Pulse dynamics of flexural waves in transformed plates
PPulse dynamics of flexural waves in transformed plates
Kun Tang Chenni Xu S´ebastien Guenneau Patrick Sebbah*
Dr. Kun Tang, Dr. Chenni Xu, Prof. Patrick SebbahDepartment of Physics, The Jack and Pearl Resnick Institute for Advanced Technology, Bar-Ilan Univer-sity, Ramat-Gan 5290002, Israel.Email Address:[email protected]. Chenni XuZhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhe-jiang University, Hangzhou, 310027, Zhejiang, China.Prof. S´ebastien GuenneauUMI 2004 Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ, United Kingdom.Keywords:
Mechanical metamaterials, transformation elastodynamics, homogenization, pulse dynamics,waveguide modes, cloaking, mirage effect.
Coordinate-transformation-inspired optical devices have been mostly examined in the continuous-wave regime: the performance of aninvisibility cloak, which has been demonstrated for monochromatic excitation, would inevitably deteriorate for short pulses. Here weinvestigate pulse dynamics of flexural waves propagating in transformed plates. We propose a practical realization of a waveshifterand a rotator for flexural waves based on the coordinate transformation method. Time-resolved measurements reveal how the waveshifterdeviates a short pulse from its initial trajectory, with no reflection at the bend and no spatial and temporal distortion of the pulse.An invisible cloak for flexural waves is proposed based on this elementary transformed device. Extending our strategy to cylindricalcoordinates, we design a perfectly invisible wave rotator. We demonstrate experimentally how a pulsed plane wave is twisted insidethe rotator, while its wavefront is recovered behind the rotator and the pulse shape is preserved, with no extra time delay. We pro-pose the realization of the dynamical mirage effect, where an obstacle is seen to be oriented in a deceptive direction.
In two independent proposals, Pendry et al. [1] and Leonhardt [2] showed that a transformation of coor-dinates can map a particular distortion of the electromagnetic field onto a change of material properties-inhomogeneous and anisotropic-, unveiling unlimited possibilities for the design of metamaterials withnew functionalities to control the flow of light. This new concept was originally proposed to design aninvisibility cloak that was first validated for electromagnetic waves [3] and thereafter extended to othertypes of waves including acoustic [4], hydrodynamics [5] and water waves [6]. In all these cases, the gov-erning equations are form invariant. When one moves to the area of elastic waves however, the elasticityequations are not form-invariant under a general coordinate transformation [7, 8]. Consequently, if cloak-ing exists for such a class of waves, it would be of a different nature than its acoustic and electromag-netic counterparts. Researchers resort to studying the special case of flexural waves in thin plates, whichare described by the the fourth order Kirchhoff-Love equation. Over the past ten years, there have beenvarious theoretical proposals to the design of elastic invisibility cloaks for flexural wave [9, 10, 11, 12, 13,14], followed by their experimental validations [15, 16, 17, 18, 19, 20]. However, experimental investiga-tions of cloaking have been mostly restricted to continuous-wave (CW) excitation [3, 21, 15, 16, 22, 23].Cloak invisibility to short pulses has been rarely tested [24, 22], since these inhomogeneous, magnetic,and anisotropic metamaterial structures are subject to inherent frequency dispersion, which inevitablydistorts the pulse both in space and time and makes its reconstruction challenging [25, 26]. Broadbandcloaking has been realized in various systems, including acoustic [23], elastic [15, 16], and water waves[6]. But achieving broad spectral range operation does not guarantee that a pulse propagating throughthe transformed medium will remain undistorted.The coordinate transformation used in the design of electromagnetic cloaks with cylindrical geometryleads to a gradient distribution of permittivity and permeability, which necessitates engineering the mag-netic response of materials hardly available in the optical range [1, 3]. To solve this issue, a reduced setof parameters has been proposed, which provides with non-magnetic structures, while preserving thecloaking performances, but which suffers from reflection and scattering due to impedance mismatch atthe outer boundary [21]. Another approach to avoid magnetic materials and preserve the invisibility of a r X i v : . [ phy s i c s . c l a ss - ph ] A ug he device itself is to maintain the determinant of the Jacobian transformation tensor at unity, thereforepreserving the volumes throughout the space [27, 28, 29, 30]. Compared to cylindrical cloaking, this so-called non-magnetic geometrical transformation is continuous i.e. adiabatic [31]. The space is not abruptlystretched or compressed and the topology is conserved during the transformation process [32], ensuringperfect impedance matching at the boundaries. This volume-preserving method has never been consid-ered for elastic waves, where it could meet the challenge of designing intrinsically reflection-less elasticdevices. [15, 16, 17].Transformation optics actually has not been limited to the design of invisibility cloaks but has led tothe development of novel wave-manipulation devices [33, 34, 35, 32]. Among them, the waveshifter, thebuilding-block of fundamental steering optical components such as the wave splitter, and the rotator, aninvisible device capable of twisting and restoring waves, creating at the same time a mirage effect [36],have been proposed to control electromagnetic waves [34], as well as scalar acoustic waves [37, 38], wa-ter waves [39, 40] or hydrodynamic flows [41]. Surprisingly, these new classes of devices have never beenconsidered for elastic waves.In this article, we adapt the non-magnetic geometrical transformation to elastic waves and call on forits continuous and volume preserving character to design a reflection-less waveshifter and an invisiblewave rotator for flexural waves. Here, we investigate the pulse dynamics of flexural waves propagatinginside a 3D-printed transformed plate by mapping the spatio-temporal field distribution in response to ashort pulse excitation. The elastic pulse is shown to perfectly follow the 20 ◦ bent of the waveshifter withminor back reflections at the bend and negligible spatial and temporal dispersion of the incident shortpulse. A modal analysis shows how higher modes of the bent waveguide are prevented from being ex-cited. We illustrate the importance of such a transformed device by using it as the building block of atruly invisibility cloak for elastic pulses. The elastic wave rotator is designed following the same strategy.Most remarkably, a short pulse plane-wave excitation is shown to be restored, after experiencing a 30 ◦ twist within the rotator. The pulse is shown to cross the device with no extra time delay, as if it prop-agated through “free space”. To the best of our knowledge, this has never been observed before in thecontext of elastic waves, and never in the time domain with electromagnetic waves [34], acoustic waves[37, 38], water waves [39, 40] or hydrodynamic flow [41]. Three dimensional full-elasticity simulationssupport our experimental observations. Finally, this rotator is proposed to demonstrate the mirage effectwith elastic waves, where a scattering object is seen to radiate from a deceptive direction. In the limit of plate thickness much smaller than the wavelength, the phase velocity c of flexural wavescan be described within the KirchhoffLove plate theory as c = (cid:113) Dωρh , where D = Eh − ν ) is the flexuralrigidity of the plate, ρ its mass density, h its thickness and E its Young’s modulus, ω being the angularfrequency. By analogy with layered anisotropic electromagnetic media [31], anisotropic phase velocityfor flexural waves can be achieved by alternating layers of materials with different elastic properties. Apossible approach is to vary the Young’s modulus or the density of the successive materials, as success-fully demonstrated by the group of Wegener [15]. But from a practical point of view, varying the thick-ness of the plate, rather than modifying its intrinsic elastic parameters, turns out to be a much simplerstrategy to introduce anisotropy. This approach has been validated for adiabatically varying thickness,e.g. for the design of elastic lenses [42, 43, 44]. Here, we demonstrate that, in contrast to an adiabati-cally varying thickness, a periodic change of plate thickness on a subwavelength scale can be utilized todesign effective anisotropic metamaterials for flexural waves. A significant advantage in terms of realiza-tion is that a single material is required and that the subwavelength structuration is easily implementedby surface machining or, more conveniently, by 3D printing, a technique now readily available for manytypes of materials, including ceramics and metals. Here we illustrate our novel approach with the designof a waveshifter.Consider a walker progressing along a horizontal path. Further consider the space transformation whichtransforms this flat path into a staircase. The walker now finds himself going upstairs, at an angle θ with Dynamics of pulse propagation in the waveshifter. (a) Sketch of the coordinate mapping from a hori-zontal waveguide in virtual space (X,Y) to an oblique waveguide in the real space (x, y). (b) Top view of the waveshifter:the waveguide is a 100 mm-long, 10 mm-wide 3D-printed ceramic plate, with a bending angle θ = 20 ◦ . Thickness is h =0.5 mm for the left arm. The corrugation in the right arm (period 2 mm) alternates thicknesses h a = 0.785 mm and h b =0.318mm, and form an angle α = 40 ◦ with the waveguide axis. A piezoelectric diaphragm is attached to the left end of thewaveguide. (c) Experiment: Snapshots of the out-of-plane velocity field in response to a Ricker pulse (see [ExperimentalSection]) with central frequency 20 kHz, measured at times 0.065 ms (top), 0.115 ms (middle), and 0.165 ms (bottom).The red dashed line outlines the physical limit of the structure. (d) Full-3D numerical simulations: Snapshots of the out-of-plane velocity field calculated at same times as (c), using Finite Element Method in Comsol Multiphysics. (e) Exper-iment: Pulse profile measured at two spatial positions, before and after the bend, marked in (b) by black and red dots,respectively. (Top) Actual measurement showing the time delay accumulated during propagation. (Bottom) The transmit-ted pulse (red) has been time-shifted to show the coincidence with the incident pulse (black). (f) Numerical simulations:Pulse profiles calculated at same positions as in (e). (Top) and (Bottom) same as in (e). The dashed red line is the pulseprofile after propagation in a straight plain waveguide (no bend and no corrugation). The perfect overlap between the 2red curves demonstrates that the residual deformation of the pulse is solely due to the natural dispersion of flexural wavesin thin plates. 3igure 2: Single-frequency wave steering across the waveshifter. (a) Experimentally measured out-of-plane velocityfield at two different frequencies 16kHz (left column), and 20kHz (right column); (b) Corresponding numerical results; (c)For comparison, the numerical simulations are shown for an empty bent waveguide (no metamaterial structure). the horizontal axis, while maintaining an upright position, as if nothing had changed for him (see illus-tration in
Figure. 1(a) ). To describe the associated coordinate transformation, we consider the map-ping which transforms a point (
X, Y ) in the virtual space (the flat path) into a point ( x, y ) of the obliqueregion in the real space (the staircase). This change of coordinates can be expressed as: (cid:40) x = Xy = X tan θ + Y , (1)where θ is the steering angle.The Jacobian Matrix of the above geometrical transformation is F = (cid:18) t (cid:19) , (2)where t = tan θ . For a given angle θ , F is independent of space coordinates. Note also that the determi-nant is unity, J = | detF | = 1, so that the transformation we have defined preserves the area throughoutspace.Based on geometrical transformation method, mapping of coordinates from the homogeneous virtualspace to the real space, results in a change of material parameters in the transformed wave equations.In the tracks of [9, 14], the transformed Kirchhoff-Love equation results in an anisotropic flexural rigidityof the form D = D F F T F F T J − . (3)The flexural rigidity becomes tensorial and accounts now for the anisotropy of the transformed medium.In contrast to elastic cloak where it varies radially [15, 16], the rigidity tensor in our case is independentof the space coordinates. The mass density itself, ρ = ρ J − = ρ , remains unaffected by this coordinatetransformation. This greatly simplifies its implementation in an actual device, as we proceed to explain.A medium with this particular anisotropic rigidity D can then be realized with a simple subwavelengthstructure by invoking effective medium theory. Here we propose to approximate the homogeneous anisotropicmedium by a bi-layered structure consisting of an alternation of two materials with identical widths butdifferent flexural rigidities D a and D b . The normal to the layered structure defines the direction of anisotropy.If in addition this direction makes an angle α with the direction of propagation, the effective rigidity ensor is then given by D oblique = R Tα (cid:20) D (cid:107) D ⊥ (cid:21) R α , (4)with D (cid:107) = 2 D a D b / ( D a + D b ), D ⊥ = ( D a + D b ) / R α = (cid:20) cos α − sin α sin α cos α (cid:21) the conventional rotationmatrix.Identifying Equation. (3) and (4), we obtain the rigidity profile for the transformed region, and a generalexpression for the angle α , D a and D b , in terms of parameter t = tan θ : t = 2 cot (2 α ) D a /D = cot ( α ) + (cid:112) − ( α ) D b /D = cot ( α ) − (cid:112) − ( α ) . (5)Keeping all elastic parameters constant, the rigidity for each layer, D a and D b , is easily implemented byadjusting the thickness, h a and h b , of the plate at each layer, following the definition of D [16, 43, 42,44]. According to Equation. (5), the thickness difference h a − h b increases steeply for θ above 40 ◦ (seeFigure. S1 in [Supplementary Information]). We therefore choose for the experimental realization an an-gle θ = 20 ◦ , to limit the geometrical constraints for the 3D-printing. Nevertheless, the design is shown inthe [Supplementary Information] to be effective up to θ = 60 ◦ .Following our method, we design a waveguide with total length L =100 mm, width W =10 mm, and adeviation angle θ =20 ◦ , as shown in Figure. 1(b). In the left section of the waveguide, plate thickness is h =0.5 mm, while the bent (right) section is corrugated at an angle α =40 ◦ with respect to the waveg-uide axis, as calculated from Equation. (5) and shown in Figure. 1(b). The period of the square cor-rugation is 2 mm, with alternating plate thicknesses h a =0.785 mm and h b =0.318 mm, to be comparedwith the wavelength, λ =18.7 mm at 16 kHz. We chose deliberately to design our waveshifter in a waveg-uide geometry (rather than “free space” propagation) in order to investigate the modal coupling at thebend while the energy flow is steered in the oblique direction, as this will be discussed later. Figure 1(b)depicts the waveshifter device, 3D-printed in Zirconium dioxyde ceramic. To investigate the pulse dy-namics along the bent waveguide, we examine its temporal response to a Ricker pulse (see definition in[Experimental Section]) centered at 20 kHz, launched from its left section. A laser vibrometer is usedto map the spatio-temporal evolution of the elastic field along the waveguide. The experimental setupis described in full details in [Experimental Section]. In Figure. 1(c), we present snapshots of the ve-locity field distribution at three different times. The pulse is seen to propagate smoothly without de-formation along the waveguide while it is deflected, with almost no reflection at the waveguide bend.Most striking, the wavefront remains vertical, in the same direction as the incident wave, like the walkerin Figure. 1(a). This is the main feature expected from the waveshifter: The anisotropic section of thewaveguide as we designed it maintains perfectly the initial wavefront direction, giving the illusion thatthe wave comes from the same direction, although its energy has been deviated. This perfectly realizesthe mapping of Figure. 1(a). The experiment is successfully compared against time-domain full-3D elas-tic wave simulations, as shown in Figure. 1(d). The complete movie of the pulse propagation is availablein [Supplementary Information].The investigation of the pulse dynamics reveals however a new remarkable feature of the waveshifter:that spatial and temporal dispersion of the pulse is negligible during propagation and deflection. To demon-strate it, we measure the pulse profile before and after the bend, at positions indicated by black and reddots in Figure. 1(b). Figure. 1(e) shows the pulse before (top) and after (bottom) time-shifting the trans-mitted pulse (red). We find that the two signals almost perfectly overlap. Surprisingly, the coincidencebetween the two signals turns out to be even better in the experiment than in the numerical simulationsshown in Figure. 1(f). Actually, the small residual temporal deformation observed in Figure. 1(e) and1(f) is solely due to the natural dispersion of the flexural waves during propagation. This is demonstratedin Figure. 1(f), where a comparison is made with a pulse propagating through an empty straight waveg-uide (red dashed line). If a time delay exists between “free” propagation and propagation in the waveshifter Modal analysis before and after the bend at frequency 16 kHz. (a) Mode profiles (y-dependence) of thefirst two even (left panel) and odd (right panel) flexural eigenmodes. Black and red colors correspond to the 0 th and 2 nd even modes, while cyan and magenta correspond to the 1 st and 3 rd odd modes. Modal components η i ( X ) ( i = 0 , , , X = 0), (b) for measured results, (c) for numerical simulations and (d)for the empty bent waveguide without corrugations. The left panels describe even modes, while the right panels are for theodd modes. (Black) zeroth-order modes; (red) second-order modes; (cyan) first-order modes; (magenta) third-order modes;Each row is normalized with the maximum amplitude of the 0 th -order even mode, η ( X ). Note the magnified ( ×
20) verti-cal axis for odd modes in right panel (b). (top), the two pulses perfectly overlap (bottom), showing that there is no pulse distortion due to thetransformed device.We now investigate the waveshifter in the spectral domain. To do so, the time response to a chirp signalbetween 10 kHz and 30 kHz is Fourier transformed to recover the spectral response.
Figure 2(a) showsthe real part of the out-of-plane velocity wavefield along the waveshifter, measured at two different fre-quencies, 16 kHz and 20 kHz. The experiment is compared to full-3D simulations at same frequenciesfor the same waveshifter (Figure. 2(b)), as well as for a uniform bent waveguide without the transformedmedium (Figure. 2(c)). The 0 th -order even mode of the waveguide efficiently converts into a new slanted“0 th -order” mode of the anisotropic waveguide, with the wavefront oriented in the same direction as theincident mode. This is in stark contrast to the simulations performed in the bent waveguide without cor-rugation (Figure. 2(c)), where the incident mode converts into a combination of even and odd higher-order waveguide modes, and any information on the initial wavefront is lost.To better quantify the efficiency of the mode conversion in our waveguide shifter, we carry out a modalanalysis of the measured and simulated fields inside the waveguide, before and after the bend [39]. Themodal decomposition is performed in the virtual space ( X, Y ): η ( X, Y ) = n (cid:88) i =0 η i ( X ) ψ i ( Y ) , (6) here η i ( X ) = (cid:82) + W/ − W/ η ( X, Y ) ψ i ( Y ) dY refers to the integration across the width W of the waveguide ofthe i th -order transverse component, ψ i ( Y ). The first orders, even and odd, transverse modes are calcu-lated in [Supplementary Information] for free boundary conditions and shown in Figure. 3(a) . Beforethe bend, we simply have ( X = x, Y = y ). After the bend, the field η ( X, Y ) is interpolated on a grid(
X, Y ), using the inverse geometrical transformation X = x and Y = y − x tan θ .Figure. 3(b) and 3(c) show the measured and calculated first even and odd modes at f=16 kHz. Resultsconfirm the efficiency of the energy transfer. The modal content of the incident signal, which is essen-tially the 0 th -order even transverse mode, is perfectly preserved after deflection by the bend. This modetranslates in the real space into the slanted mode seen in Figure. 2(c) with k -vector in the forward di-rection, x , but energy flow along the bent waveguide. Note that the scale for measured odd modes is 20times smaller than that for even modes. After the bend, the conversion from incident mode to first oddmode due to the asymmetry of the designed structure, remains negligible. This contrasts with the emptywaveguide (Figure. 3(d)) where the sharp change of direction couples the excitation to higher modes: themagnitude of first-order odd mode becomes comparable to that of mode 0 (Figure. 3(d)). This demon-strates the efficiency of our design, which deflects the 0 th -order even mode into a slanted 0 th -order mode,preserving the wavefront direction while preventing higher modes from being excited. The waveshifter is an interesting device which can be used as the building block of a variety of new func-tional components, including wave splitters, combiners and invisibility cloak [32]. This is illustrated herewith a cloaking device based on four waveshifters, arranged in a symmetric way around a diamond-shapedhole (see
Figure. 4 ). Such a cloaking device has been proposed in [31] for CW lightwaves and is demon-strated here for pulsed elastic waves. We use the parameters calculated earlier, θ , h a , h b , h , and α , butinstead of limiting the transformed region to a waveguide geometry, we simulate an incident pulse (Rickerpulse centered at 20 kHz) with a 160 mm-wide Gaussian wavefront propagating in a wide corrugatedarea around the diamond-shaped hole. Figure 4 compares the field distribution of the vertical elastic ve-locity resulting from the scattering by a bare hole with free boundaries without and with the cloakingdevice, at three different time steps. The Gaussian wavefront splits as it hits the hole and the wavefrontrapidly breaks apart. With the cloak however, the wavefront is maintained while following the edges ofthe obstacle and recombines after the hole. Beyond the hole, the initial Gaussian profile is restored. Inaddition to the cloaking effect, the cloak itself is invisible, with negligible back reflection. We checkedthat the temporal elongation of the pulse is solely due to flexural waves dispersion, as it would occurnaturally in a plain plate without the obstacle. The extend to which the diamond hole dimensions canbe increased is discussed in [Supplementary Information] This cloaking effect is however restricted by construction to a single direction of incidence. Here, we pro-pose to design an isotropically invisible device, the so-called wave rotator, by transposing the concept ofthe waveshifter from the Cartesian coordinate system to the polar coordinate system. By analogy withEquation. (1), we define the following transformation (cid:40) r = Rθ = ϑ f ( R ) + ϑ , (7)where f ( R ) is an arbitrary continuous function and ϑ the rotating angle of the device. Instead of thetranslation at fixed angle θ achieved by the waveshifter (Figure. 1a), this new coordinate transform ro-tates by an angle ϑ the wavefront incident from any direction, as it penetrates an annular region a ≤ r ≤ b . When the wave exits the annulus, the rotation effect is reversed and the wavefront direction isrestored. A necessary condition on f ( R ) is therefore f ( b ) = 0 and f ( a ) = 1. To push the analogy with Cloaking flexural waves with 4 waveshifters . Full-3D elastic-wave transient simulations of elastic field dis-tribution resulting from the propagation of a pulsed gaussian wave (Ricker pulse centered at 20 kHz) with 160 mm-widetransverse profile, at times t=0.15 ms, t=0.25 ms, t=0.38 ms. (left panel) Bare diamond-shaped hole; (right panel) dia-mond cloak. The cloak is composed of 4 corrugated regions around a diamond-shaped hole with θ =20 ◦ , as defined in thefigure. Geometric parameters of the corrugated region 1 are identical to those of the waveshifter of Figure. 1. Region 3 isthe mirror image of Region 1 with respect with the horizontal axis. Regions 2 and 4 are mirror images of Region 1 and 3with respect to the vertical axis. One notes that scattering off the hole is nearly suppressed by the corrugated regions attime step t = 0 . θ .8igure 5: Dynamics of pulse propagation in the rotator. (a) Top view and (b) magnified view of the 3D-printedrotator of uniform plate thickness h =1 mm inside and outside the rotating annulus with inner radius a =15 mm and outerradius b =30 mm. The ring consists of spiraled corrugations of alternate heights h a =2 mm and h b =0.5 mm. (c) Experi-ment: Snapshots at time t = 0.4 ms (left), 0.55 ms (middle), and 0.65 ms (right) of the measured out-of-plane velocity field,which shows the plane wavefront of the Ricker pulse propagating before (left), during (middle) and after (right) the rotator(materialized by the red drawing). The rotation experienced by the elastic wave within the rotator is ϑ = 30 ◦ . (d) Numer-ical simulations. (e) Experiment: temporal velocities at three spatial positions before, inside and after the rotator, markedby colored dots in (a). The red and blue curves in the lower panel are time-shifted to bring the three peaks in coincidence.(f) Same as (e) for numerical simulation: rotator plate (solid lines) and plain plate (dashed lines).9 he waveshifter a step further and allow a direct transposition of the design method proposed earlier, weassume that the Jacobian matrix associated with this new coordinate transform has the same form as inEquation. (2), but defined this time in polar coordinate system: F = (cid:18) ∂r∂R R ∂r∂ϑ − R ∂θ∂R ∂θ∂ϑ (cid:19) = (cid:18) t (cid:48) (cid:19) , (8)where t (cid:48) is a constant to be defined. A necessary condition to satisfy this equality is − R ∂θ∂R = t (cid:48) . (9)From this condition, we obtain (cid:40) f ( R ) = ln( b/R )ln( b/a ) t (cid:48) = ϑ ln( b/a ) (10)For given rotating angle ϑ and radii a and b , the Jacobian matrix F is constant, independent of spacecoordinates and its determinant is unity so that the transformation preserves the volumes. Under theseconditions, the change of material parameters which realizes the distortion of the wave of Equation. (7),is obtained by following step by step the procedure proposed to design the waveshifter. The rigidity ten-sor D is again given by Equation. (3), while the mass density remains unchanged ρ = ρ . The anisotropyis introduced in the same way, by alternating layers with different flexural rigidities. The angle α formedby these layers with the local polar frame is constant, as it was with the tilted Cartesian frame of thewaveshifter. By definition, the curve defined in polar coordinate by a constant tangential angle α is alogarithmic spiral, r = a.e k ( θ − β ) , with k = − tan α , and β is an arbitrary initial angle for r = a . The an-gle α is given by Equation. (5), where t must be replaced by t (cid:48) = ϑ ln( b/a ) . The subwavelength anisotropicstructure is realized by choosing N = 24 initial angles β = 0 , πN , ... ( N −
1) 2 πN , (11)which defines N logarithmic spirals (see Figure 5(a) and 5(b) ). Regions between 2 successive spiralsdefine regions with alternating flexural rigidities D a and D b , as defined in Equation. (5). Following themethod used for the design of the waveshifter, this is implemented practically by varying the thicknesses h a and h b of the plate in these regions. The resulting structure is similar to [16, 42, 43, 44].A rotator with rotation angle ϑ = 30 ◦ has been designed and 3D-printed on a stiff photo-resin in a18 cm ×
18 cm square plate with thickness h =1 mm. The inner and outer radius for the rotating an-nulus are a = 15 mm and b = 30 mm, respectively. A close-up on the rotator (Figure. 5(b)) shows thespiral-like corrugation with alternating thicknesses h a =2 mm and h b =0.5 mm. We propagate a shortpulsed plane wave (Ricker pulse centered at 4kHz) across the rotating device. Figure 5(c) presents suc-cessive snapshots of the wavefield vertical velocity at three different times, while a complete movie isavailable in [Supplementary Information]. This shows clearly how the incident wave acquires a 30 ◦ anti-clockwise twist as it penetrates the rotator. As it exits the device, the wavefront is rapidly restored afterabout one wavelength and the plane wave continues its journey as if nothing happened. Note that thisillusion of total invisibility is also achieved in the backward direction where almost no perceptible energyis being scattered. Besides, rotational symmetry ensures that invisibility is achieved from any directionthe device is looked at. Full-3D time-domain simulations confirms this behavior (Figure. 5(d)). Actually,not only the wavefront is preserved but the temporal shape of the pulse does not experience any spatialor temporal distortion. This is demonstrated in Figure. 5(e) and 5(f) where we compare the temporalpulse profile measured at three different positions, on both sides and inside the rotator, as marked bythe colored dots in Figure. 5(a). By time-shifting the peaks, we show good temporal overlap for the ex-periment and excellent coincidence for the numerical simulations: the pulse profile has been preserved.Even more striking, we found that no phase delay is accumulated during propagation across the rotat-ing device. This is shown by comparing the pulse after crossing the rotator (full red line) to a pulse mea-sured at the same position in a plain plate without the rotator device (dashed red line). The perfect over-lap of the two time signals demonstrates that cloaking provided by the rotator is perfect. This is in stark Dynamic mirage effect in transient regime . Full 3D elastic-wave simulations showing the propagationacross the wave rotator of a flexural plane wave (Ricker pulse centered at 4 kHz) launched from left edge. (a) Homoge-neous plate with a clamped rectangular scatter tilted at 30 ◦ with respect to the wavefront of the incident plane wave; (b)Plate with the transformed rotator enclosing the same tilted scatter; (c) Plate with a scatter in vertical position with norotator. contrast with other cloaks based on resonant dispersive structures, where pulse experiments would in-evitably disclose the presence of the device [25]. Actually, the wave rotator is not a cloaking device per se as it does not hide an object. It presents how-ever the surprising ability to create a mirage effect, by giving the illusion that the object inside the de-vice is located in a deceptive position. This is tested numerically here for flexural waves in the time do-main, based on the proposed design. We clamp, within the annulus of the wave rotator studied earlier,a rectangular obstacle at an angle ϑ = 30 ◦ with respect to the wavefront of the incident pulsed planewave. The field distribution of the flexural mode is recorded at different times, as shown in Figure. 6(a) .The tilted obstacle preferentially reflects back the elastic field at an angle − ϑ . When the rotator struc-ture is added around the obstacle, the field is now backscattered horizontally, as if the rectangular objectwas perfectly aligned with the incident wavefront (Figure. 6(b)). This is confirmed by comparing thefield reflected by the tilted obstacle in the presence of the cloak (Figure. 6(b)) and the field reflected bythe obstacle in a vertical position without the cloak (Figure. 6(c)), which shows the same spatial distri-bution. A more quantitative analysis is proposed in Figure. 7 by comparing the scattering diagrams attime step 1.3 ms and at a distance of 60 mm from the center of the rectangular scatterer. Good overlapis found in the backward direction (0 ◦ ) of the scattered-field distributions induced by the bare verticalscatterer (blue line) and the cloaked tilted scatterer (red line), while in contrast, the bare tilted scattererreflects the wave at − ϑ = − ◦ (black line). One can say that the object has deceitfully been straight-ened in vertical position by the rotator, with the impression for a distant observer that scattered lightcomes from an unexpected direction. Scattering diagram of the mirage effect . Scattered-field angular distribution calculated from Figure. 6 attime step 1.3ms and at a distance of 60 mm from the center of the scatterer for (blue line) the bare vertical scatterer; (redline) the cloaked scatterer tilted at 30 ◦ with respect to the wavefront of the incident plane wave; and (black line) the baretilted scatterer. 0 ◦ represents the backward direction. In conclusion, we have shown that coordinate transformation can be adapted to design new devices forflexural waves. This leads us to an interesting strategy for the design of transformed elastic devices. Weproposed and successfully tested experimentally a waveshifter and a wave rotator. To shape the anisotropictransformed space, we have taken the option to carefully engineer a single material by corrugating thesurface of the plate on a subwavelength scale, a design well-suited for practical realization. The magicof the coordinate transformation manifests itself in the simply-designed reflectionless waveshifter, wherethe 0 th -order elastic mode maintains the direction of its wavefront beyond the bend, while its energy flowis deviated in a different direction. We demonstrate that both devices work for short pulses, with virtu-ally no spatial or temporal dispersion. The rotator turns out to be a truly transparent device since thephase delay accumulated across the device is the same as it would be in free space. To the best of ourknowledge, this has never been observed, with supposedly invisible devices. The usefulness of analogieswith flexural waves in plates, but also with wave optics and water waves for the control of surface seis-mic waves was pointed out in [45]. We believe that our design approach can offer an interesting route tothe control of e.g. surface Rayleigh waves in soils, structured in a similar fashion to elastic plates, usingan alternation of trenches and walls, in order to control quakes in the time domain. Measurement setup : A 3D-printer based on nanoparticle jetting technology (Xjet, Carmel 1400) has beenused to fabricate the waveshifter waveguide of Figure. 1(b). The device was printed on zirconia ( Z r O ),a ceramic with the following elastic parameters: Young’s modulus E =207 GPa, mass density ρ =6040 kg/m and Poisson’s ratio ν = 0.32. A piezoelectric diaphragm (Murata 7BB-12-9) located on the flat arm ofthe waveguide is used to excite the fundamental mode of the waveguide. Both ends of the waveguide are overed with blu-tack on both sides to reduce reflections.The rotator was manufactured using PolyJet (Stratasys Objet 24), a technology based on photo-polymer3D printing. Here we used Vero PureWhite RGD837, a stiff resin with Young’s Modulus E= 2.5GPa,mass density ρ = 1180 kg/m and Poisson’s ratio ν = 0.25. Eighteen piezoelectric diaphragms (Murata7BB-12-9) were bonded along one edge of the 180 cm ×
180 cm square plate. All transducers are excitedsimultaneously with the same signal to generate a plane wave. Blu-tack was also used to reduce reflec-tion at the edge on the plate. In both cases, a Ricker pulse was generated at each transducer by an arbi-trary function generator (Agilent 33220A) with the addition of a high-voltage amplifier. A laser vibrom-eter (Polytec sensor head OFV534, controller OFV2500) was scanned on the flat surface of the device tomeasure the spatio-temporal velocity field of the flexural waves (1mm step grid for the waveshifter and2 mm step grid for the wave rotator). The images are processed using a Hampel filter and a cubic inter-polation.
Ricker pulse : We recall that a Ricker pulse is the second derivative of a Gaussian function and is definedin the time domain by: A = (cid:2) − π f ( t − /f ) (cid:3) e − π f ( t − /f ) . (12)Note that this zero-mean symmetric pulse is solely defined by a single parameter, its most energetic fre-quency f . Numerical simulations : All 3D full-wave simulations were conducted with the Solid Mechanic Module ofthe finite element software COMSOL Multiphysics 5.3. Low-reflection boundary were imposed in the fre-quency domain, on the left and right ends of the waveshifter and rotator plate, and on the outer bound-aries of the cloaking plate. The largest mesh-element was set to be smaller than one-tenth of the lowestwavelength. A finer mesh was used where the domain geometry changes abruptly.
Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.
Acknowledgements
The authors thank Agnes Maurel from the Langevin Institute, ESPCI-CNRS, for fruitful discussions onhomogenization of elastic systems, Avi Cohen from X-jet for the waveshifter fabrication, and Shai Ing-ber from SU-PAD for the rotator fabrication. This research was supported in part by The Israel ScienceFoundation (Grants No. 1871/15 and 2074/15) and the United States-Israel Binational Science Founda-tion NSF/BSF (Grant No. 2015694). P. S. is thankful to the CNRS support under grant PICS-ALAMO.
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Dr. Kun Tang, Dr. Chenni Xu, Prof. Patrick SebbahDepartment of Physics, The Jack and Pearl Resnick Institute for Advanced Technology, Bar-Ilan Univer-sity, Ramat-Gan 5290002, Israel.Email Address:[email protected]. Chenni XuZhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhe-jiang University, Hangzhou, 310027, Zhejiang, China.Prof. S´ebastien GuenneauUMI 2004 Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ, United Kingdom. θ Figure S1:
Geometric parameters of the waveshifter dependence on bending angle θ . Plot of α , D a ( D b ), h a ( h b ) vs. θ , using Equation. (5) of the main text. In the experimental investigation and numerical analysis presented in the main text, the bending angle θ of the waveshifter was chosen to be 20 ◦ . Here, we check the validity of our approach for larger values of θ . The dependence on θ of the plate rigidities D a , D b , the rotating angle α and the plates thicknesses h a , h b , are calculated following Equation. (5) and shown in Figure. S1, see also Figure. 1(b). When θ variesbetween 0 and 90 ◦ , α decreases gradually, while D a and h a diverge rapidly near 90 ◦ (notice the logarith-mic scale in Figure. S1). This indicates that stronger anisotropy is required for sharper bending angle θ .Following the method presented in the main text, we design a waveguide with the same geometry as thewaveshifter shown in Figure. 1, but with different bending angles θ , ranging from 30 ◦ to 70 ◦ . The cor-responding values of α , h a and h b are directly obtained from Equation. (5). The out-of-plane velocityfield distribution is calculated at 16 kHz and shown in Figure. S2 with a comparison to the uniform bent Waveshifter with different bending angles θ . Left column: Simulated out-of-plane velocity field at f = 16 kHz for the waveshifter with different bending angles (a) θ = 30 ◦ , (b) θ = 40 ◦ , (c) θ = 50 ◦ , (d) θ = 60 ◦ and(e) θ = 70 ◦ . Right column: Numerical simulations for the corresponding empty waveguides with no corrugation. waveguide without corrugation. For values of θ between 30 ◦ and 60 ◦ , the 0 th -order mode of the waveg-uide is efficiently transmitted beyond the bend, with the direction of its wavefront well preserved. Incontrast, the same incident mode is seen to convert into a combination of even and odd higher-ordermodes in the uniform waveguide without corrugation. Beyond these values ( θ ≥ ◦ ), the waveshifterlooses its characteristics: the wavefront direction is no longer preserved leading to mode conversion andback-reflection. At these angles, the homogenization formula breaks down, which assumes a low, or at east moderate, contrast in material parameters.We present in Figure. S3 snapshots of the velocity field distribution at three different times in responseto a short incident pulse, for bending angles ranging from 30 ◦ to 50 ◦ . The pulse is seen to propagate s-moothly across the waveguide bend with negligible back-reflections, while the wavefront remains ver-tical, in the same direction as the incident wave. This confirms the performance for short pulses of ourdesigned waveshifters for large bending angles, up to 50 ◦ . For larger angles θ and larger contrast in ma-terial parameters, local resonances occur in the long wavelength limit and the resulting effective param-eters become dispersive. The metamaterial waveguide no longer approximates a dispersionless (frequen-cy independent) transformed medium, and this has a deleterious impact on the metamaterial efficiency.In fact, the larger the contrast in material parameters, the more dispersion in effective parameters, andthus the more impact on the pulse propagation. The choice of θ = 20 ◦ in the experimental demonstra-tion presented in the main text is a good compromise between experimental constraints and large valuesof θ , the main restriction being the resolution of the 3D printing machine and the thickness h b , whichdecreases rapidly with θ (see Figure. S1).Based on these results, we construct a cloaking device following the design proposed in the main text(Figure. 4), with four anisotropic corrugated regions arranged in a symmetric fashion around a diamond-shaped hole. We use the parameters calculated in Figure. S2, h a , h b , h , and α , for different values of θ .A Ricker pulse centered at 20 kHz with a 160 mm-wide Gaussian transverse profile is propagated in thedirection of the hole. The field distribution of the out-of-plane velocity computed at time step 0.38ms, isshown in Figure. S4 for θ = 20 ◦ , θ = 30 ◦ , θ = 40 ◦ and θ = 50 ◦ , for the diamond-hole with and withoutthe cloaking corrugation. We find that the outgoing wave points in the same direction as the incidentwave and that the incident wavefront is well restored for metamaterial cloaks with bending angles up to θ = 40 ◦ . This is in striking contrast with the field distributions resulting from the scattering by a barehole (2 nd row in Figure. S4), where the incident plane wavefront is seen to scatter off into multiple direc-tions, and any information on the initial wavefront is lost. The cloaking device is therefore clearly effec-tive to cloak even large diamond-shape holes, at least for diamond angles up to θ = 40 ◦ . For θ ≥ ◦ , themetamaterial cloak still refocuses efficiently the wavefront in the incident direction, but starts to yieldsignificant wavefront distortions.To better quantify the cloaking efficiency, we perform a spatial Fourier transformation to obtain the ve-locity amplitude in wave-vector ( k ) space (3 rd and 4 th row in Figure. S4). The simulated wave field in k space for metamaterial cloak is mainly localized around two sparkling lines, similar to wave propagationinside the plain plate, which indicates most of the energy points at the forward direction. However, thescattered wave by the bare hole is composed of a wider range of wave vectors located within an annu-lus regime. It means that the scattered wave no longer remains a forward plane wavefront and orient inmultiple directions. This confirms that the metamaterial cloak refocuses efficiently the wavefront in theincident direction for bending angles up to θ = 50 ◦ .The waveshifter can thus be viewed as a building block for certain types of diamond shaped cloaks, suchas shown in Fig. S4. Nonetheless, in the similar way to the waveshifter, a diamond cloak design is con-strained by the requirement of a moderate contrast in material parameters that approximate the trans-formed plate medium, which is dictated by the angle θ inside the diamond shaped hole. The increasingdispersion of the effective medium induced by the increasing contrast in material parameters (here thelayers’ thicknesses), leads to an increased distortion of the Ricker pulse for large θ , as becomes clear for θ = 50 ◦ . In fact, the physical phenomenon of effective dispersion induced by large contrast in material-s parameters of periodic structures occurs for any type of waves, so similar constraints exist for acousticand electromagnetic cloaks: The price to pay for a cloak suppressing the scattering of a large object isthe requirement of a large anisotropy of the transformed medium, which will then be approximated byperiodic structures with large contrast in material parameters. Therefore such cloaks might work well intheory throughout a large frequency band when using homogenization approaches to mimic the trans-formed media, but their efficiency will in practice remain moderately low for Ricker pulses due to the in-herent dispersion in the effective parameters. Dynamics of pulse propagation in the waveshifter with different bending angles θ . (a) Full-3Dnumerical simulations: Snapshots of the out-of-plane velocity field calculated in response to a Ricker pulse with central fre-quency 20 kHz, at times 0.065 ms (left panel), 0.115 ms (middle panel), and 0.165 ms (right panel), for waveshifters withbending angles (a) θ = 30 ◦ , (b) θ = 40 ◦ , and (c) θ = 50 ◦ . Here we provide a short explanation on calculating the flexural eigenmodes supported by homogeneousplate with stress-free boundaries. The details can be found in ref. [1]. As Kirchhoff-Love plate equationis of fourth order, there exist two sets of modes at each frequency w ( e ) ( y ) = A h cosh (cid:18) χ m W (cid:19) cosh( χ p y ) − k ν − χ p k ν − χ m cosh (cid:18) χ p W (cid:19) cosh( χ m y ) i (S1) w ( o ) ( y ) = A h sinh (cid:18) χ m W (cid:19) sinh( χ p y ) − k ν − χ p k ν − χ m sinh (cid:18) χ p W (cid:19) sinh( χ m y ) i (S2)Where χ p = √ k + K , χ m = √ k − K , K = ω p ρh/D , and A is a normalization constant suchthat R W/ − W/ dy | w ( o,e ) ( y ) | = 1. The dispersion relations ω ( k ) can be derived by solving the followingtranscendental equations (cid:2) K + (1 − ν ) k (cid:3) χ m tanh ( χ m W/
2) = (cid:2) K − (1 − ν ) k (cid:3) χ p tanh ( χ p W/
2) (S3)for even modes, and (cid:2) K + (1 − ν ) k (cid:3) χ m coth ( χ m W/
2) = (cid:2) K − (1 − ν ) k (cid:3) χ p coth ( χ p W/
2) (S4)for odd modes.It yields the wavenumber k ( o,e ) i associated to each mode w ( o,e ) at frequency ω . The y -dependence of thefirst two even and odd eigenmodes is shown in Fig. 3(a). The mode profile of first even mode (mode 0)is close to a plane wavefront. The pulse dynamic propagation of the measured and simulated wave patterns inside waveshifter and ro-tator are shown in the enclosed videos.
Diamond-shape cloak with increasing angle θ . Full-3D elastic-wave transient simulations of elastic fielddistribution at time step 0.38 ms: (1 st column) for the plain plate; (1 st row) for the diamond cloak composed of four cor-rugated regions, (2 nd row) for the bare diamond-shaped hole, with bending angles (2 nd column) θ = 20 ◦ , (3 rd column) θ = 30 ◦ , (4 th column) θ = 40 ◦ , (5 th column) θ = 50 ◦ . Spatial Fourier transformation is performed on the velocity fieldshown in the 1 st and 2 nd rows to obtain the velocity amplitude in wave-vector space shown in 3 rd and 4 th rows.5 Video: “waveshifter-experiment”, “waveshifter-simulation”.Measured and simulated time evolution of the spatial distribution of the displacement for wave prop-agation through designed waveshifter. The incident pulse is a Ricker pulse with peak frequency f =20 kHz . • Video: “rotator-experiment”, “rotator-simulation”.Measured and simulated time evolution of the spatial distribution of the displacement for wave prop-agation through designed rotator. The incident pulse is a Ricker pulse with peak frequency f =4 kHz . • Video:“mirage effect-vertical scatter”,“mirage effect-rotated scatter”, “mirage effect-rotated scatterwith outside rotator”.Simulated time evolution of the spatial distribution of the displacement through bare vertical scat-ter, rotated scatter, and rotated scatter with outside rotator. The incident pulse is a Ricker pulsewith peak frequency f = 4 kHz . • Video:“1D cloak-void”,“1D cloak-void with a cloak”.Simulated time evolution of the spatial distribution of the displacement for wave propagation througha bare diamond-shaped void or void coated with a one-dimensional cloak. The incident pulse is aRicker pulse with peak frequency f = 20 kHz . References [1] M. C. Cross, R. Lifshitz,
Phys. Rev. B ,085324.