Conformal hyperbolic optics

Hyperbolic metamaterials have attracted considerable interest in the research community for their peculiar ability to enhance control of electromagnetic waves' propagation. Although conformal transformation optics provides a unique platform for metamaterials design, this method has not been used for hyperbolic metamaterials yet. This comes from the lack of a well-defined mathematical structure. We extend conformal transformation optics to hyperbolic metamaterials, by applying Clifford algebra to analyze light propagation. We will show that the effective line element of a trajectory of light propagation conforms to ultrahyperbolic—not Euclidean—geometry. We also, by using conformal hyperbolic mapping, obtain all conformal spaces with Minkowski space-time. Finally, we employ this theory to study the electric-field pattern of dipoles.

Figure 1

The hyperbolic space-time which describes the isotropic HMM is illustrated. Their associated typical trajectory rays are demonstrated by the yellow line $(z=0.2y)$ and the green line $(z=1.8y)$, while the red lines characterize the horizon $\left|z\right|=\left|y\right|$.

Figure 2

Dipole fields ${\lambda }^3{E}_z(0,y,z)$ and ${\lambda }^3{E}_y(0,y,z)$ in the HMM with ${\epsilon }_{\perp }=-1$ and ${\epsilon }_{\parallel }=3$, and ${\epsilon }_{\perp }=1$ and ${\epsilon }_{\parallel }=-3$ in (a) and (b), respectively.

Figure 3

The conformal space-times with conformal mapping $\chi ={\zeta }^2$ and its associated typical trajectory rays are demonstrated by the yellow $(z=0.2y)$ and green $(z=1.8y)$ lines while the red lines characterize the horizon in (a); dipole fields ${\lambda }^3{E}_z(0,y,z)$ and ${\lambda }^3{E}_y(0,y,z)$ are illustrated in (b) and (c), respectively. (d)–(f), (g)–(i), and (j)–(l) demonstrate conformal mappings $\chi ={\zeta }^3$, $\chi =exp\kappa \zeta$, and $\chi =arctan\zeta$ and associated rays and dipole field, respectively. In all plots we consider ${\epsilon }_{\perp }=-1$ and ${\epsilon }_{\parallel }=3$ for $E_z$ and ${\epsilon }_{\perp }=1$ and ${\epsilon }_{\parallel }=-3$ for $E_y$.

Figure 4

Numerically simulated field distributions of the (hyperbolic) Bessel function of order $l=0$ in conformal space with conformal maps $\chi =\zeta +R\left(\varphi \right)\zeta$, for $\varphi =6^{\circ }$, ${30}^{\circ }$, and ${60}^{\circ }$ in (a), (b), and (c), respectively; (d)–(f) demonstrate the (hyperbolic) Bessel function of order $l=1$ for the same angles.

Figure 5

Distribution of the (hyperbolic) Bessel function of order $l=0,1,2$, in the ultrahyperbolic space in (a), (b), and (c), respectively; the same distributions are plotted in (d)–(f), (g)–(i), (j)–(l), and (m)–(o) in conformal spaces with conformal maps $\chi ={\zeta }^2$, $\chi ={\zeta }^3$, $\chi =exp\kappa \zeta$, with $\kappa =0.7$, and $\chi =arctan\zeta$, respectively.