation? At what frequency is Re{n} = −1? (iv) Plot βc versus f over the domain 4 f 11 GHz, and comment on the sign of the phase and group velocities over frequencies where propagating solutions exist. (b) Repeat part (a) using κe = κm = 0.05ωpe .