Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials

A metamaterial (MM) differs from an ordinary optical material mainly in that it has a dispersive magnetic permeability and offers greatly enhanced design freedom to alter the linear and nonlinear properties. This makes it possible for us to control the propagation of ultrashort electromagnetic pulses at will. Here we report on generic features of ultrashort electromagnetic pulse propagation and demonstrate the controllability of both the linear and nonlinear parameters of models for pulse propagation in MMs. First, we derive a generalized system of coupled three-dimensional nonlinear Schrödinger equations (NLSEs) suitable for few-cycle pulse propagation in a MM with both nonlinear electric polarization and nonlinear magnetization. The coupled equations recover previous models for pulse propagation in both ordinary material and a MM under the same conditions. Second, by using the coupled NLSEs in the Drude dispersive model as an example, we identify the respective roles of the dispersive electric permittivity and magnetic permeability in ultrashort pulse propagation and disclose some additional features of pulse propagation in MMs. It is shown that, for linear propagation, the sign and magnitude of space-time focusing can be controlled through adjusting the linear dispersive permittivity and permeability. For nonlinear propagation, the linear dispersive permittivity and permeability are incorporated into the nonlinear magnetization and nonlinear polarization, respectively, resulting in controllable magnetic and electric self-steepening effects and higher-order dispersively nonlinear terms in the propagation models.

Figure 1

(Color online) Refraction index $n$, normalized GVD ${\beta }_2$, and the ratio of phase velocity to group velocity $v$ in an ideal lossless MM versus normalized frequency $\varpi$ for ${\varpi }_p=0.8$ (a), 1 (b), and 1.2 (c). ${\beta }_2$ is calculated in units of $1∕\left(c{\omega }_{pe}\right)$. The gray regions are the stopband in which the electromagnetic waves cannot propagate.

Figure 2

(Color online) $g_1$ and $q_1$ versus normalized frequency $\varpi$ for ${\varpi }_p=0.8$ (a), 1 (b), and 1.2 (c) for $f=5$ and 10. In (b), the curves for $g_1$ and $q_1$ are overlapped.

Figure 3

(Color online) Self-steepening parameters, including the conventional SS, the total electric SS, and the total magnetic SS parameters versus normalized frequency $\varpi$ for $f=10$ and ${\varpi }_p=0.8$ (a), 1 (b), and 1.2 (c). In (b), the curves for electric and magnetic SS parameters are overlapped.