PHYSICAL REVIEW APPLIED 15, 034039 (2021) Singular Lenses for Flexural Waves on Elastic Thin Curved Plates Dongwoo Lee , 1 Yiran Hao , 2 Jeonghoon Park , 1 In Seok Kang , 3 Sang-Hoon Kim , 4 Jensen Li , 2, * and Junsuk Rho 1,3, † 1 Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea 2 Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 3 Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea 4 Division of Marine Engineering, Mokpo National Maritime University, Mokpo 58628, Republic of Korea (Received 2 September 2020; revised 10 January 2021; accepted 10 February 2021; published 15 March 2021) 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. DOI: 10.1103/PhysRevApplied.15.034039 I. INTRODUCTION Transformation optics (TO) allows us to achieve coun- terintuitive phenomena such as invisibility cloaking in an inhomogeneous medium and can be extended to a broader context of illusion optics [1–4]. The general recipe of using coordinate transformations and inhomogeneous media with TO can be further used to realize gradient- index (GRIN) devices such as the Luneburg lens, the Maxwell fish-eye lens, the field concentrator, and the Eaton lens [5–16]. In particular, the Eaton lens gives U-turn transport of light; this turn corresponds to a nonreserving mirror effect that yields a flipped image while performing an unusual type of retroreflection on all rays that come in and out through the same side. However, this device requires an extremely large refractive index n, i.e., a sin- gularity at the center of the device. For this reason, among others, the device is impractical. Progress in understanding * jensenli@ust.hk † jsrho@postech.ac.kr of the theory of TO has not removed the problem of achieving this singularity in practice. Alternatively, nonsingular properties have been theoret- ically proposed by transmuting the singularity of n in the Eaton lens [17] and have been experimentally proven in the microwave domain using metamaterials constructed of double split-ring resonators [18]. Although not universal, such a technique can be applied to certain types of sin- gularity. The transmutation relaxes the singularity to an anisotropic medium in a principal axis. The requirement for the anisotropic medium imposes design limitations that impede the development of a simple methodology. With- out use of the transmutation method, an Eaton lens that induces 90 ◦ bending has been demonstrated experimen- tally by controlling an isotropic n while modulating the height of PMMA on a gold film for surface plasmon polari- tons [8]. Given that the maximum n was restricted to 1.54, the previous proof-of-concept study did not fulfill the need of a singularity in n. A later model of the same 90 ◦ ben- der has used woodpile photonic crystals for n ≤ √ 3 and compound liquids as a special treatment to achieve √ 2.5 ≤ n ≤ √ 40 in gigahertz waves [19]. An effect similar to the 2331-7019/21/15(3)/034039(9) 034039-1 © 2021 American Physical Society