Description:

Starting with the Maxwell equations and the corresponding nonlinear wave equation, a pulse evolution equation for a metamaterial with cubic (Kerr) nonlinearity has been derived. The given model equation goes beyond the commonly employed slowly evolving wave approximation (SEWA). The dispersive properties of the dielectric susceptibility and magnetic permeability are accounted for in accordance with the Drude model. Using this model equation, propagation properties of a single-cycle pulse and a 4-cycle pulse are studied numerically in a left-handed metamaterial (LHMM) with third-order nonlinearity.
Figure.1. (i) Frequency dependence of the real parts of the dielectric permittivity, Re(ε), and the magnetic susceptibility, Re(μ), and the real and imaginary parts, Re(n) and Im(n), of the refractive index for γ=5 × 10−4. (ii) The dimensionless electric field, E, of the input (solid line) and the output (dotted line) single-cycle pulse as a function of the dimensionless time, t, after a propagation distance of 7λp. The solid line corresponds to the input pulse, while the dashed and dotted lines represent the output pulse with ωm∕ωp =0.8 and ωm∕ωp =1.2, respectively. (iii) Input (dashed line) and the output (solid line) dimensionless electric field of 4-cycle pulse after a propagation distance of 0.9λp for ωm∕ωp =0.8. In (ii) and (iii) both χ(3) =10−10 esu, and E0 =3 × 104 Statvolts∕cm.
Figure.2. (a) The dimensionless intensity, |E|2, as a function of the dimensionless transverse radial distance, r, and the dimensionless propagation distance, ξ, of a single-cycle pulse after a propagation distance of 4 × 10−4λp (b) The dimensionless intensity, |E|2, as a function of the dimensionless time, t, and the dimensionless propagation distance, ξ, of a single-cycle pulse after a propagation distance of 4 × 10−4λp.
Figure.3. (a) Same as Fig.2(b) for a 4-cycle pulse after a propagation distance of 8 × 10−4λp. (b) The dimensionless intensity, |E|2, as a function of the dimensionless transverse coordinates, x and y, of a 4-cycle pulse after a propagation distance of 8 × 10−4λp.
Figure.4. (a) Same as Fig.2(b) after a propagation distance of 8 × 10−4λp. (b) Same as Fig.3(b) for of a single-cycle pulse.
In figures (2), (3) and (4) ωm/ωp =0.8, χ(3) =10−10 esu, and E0 = 3 × 104 Statvolts∕cm.