Theory of electromagnetic wave frequency upconversion in dynamic media

Frequency upconversion of an electromagnetic wave can occur in ionized plasma with decreasing electric permittivity and in split-ring resonator-structure metamaterials with decreasing magnetic permeability. We develop a general theory to describe the evolution of the wave frequency, amplitude, and energy density in homogeneous media with a temporally decreasing refractive index. We find that upconversion of the wave frequency is necessarily accompanied by partitioning of the wave energy into low-frequency modes, which sets an upper limit on the energy conversion efficiency. The efficiency limits are obtained for both varying permittivity and varying permeability.

Figure 1

(a) Frequency of each mode ${\omega }_1$ (solid curve) and ${\omega }_2$ (dashed curve) normalized to the initial wave frequency ${\omega }_{\mathrm{i}}$ and (b) energy density of each mode $W_1$ (solid curve), $W_2$ (dashed curve), and the total energy density $W_{\mathrm{total}}=W_1+W_2$ (thin black curve) normalized to the initial wave energy density $W_{\mathrm{i}}$ as a function of the coupling strength $\sqrt{N{e}^2/\left(m_e{\varepsilon }_0\right)}$ normalized to ${\omega }_{\mathrm{i}}$. The three pairs of curves correspond to ${\omega }_0^2/{\omega }_{\mathrm{i}}^2=0$ (black curve), 0.3 (gray curve), and 0.5 (light gray curve), respectively.

Figure 2

(a) Frequencies of each mode ${\omega }_1$ (solid curves) and ${\omega }_2$ (dashed lines) normalized to the initial wave frequency ${\omega }_{\mathrm{i}}$ and (b) energy density of each mode $W_1$ (solid curves), $W_2$ (dashed curves), and the total energy density $W_{\mathrm{total}}=W_1+W_2$ (thin black curve) normalized to the initial wave energy density $W_{\mathrm{i}}$ as a function of the filling fraction $F$. The three pairs of curves correspond to ${\omega }_0^2/{\omega }_{\mathrm{i}}^2=0.1$ (black curves), 0.3 (gray curves), and 0.5 (light gray curves), respectively.