Caustics due to a Negative Refractive Index in Circular Graphene $p\mathrm{\text{−}}n$ Junctions

We show that the wave functions form caustics in circular graphene $p\mathrm{\text{−}}n$ junctions which in the framework of geometrical optics can be interpreted with a negative refractive index.

Figure 1

Incident plane wave of electron in single layer graphene is scattered by a rotational symmetric potential $V(r)$ (circular region).

Figure 2

$|\Psi {|}^2$ (in scale of logarithmic to base 10) is plotted inside and outside, close to the scattering region (the dashed line shows the boundary of the $p\mathrm{\text{−}}n$ junction). Here $k_{\mathrm{in}}R=300$ and $k_{\mathrm{out}}R=300$ corresponding to $n=-1$, and $x$ and $y$ are in units of $R$. The solid (dotted) line corresponds to the caustic for $p=1$ ($p=2$) (see the text).

Figure 3

The same as in Fig. 2 with $k_{\mathrm{in}}R=300$ and $k_{\mathrm{out}}R=200$ corresponding to $n=-1.5$.

Figure 4

The basic geometry of ray paths inside the circular $p\mathrm{\text{−}}n$ junction. The incident ray from left with impact parameter $b$ and angle of incidence $\alpha$ is refracted at the boundary of the junction with angle $\beta$ and then after $p-1$ internal reflections exits from the junction. Here $p=3$ and we set $n=-1.3$ and $\alpha =60°$.

Figure 5

Caustic inside the $p\mathrm{\text{−}}n$ junction formed from the envelope of the refracted ray paths of different incident rays. The thick solid line is the caustic curve calculated from (9). Here $p=1$ and $n=-1$.