Realization of Ultrathin Waveguides by Elastic Metagratings

Guiding transports of classical waves has inspired a wealth of nontrivial physics and momentous applications in a wide range of fields. To date, a robust and compact way to guide energy flux travelling along an arbitrary, prescheduled trajectory in a uniform medium is still a fundamental challenge. Here we propose and experimentally realize a generic framework of ultrathin waveguides for full-angle wave trapping and routing. The metagrating-based waveguide can totally suppress all high-order parasitic diffractions to efficiently route guided elastic waves without leakage. Remarkably, the proposed waveguide protype works in a broad frequency range from 12 to 18 kHz and regardless of the incident angle. An analytical slab-waveguide model is further presented to predict and tailor the diffracted patterns in the metagrating-based waveguide. Compared with existing methods based on topological edge states or defected metamaterials, our meta-waveguide strategy exhibits absolute advantages in compact size, robust performance, and easy fabrication, which may provide a new design paradigm for vibration control in solids, wave steering in electromagnetics, acoustics and other waves.


Introduction
Guiding waves in a desired manner plays an essential role in long-distance energy transport and thus remains as a hot topic in both fundamental physics and applied science 1 . The origin of waveguide can be traced back to the first "Light Fountain" experiment conducted by Jean-Daniel Colladon in 1842 2,3 . Due to the total-internal-reflection (TIR) beyond a critical incidence angle, the light was trapped inside the "water channel" with a higher refractive index than the surrounding air. This light-guiding fancy has inspired not only modern telecommunication and sensing devices like optical fibers 3,4 , but also the ongoing developments of guided-mode physics for slab-waveguides 5 , nonlinear optics 6 and optical cavities 7 . However, such a high-index waveguide paradigm is only available for partial incidence angles. In contrast, an omnidirectional total-wave-trapping in highly confined waveguide can be only realized by introducing extremely mismatched (soft/hard) impedance boundaries 5,8 . For example, in an airborne waveguide, air and solids serve as the waveguiding and extremely hard surrounding media, respectively. It seems unprocurable to realize a waveguide between arbitrary media with similar impedances or even in an uniform medium.
To this end, guiding waves along an arbitrary path has been enabled by artificially exotic media, such as acoustic/elastic metamaterials with well-engineered defects [9][10][11] . By underpinning band gaps and periodically arranged defects in sonic crystals or metamaterials, we can confine wave paths in the defected area and guide wave propagation along a prospective route, even regardless of sharp corners. Nevertheless, the inherent narrow working band, energy leakage and large footprints still pose insurmountable obstacles for the popularization of this method. As another emerging scheme, nontrivial topological phases have rapidly ushered in a significant revival of waveguide fashion in very recent decade [12][13][14][15] . Achieved by the analogues of quantum/quantum-spin Hall effect 14 , Dirac cones 13,16 , gapless states 17 or Weyl systems 18,19 , topological edge states have realized robust acoustic/elastic waveguides, which are stable against local perturbations and capable of backscattering immune. However, apart from poor coupling with the background media, the topological waveguide framework still suffers from the challenges of extremely narrow bandwidth, rigorously exquisite complexity and bulky volume. Such severe constraints are so general in all existing designs that undeniably hinder the technical availability and fabrication feasibility to a large extent.
Without losing functionality, metasurfaces are capable of great flexibility and rich wave phenomena, such as wave shaping [25][26][27][28] , absorbing 29-31 and isolating 32,33 . These versatile modulation abilities arise from the well-tailored abrupt phase variation, which can be programmed by the generalized Snell's law (GSL) 20 , ( − sin ) 0 = , where and are the angles of incident and outgoing waves, = ⁄ is the phase gradient and the integer denotes the diffraction mode. Based on the GSL, it is easy to see that the 0 ℎ transmitted mode ( = 0) is always existing, which indicates that the waves can transmit unavoidably from one side to another side. As a result, an efficient total-wave-blocking seems impossible, let alone a further waveguiding using metasurfaces. Excitingly, metagrating was very recently demonstrated promising in remedying this capability limitation of metasurface [34][35][36] . Emphasizing on the unity efficiency, metagratings can modify the GSL with a supercell concept, which consists of only a few (or even single) unit cells. By doing this, anomalous refractions or/and reflections of every higher-order diffraction mode could be coherently assigned 36,37 . It is noteworthy, as an essential factor, each sub-unit in the metagrating should be high-efficient to guarantee the whole steering efficiency. However, more to the point of total-wave-trapping, the existing local-resonance-based designs are not sufficient to satisfy such criteria 33,38 . To the best of our knowledge, no work has been reported on a high-efficient, broadband, omnidirectional wave-blocking by metagratings or metasurfaces. Thus far, a physical realization of compact waveguide for all-directional wavetrapping and guiding remains an outstanding fundamental challenge.
In this work, by employing a bipartite metagrating paradigm, we propose an ultrathin waveguide framework for omnidirectional trapping and efficient routing of elastic-waves under a full-angle source excitation. Only using two unit-cells as a subwavelength supercell, we present theoretically and experimentally that the ultrathin waveguide can isolate all wave orders and guide elastic-waves along an arbitrary path. Moreover, an analytical slab-waveguide model is proposed to predict and steer the guided-mode physics. Fundamentally distinct from the topological edge states or defected metamaterials, we demonstrate that the proposed waveguide system exhibits a nontrivially ultrathin, robust and broadband guiding feature.

Results
Design paradigm and mechanism. Figure 1a illustrates the schematic of an ultrathin waveguide routing along an arbitrary path for flexural waves in a plate. It consists of two layers of thin elastic metagratings to function as curbs of the propagation path. To achieve efficient wave routing without leakage, the unique property of our metagratings is that they can totally reflect incident waves from any directions, as illustrated in Fig. 1b. To this end, we design the metagratings based on the following mechanism. We first introduce a passive elastic metagrating with periodically where is the wavelength, is the diffraction order, subscripts and represent the transmitted and reflected waves, respectively. It indicates that the selection of transmission or reflection performance for the outgoing waves can be determined by the integer-parity design of metagratings, which has been experimentally verified recently in both acoustics 36 and elastodynamics 37 . Based on Eq. (1), we then introduce the following design criteria to the elastic metagrating, By doing this, only = 0 is permitted under such design, all the other diffraction orders can be totally suppressed. Recalling that = 2 is selected as an even number here, it is evident that an efficient specular reflection with = can be achieved regardless of the incident angle. In this way, by assembling two omnidirectionally reflected metagratings along an arbitrary route, we can totally trap and guide waves at will.
The omnidirectionally specular reflection of our metagratings can also be interpreted from a diffraction-based perspective in the -space. As illustrated in Fig. 1c, upon the incident wave, the metagrating with period and wave number = 2 / generates multiple diffraction modes, which are depicted by the green arrows. Due to the integer-parity design of metagratings ( = 2 here), the directions of arrows are different for different modes based on Eq. (1). It is important to note that, although infinite modes can be excited according to the diffraction theory, most modes are evanescent within the near field. As a result, only a few diffraction modes in the range of ∈ [− 0 , 0 ] can be propagated into the far field, where 0 = 2 / is the wave number of the background medium. It is readily to see that, in the case of > 2 0 (corresponding to / > 2 ), there exists only one propagating mode, i.e., the 0 ℎ order mode referring to specular reflection. At the end of this part, we would like to stress that the proposed waveguide is composed of two thin layers only, which exhibits great advantages in compact size than previous strategies of  To evaluate the robustness of the proposed metagrating, we further investigate the transmittance | | at different frequencies and in Fig. 2b. Remarkably, | | maintains a stably small value (less than 0.32, corresponding to 10% of energy) for all incident angles within a wide operating frequency range of 8~18 kHz, except for an abrupt disturbance near 12 kHz. The broadband robustness of the omnidirectional reflection stems from the relatively stable phase shift between the two unit cells, as illustrated in Fig. 2c. It is noted that Δ maintains an approximate value to in the frequency range of 8~18 kHz, although the supercell is designed exactly for 15 kHz. The slight variation roughly satisfies the design criterion of 2 -phase-wrap per supercell, leading to a broadband performance of omnidirectional reflection. Furthermore, it is evident that there is an abrupt peak at 12 kHz, which corresponds to an ultra-narrow disturbance in the  Supplementary Fig. 6. If the point source 1 (or 2 ) is placed outside (or inside) the enclosed region in Fig. 2f (or Fig. 2g), almost all waves are bounced back by the metagrating boundary (or confined in the triangle region). Therefore, the metagrating acts like a "cage" to almost totally trap the wave energy and isolate it from outside. Significantly, this isolation performance is omnidirectional, and no wave leakage occurs in a full-angle range. We further quantitatively evaluate the normalized energy along a straight line across the "cage" boundary (the yellow dashed line in Fig. 2g). As shown in Fig. 2h, the square of the displacement amplitude  highly efficient transmission even after continuous sharp corners, like a N-shape path (Fig. 4a). It could also route the omnidirectional waves to make a smooth U-turn with almost no leakage (Fig.   4b). Moreover, the waveguide framework could realize multiple elastic-wave propagation paths.
As illustrated in Fig. 4c, two incident waves from point sources 1 and 2 can travel along two parallel paths without any influence with each other even the adjacent distance is less than one waveguide width. Furthermore, as shown in Fig. 4d, we can divide one elastic wave beam into two branches by using the waveguide junction for more functional switching routes. Such robust routing performance allows us to demonstrate a novel and highly efficient waveguide strategy with outstanding ultrathin footprint, which is totally distinct from topological states or defected metamaterials. channel width . Specifically, single or multiple rows of discrete standing wave packets are generated in waveguides with different normalized widths / . We here show that such interesting field spectrums can be interpreted by the guided-mode theory. In consideration of the distinct difference from classical slab waveguides 5 , we propose a modified guided-mode theory to predict the guided-mode pattern in the metagrating-based waveguide for a full-angle wave incidence.
As illustrated in Fig. 5a, we first consider a pair of wave beams generated from the point source and then propagating in the waveguide with incident angles ± , respectively.
Different from classical waveguides using high-index heterogeneous materials 5 , the wave number in the proposed waveguide is equal to that in the host plate. As discussed in previous sections, the wave beam is totally reflected at the metagrating and then travels in a zig-zag path. Once touching the metagrating, the wave beam will be mirrored back to the waveguide but with a phase shift . To have a guided mode, point before boundary reflection and point after boundary reflection possess the same phase with their projections and , respectively. Namely, the round-trip phase acquired by the ray must be an integer multiple of 2π, where is an integer corresponding to the order of the guided mode = + 1. Noting that − = 2 cos , we have As a special case, if we place the point source in the middle of the waveguide, points and also have the same phase due to symmetry. Thus, the phase shift from to should also be an integer multiple of 2 , / is less than a certain value 2 , the guided mode will disappear. This phenomenon is numerically verified in Supplementary Fig. 5, where we find that waves are prohibited from travelling along the waveguide when / is less than 0.2 (i.e., = 0.4 ).

Discussion
Here, as a further step, we would like to show that the guided-mode can be efficiently controlled and converted by designing waveguides with variable sections. Based on our proposed guide-mode theory (Eqs. (4) and (5)), by changing the width in Fig. 6a, such a gradient waveguide is capable of converting the guided-mode pattern from = 3 to = 1 smoothly.
It is interesting to note that, as shown in Fig. 6b, the waveform in the part of = 1 is quite regular with a wavelength and has an almost constant amplitude. This is a direct evidence that the waveguide can rectify distorted incident waves by reducing the width to permit the singlemode = 1. It has been a generally acknowledged fact that a single guided-mode can realize more robust and lower-loss energy propagation than multiple-modes, especially during a longdistance communication 39,40 . Therefore, it unambiguously presents that our metagrating-based  In conclusion, we present the first realization of an ultrathin waveguide strategy constructed by metagratings. Compared with the topological metamaterials and defected phononic crystals, the most notable paradigm in this work is that the metagrating-based waveguide shows great advantages on ultrathin footprint, broadband performance, and full-angle availability. By designing a bipartite supercell, the proposed metagrating can completely suppress non-zero diffraction orders and transmitted waves. Highly confined omnidirectional-wave-trapping and routing devices are experimentally realized by the proposed waveguide protype. A modified guided-mode theory is further presented and verified to predict the diffracted guided-mode patterns in waveguides with different values of / . Robust guided-mode tailoring and rectification capability, such as from distorted multiple-mode pattern to uniform single-mode transport, is unambiguously performed. The proposed ultrathin metagrating-based waveguide