Light scattering in pseudopassive media with uniformly balanced gain and loss

We introduce a class of metamaterials with uniformly balanced gain and loss associated with complex permittivity and permeability constants. The refractive index of such a balanced pseudopassive metamaterial is real. An unbounded uniform pseudopassive medium has transport characteristics similar to those of its truly passive and lossless counterpart with the same real refractive index. However, bounded pseudopassive samples show some unexpected scattering features which can be further emphasized by including such elements in a photonic structure.

Figure 1

The electric $E\left(z\right)$ $(y\text{−}z$ plane) and the magnetic $H\left(z\right)$ $(x\text{−}z$ plane) field distribution (amplitude profiles) of the resonance localized mode ${\lambda }_r=1.256$ $\mu \text{m}$ for a Bragg grating with an embedded PP defect (green) (or narrow dark strip at the center, in grayscale print version). The grating consists of 20 lossless bilayers (orange and white) at each side of the defect. Their indexes of refraction are $n_1=1.45$ and $n_2=1.755$ and their widths are $d_1=0.2167$ $\mu \text{m}$ and $d_2=0.179$ $\mu \text{m}$. The PP defect layer has an index of refraction of $n_p=3.48$ corresponding to constituents ${\epsilon }_p=3.48-i0.01$ and ${\mu }_p={\epsilon }_p^{*}$ while its width is $d_p=0.0542$ $\mu \text{m}$.

Figure 2

(a) The transmittance $\mathcal{T}\left(\omega \right)$, reflectance $\mathcal{R}\left(\omega \right),$ and absorption $\mathcal{A}\left(\omega \right)$ for the setup of Fig. 1 $\left({\epsilon }^{\prime \prime }<0\right)$. Because $\mathcal{A}<0$ (amplification) we plot in the subfigure the $\left|\mathcal{A}\right|$. A superunitary transmittance and reflectance is evident in the frequency domain around the resonance frequency. (b) The amplification (absorption) coefficient $\mathrm{\Theta }$ versus frequency has a singular point at the resonance frequency which signifies a lasing action. (c) The same setup as in Fig. 1 for the case of the CPA frequency. In this case the electric field has a minimum at the defect layer while the magentic field has a maximum. (d) The amplification (absorption) coefficient $\mathrm{\Theta }$ versus frequency shows a zero at the CPA frequency.

Figure 3

The electric $E\left(z\right)$ (blue line at the $y\text{−}z$ plane) and the magnetic $H\left(z\right)$ (red line at the $x\text{−}z$ plane) fields (amplitude profiles) of two representative FP modes which reside at the upper (left inset) and lower (right inset) band edges. The grating consists of 50 bilayers (pink color indicates a PP layer and white a lossless passive layer) (or dark and bright layers in grayscale print version, respectively). In the insets we show the field profiles for the first half of the structure, i.e., 25 layers. Left (right) inset correspond to the FP resonance at the upper (lower) edge of the lower (upper) band. The constituents of the PP layers are ${\epsilon }_p=4-0.001i,{\mu }_p={\epsilon }_p^{*}$ with a width layer $d_p=0.1309$ $\mu \text{m}$. The lossless passive layers have constituents $\epsilon =3$ and $\mu =1$ and width $d=0.3023$ $\mu \text{m}$. The transmittance $\mathcal{T}\left(\omega \right)$ is shown in the main panel.