2.10 Metamaterial

Metamaterial is a metamaterial medium (negative index material, left-handed material LHM). For metamaterial medium both, complex relative permittivity and complex relative permeability are given by one of three dispersion models: Drude, Debye, or Lorentz. QuickWave allows for considering single-, dual-, and triple-pole electric and magnetic dispersion, which is given by arbitrarily weighted composition of the above dispersion models. For this medium type:

Sigma = s [S/m] - electric conductivity

and

SigmaM = s_m/(k)⁴ [W/m] - magnetic loss

are defined analogously as for dielectric isotropic.

Its complex relative permittivity (including series losses) is given by Debye, Drude or Lorentz dispersion model with user-specified parameters:

Drude:             er(w) = eps_inf  + (2p f_p)² / (j w 2p v_c  - w²)

Debye:             er(w) = eps_inf  + (eps_s - eps_inf) / (1 + j w tau)

Lorentz:          er(w) = eps_inf  + (eps_s - eps_inf) (2p f_p)² / ((2p f_p)²  + j w 2p v_c - w²)

where:

eps_inf - relative permittivity at infinite frequency [unitless],

eps_s - static relative permittivity [unitless],

tau - relaxation time [ns],

v_c - inverse of the relaxation time (collision frequency) [GHz],

f_p - pole frequency (plasma frequency) [GHz],

amp - weight coefficient of the corresponding dispersive pole (relevant for dual- and triple-pole dispersive medium models only) [unitless].

For Debye and Lorentz, eps_s < eps_inf is not allowed, and eps_s = eps_inf causes that the medium is considered as dielectric isotropic with frequency-independent permittivity eps_inf and conductivity Sigma.

For Drude and Lorentz, f_p =0 causes that the medium is considered as dielectric isotropic with frequency-independent permittivity eps_inf and conductivity Sigma.

For Debye, tau=0 causes that the medium is considered as dielectric isotropic with frequency-independent permittivity eps_s and conductivity Sigma.

There are also limits on the admissible level of losses with respect to the FDTD time step Δt. For Debye, the requirement is tau > 0.5 Δt; for Drude and Lorentz, (2p v_c)^-1 > 0.5 Δt.

Its complex relative permeability (including series magnetic loss) are given by Debye, Drude or Lorentz dispersion model with user-specified parameters:

Drude:             m_r(w) = mu_inf  + (2pf_p)² / (j w2pv_c  - w²)

Debye:            m_r(w) = mu_inf  + (mu_s - mu_inf) / (1 + j wtau)

Lorentz:          m_r(w) = mu_inf  + (mu_s - mu_inf) (2pf_p)² / ((2pf_p)²  + j w2pv_c - w²)

where:

mu_inf  - relative permeability at infinite frequency [unitless],

mu_s - static relative permeability [unitless],

tau - relaxation time [ns],

v_c - inverse of the relaxation time (collision frequency) [GHz],

f_p - pole frequency (plasma frequency) [GHz],

amp weight coefficient of the corresponding dispersive pole (relevant for dual- and triple-pole dispersive medium models only) [unitless].

For Debye and Lorentz, mu_s < mu_inf is not allowed, and mu_s = mu_inf causes that the medium is considered as dielectric isotropic with frequency-independent permeability mu_inf and magnetic loss SigmaM.

For Drude and Lorentz, f_p =0 causes that the medium is considered as dielectric isotropic with frequency-independent permeability mu_inf and magnetic loss SigmaM.

For Debye, tau=0 causes that the medium is considered as dielectric isotropic with frequency-independent permeability mu_s and magnetic loss SigmaM.

There are also limits on the admissible level of losses with respect to the FDTD time step Δt. For Debye, the requirement is tau > 0.5 Δt; for Drude and Lorentz, (2p v_c)^-1 > 0.5 Δt.