Electromagnetic wave transformation in an oscillator medium with growing density

The model of a medium made of oscillators can describe various materials, including artificial ones made of meta-atoms. Here the transformation of an electromagnetic wave in such a medium with rapidly growing oscillator density is studied. The initial polariton is shown to be transformed not only into new polariton modes but also into natural oscillations of the existing and created oscillators. The oscillations produce zero net polarization but consume significant amount of the initial polariton energy. Although the boundary conditions at the temporal jump are sufficient to find the new polaritons, the description of the polarization oscillations requires integrating the material equations because the oscillations are uncoupled from the fields. Under some conditions, the spatial dependence of the amplitude of the oscillations can provide a snapshot of the electromagnetic field distribution at the moment of rapid density growth. Some subtle issues related to the continuity conditions at the density jump are also discussed.

Figure 1

Illustration of wave transformation in an oscillator medium with growing density. Initially, $t<0$, a plane wave propagates inside the medium with some oscillator density (blue oscillators). At $t=0$, new oscillators are added (light blue oscillators). This results in the formation at $t>0$ of transmitted and reflected waves at frequencies that differ from that of the initial wave.

Figure 2

Dispersion $k\left(\omega \right)$ for the waves in the oscillator medium at several values of the density parameter ${\mathrm{\Omega }}_p$. The dots labeled 1, 2, 3 on the dispersion curves illustrate how the conservation of the wave number during the density growth from ${\mathrm{\Omega }}_p/{\mathrm{\Omega }}_0=0.5$ to ${\mathrm{\Omega }}_p/{\mathrm{\Omega }}_0=1$ transforms the initial wave defined by dispersion point 1 into two other waves defined by 2 and 3.

Figure 3

Energy density partition $W_{E,H,P}/W_t$ in a polariton for two values of the oscillator density (a) ${\mathrm{\Omega }}_p/{\mathrm{\Omega }}_0=0.2$ and (b) ${\mathrm{\Omega }}_p/{\mathrm{\Omega }}_0=1.2$. The gray regions in (a) and (b) show the frequency ranges forbidden for propagation.

Figure 4

Frequencies ${\omega }_{2,3}$ of the modes excited by a plane wave at frequency ${\omega }_1$ as functions of the density of the created oscillators (a) without any initially existing oscillators ${\mathrm{\Omega }}_{p1}=0$ and (b) with some initially existing oscillators ${\mathrm{\Omega }}_{p1}/{\mathrm{\Omega }}_0=0.5$. The dots show the initial frequencies in each case.

Figure 5

Schematics of polarization compensation at the natural frequency ${\mathrm{\Omega }}_0$. The existing and created oscillators induce polarizations ${\mathbf{P}}_A$ and ${\mathbf{P}}_B$, respectively, which are equal in magnitude but counter-directed at any time moment.

Figure 6

Spatial distribution of the energy of free oscillations $W_{\mathrm{fo}}\left(x\right)=W_A\left(x\right)+W_B\left(x\right)$, its components $W_{A,B}\left(x\right)$, and the absolute value of the true initial electric field $E_y(x,t=0)=2E_{1}cos\left(k_{1}x\right)$. The parameters are ${\omega }_1/{\mathrm{\Omega }}_0=0.2, {\mathrm{\Omega }}_{p1}/{\mathrm{\Omega }}_0=0.5, {\mathrm{\Omega }}_{p2}/{\mathrm{\Omega }}_0=1$ for which the averaged energies obey $W_A^{-}/W_A^{+}=W_B^{-}/W_B^{+}=4/9, W_A/W_B=3, W_{\mathrm{fo}}/W_1=0.0832$. The $x$ values are normalized to ${\lambda }_1=2\pi /k_1$.

Figure 7

Normalized energy densities of the modes excited by a plane wave at frequency ${\omega }_1$ as a function of the density of created oscillators. (a) Energy density of the modes $W_{2,3}^{\pm }/W_1$ in the case without any initially existing oscillators ${\mathrm{\Omega }}_{p1}=0$. (b, c) Energy density of the modes $W_{2,3}^{\pm }/W_1, W_{\mathrm{fo}}/W_1$ in the case with initially existing oscillators at densities (b) ${\mathrm{\Omega }}_{p1}/{\mathrm{\Omega }}_0=0.5$ and (c) ${\mathrm{\Omega }}_{p1}/{\mathrm{\Omega }}_0=1.2$. In panel (c) the energies $W_2^{\pm }$ were multiplied by 20 to make them visible.