Realization of ultrathin waveguides by elastic metagratings

Guiding classical waves has inspired a wealth of nontrivial physics and significant applications. To date, a robust and compact way to guide energy flux traveling 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 omnidirectional wave trapping and efficient routing. The metagrating-based waveguide can totally suppress all high-order parasitic diffractions to route guided elastic waves without leakage. The proposed waveguide protype works in a broad frequency range under a full-angle radiated source. An analytical slab-waveguide model is presented to predict and tailor the diffracted patterns. 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 design paradigm for vibration and noise control, energy harvesting, microfluidics, wave steering in acoustics and other waves. Guidance of classical waves is key to many technologies, but high-efficiency, omnidirectional performance is difficult to achieve. Here, an ultrathin, broadband elastic metagrating is proposed for suppression of parasitic diffraction and guiding waves along an arbitrary path.

G uiding 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 is 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 slabwaveguides 5 , nonlinear optics 6 and optical cavities 7 . However, such a high-index waveguide paradigm is only available for partial incidence angles, which will unavoidably suffer from energy leaking under a full-angle source excitation. 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 acoustic waveguide, air and solids serve as the waveguiding and extremely hard surrounding media, respectively. Although extremely mismatched impedance can realize elastic waveguide, for instance, a curved beam, infinite slot will definitely diminish the structural stiffness seriously. Furthermore, the closed waveguide will prevent the interaction and information exchange between the interior and exterior. Such standard closed waveguides have been investigated adequately in the past century 9 , which do not fall within the scope of our concern in this research. Recently, a shadow waveguide was proposed to achieve an acoustic tweezer in an open chamber without physical boundary, which can be regarded as an open waveguide paradigm 10 . Avoiding damaging the integrity of structure, this open waveguide allows substance exchange, which brings more possibilities for complicate wave manipulation and potential applications in biomedical and microfluidics. However, how to design such an open waveguide to route elastic waves between arbitrary media with similar impedances or even in a uniform medium seems unprocurable.
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 [11][12][13] . 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 [14][15][16][17] . Achieved by the analogs of quantum/quantum-spin Hall effect 16 , Dirac cones 15,18 , gapless states 19 or Weyl systems 20,21 , 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. Furthermore, by digging periodic metallic-post structures as bilateral edge walls, substrate integrated waveguide 22,23 in planar circuits has been proved a practical scheme for guiding TE 10 mode at microwave frequencies. However, how to design an ultrathin and compact waveguide to route elastic waves in a uniform medium seems unprocurable.
Alternatively, the recently proposed metasurface promises a thriving future for wavefront steering with a more compact footprint and thus an easy-fabricated low-dimension design [24][25][26][27][28][29][30] . Without reducing functionality, metasurfaces are capable of great flexibility and rich wave phenomena, such as wave shaping 25,27,31,32 , absorption [33][34][35] , and isolation 36,37 . These versatile modulation abilities arise from the well-tailored abrupt phase variation, which can be programmed by the generalized Snell's law (GSL) 24 , sinθ o À sinθ i À Á k 0 ¼ ξ, where θ i and θ o are the angles of incident and outgoing waves, ξ ¼ dϕ=dx is the phase gradient. For high-order wave modes generated by periodicity, the GSL is not sufficient to predict the diffraction waves. It is further developed as a more general form 27 , where the integer n denotes the diffraction mode. Based on Floquet theorem, the 0th transmitted mode (n = 0) is always existing, which indicates that the waves can transmit unavoidably from one side to another side. Some methods including using auxiliary fields 38,39 and bianisotropic metasurface 40,41 are proposed to realize perfect reflection with high efficiency in a passive lossless structure, but they are only available for specific incident angles. As a result, an efficient totalwave-blocking for full-angle incident waves seems impossible, let alone a further waveguiding using metasurfaces. Excitingly, metagrating was very recently demonstrated promising in remedying this capability limitation of metasurface [42][43][44] . 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. Distinguished from gradient index metasurface, metagrating combines the gradient unit cells with the concept of grating consisting of periodic supercells. It can harness phase modulation of gradient unit cells in a supercell, also utilize the diffraction effect of the grating's periodicity to manipulate the high-order wave modes, leading to richer diffraction phenomena. By doing this, anomalous refractions or/and reflections of every higher-order diffraction mode could be coherently assigned 44,45 . It is noteworthy that each sub-unit in the metagrating should be highly efficient to guarantee the whole steering efficiency. However, more to the point of total-wave-trapping, the existing localresonance-based designs are not sufficient to satisfy such criteria 36,46 . To the best of our knowledge, no work has been reported on a high-efficient, broadband, omnidirectional waveblocking by metagratings or metasurfaces. Thus far, a physical realization of compact waveguide for all-directional wave-trapping and guiding remains an outstanding fundamental challenge.
In this work, we propose an ultrathin waveguide framework for omnidirectional trapping and efficient routing of elastic-waves under a full-angle source excitation. Using only 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 repeated supercells. Each supercell, composed of only two unit-cells (m = 2) and with a period length L, leads to a phase gradient ξ ¼ 2π=L. Reflection or transmission of the higher order wave modes depends on the integer parity of multiple reflections, i.e., the number of times wave propagating inside the metagrating. For a gradient-index metagrating, the phase difference between adjacent unit cells is expressed as 2π/m, where m is the unit cell number in a supercell. With r times of multiple reflections, phase difference between adjacent unit cells becomes r 2π m : From another perspective, the nth-order mode induced by grating diffraction has a phase shift of 2nπ in a supercell, thereby the phase difference between two adjacent unit cells can be also expressed as n 2π m . When a wrap of 2π is applied to n 2π m , the phase difference become equivalent. Therefore, the corresponding relation can be further expressed as n 2π m þ 2π ¼ r 2π m . Then we can draw the conclusion that the number of multiple reflections r is related to the diffraction order n and the number of unit cell m in a supercell 44,45 , r = m + n. Based on the integer-parity principle, the modified GSL distinguishing transmission and reflection was proposed by Fu et al. 44 in 2009, which can be expressed as, where λ is the wavelength, n is the diffraction order, subscripts t and r 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 44 and elastodynamics 45 . Based on the diffraction principle of grating, we then introduce a relation between wavelength λ and grating constant L. It has been found that when λ/L > 2 is satisfied, only n = 0 mode is permitted under such design, all the other diffraction orders can be totally suppressed 23,47 . Recalling that m = 2 is selected as an even number here, it is evident that an efficient specular reflection with θ r = θ i 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 k-space. As illustrated in Fig. 1c, upon the incident wave, the metagrating with period L and reciprocal lattice vector ξ = 2π/L generates multiple diffraction modes, which are depicted by the green arrows. Due to the integer-parity design of metagratings (m = 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 k x 2 ½Àk 0 ; k 0 can be propagated into the far field, where k 0 ¼ 2π=λ is the wave number of the background medium. Actually, this rule is a standard result stemming from Floquet theory 48,49 . The tangential wave vector k x of the nth order diffracted wave has the relationship with incident wave vector k i ð¼ k 0 sinθ i Þ and the Fig. 1 Metagrating-based ultrathin waveguide. a Schematic of the proposed waveguide constructed by two layers of elastic metagratings to confine the waves along an arbitrary route. The supercell, consisting of only two unit-cells, is periodically distributed in each metagrating. b The metagrating can omnidirectionally block flexural waves in a thin plate in the form of specular reflection. The interval of periodically repeated supercells L is less than half of the wavelength. c A diagram for how the metagrating omnidirectionally reflect an incident wave. k x denotes the wavenumber along the metagrating. ξ and n denote reciprocal lattice vector and diffraction order, respectively. The black arrows represent wave vector k x of incident and scattering waves, while green arrows represent wave vector k x from metagrating diffraction. Upward (downward) arrows represent the reflection (transmission) of waves. The green box is the regime of propagation modes. The solid arrow in the green box indicates that the 0th order waves can propagate into far field, while the dashed arrows beyond the green box indicate other-order diffraction waves will decay evanescently in the near field. The y-axis represents the direction of wave vector k y . reciprocal lattice vector ξ(= 2π/L) as k x ¼ k i þ nξ. The corresponding normal wave vector k y can be expressed as . It can be easily derived that in the case of ξ > 2k 0 (corresponding to λ/L > 2), only the 0th diffraction order referring to either specular reflection or direct transmission exists as propagating mode, whereas other-order diffraction waves decay evanescently in the near field, no matter what the incidence angle is. As the number of unit cells m in a supercell is even (here m = 2) here, the 0th diffracted waves give rise to a total specular reflection according to Eq. 1.
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 compared with previous strategies of defected phononic crystals or topological metamaterials. It should also be mentioned that, although this metagrating-based waveguide focuses on flexural waves in plates here, it is also applicable to other types of waves, such as the waves in plasma, electromagnetism, and acoustics.
Realization of the omnidirectionally reflected metagrating. As depicted in the inset of Fig. 1a, for each supercell, we propose a straight beam and a zigzag-type one as unit cells I and II, respectively. The phase shift for flexural waves (A0-mode lamb waves) between the two unit-cells is set as π, i.e., Δϕ = π, to realize a phase wrap of 2π per supercell. Unit cells are built by digging holes in thin steel plates with thickness h = 1.5 mm throughout the study. The length of the supercell L = 14 mm is designed as 0.46λ at 15 kHz (λ = 30.6 mm), and the metagrating width H is 0.67λ. The slot proportions of cell Ι and cell ΙΙ are 78.6 and 47.8%, respectively. See the Supplementary Fig. 1 and Supplementary Note 1 for more details. Figure 2a illustrates the transmittance |t| as a function of θ i at the operating frequency of 15 kHz for each individual unit cell and the supercell after combination. Due to the set-up limitation, the range of the incident angle θ i 2 À80 ; 80 Â Ã is exhibited. A nearly total transmission ( t j j ffi 1) is always observed for both individual cell I and cell ΙΙ. On the contrary, a low transmittance (|t| < 0.25) is performed for the supercell with an alternate arrangement of the two unit-cells, regardless of the incident angle θ i . Such astonishing reversal of transmission under a full range of incident angles clearly demonstrates the feasibility of our design strategy of omnidirectionally reflected metagrating. On the other hand, the reflection phase of metagrating φ as the function of incident angle θ i is displayed in Fig. 2a. It shows a stable value around 0.4π for different incident angles θ i 2 ðÀ75 ; 75 Þ.
To evaluate the robustness of the proposed metagrating, we further investigate the transmittance |t| at different frequencies f and θ i in Fig. 2b. It 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. Different from locally resonant designs, the phase difference between the two zig-zag unit cells is generated by changing the propagating length of elastic waves. It can be guaranteed that the phase change keeps around π within a broad frequency band (see Fig. 2c). It is noted that Δϕ maintains an approximate value to π in the frequency range of 11-18 kHz (deviation from π is no more than 20%), although the supercell is designed exactly for 15 kHz. The slight variation roughly satisfies the design criterion of 2π-phase-wrap per supercell, leading to a broadband performance of omnidirectional reflection. Significantly, in the frequency range of 8-11 kHz, where phase difference has a dramatic derivation from π, metagrating still maintains a good performance of omnidirectional total-reflection. To illustrate it, the transmittance of some other supercells with a significant phase difference from π is further analyzed (see Supplementary Fig. 2 and Supplementary Note 2). It is verified that the omnidirectional total-reflection of the proposed metagrating is less sensitive to phase difference. The inherent insensibility also contributes to the broadband performance. Furthermore, it is evident that there is an abrupt peak at 12 kHz, which corresponds to an ultra-narrow disturbance in the transmittance contour. This point is induced by the twisting resonance of unit cell II (see further discussions in Supplementary  Fig. 3 and Supplementary Note 3). It is worth noting that the transmittance |t|for only one kind of unit cell (cell I or cell II) keeps a very high value (≈1) in a broad frequency range (see Supplementary Fig. 4). It can be also demonstrated theoretically based on our previous work 32 . On account of the existence of evanescent waves, the actual impedance Z is much closer to value 1, which has more matched impedance compared to acoustic case (see detailed discussion in Supplementary Note 4). According to the expression of transmittance jtj ¼ 2 ffiffi ffi Z p Zþ1 , the theoretical value of |t| is 0.956 in our design, which can be regarded as a total transmission of flexural waves. It is therefore illustrated that the broadband characteristic comes from metagrating rather than the mismatched impedance of slots. Full-wave simulations of the displacement fields for flexural waves (A 0 -mode) at 15 kHz in a thin steel plate are visualized in Fig. 2d. Different oblique incidences (θ i ¼ 0 ; 30 ; 60 ) are performed as examples, respectively. Obviously, a nearly total reflection is achieved regardless of the incident angle, which agrees well with the results in Fig. 2a. At all incident angles, the specular reflection is observed, which indicates that only the diffraction mode n = 0 is generated by the proposed metagrating. In all cases, the numerical field distributions show that the propagation directions of the scattered waves can be well theoretically predicted by Eq. 1.
The omnidirectional reflection of the metagrating is further proposed to trap the elastic waves and vibrations. As illustrated in Fig. 2e, the metagrating can be arranged to construct an enclosed region with an arbitrary shape, for example, a triangular area. The cases of other exotic shapes are also performed in the Supplementary Fig. 5 and Supplementary Note 5. The metagrating is fabricated by wire electrical-discharge machining in a steel plate with thickness 1.5 mm. Two point sources S 1 and S 2 , excited by piezoelectric wafers, are confined outside and inside the enclosed region, respectively. A laser Doppler vibrometer (Polytec, NLV-2500) is used to capture and visualize the full wave fields of flexural wave. Detailed experimental setups are described in Supplementary Fig. 6. If the point source S 1 (or S 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. The similar functionality for electromagnetic waves has also been realized by gradient metallic gratings 50 . Significantly, this isolation performance is omnidirectional, and no wave leakage occurs in a full-angle range. Although the periodicity has been lost at the corner of cage, it still works well if the empty is filled by plate medium. The relevant simulated results are provided in Supplementary Fig. 7. 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 |w| 2 has a striking drop from oscillated high values inside the metagrating (122 mm < x < 309 mm, shaded in blue) to less than 0.01 outside. In general, the experimentally measured results (marked as green dots) are consistent with the simulated profile (marked as the magenta curve), except for some deviations in the closed region (122-309 mm). Such deviations are mainly induced by the repeating multi-reflections and irregular wave interference in the closed region, where a disordered field pattern is generated. But it is evident that the energy inside is dramatically higher than that outside for both experimental and numerical observations. It allows us to demonstrate superior omnidirectional-isolation feature. Moreover, we would like to stress that the cage effect is quite robust to the frequency and works once the wave touches the metagrating. To unambiguously illustrate this point, the corresponding transient responses of the whole wave propagation are numerically and experimentally captured (see Supplementary Movies 1-4). The omnidirectional cage effects at other frequencies (13)(14)(15) are also performed in the Supplementary Fig. 8 and Supplementary Note 6.
Arbitrary waveguide routing. We further construct a series of ultrathin waveguides with arbitrary routing paths by utilizing the omnidirectionally reflected metagratings as curbs of the waveguides. As shown in Fig. 3a, a "L"-shaped waveguide with channel width l = 45 mm is firstly fabricated by parallel alignment of two designed metagratings. The turning part of the waveguide sample is enlarged as the inset. Placing a piezoelectric wafer at the inlet point S as a full-angle point source, one can see that almost all waves are restricted and trapped in the waveguide to propagate along the pre-designed channel with little leakage (Fig. 3b). It means that the metagrating-based waveguide is capable of guiding waves via omnidirectional reflection, like an optical slabwaveguide. The robust waveguiding feature is verified by the good agreement between the numerical and experimental energy fields at several adjacent frequencies from 13 to 15 kHz. Some discrepancies for the wave filed pattern at 15 kHz mainly come from the sensitivity of waveguide to the transition frequency and the effect of loss factor (see Supplementary Fig. 9 and Supplementary Note 7 for detailed discussion). In a more vivid way, the numerically and experimentally captured transient responses of the whole waveguiding process are provided in Supplementary Movies 5 and 6.
We then conduct a quantitative analysis of the waveguiding efficiency by calculating the averaged displacement amplitude jwj ave ¼ 1 k ∑ k j¼1 jwj j in three selected square regions (A, B, and C in Fig. 3a). Region A is placed inside the waveguide before the turning corner with a distance r A = 95 mm to the point source S, while region B is in the vertical channel after the turning corner with a straight-line distance r B ¼ 241 mm to S. Region C is placed adjacent to B but just beyond the waveguide boundary, with   Fig. 3c), respectively. A higher value of η than the cases without waveguides indicates a better performance of the waveguide. The profiles of tested η as a function of frequency f are presented in Fig. 3c. In the thin plate with waveguide (w/), the value η(A/C) keeps a stable high value of over 9.2 in the frequency range of 12-18 kHz and reaches a peak of around 17.8 at 17.2 kHz, implying a robust broadband feature. Thus, the contrast ratio has a striking enhancement by virtue of the metagrating-based waveguide. Furthermore, the tested value η(B/C) is always over the value without the waveguide in the frequency range 12-18 kHz. Specifically, in most of the frequency range (12-14.4 kHz and 16.4-17.4 kHz), η(B/C) keeps over 8.0, although the region B is further away from the source than C. The significant enhancements in the contrast ratios clearly manifest the efficient routing and guiding performance of our waveguide strategy over a long travel distance and in a wide frequency range. The insertion loss of the waveguide system is also evaluated in Supplementary Fig. 10 and Supplementary Note 8. It shows that energy dissipation denoted by η(B/A) is stably kept around −3 dB under 14.3 kHz. Compared with the energy dissipation of −5 dB above 15 kHz, it indicates that single guided-mode can efficiently avoid energy dissipation. Experimentally measured η(B/A) is less than the simulated results, which can be attributed to inherent material loss. Moreover, a straight waveguide without backward scattering is more beneficial for long-distance energy transportation than a curved waveguide. In practical applications, singlemode and straight-channel paradigms are preferred to avoid significant energy leakage. Some optimizations should be adopted around the turning corners for bent waveguides.
A series of waveguides of various shapes are further designed to evaluate the robustness of the proposed strategy, as depicted in Fig. 4. It is verified that the waveguide could maintain a highly efficient transmission even after continuous sharp corners, like a "N"-shaped 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 S 1 and S 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 highly efficient waveguide strategy with outstanding ultrathin footprint, which is totally distinct from topological states or defected metamaterials.
Guided-mode theory in metagrating-based ultrathin waveguides. It is observed in Fig. 4 that the guided wave fields through the metagrating-based waveguides have strong dependency on the channel width l. Specifically, single or multiple rows of discrete standing wave packets are generated in waveguides with different normalized widths l/λ. We here show that the 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 firstly consider a pair of rays from the point source S and then propagating in the waveguide with incident angles ± θ i , respectively. The wavenumber k 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 impinging upon the metagrating, the wave beam will be mirrored back to the waveguide but with a phase shift φ. To have a guided mode, according to the classical Fabry-Perot resonance conditions for non-ideal mirrors 51 , wave fields in points A and point C have the same phase with their projections B and D, respectively. Namely, the round-trip phase acquired by the ray must be an integer multiple of 2π, where β is an integer corresponding to the order of guided mode. Noting that AC À BD ¼ 2lcosθ i , we have As a special case, if we place the point source S in the middle of the waveguide, wave fields in points A and E also have the same phase due to symmetry. Thus, the phase shift from E to B should also be an integer multiple of 2π, where γ is an integer. Compared Eq. 2 with Eq. 4, we have β = 2γ with γ = 0,1,2…, which means that only even modes can be generated in the case of symmetric point source. Considering that θ i 2 ½À π 2 ; π 2 , Eq. 3 becomes l λ ¼ 2γπþφ It infers that when l/λ is less than a certain value φ 2π , the guided mode will disappear. This phenomenon is numerically verified in Supplementary Fig. 11, where we find that waves are prohibited from traveling along the waveguide when l/λ is less than 0.2 (i.e., φ = 0.4π). This result is consistent with an expected reflection phase φ ≈ 0.4π for different incident angles, as shown in Fig. 2a. Figure 5b shows the variation of β versus the ratio l/λ and the incident angle θ i based on the guided mode theory when the point source S is placed at the center of the waveguide. A general case of arbitrarily placed S is also supplied in Supplementary Fig. 11 and Supplementary Note 9. As illustrated, the guided-mode pattern can be manipulated by the waveguide width and the excitation frequency. It is necessary to note that for the specific width l, only the incident waves satisfying Eq. 3 can generate guided mode by Fig. 4 The proposed waveguide strategy for arbitrary routes at 15 kHz. a Simulated energy fields of a "N"-shaped waveguide with width l = 0.82λ. The radius of curvature is 2.8λ. S refers to the location of point source. b "U"-shaped waveguide with width l = 1.47λ. The radius of curvature is 3.3λ. S refers to the location of point source. c Two adjacent waveguide paths with a distance of 2λ (l = 0.82λ). The radius of curvature is 2.9λ. Two point sources are placed on S 1 and S 2 . d Wave splitting by a waveguide junction (l = λ). The radius of curvature is 3.2λ. S refers to the location of point source. constructive interference. For the value l/λ ∈ (0.2, 1.2], only the branch β = 0 can be found in Eq. 3, inferring that only the guidedmode order β = 0 exists. When the ratio is set as l/λ ∈ (1.2, 2.2], we have β = 0 and β = 2 under the full-angle incidence. In this case, the guided modes order β = 0 and β = 2 exist simultaneously. Similarly, we can predict β = 0, 2, 4 for l/λ ∈ (2.2, 3.2]. As verifications, we checked the field patterns of three cases l/λ = 0.5, 1.5, 2.5 in Fig. 5c, respectively. As expected, the guided-mode orders are concordant with our analytical predictions. It is clearly presented that there are one, three, and five rows of standing wave beams within the waveguide accordingly. A yellow dashed line across the waveguide is further selected to quantify the wavepackets. Except the two distinct crests induced by the metagrating boundaries, the wave-packet number is exactly 1, 3, and 5 in corresponding waveguides, respectively. Combining wave equation with metagrating boundary conditions, the analytical mode profiles can be derived, which are provided in Supplementary  Fig. 12 and Supplementary Note 10.

Discussion
Here, as a further step, we would like to show that the guidedmode can be efficiently controlled and converted by designing waveguides with variable sections. Based on our proposed guidemode theory (Eqs. 3 and 4), by changing the width l in Fig. 6a, such a gradient waveguide is capable of converting the guidedmode pattern from multiple modes to single mode smoothly 52,53 . As shown in Fig. 6b, the waveform in the part of β = 0 is quite regular with a wavelength λ and has an almost constant amplitude. This is direct evidence that the waveguide can rectify distorted incident waves by reducing the width to permit the single-mode. The robust rectification functionality still works if the wave source is placed in a general case (see Supplementary Fig. 13 and Supplementary Note 11). It has been a generally acknowledged fact that a single guided-mode pattern can realize more robust and lower-loss energy propagation than multiple-modes, especially during a long-distance communication 54,55 . Making full use of single guided-mode and avoiding sharp turning corner are desirable for long-distance energy transport. Therefore, it unambiguously presents that our metagrating-based waveguide possesses rectifying capacity, accurately predictable manipulation, and potential tunability for a low-loss, long-distance energy transport.
In conclusion, we present the realization of an ultrathin waveguide strategy constructed by metagratings for elastic waves. 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 in 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-wavetrapping and efficient routing devices are experimentally realized by the proposed waveguide protype under a point source excitation. A modified guided-mode theory is further presented and verified to predict the diffracted guided-mode orders in waveguides with different values of l/λ. Robust guided-mode tailoring and rectification capabilities, such as from distorted multiplemode pattern to uniform single-mode transport, which are similar to the power splitting and mode conversion functionalities in electromagnetic waves guiding systems, are unambiguously performed. The proposed ultrathin metagrating-based waveguide opens a prototype for designing compact, low-loss transport devices for mechanical energy, which may also stimulate analogous research in acoustics and other wave physics.

Methods
Numerical simulations. We use COMSOL Multiphysics to conduct a series of fullwave simulations throughout the paper. A thin steel plate with a same thickness of 1.5 mm is investigated as the surrounding medium and the metagratings have the same material parameters with steel plate: Young's modulus E = 200 GPa, density ρ = 7930 kgm −3 , and Poisson's ratio ν = 0.3. In all simulations, we apply Perfect Matched Layers (PML) in peripheral boundaries of thin plate to eliminate the effect of reflection. In the simulations of unit cells to obtain phase shift and transmitted amplitude, periodic boundary condition is applied at the top and bottom boundaries.
Experimental setup. The actual waveguide sample is a thin steel plate with dimension of 800 × 800 × 1.5 mm 3 , which is fabricated by wire electrical discharge machining. A 3-cycle tune burst is generated by RIGOL DG4062 signal generator, which is denoted as F t ð Þ ¼ ½1 À cosð 2πf c t 3 Þ sin ð2πf c tÞ with a central frequency f c = 15kHz. Through the power amplifier (ATA-2022H) the signal is magnified and then transfers to PZT which is bonded on the entrance of waveguide. A Polytec NLV-2500 Laser vibrometer is used to capture the out-of-plane velocity fields by laser beam point by point. The scanning head is fixed on the two-axis motorized translation stage and its moving step Δd = 4 mm, which means the spatial resolution is about 8 points per flexural wavelength at 15 kHz. The time-domain response of every point is stored into computer. The sampling rate is set as 1.19 Ms −1 . To improve signal-to-noise ratio, every scanning point is measured 20 times and averaged. In all surrounding boundaries of plate, we use visoelastic damping material (Blu-Tack) to minimize the effect of reflection. For the sake of comparison with simulation, normalized out-of-plane displacement fields (300 × 240 mm 2 in the region of waveguide) at different frequencies are obtained through fast Fourier transform based on the acquired data in experiment. Fig. 6 Flexural wave pattern excited by a point source in a waveguide with gradient width. a The out-of-plane displacement wave fields. The point source S is placed in the middle of waveguide. β refers to the guidedmode order. With the decrease of waveguide width, the guided-mode orders transit from β = 0, 2 to β = 0 smoothly. b The out-of-plane displacement w along the central line of the waveguide (the white dashed line in a). The waveform in the part of β = 0 is quite regular with a wavelength λ and has an almost constant amplitude. It is illustrated that the waveguide can rectify distorted incident waves to permit the single-mode.

Data availability
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