Negative refraction of Lamb waves: From complementary media to elastic superlens
François Legrand, Benoît Gérardin, François Bruno, Jérôme Laurent, Fabrice Lemoult, Claire Prada, Alexandre Aubry
NNegative refraction of Lamb waves:From complementary media to elastic superlens
Fran¸cois Legrand † , Benoˆıt G´erardin † , Fran¸cois Bruno, J´erˆome Laurent,Fabrice Lemoult, Claire Prada, and Alexandre Aubry ∗ Institut Langevin, ESPCI Paris, PSL University,CNRS, 1 rue Jussieu, F-75005 Paris, France † These authors equally contributed to this worke-mail: [email protected] (Dated: February 18, 2021)We report on experimental and numerical implementations of devices based on the negative re-fraction of elastic guided waves, the so-called Lamb waves. Consisting in plates of varying thickness,these devices rely on the concept of complementary media, where a particular layout of negativeindex media can cloak an object with its anti-object or trap waves around a negative corner. Thediffraction cancellation operated by negative refraction is investigated by means of laser ultrasoundexperiments. However, unlike original theoretical predictions, these intriguing wave phenomenaremain, nevertheless, limited to the propagating component of the wave-field. To go beyond thediffraction limit, negative refraction is combined with the concept of metalens, a device convertingthe evanescent components of an object into propagating waves. The transport of an evanescentwave-field is then possible from an object plane to a far-field imaging plane. Twenty years afterPendry’s initial proposal, this work thus paves the way towards an elastic superlens.
For the last 20 years, negative refraction received a considerable attention, either for wave focusing , lensing ,imaging , or cloaking purposes. In a negative index material, the energy flow as dictated by the Poynting vectoris in the opposite direction to the wave vector . This peculiar property implies that, at an interface between positiveand negative index material, waves are bent the unusual way relative to the normal. Any negative refracting slabthus forms a flat lens which does not suffer from any spherical aberration . Better yet, negative refraction has pavedthe way towards the notion of perfect lens and the ability of overcoming the diffraction limit in wave imaging. Moregenerally, it has also given rise to the notion of complementary media and the ability to cancel the propagation ofwaves by adjoining two mirror regions of opposite refractive indices, thereby paving the way towards the concept ofinvisibility cloak .These notions of perfect lens and complementary media have drawn considerable attention in the physics community.Yet, the experimental implementations of those concepts have remained relatively limited so far. Many attempts havebeen proposed to build artificial media, such as photonic/phononic crystals or metamaterials, which can bedescribed by a negative index material . Yet, these man made media first suffer from the inherent periodicity whichimposes a limitation on the negative refracting lens resolution . Also, the intrinsic resonant nature of many of thedesigns induces strong energy dissipation losses, which limit the depth-of-field and the resolving power of the devices.All these features, in addition to the manufacturing imperfections, explain the relatively poor imaging performanceof the implemented superlenses compared to the initial expectations.More recently, an alternative route has been proposed for the negative refraction of guided elastic waves. Indeed,an elastic plate supports an ensemble of modes, including the so-called Lamb waves, which exhibit complex dispersionproperties. Interestingly, some Lamb modes, often referred to as backward modes, display a negative phase veloc-ity . This particularity results from the repulsion between two dispersion branches with close cut-off frequencies,corresponding to a longitudinal and a transverse thickness mode of the same symmetry. The lowest branch exhibitsa minimum corresponding to a zero-group velocity (ZGV) point . At slightly higher frequency, exploiting theexistence of a negative phase velocity mode, negative refraction of Lamb waves has been achieved without any meta-material. The underlying mechanism consists in a mode conversion between forward and backward propagating modes(or vice versa) either at a step-like thickness discontinuity or at the interface between two different materials withan adequate acoustic impedance mismatch . More recent studies investigated the negative reflection of Lamb wavesat a free plate edge and the conversion of propagating modes at a thickness step in order to optimize the negativerefraction effect .Building on these previous works, we investigate the concept of complementary media through the realisation oftwo devices consisting in duralumin plates of varying thickness. The first one exploits the idea of anti-object inorder to cloak a region of interest. This is done by adjoining a mirror region of opposite index that cancels thediffraction undergone by the waves inside the first region (see Fig. 1a). In a second step, we show how such acomplementary medium can behave as a trap if the waves are generated inside it . Diffraction cancellation via ∗ [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] F e b negative refraction makes the wave circulate for ever around negative corners (see Fig. 1b). In this work, the design a n=-1n=1 metalens n=1n=n n=-1n=-n objobj b ACD B n = -1n = 1 a c
FIG. 1.
Taming waves with complementary media . a Cloaking by an anti-object: Two slabs of equal thickness andplaced adjacent to one another cancel wave diffraction if both display anti-symmetric refractive index distributions with respectto their common interface. Based on this principle, any object can be made invisible by its complementary counterpart, theso-called anti-object. b Wave trapping by a double perfect corner: A subset of rays diverging from a point source is returnedto the source by means of successive negative refraction events. These waves then continue to circulate around the cornerbefore being eventually absorbed. c The combination of a negative refraction slab with metalenses yields a superlens: ( i ) Theevanescent components of a point source is converted into propagating waves by interacting with the sub-wavelength resonatorsof a metalens; ( ii ) This propagating wave-field is replicated in a far-field imaging plane by means of a negative refracting lens;( iii ) Introducing a second metalens, identical to the first one, in the imaging plane back-converts these propagating componentsinto an evanescent wave-field that shall give rise to a sub-wavelength image of the initial source by spatial reciprocity. of both devices is based on a semi-analytical model and optimized thanks to finite difference time-domain (FDTD)simulations . The propagation of Lamb waves across such plates accordingly is then investigated experimentally bymeans of laser interferometry. On the one hand, diffraction cancellation is assessed by the Strehl ratio , a parameterthat quantifies wave distortions at the output of the complementary cloak. On the other hand, wave circulation aroundnegative corners is highlighted by investigating the time-dependence of the wave-field. However, unlike Pendry’s initialproposal , only the propagating component of the wave-field is properly recombined in each device.The third system under study aims at extending these wave phenomena to the evanescent component of the wave-field. This is done by coupling the negative refraction phenomenon with the concept of metalens. Introduced byLemoult et al. ten years ago , this arrangement of sub-wavelength resonators acts as a converter of the evanescentwave-field into propagating waves. By placing such a metalens in the vicinity of both the object and imaging planesof the negative refracting lens (see Fig.1c), the evanescent wave-field (or at least a part of it) can be transported formthe object to the imaging plane. Again, our semi-analytical model and a numerical simulation enable the design ofsuch an elastic superlens. The evanescent wave-field gives rise to a sub-wavelength focal spot of λ / Results
Harnessing forward and backward Lamb modes.
Whereas an infinite medium supports two - longitudinaland transverse - waves traveling at distinct but unique velocities, plates support two infinite sets of dispersive Lambmodes. The latter ones are elastic waves whose particle motion lies in the sagittal plane that contains the directionof wave propagation and the normal of the plate. The deformation induced by these modes can either be symmetric(referred to as S i ) or antisymmetric (referred to as A i ) with respect to the median plane.In this article, the propagation of elastic waves across a duralumin plate (aluminium alloy) is studied. The materialdensity is ρ = − . Its longitudinal and shear wave velocities are c L = − and c T = − ,respectively. The dispersion curves of symmetric Lamb modes in a 1 mm-thick plate are displayed in green in Fig. 2.The symmetric zero-order mode S is the extensional mode of the plate. It exhibits free propagation to zerofrequency, whereas the higher order modes admit a cut-off frequency. The dispersion branch S displays severalcrucial features. Its repulsion with the S -mode gives rise to a ZGV point ( f ZGV , k ZGV ) (see Fig. 2). Betweenthis ZGV resonance frequency f ZGV and the cut-off frequency, coexist a backward mode, referred to as S b , and theforward mode S .If one consider a single Lamb mode impinging at an interface between two plates of different thickness, one can expecta large number of reflected and transmitted modes which can either be propagative, inhomogeneous or evanescent One striking phenomenon is that, at the crossing point between the forward mode in the thickest part (red line inFig. 2) and the backward mode in the thinnest part (blue line in Fig. 2), the S incident mode is mainly transmittedinto the S b mode. The thickness step should be made symmetric with respect to the mid-plane of the plate in orderto avoid conversion into anti-symmetric Lamb modes. A semi-analytical study of this phenomenon has allowed the F r e q u e n c y ( M H z ) S S S S (mm -1 ) S S S S f c λ c −1 −λ c −1 f zgv f zgv FIG. 2.
Forward and backward Lamb modes . Dispersion curves of the symmetric Lamb modes in duralumin plates of d = d = . S mode (red) in the thick part and the backward mode S b (blue) in the thin part are of special interest for negative refraction.Their phase velocity coincide at crossing frequency f c . optimization of the transmission coefficient between the S and S b modes. In such a geometry, the optimal thicknessstep ratio is close to 0 .
9. The corresponding amplitude transmission coefficient is T = .
93. Furthermore, this hightransmission is effective over a large angular spectrum ( T > . ○ ).This phenomenon arises from the equality of S and S b absolute wave numbers at the crossing point. This impliesthat these two modes are associated with similar stress-displacement fields and only differ by their opposite wavevectors. In consequence, the negative refraction process is particularly efficient at such a thickness step. In thispaper, we will show this peculiar property holds in more complex geometries and how to harness negative refractionto compensate the diffraction or trap the waves through the use of the complementary media concept. Cloaking by an anti-object.
For a complex slab made of alternatively positive and negative index areas, thejuxtaposition of a complementary mirror slab with an opposite refractive index distribution allows the cancellation ofwave diffraction . This concept may be extended to various kinds of index distributions. For instance, the scatteredwave-field induced by an object can be hidden by adjoining a so-called “anti-object” as illustrated by Fig. 1a. Theanti-object is the negative mirror image of the object, the mirror being taken to lie on the interface between the twocomplementary slabs. The object is thus hidden for one observer downstream to the anti-object as the wave-fields atthe input and output of the complementary slabs shall be identical.Our goal is here to implement this idea for guided elastic waves. The system studied here is a 1 mm-thick plateexcited by a line source that emits the forward S mode as a plane wave. The designed object is a 7 . . S modeis converted into the backward S b mode. For the proof-of-concept, two plate configurations are considered : thereference one with the object [see Fig. 3a] and the other one with the object and its anti-object [see Fig. 3b].These devices are first investigated numerically using a FDTD code . The simulation parameters are describedin the Methods section. A normal displacement pulse is applied to the line source. The normal displacement inducedat the plate surface is recorded over a time length ∆ t = µ s. A spatio-temporal filter described in the Methodssection is then applied to the recorded wave-field in order to isolate the S and S b mode contributions.Figure 3c shows the corresponding wave field in the reference plate at the crossing frequency f c = .
32 MHz. Thecorresponding wavelength λ c is of 12.2 mm. Interferences between the incident and scattered waves result in phaseand amplitude distortions, thus revealing the object to a downstream observer. To quantify the impact of the objecton the phase of the recorded wave-field, a relevant parameter is the Strehl ratio S . Often used in adaptive optics, S quantifies the phase distortions of a wave-front. In the experimental configurations displayed in Fig. 3a and b, itcan be defined as follows: S( x ) = ∣< e iφ ( x,y ) > y ∣ , (1)where φ ( x, y ) is the phase of the wave field at the surface of the plate and the symbol ⟨⋯⟩ y stands for a spatialaverage along the y − direction. The parameter S is equal to unity if the wave-field is a plane wave propagating alongthe x − direction, hence coinciding perfectly with the incident wave-field. On the contrary, the parameter S tendstowards zero for a fully incoherent wave-field.Figure 3e displays the evolution of the Strehl ratio computed from the wave-field supported by the reference plate(Fig. 3c). Upstream to the object, the incident wave-field is not impacted ( S ∼ .
98) meaning that the object does notinduce any significant back scattering. Its transverse cross-section is indeed larger than the wavelength. The incidentwave-field is thus scattered in the forward direction. This is quantified by the abrupt decay of the parameter S from0.98 to 0.2 at the object’s boundary. After the object, the wave-field remains distorted (Fig. 3c) with a Strehl ratio S remaining below 0 . d= 1.0 mmd= 0.9 mm P i e z o e l e c t r i c s t r i p Interferometer
20 mm c df g (h) a b x [mm] he x [mm] S t r e h l r a t i o S t r e h l r a t i o FIG. 3.
Cloaking by an anti-object. a , b Experimental configuration for the reference and complementary plates, respec-tively. c - e FDTD numerical simulation. The real part of the wave-field obtained in the reference ( c ) and complementary ( d )plates is displayed at the crossing frequency f c (see Fig. 2). e Evolution of the Strehl ratio S( x ) (Eq. 1) across the reference(blue) and complementary (red) plate. f - h Experimental result. The real part of the wave-field obtained in the reference ( f )and complementary ( g ) plates is displayed at the crossing frequency f c . h Evolution of S( x ) across the reference (blue) andcomplementary (red) plate. The addition of a complementary slab including the anti-object can compensate for these strong wave-distortions(Fig. 3b]). The displacement field is calculated for the complementary plate using the FDTD simulation and displayedin Fig. 3d at the crossing frequency f c = .
32 MHz. The phase distortions accumulated by the wave through theobject’s band are now perfectly compensated by the anti-object’s band. This phenomenon is induced by the conversionbetween the forward and backward modes at the interface between the two bands. The wave-front at the output of thecomplementary band retrieve the shape of the incident wave-front at the input, as if the object and complementarybands had disappeared. The anti-object thus enables the cloaking of the object to a downstream observer.The evolution of the Strehl ratio across the complementary plate provides a quantification of the cloaking perfor-mance (red curve in Fig. 3e). Upstream to the object, S fluctuates around a value of 0.9. These oscillations area manifestation of the spurious reflections arising at each thickness step. As mentioned previously, the conversionbetween the S and S b modes is not perfect and residual back-reflections take place at each interface. The parameter S then abruptly decays in the complementary slabs to reach a value of 0.3, before retrieving a close to ideal value( S ∼ .
98) at the device output. This excellent Strehl ratio confirms that the object is efficiently cloaked by itsanti-object in transmission, while spurious reflections prevent from a perfect cloaking in reflection.Going further, it is interesting to notice that the theoretical energy transmission coefficient at each thicknessstep is T ∼ . and thus the global transmission coefficients through the six steps crossed by the wave shouldbe T ∼ .
38. Yet, one can observe that the transmission through the devices is remarkably good. This strikingobservation comes from the elegant physics of complementary media: the reflections at the interfaces in the first bandinterfere destructively with the same reflections in the complementary band .Following this numerical study, the devices displayed in Fig. 3a and b have been fabricated using a 1 mm-thickplate whose negative index areas have been etched by means of several engraving techniques described in the Methodssection. The thickness map of each plate is provided in Supplementary Fig. S1. A piezoelectric strip is glued on thethick part of the plate to generate an incident plane wave (forward S mode). The normal displacement at the platesurface is measured by means of a photorefractive interferometer (see Methods section).Figures 3f and g show the corresponding wave-fields at the crossing frequencies in the reference and complementaryplates, respectively. These wave-fields show some difference compared to their numerical counterparts (Figs. 3c andd). This is partly explained by thickness maps that slightly differ from the initial design (see Supplementary Fig. S1).It also appears that the incident wave-fields are not perfectly linear. The glue layer below the piezoelectric strip isnot perfectly homogeneous. Nevertheless, the distortion of the incident wave-field can be taken into account in the a (b)
20 mm a b d= 1.0 mmd= 0.9 mm c TransducerInterferometer
FIG. 4.
Wave focusing in a perfect corner. a
Experimental configuration. b Real part of the normal displacement wavefield simulated at the surface of the plate at the crossing frequency f c = .
32 MHz. c Real part of the normal displacementwave field experimentally measured at the surface of the plate at the crossing frequency f c = .
46 MHz. computation of the Strehl ratio. Indeed a complementary medium shall, in theory, reproduce exactly at its outputthe wave-field at its input. Hence, by subtracting the incident phase to the recorded wave-field, the Strehl parametercan be made independent of the incident wave-field: S( x ) = ∣< e i [ φ ( x,y )− φ ( x = ,y )] > y ∣ , (2)The result is displayed in Fig. 3h. The experimental Strehl ratios exhibit an evolution similar to the numericalpredictions (Fig. 3e). However, in the complementary plate, it does not reach the expected value of 1 but rathersaturates around S ∼ .
7. The cloaking effect is thus less spectacular in the experiment than in the numericalsimulation. Several experimental limitations can explain this difference (see Supplementary Fig. S1). First, thethickness of the negative index areas is not perfectly homogeneous. Second, the engraved anti-object and object arenot exactly mirror from each other. Hence, the wave-front recombination at the device output cannot be optimal.These devices suffer from the same limitations than previous cloaking realisations reported in the literature : Thecloaking performance is hampered by manufacturing imperfections.Nevertheless, the results shown here constitute a first experimental proof of concept of the scattering cancellationthrough complementary media for elastic waves and even, to our knowledge, in wave physics. Furthermore, theuse of natural backward Lamb waves shows the advantage of being free of the meticulous conception of a resonantmetamaterial. This strategy has allowed us to overcome the energy dissipation issues generally encountered in suchman-made media.An alternative cloaking application of negative refraction is demonstrated in Supplementary Fig. S2 with the useof complementary bands . Each band is shown to annihilate wave diffraction from the other. The overall effect is asif a section of space was removed from the experiment. Another interesting ability of cloaking is also to isolate anobject from the external environment with no possible interaction. In other words, no wave can access the object fromthe outside, and no wave generated by the object can get out of the cloak. In the next section, we show that this ispartially the case in complementary media. A part of the wave generated by a source inside a complementary mediumremains trapped inside. To highlight this striking phenomenon, Lamb wave propagation is investigated around adouble perfect corner (Fig. 1b). Trapping by a double perfect corner.
The double perfect corner device is composed of four quadrants of oppositerefractive index (Fig. 4a). A point source at point A emits a diverging wave in one of the positive index quadrant.This wave-field is then negatively refracted at both interfaces in the negative index quadrants. Thus the wave, bysuccessive negative refraction phenomena, travels around the double corner creating images of the source in eachquadrant, named A, B, C and D in Fig. 1c. This trapping phenomenon has been highlighted, both theoretically andnumerically, for electromagnetic waves and for flexural waves .A numerical simulation of the double perfect corner for Lamb waves is first performed (see Methods). As before,a duralumin plate is considered, with positive and negative index parts of thickness 1 mm and 0.9 mm respectively. Anormal displacement point source is placed on the top right quadrant at 15 mm from both interfaces. It generates acylindrical forward S Lamb wave. At each interface, a conversion into the backward mode S b is expected. Figure 4bshows the simulated displacement field at the crossing frequency. The trapping of Lamb modes around the corner isillustrated by a refocusing of waves in each quadrant. Admittedly, the focusing in the third quadrant is not as wellresolved as in the other quadrant. This unwanted effect is inherent to the device as explained by a ray approachdepicted in Fig. 1b: (i) Part of the emitted rays straightly propagate towards the edges of the plate and get absorbedby perfectly matched layers (red arrows); (ii) Part of the negatively refracted rays are also lost in the second andthird quadrants (orange arrows). Therefore, only half of the angular spectrum of the incident wave-field (blue arrows)contribute to the refocusing in the the opposite quadrant.
20 mm a b c d e f g h S i m u l a t i o n E x p e r i m e n t FIG. 5.
Trapping a wave-packet around a perfect corner . Snapshots of a wave packet (∆ f ∼ . MHz ) propagatingaround the double perfect corner. a - d Numerical simulation ( f c = .
32 MHz). e - h Experiment ( f c = .
46 MHz).
Following this numerical study, a double perfect corner is manufactured on a duralumin plate by chemical etching(see Methods). The erosion process being imperfect, a control of the thickness is necessary and the correspondingthickness map is shown in Supplementary Fig. S1. The source is a piezoelectric transducer glued on the top rightpositive index quadrant at 15 mm from both interfaces (see Methods). The normal displacement is measured at theplate surface using a photorefractive interferometer. The wave-field filtered at the crossing frequency is shown inFig. 4c. As predicted, refocusing occurs in each quadrant. The resulting wave field is closely similar to the numericalresult displayed in Fig. 4b obtained with perfectly matched layers. The reflections on the plate edges are thereforeinsignificant in the experiment because of the overall dimension of the plate (see Methods). The residual discrepancybetween the numerical and experimental wave-fields is explained by: ( i ) a different thickness ratio ( d / d = . f c = .
46 MHz instead of 3.32 MHz) and corresponding wavelength( λ c =
10 mm instead of 12.2 mm); ( ii ) thickness fluctuations in the eroded areas (see Supplementary Fig. S1).While a study of the wave-field in the frequency domain nicely shows the refocusing of waves in each quadrant, itis interesting to observe the field in the time domain in order to assess the trapping of Lamb waves around the doubleperfect corner. To that aim, an inverse Fourier transform of the wave-field is performed over a frequency bandwidthof 0 . t =
1, 20, 40 and60 µ s in Fig. 5. The images of the source located in quadrant A are observed in quadrants B and C at time t = µ s(Fig. 5b), before recombining in quadrant D at t = µ s (Fig. 5c). As expected, the negatively refracted waves comeback to the initial source location in quadrant A at time t = µ s (Fig. 5d.The temporal behavior of the wave-field around the double perfect corner is now investigated experimentally(Fig. 5e-h). The emitted wave [ t = µ s, Fig. 5e] refocuses in quadrants B and C [ t = µ s, Fig. 5(f]. However,the displacement amplitude is twice lower in quadrant C than in quadrant B. This is explained by the fact thatthe thicknesses of quadrant B and C slightly differ (see Supplementary Fig. S1), resulting in different transmissioncoefficients at the interface. Moreover, some step irregularities probably induce spurious reflections at the interfacebetween quadrants A and C. Nevertheless, negatively refracted waves recombine in quadrant D ( t = µ s, Fig. 5g)but dissipation losses prevent us from observing, at the return trip time, the refocusing at the initial source location(see Fig. 5h).Lamb wave focusing by negative refraction, as observed for the flat lens or for the perfect corner, is diffractionlimited. Unlike Pendry’s initial proposal , the dimension of the focal spots remains limited to a half-wavelength.Indeed, the negative refraction Lamb mode focusing only involves propagating modes of wavelength much larger thanthe plate thickness. In the next section, a strategy is proposed to design an elastic superlens by combining the negativerefraction phenomenon with the metalens concept . Superlensing by combining negative refraction and metalens concepts.
Lemoult et al. introduced theconcept of metalens , an arrangement of sub-wavelength resonators that allow to convert the evanescent componentsof a source into propagating waves . Initially coupled to a time-reversal mirror in order to produce sub-wavelengthfocusing at the source location, our idea is here to couple the metalens concept to the negative refraction phenomenon.Time reversal and negative refraction are actually intimately linked processes . Nevertheless, one main difference isthat the object and imaging planes are distinct in a negative refraction scheme: The evanescent component of theobject wave-field is thus recombined at another imaging plane .In this paper, the following strategy is adopted: Introducing two identical metalenses in the vicinity of the objectand imaging planes (see Fig. 1c). The first metalens will convert the evanescent components of the object into e k y /(2 π ) F o u r i e r s p e c t r u m ( d B ) wave-field imaging plane FT high-pass filter ab
01 50 1000 150 c y ( mm ) dd/2 x(mm) e v a n c e s c e n t � i e l d m a g n i t u d e ( a . u ) m a g n i t u d e ( d B ) d λ /2 f λ /6 Evanescent Evanescent Evanescent EvanescentPropagative Propagative k y /(2 π ) g FIG. 6.
Transporting an evanescent wave-field in the far-field by means of an elastic superlens. a , b Normaldisplacement magnitude at plate surface without ( a ) and with ( b ) metalenses. c Evolution of the evanescent component, k y = .
11 mm − , of the wave-field along the x -direction without (blue) and with (red) metalenses. d Normal displacementmagnitude along the y -axis in the imaging plane ( x =
120 mm) (blue) without and (red) with metalenses. e Spatial Fouriertransform of the image wave-field without (blue) and with (red) metalenses. f High-pass filter applied to the image wave-fieldin order to point out the evanescent component contribution to the focusing. g Spatial Fourier transform of the filtered imagewave-field. In panels d - g , the displacement magnitude is shown in dB. propagating waves. The corresponding wave-field is reproduced in an imaging plane thanks to a negatively refractingslab. By virtue of spatial reciprocity theorem, the second metalens in the imaging plane shall back-convert thepropagating waves into the evanescent components of the object in this imaging plane.To that aim, a meta-lens should be designed for the incident forward S Lamb mode. As demonstrated in previousworks , an efficient sub-wavelength resonator for Lamb waves is the resonant blind hole. The perforation mustbe symmetric in order to avoid any conversion into anti-symmetric modes. The holes depth are chosen to maximizethe reflection of S mode in itself. An optimal thickness ratio of 0.76 between the plate and the holes is found usingthe semi-analytical model described in Ref. (see Supplementary Fig. S4). The holes diameters are set to 2 mm inorder to meet the following criteria: ( i ) being sufficiently small compared to the S mode wavelength ( λ c = . ii ) beinglarge enough to maintain a sufficiently high scattering cross-section. The choice of the inter-hole distance (1 mm) ismade accordingly.This metalens is placed both in the object and imaging planes of a negative refracting lens of 0.92 mm thicknessand width 50 mm (Fig. 1c). The object and imaging planes are at 25 mm from the negative refracting slab interfaces.A point source is placed in the vicinity of the object plane ( x =
20 mm). This system is simulated by meansof a FDTD code (see Methods). For sake of comparison, a reference numerical simulation is performed. Itconsists in simulating the same system without the metalenses. Figures 6a and b display the modulus of the normaldisplacement for the reference and metalenses devices, respectively, at the crossing frequency. As already investigatedexperimentally , the negative refracting slab in Fig. 6a shows a double focusing process in the slab and after it. Thespatial profile of the focal spot in the imaging plane is displayed in Fig. 6d. Its full width at half maximum (FWHM) δ is of 11.2 mm ( δ ∼ λ c ). This spatial extension is limited by the numerical aperture of the negative refracting lens: δ ∼ λ c /( θ ) where θ ∼ o is the maximum angle under which the lens aperture is seen from the point source. TheFourier transform of the wave-field along the y -direction confirms this limited collection angle (Fig. 6e). It is boundedbetween − k sin ( θ ) and k sin ( θ ) (with k = π / λ c the wave number) which confirms the absence of evanescent wave-fieldin the imaging plane.In presence of metalenses, a double refocusing process is also observed in Fig. 6b. As before, the focal spot insidethe negative refracting lens seems diffraction-limited. However, the focal spot in the imaging plane looks much thinnerin presence of the second metalens, compared to the reference wave-field in Fig. 6a. This observation is confirmed byFig. 6b that displays the displacement magnitude of this focal spot across the imaging plane. Its FWHM δ is close to λ c which implies a numerical aperture sin θ ∼ y − direction shows that both propagating components ( ∣ k y ∣ < k )and evanescent components ( ∣ k y ∣ > k ) contribute to this wave-field (Fig. 6e). This evanescent wave-field can beaccounted for by the effect of the second metalens. Not surprisingly, this array of sub-wavelength resonators gives riseto large spatial frequencies in its near-field. The question is to determine if this evanescent wave-field contributes tothe focusing process in the imaging plane. To do so, a spatial high-pass band filter has been applied to get rid off thepropagating part of the wave-field and only focus on its evanescent part [see Fig. 6g]. The corresponding focal spotis obtained by an inverse Fourier transform and its magnitude is displayed in Fig. 6f. In presence of metalenses, asharp focal spot is obtained exactly at the location of the source image with a FWHM δ ∼ λ c /
6. The strong secondarylobes at -3 dB are induced by the cut-off frequency of the high pass filter. Nevertheless, this result shows that thenegative refraction and metalens phenomena are perfectly complementary. Indeed, they provide: ( i ) a transport ofthe evanescent wave-field over a distance much larger than the wavelength ( ∼ λ c in the present case); ( ii ) a properrecombination of the evanescent components in the imaging plane, thereby leading to sub-wavelength focusing. Thisnumerical proof-of-concept thus paves the way towards a future experimental implementation of an elastic superlens. Discussion
In this paper, backward Lamb modes have been taken advantage of to design and implement devices based on thephysics of negative refraction and complementary media. A previous analytical study and numerical simulations haveallowed the optimization of such devices and demonstrated the merits of this strategy. In particular, a material likeduralumin displays an excellent conversion between forward and backward Lamb modes at a thickness step, whileexhibiting low dissipation losses in contrast with meta-materials. These performances have been qualitatively con-firmed by experimental realizations of such devices by means of laser interferometry. Nevertheless, these experimentshave also pointed out the extreme sensitivity of complementary media to manufacturing imperfections. The etchingmethods used in this paper, namely chemical etching and die sinking electrical discharge machining, have shown somelimits, in particular for tailoring sharp corners and guaranteeing a perfectly homogeneous plate thickness over a fewdecimeters. To cope with these issues, additive manufacturing, in particular selective laser melting (or laser powderbed fusion), seems to be particularly promising. Other micro-machining methods used for semiconductors or MEMS,such as laser-assisted wet etching , could provide a sharper design but they would require to work on smaller platedimensions, hence higher frequencies and larger dissipation losses. In that perspective, alternative materials moreadapted to each manufacturing technique could be used such as silicon or aluminium . Note, however, that thischange of material can be detrimental to the conversion efficiency between forward and backward Lamb modes ateach thickness step .On a more fundamental side, the striking properties of complementary media can be leveraged by means of transfor-mation optics . Indeed, the ratio of plate thickness d to wavelength λ determines the effective stiffness of the plateand the phase velocity of the mode. As shown by Lefebvre et al. for flexural waves, one can tune the local phasevelocity with the thickness of the plate. Another option consists in using surface bonded slice lenses . Graded indexdevices can thus be designed by means of conformal transformations . Applied to any complementary medium, itcan lead to the design of a whole set of cloaking devices .With regards to the superlens, one open question remains the existence of resonant sub-wavelength Lamb modesat a thickness step. Analogous to the role of surface plasmons in the perfect lens of Pendry , they would allowthe transport of the evanescent components across a negative refracting lens made of two thickness steps. Thesesurface resonant states do exist, in theory, for acoustic waves at the interface between two fluids of opposite refractiveindex . To the best of our knowledge, their existence has not been pointed out, so far, for elastic waves. For themore specific case of Lamb waves, such resonant states seem to arise at the thickness steps in the elastic superlensinvestigated in Fig. 6b. An enhancement of the evanescent components of the wave-field is actually observed at eachthickness step of the negative refracting lens in presence of metalenses (see Fig. 6c). Further investigations are thusneeded to understand the nature and role of these resonant states at the interface of the elastic superlens. On thetheoretical side, the existing analytical model should be extended to the case of evanescent incident wave-fields. Onthe numerical side, finite element modeling would be more adapted than FDTD simulations because of the resonantfeature of a perfect lens. An adaptive meshgrid would also enable a finer spatial sampling in the vicinity of thicknessdiscontinuities. On the experimental side, a monochromatic and spatially-selective generation of Lamb waves wouldbe required to excite selectively such resonant modes.In this work, we have highlighted both numerically and experimentally the elegant physics of complementary mediain the context of guided elastic waves. Lamb waves are actually perfect candidates to observe negative refractionphenomena since, under certain conditions, thickness steps on a plate can generate a very efficient conversion betweenforward and backward Lamb modes. By designing complementary media as an arrangement of thickness steps overa plate, the ability of cloaking a part of the space or of trapping waves around some singular points has beendemonstrated. Despite manufacturing imperfections, this is a first experimental proof-of-concept of complementarymedia for elastic waves. At last, a numerical study has paved the way towards a new route for the design of anelastic superlens. By combining negative refraction with the metalens concept, the evanescent field of a sourcecan be transported in the far-field and recombined in an imaging plane by means of a negative refracting lens.However, two main issues remain to be solved before implementing experimentally such an elastic superlens: ( i ) Amore elaborated manufacturing strategy for sharply tailoring the metalens and the thickness steps ; ( ii ) A bettertheoretical understanding and experimental harnessing of resonant surface modes that have been highlighted at eachthickness step. From a more applied point-of-view, Lamb waves draw nowadays increasing attention for the design ofnew electro-acoustic devices in electrical engineering or acoustic sensors in MEMS technology. They are also commonlyused for non-destructive testing in the aviation, automobile or nuclear power industries. For all these applications,the control of Lamb waves is essential. In that perspective, the ability to cancel their propagation in some parts ofthe space or to focus them at a deep sub-wavelength scale is of a great interest whether it be for imaging, sensing orfiltering applications. Methods
Numerical simulations.
Numerical simulations have been performed by means of the FDTD Simsonic software to studythe propagation of elastic waves across a duralumin plate of density ρ = − , longitudinal wave speed c L = − and transverse wave velocity c T = − . The plate dimension and mesh size are given in Tab. I. Perfectly matched layers(PML)s are applied at the edges of the plate in order to avoid spurious reflections.Simulated Complementary Superlensdevice mediaDimension [mm ] 200 ×
200 150 × Simulation Parameters .Depending on the simulated device, the source is either point-like (double perfect corner, superlens) or linear (anti-object).It consists in a normal displacement source applied at the surface of the plate. The excitation function is a pulse of 3.3 MHzcentral frequency with -6 dB bandwidth of 100%. The normal displacement of the induced wave-field is then recorded over110 µ s at the surface of the plate with a spatial period δ = . Experiments.
For each experiment, a 200 ×
200 mm duralumin plate is used. Such a large dimension minimizes reflectionson the edge of the plate during the recording time. The plate, which initially displays an homogeneous thickness of 1 mm, isengraved in the negative index parts to obtain a thickness of 0 . µ s-long linear chirpsweeping a frequency spectrum ranging from 3 . . Data post-processing.
A temporal Fourier transform is then applied to the recorded wave-fields in order to study each deviceat the crossing frequency f c between the forward and backward modes: f c = .
32 MHz (in numerical simulations), f c = . f c = .
29 MHz for the anti-object cloaking plate, and f c = .
46 MHz for the perfect cornerplate. This wave field is then filtered spatially using a low pass filter in order to isolate the contribution of the S and S b modes. The spatial frequency cutoff of this filter is of 0 .
25 mm − .In the superlens simulation, the propagating components of the wave-field are filtered by means of an additive high-passfilter whose spatial frequency cutoff is of 0.085 mm − [see Fig. 6(e)]. The attenuation applied to the propagating component ofthe wave-field is of 20 dB. Author contributions
A.A. initiated and supervised the project. B.G., J.L., C.P. and A.A. designed the complementary devices. Fr.L.,Fa.L., C.P. and A.A. designed the elastic superlens. Fr.L. and B.G. performed the numerical simulations. Fr.L.,B.G., F.B. and J.L. performed the experiments. All authors analyzed and discussed the results. Fr.L., C.P. and A.A.prepared the manuscript. All authors contributed to finalizing the manuscript.
Acknowledgements
The authors thank Wei Guo and Samuel M´etais for their works on the numerical aspects of the anti-oject andperfect corner devices, respectively, during their internship. The authors are also grateful for funding provided bythe Agence Nationale de la Recherche (No. ANR-15-CE24-0014-01, Research Project COPPOLA) and by LABEXWIFI (Laboratory of Excellence within the French Program Investments for the Future, No. ANR-10-LABX-24 andNo. ANR-10-IDEX-0001-02 PSL*). B.G. acknowledges financial support from the French Direction G´en´erale del’Armement (DGA).1
Supplementary Information
This document provides further information on: ( i ) the mapping of the plate thickness in each experimental device;( ii ) the study of complementary bands; ( iii ) the design of blind holes in the numerical study of the superlens. S1. PLATES THICKNESS MAPPING
In order to evaluate the quality of the manufactured plates, a local measurement of the plate thickness shouldbe performed. To that aim, a map of the ZGV resonance frequency (Fig. 2) is achieved by means of a heterodyneinterferometer and a pulsed Nd:Yag laser whose wavelength is 1064 nm (Centurion, Quantel) .For the cloaking plates, the ZGV frequency measured on the thick part of the reference plate ( f + zgv = .
888 MHz)is higher than the one measured on the complementary plate ( f + zgv = .
812 MHz). The reference plate is thus thinnerby a factor 0 .
97. This difference implies a slightly different wave-length λ c exhibited by the S and S b modes at thecrossing frequency in each plate (see Fig. 3 of the accompanying paper).Figures S1a and b display the ZGV frequency across the reference and complementary plates normalized by theirrespective f + zgv . This quantity thus yields the thickness across the plate normalized by its value obtained from anaverage over the black dotted rectangle. In each case, the thickness ratio between the thin and thick parts of theplate ranges between 0.88 and 0.91. Note that, in the complementary plate, the object and anti-object are not strictlymirror from each other. This partly explains the lower cloaking performance obtained experimentally compared tothe numerical prediction (see Fig. 3 of the accompanying paper). a b
10 mm c FIG. S1.
Maps of the plates thickness. a
Reference plate. b Complementary plate. c Double perfect corner plate. In eachcase, the measured thickness is normalized by its mean value ( d = A relative thickness map of the perfect corner device has been measured with the same technique. The result isdisplayed in Fig. S1c. The mean thickness ratio is d / d = .
86. This value implies a crossing frequency f c = .
46 MHzbetween the forward S mode in the thick part and the backward S b mode in the thin part of the plate. S2. COMPLEMENTARY BANDS
A negative refracting slab can be as a piece of negative space: It annihilates the diffraction of waves propagatingover an equal thickness of opposite index. This striking property gave birth to the concept of complementary media.Two slabs of material of equal thickness and placed adjacent to one another annihilate each other if one is the negativemirror image of the other, the mirror being taken to lie on the interface between the two slabs.The first configuration proposed by Pendry consisted in associating two complementary bands of thickness L , anexample of which is given by Fig. S2b. The phase accumulated by the wave during its travel through the first bandis exactly compensated by the complementary band. At the output of this system, the transmitted wave-front isperfectly analogous to the incident one. The overall effect is as if a layer of space of thickness 2 L had been removedfrom the experiment (Fig. S2c). This property holds whatever the incident wave-field. For the particular case of apoint-like source, the complementary bands translate this source into a virtual one shifted by a distance 2 L (Fig. S2b).In this work, we have implemented this idea for guided elastic waves. The system studied here is a 1 mm-thickduralumin plate excited by a piezoelectric transducer that emits the forward S mode as a cylindrical wave (Fig. S2a).The designed complementary bands consists in an arrangement of thickness steps over the plate. The positive andnegative index areas corresponds to a plate thickness of 1 and 0.9 mm, respectively. At the crossing frequency and ateach thickness step, there is a conversion between the forward S and backward S b modes (see Fig. 2). The overallthickness 2 L of the complementary bands is of 40 mm.2 y [ mm ]
25 50-25-50 0 -50-2502550 x [mm] y [ mm ] -50-2502550 da g i x [mm] f InterferometerTransducer
25 50-25-50 0 x [mm] s o u r c e p o n c t u e ll e s o u r c e v i rt u a l s o u r c e s o u r c e p o n c t u e ll e b c N u m e r i c a l s i m u l a � o n E x p e r i m e n t eh d= 1.0 mmd= 0.9 mm n = 1n = -1 −π+π0−110 FIG. S2.
Cancelling wave diffraction over a region of space by means of complementary bands. a
Experimentalconfiguration. b An alternative pair of complementary bands annihilates the effect of the other. Wave does not follow a straightline path in each layer, but the overall effect is as if a section of space thickness 2 L were removed from the experiment. c The latter effect is highlighted by juxtaposing the wave-fronts upstream and downstream to the complementary bands. d - f Numerical simulation of the complementary bands: Real part ( d ) and phase ( e ) of the normal displacement wave-field at thecrossing frequency f c = .
32 MHz. ( f ) Juxtaposition of the wave-fronts at the input and output of the complementary bands asin c . d - f Experimental implementation of the complementary bands: Real part ( d ) and phase ( e ) of the normal displacementwave-field recorded at the crossing frequency f c = .
37 MHz. f Juxtaposition of the wave-fronts at the input and output of thecomplementary bands as in c . These devices are first investigated numerically using a FDTD code . The simulation parameters are describedin the Methods section of the accompanying paper. A normal displacement pulse is applied to the point source placedat a distance of 25 mm from the complementary bands. The normal displacement induced at the plate surface isrecorded over a time length ∆ t = µ s. A spatio-temporal filter described in the Methods section is then applied tothe recorded wave-field in order to isolate the S and S b mode contributions. Figure S2d and e show the correspondingwave field and its phase at the crossing frequency f c = .
32 MHz. The incident wave is negatively refracted in thethinnest parts of the first complementary band, resulting in a strongly distorted wave-front at its output ( x = x >
20 mm). The latter one is associated with a virtual source shifted by a distance2 L compared the real source location. It thus appears to be inside the complementary bands to a downstream observer.At the output of the device, wave propagates as if the space containing the complementary bands had disappeared.This effect is highlighted by juxtaposing the wave-fronts in the areas upstream and downstream to the complementarybands (Fig. S2c).However, as shown by Fig. S2b, the magnitude of the wave-field is not perfectly homogeneous at the output. Ashaded area occurs in the superior part of the plate for y ∈ [− − ] mm. A first reason is the decrease of theconversion coefficient between the forward S and backward S b modes for large angles of incidence . In addition, afraction of the incident wave is reflected by the corners between the positive and negative index areas (see for instance3 r S r S t S r S ∑ r i i R e � l e c t i o n a n d T r a n s m i ss i o n C o e f� i c i e n t s Hole thickness ratio x2rz a b
FIG. S3.
Design of the metalens . a Geometry of a single blind hole. b Theoretical reflection and transmission coefficientsof the incident S mode as a function of the thickness ratio in a 1 mm thick duralumin plate at the frequency f = .
31 MHz. the red arrow in Fig. S2b). These spurious reflections induce an heterogeneous angular distribution of the transmittedwave-front across the complementary bands. This effect had already been noticed by Pendry but, in an ideal case,those spurious reflections would be eliminated by destructive interferences between complementary corners via anevanescent coupling between them. In the present case, such a mechanism does not occur for Lamb waves and thereflections induced by the complementary corners give rise to a shadow zone behind the complementary bands.Following this numerical study, the complementary band plate is manufactured on a duralumin plate by electricaldischarge machining. The source is a piezoelectric transducer glued on the plate at 25 mm from the complementarybands. The normal displacement is measured at the plate surface using a photorefractive interferometer (see Methodsof the accompanying paper). The wave-field filtered at the crossing frequency and its phase are shown in Fig. S2g andh, respectively. The crossing frequency f c is here of 3 .
37 MHz and slightly differs from its theoretical value ( f c = . S3. DESIGN OF THE METALENS
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