Localized $\mathcal{PT}$-symmetric directionally invisible scatterers

We demonstrate how to create localized $\mathcal{PT}$-symmetric directionally invisible scatterers directly from a governing wave equation. Moreover, we can construct general non-$\mathcal{PT}$-symmetric objects which exhibit directional invisibility, though such an effect is typically associated with $\mathcal{PT}$-symmetric objects. Whereas previously the determining condition for an optical $\mathcal{PT}$-symmetric device was a $\mathcal{PT}$-symmetric complex refractive index, we show that a broader condition, requiring only the scattering potential to be $\mathcal{PT}$-symmetric, leads to the same behavior. This enables the construction of $\mathcal{PT}$-symmetric objects without a $\mathcal{PT}$-symmetric complex refractive index, effectively doubling the number of possible invisible objects. Consequently, the set of gain-loss invisible objects is much broader than previously realized, and several examples are shown.

Figure 1

Real part (a) and imaginary part (b) of the scattering potential $F\left(\mathbf{r}\right)$ associated with Eq. (7), $(a=1)$.

Figure 2

Real part of the total field (incident + scattered) when the incident plane wave ($\lambda =1$) propagates in (a) the positive $\widehat{\mathbf{x}}$ direction $(\theta =0^{\circ })$ and (b) the negative $\widehat{\mathbf{x}}$ direction $(\theta ={180}^{\circ })$. The circles indicate the domain of the scatterer, which has a radius $a=1$. $U_0$ is the incident field amplitude.

Figure 3

Extinction and scattering cross sections as a function of incident angle [$\mathcal{PT}$-symmetric example, Eq. (7)].

Figure 4

Real (a) and imaginary (b) parts of the scattering potential for a non-$\mathcal{PT}$-symmetric nonscattering scatterer $(a=1)$.

Figure 5

Real part of the total field (incident + scattered) when the incident plane wave propagates in (a) the positive $\widehat{\mathbf{x}}$ direction ($\theta =0^{\circ }$) and (b) the negative $\widehat{\mathbf{x}}$ direction ($\theta ={180}^{\circ }$). The fields are scaled to the incident field amplitude $U_0$. The circles indicate the domain of the scatterer, which has a radius of $a=1$. The extinction and scattering cross sections are shown in (c) for all angles $\theta$ or incidence.