2.8    Dielectric Dispersive Nonlinear

Dielectric dispersive nonlinear is a dispersive dielectric medium with third-order nonlinear polarisation. Its complex permittivity is given by Lorentz dispersion model (single-, dual-, and triple-pole) and the third-order nonlinear polarisation, by Kerr-Raman model. For this medium type:

Mu = m_r [dimensionless] – relative magnetic permeability,

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 Lorentz dispersion model with user-specified parameters:

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

where:

eps_inf [dimensionless] - relative permittivity at infinite frequency,

eps_s [dimensionless] - static relative permittivity,

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

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

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

For Lorentz, eps_s < eps_inf is not allowed. The admissible level of losses with respect to the FDTD time step Δt is:

(2p v_c)^-1 > 0.5 Δt.

The third-order nonlinear polarization of the medium is given by Kerr-Raman model with user-specified parameters:

(Eq. 26-28 in M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron., vol. 40, no. 2, pp. 175-182, 2004)

where:

tau1 = t₁ [fs] - inverse of the characteristic frequency,

tau2 = t₂ [fs] - damping time constant,

hi3 =c₀ [(m/V)^2] - strength of the third-order nonlinearity,

alpha = a [dimensionless] - Kerr contribution to the total Kerr-Raman nonlinearity.