Transformation optics, which is generically applicable to other classical waves such as acoustic and elastic waves, provides an emerging design paradigm to manipulate waves. However, some lenses and optical-transformation devices require a singular refractive index; meeting this requirement is a significant challenge. A method called transmutation can relax some types of index singularity into finite anisotropy around the singularity. Here, we show that such lenses with a singularity for flexural waves can be obtained by approaching a near-zero thickness of the plate precisely at the location of the singularity. As examples, we demonstrate a series of Eaton lenses theoretically and experimentally by projecting the refractive index in space onto the thickness in plates and by working in a broad frequency range in which impedance mismatch is negligible. This framework offers an insight into feasible methods that can be used to develop singular devices such as cloaking devices on thin flexible curved plates and can be further extended to a general methodology for shaping elastic waves. We hope that this elastic platform can also be a test bed to indirectly study unprecedented phenomena enabled by gravitational and quantum fields in terms of analog models.

The properties of the uniform Bose gas are studied within the optimized variational perturbation theory (Gaussian approximation) in a self-consistent way. It is shown that the atomic Bose-Einstein condensate with a repulsive interaction becomes unstable when the gas parameter $\gamma =\rho a^3$ exceeds a critical value ${\gamma }_{crit}\approx 0.01$. The quantum corrections beyond the Bogoliubov-Popov approximation to the energy density, chemical potential, and pressure in powers of $\sqrt{\gamma }$ expansions are presented.

We extended a previous study of the Eaton lens at specific refraction angles to the Eaton lens at any refraction angle. The refractive index of the Eaton lens is complicated and has no analytical form except at a few specific angles. We derived a more accessible form of the refractive index for any refraction angle with some accuracy using the retroreflector approximation. The finding of this study will be useful for a rapid estimation of the refractive index, and the the design of various Eaton lenses.

A three-dimensional acoustic Luneburg meta-lens has the advantage of refracting sound waves for all incident angles and focusing higher sound pressure compared to a two-dimensional lens. The lens made of plastic with a diameter of 120 mm was designed with thousands of lattice column-shaped meta-atoms to maintain its three-dimensional shape. The lens's three-dimensional focusing performance and acoustic imaging were simulated and measured in the frequency range of 5 kHz–20 kHz. The omnidirectional property was confirmed by rotating the lens to change the incident angle and measuring the sound pressure. The development of these spherical Luneburg meta-lenses is expected to improve the performance of devices that require acoustic focusing.

Eaton lens is a typical gradient index lens which has many noble properties. The refractive index of the lens has a singularity at the center of the lens. The singularity makes the generalized Eaton lens rotate waves at arbitrary deflecting angles. The analytic solutions of the generalized Eaton lens were studied herein. The solutions were obtained from a nonlinear equation of various order. Only four deflecting angles produced analytic solutions: 2π, π, π/2, and π/3. The solutions of the other angles are obtained by numerical calculations only. The refractive indices and trajectories at the four angles were plotted. The possible application areas of the lens were discussed.