Recent observation of black hole and gravitational wave has stirred up great interest of Einstein's general relativity. In optical system, the "optical black hole" has also been a key topic in mimicking black holes. Another good way to study or mimic general relativity effects is based on transformation optics. In this paper, we propose a way by utilizing transformation optics theory to directly obtain the equivalent isotropic refractive index profiles which are the analogies of some static spaces of general relativity, such as de Sitter space, Anti de Sitter space, and Schwarzschild black hole. We find that, the analogue of de Sitter space is the Poincaré disk, while Anti de Sitter space is equivalent to Maxwell's fish eye lens. In particular, we prove that the "optical black hole" actually has infinite number of photon spheres, while our black hole only has a single one, which is closer to the real black hole. We study the effect from both geometric optics and wave optics. It can also be generalized to mimic any kind of metrics. Furthermore, with the isotropic refractivity index profile, we visualize the gravitational lensing effect of black hole from our software TIM. The image not only recovers the donut-liked halo of black hole, but also shows other phenomena.

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Hyperbolic shear polaritons (HShPs) emerge with widespread attention as a class of polariton modes with broken symmetry due to shear lattices. We find a mechanism of generating quasi-HShPs(q-HShPs). When utilizing vortex waves as excitation sources of hyperbolic materials without off-diagonal elements, q-HShPs will appear. In addition, these asymmetric q-HShPs can be recovered as symmetric modes away from the source, with a critical transition mode between the left-skewed and right-skewed q-HShPs, via tuning the magnitude of the off-diagonal imaginary component and controlling the topological charge of the vortex source. It is worth mentioning that we explore the influence of parity of topological charges on the field distribution and demonstrate these exotic phenomena from numerical and analytical perspectives. Our results will promote opportunities for both q-HShPs and vortex waves, widening the horizon for various hyperbolic materials based on vortex sources and offering a degree of freedom to control various kinds of polaritons.

Transformation optics (TO) facilitates flexible designs of spatial modulation of optical materials via coordinate transformations, thus, enabling on-demand manipulations of electromagnetic waves. However, the application of TO theory in control of hyperbolic waves remains elusive due to the spatial metric signature transition from ( + , + ) to ( − , + ) of a two-dimensional hyperbolic geometry. Here, we proposed a distinct Pythagorean theorem, which leads to establishing an anisotropic Fermat’s principle. It helps to construct anisotropic geometries and is a powerful tool for manipulating hyperbolic waves at the nanoscale and polaritons. Making use of absolute instruments, the excellent collimating and focusing behaviors of naturally in-plane hyperbolic polaritons in van der Waals α – MoO 3 layers are demonstrated, which opens up a new way for polaritons manipulation.

General relativity establishes the equality between matter-energy density and the Riemann curvature of spacetime. Therefore, light or matter will be bent or trapped when passing near the massive celestial objects, and Newton’s second law fails to explain it. The gravitational effect is not only extensively studied in astronomy but also attracts a great deal of interest in the field of optics. People have mimicked black holes, Einstein’s ring, and other fascinating effects in diverse optical systems. Here, with a gradient index lens, in the geometrical optics regime, we mimic the Schwarzschild precession in the orbit of the star S2 near the Galactic Center massive black hole, which was recently first detected by European Southern Observatory. We also find other series of gradient index lenses that can be used to mimic the possible Reissner–Nordström metric of Einstein’s field equation and dark matter particle motion. Light rays in such gradient lenses will be closed in some cases, while in other cases it would be trapped by the center or keep dancing around the center. Our work presents an efficient toy model to help investigate some complex celestial behaviors, which may require long period detection by using high-precision astronomical tools. The induced gradient lenses enlightened by the gravitational effect also enrich the family of absolute optical instruments for their selective closed trajectories.