CPA-Lasing in Thin-Elastic Plates via Exceptional Points
CCPA-Lasing in Thin-Elastic Plates via Exceptional Points
M. Farhat, ∗ P.-Y. Chen, S. Guenneau, and Y. Wu † Computer, Electrical, and Mathematical Science and Engineering (CEMSE) Division,King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Department of Electrical and Computer Engineering,University of Illinois at Chicago, Chicago, Illinois 60607, USA UMI 2004 Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ, United Kingdom (Dated: July 6, 2020)We present here how a coherent perfect absorber-laser (CPAL) enabled by parity-time ( PT )-symmetry breaking may be exploited to build monochromatic amplifying devices for flexural waves.The fourth order partial differential equation governing the propagation of flexural waves leads tofour by four transfer matrices, and this results in physical properties of the PT -symmetry specific toelastic plate systems. We thus demonstrate the possibility of using CPAL for such systems and weargue the possibility of using this concept to detect extremely small-scale vibration perturbationswith important outcomes in surface science (imaging of nanometer vibration) and geophysics (im-proving seismic sensors like velocimeters). The device can also generate finite signals using very lowexciting intensities. The system can alternatively be used as a perfect absorber for flexural energyby tailoring the left and right incident wave for energy harvesting applications. In recent years, the use of resonant elements enrichedthe properties of periodic media, with the paradigm shiftof metamaterials. These are constructed from a judi-cious arrangement of physical resonators whose size isvery small compared to the typical wavelength of in-terest [1] and permit some exotic applications such asnegative refraction [2, 3] or scattering cancellation tech-nique (SCT) [4, 5]. Several research groups have workedon the extension of metamaterials and metasurfaces toelastic waves in solid structures [6, 7]. For instance,the tensorial nature of the equations governing elasticwaves requires complex analytical and numerical model-ing that takes into account the coupling between pres-sure and shear waves at solid interfaces [8]. In the samevein, a particular type of elastic solid, the thin elasticplate (TEP), has drawn a growing interest in the wavephysics community [8, 9]. The plate has a small verticaldimension (thickness) in comparison to its lateral dimen-sions and the wavelength [8], resulting in the vertical dis-placement of the plate largely determined by the flexuralmode (i.e. no shear), sometimes designed as A mode[8]. The bending of these TEPs can be described by theKirchhoff-Love equation (fourth order partial differentialequation (PDE)) and interestingly has a scalar nature inthe case of isotropic plates [8]. This feature allows for amore straightforward numerical modeling of waves prop-agating in isotropic TEPs. Subsequently, several designshave been proposed for flexural waves, including cloaking[10, 11], negative refraction [12], localized surface platemodes [13], SCT [14], elastic plate crystals [15], etc.On another side, it was shown in 1998 thatnon-Hermitian Hamiltonians with Parity-Time ( PT )-symmetry have real eigenvalues [16]. First used in quan-tum mechanics [17], this feature was then applied to op-tics because the paraxial wave equation is mathemat-ically equivalent to the Schr¨odinger equation [18, 19], leading to some remarkable properties, such as an asym-metric propagation of the modes or the existence of an ex-ceptional point (EP) where the PT -symmetry is broken[20, 21]. PT -symmetry gained a tremendous momentumamong the photonics community due to its promisingoutcomes, e.g. environmental sensing [22], on-chip opti-cal systems [23], cavity-mode selection in microring lasers[24]. In the same vein, it was shown that acoustic wavesexhibit such non-reciprocal behavior when loss and gainlayers are balanced [25, 26]. Hence, although these PT -symmetric acoustic systems are still at an early stage,several promising applications have been recently envi-sioned, e.g. unidirectional invisibility cloaking [27], invis-ible acoustic sensor [28], phononic laser [29], and acous-tic Willis coupling [30]. With regards to elastodynamicswaves, shunted piezoelectric thin materials may lead togain/loss in elastic plates, depending on the resistance ofthe shunted circuit [31–33], which was previously used torealize negative refraction [34]. Flexural waves in beamswere further shown to possess PT -symmetric effects [35].In Ref. [36], a different technique was employed to pro-duce non-reciprocal wave transmission.We show in this Letter the possibility to realize theequivalent of lasing in elastic plates, i.e., coherent per-fect absorber laser (CPAL) thanks to gain and loss valuescorresponding to the lasing threshold displaying a quan-tized behavior, which occurs due to topological characterof the system. The spectral singularity could be also usedfor coherent perfect absorber in elastic plates.Flexural waves propagating within an isotropic homo-geneous TEP obey the Kirchhoff-Love biharmonic equa-tion [8], in terms of the vertical displacement W , in thefrequency-domain regime, i.e., by assuming an e − iωt timedependence (See Supplementary Material (SM) [37] forthe general equation in heterogeneous TEPs) ∆ W − β W = 0, where ∆ is the Laplacian operator [38]. More- a r X i v : . [ phy s i c s . c l a ss - ph ] J u l FIG. 1: (a) Structure of the gain/loss device. (b) 2D plot of the eigenvalues versus frequency and the imaginary part of theYoung modulus. (c) Eigenvalues for specific values of the loss/gain in the Young modulus (0, 0.1, 0.2, 0.25, 0.4, and 0.5) GPa. over, the derivation of the transfer and scattering matri-ces of this fourth order system are detailed in SM [37].This equips us with the necessary mathematical arsenalto fully characterize such layered elastic plate systems interms of transmission and reflection. In addition to theusual propagating flexural waves, i.e., e iβx and e − iβx ,there exist evanescent (inhomogeneous) flexural waves,differentiating the TEP from its acoustic counterpart,in which only the propagating waves are considered. Inthe free propagation domain, only the propagating com-ponent survives as shown in Eq. (8) in Ref. [37], theevanescent wave is proportional to e β L x on the left prop-agating side (negative x ) and to e − β R x on the right side(positive x ). Since these evanescent waves decay expo-nentially as they travel away from their corresponding in-terfaces, they do not contribute to the transmission andreflection coefficients, which are measured in the far-field.This is similar to the calculation of the radar scatteringcross-section, considered for example in [14]. However,in order to fully characterize the transmission and reflec-tion of flexural waves, one has to take into account thecontribution of all waves at the inner interfaces (shown inFig. 1 in the SM [37]). What is more intriguing is thatevanescent waves establish propagating components, inthe presence of gain and loss. This behavior is contraryto the case of elastic plates without loss and /or gain,where the evanescent waves are confined to the interfaces.The structure which we consider (schematized inFig. 1(a)) consists of three elastic layers denoted as G,L, and P, which stand for gain, loss, and passive, respec- tively. The possible realization of gain and loss in suchelastic structures has been proposed in Refs. [33, 34].A shunted piezoelectric TEP [31, 32] may lead to an ef-fective Young modulus (of flexural rigidity) with a posi-tive (loss) or negative (gain) imaginary part, dependingon the use of an inductor and a positive (negative) re-sistor. We thus assume that the gain and/or loss canbe tuned in a reasonable range. The geometry of thestructure is given in Ref. [39]. The eigenvalues andthe reflection and transmission spectra of this structureare computed using the S -matrix (See SM [37]) when aplane flexural waves is impinging from the left and/orthe right. The results are depicted in Fig. 1(b)-(c), inthe frequency range 10-40 Hz. Since PT -symmetric wavesystems are reciprocal, the transmittance is the same forwave incident from both directions. However, for thisspecific PT -symmetric scenario, the reflectance is dras-tically different for the right ( R R ) and left ( R L ) inci-dences as shown in the SM [37] (Fig. 2). The two arerelated to the transmittance through r L r ∗ R + tt ∗ = 1.The eigenvalues of the scattering matrix ( s ± ) are ob-tained as function of the S-parameters ( t, r R , and r L ),i.e. s ± = t ± √ r L r R = t (1 ± i (cid:112) (1 / | t | − (cid:61) ( E ) (highlighted curves show the case of (cid:61) ( E ) = 0 . | t | >
1, and the flex-
FIG. 2: (a) Amplitude of the eigenvalues in the frequency domain where CPAL takes place. Points A and B indicate lasing andperfect absorption operation, respectively. (b) Transmittance and reflectance from the CPAL structure. The inset plots theoutput coefficient Ψ. (c) Snapshots of flexural energy for the PT -symmetric CPA flexural laser in (a) at operating frequency541 Hz indicated by A in Fig. 2(a), operated in the lasing mode when the incident wave impinges from the left (top) and theright (bottom). (c) Same as in (a) but for the CPA mode (indicated by B in Fig. 2(a)) at the same frequency but with both leftand right incidence related through Ω R = M s Ω L . The colorbar in (c) and (d) is normalized by the amplitude of the incidentflexural wave. ural system is thus in the so-called broken phase. Onthe contrary, for frequencies higher than 31 Hz, s ± haveboth unit-module and are non-degenerate, implying thatthe system is in the symmetric phase. Around this crit-ical frequency, a sudden phase change occurs, whencethe PT -symmetric structure flips from a broken- PT toa PT -symmetric domain: an EP takes place. This EPmeans a sudden change in the output of the elastic systemdue to spontaneous breakdown of the PT -symmetry. Fora small value of (cid:61) ( E ), the EP frequency is around 10 Hz,while for (cid:61) ( E ) = 0 . r L , that undergoes an abrupt jump of π -radians,around the same frequencies [37], validating the possibil-ity of tuning the EP location by varying the imaginarypart of the Young’s modulus of gain/loss layers. Sucha large tunability of the EP with the amount of (equal)loss (and/or gain) in Young’s modulus is somehow spe-cific to flexural waves, as in acoustics, for example, thelocation of the EP changes only slightly with (cid:61) ( ρ ) (lessthan 10% change in the EP frequency compared to 400%for the flexural case for an equivalent change in the rel- ative imaginary part; also for acoustics the frequency isredshifted with increasing imaginary part, while here itis blueshifted). More detailed analysis of the peculiar-ity of flexural PT -symmetric systems is given in SM [37]and showcases more degrees of freedom to tune and thuscontrol the location and even shape of the EP zone, incomparison to other wave systems, essentially due to itsparabolic dispersion relation and the coupling betweenpropagating and evanescent waves at the interfaces be-tween the gain/loss layers.Inspired by this behavior of EP for flexural waves, weconsider now the possibility of CPAL effect. In fact, itis well known that in optics, PT -symmetric systems canoperate as coherent perfect absorbers by totally absorb-ing the incoming energy (from impinging waves) and aslasing oscillators by emitting coherently outgoing waves[40–42]. These two phenomena can be characterized bythe overall output flexural coefficient ΨΨ = | Ω L | + | Ω R | | Ω L | + | Ω R | , (1)which accounts for the ratio of total outgoing intensity(energy that exits the system) to that of the incomingwaves (energy that impinges onto the system). As ab-sorbing and even perfectly-absorbing structures are al-ready known in elasticity and acoustics, we focus here onthe effect of lasing for these waves, and that representsthe main novelty of this Letter. In the previous results,at best the eigenvalues are offset by 15%. However, forstructures consisting of a few unit-cells shown in Fig. 6of SM [37], we can see that more offset may be observed,but the values are still too low to be considered as lasing.To obtain efficient lasing, we maintain the same struc-ture as before (i.e., the three layers shown in Fig. 1) andwe apply higher loss/gain parameter, as well as shift tohigher (blueshifted) frequencies. The result is plotted inFig. 2(a), that depicts the eigenvalues ( | s ± | ) for frequen-cies in the range 450 Hz-650 Hz, corresponding to wave-lengths between 18.8 cm and 22.6 cm (to be comparedto the width of these layers (in the x -direction) assumedto be identical and set as 20 cm). The plot is given forthree values of (cid:61) ( E g,l ) (1.49, 1.5, and 1.51 GPa) and | s + | can reach lasing regime, with | s + | > , at frequencyaround 541 Hz (denoted A). The complementary eigen-value | s − | ≈
0, which is reminiscent of CPA regime (as infact we have s − = 1 /s ∗ + ). Thus, at the same frequency,one has the lasing regime ( s + ) where the outgoing en-ergy is hugely amplified and CPA regime ( s − ) where theoutgoing energy is cancelled (i.e., all impinging signal isabsorbed by the plate system).To get a clearer picture, we consider the behavior ofthe CPAL system operating at the frequency of point A(highlighted in Fig. 2(a)) under an incident flexural waveof unit-amplitude | Ω , L,R | = 1. The transmitted and re-flected flexural energy T and R are depicted in Fig. 2(b).At the CPAL point, both T and R reach extremely highvalues. The coefficient Ψ, shown in the inset of Fig. 2(b),gives a better picture of the lasing efficiency of the de-vice. At frequency 541.4 Hz the spike reaching 10 isa clear evidence of the lasing effect. To further demon-strate this effect, in Fig. 2(c) we plot the flexural energyin the vicinity of the CPAL device where a normally in-cident flexural wave of unit-amplitude is impinging fromthe left (top panel) and the right (bottom panel). In bothcases, the outgoing waves (reflected and transmitted) aresignificantly amplified (in the range of 10 to 10 depend-ing on the incidence region). However, for one scenario(left incidence) the transmittance is higher than the re-flectance. For the other scenario (right incidence) it isthe reflectance which is higher as can be seen from theplots.These results demonstrate the potential of using sim-ple and compact (60 cm width and 2 cm thickness δ )flexural systems to achieve the equivalence of a flexurallaser that we might call a FLASER. Consider a flexuralwave with very small vertical displacement, of amplitude | W | ≈ µ m (i.e., | W | (cid:28) δ ) incident on the CPAL. Al-though this signal is small, it will be amplified by the CPAL flexural device, and the output displacement willbe in the range of 1 cm, i.e., | W | ≈ δ . Now to relate thiseffect to the transfer matrix (See SM [37]) it is straight-forward to see that lasing may occur, when we have fi-nite (propagating) outgoing signals Ω L and Ω R for verysmall incoming signals. This may occur for M s = 0(and M s = Ω R / Ω L ) if we ignore the evanescent fields,by inspection of Eqs. (21)-(24) in the SM [37]. However,this is generally not possible, as the evanescent fields Ω L and Ω R cannot be assumed to be zero at the boundaryof the system. Hence, the reduced system (in terms ofreflection coefficients) to be satisfied in the general case,i.e., by asking that the incident signal has unit-amplitude(Ω L = 1) and that the outgoing signals diverge, can b ex-pressed as (cid:18) M s M s M s M s (cid:19) (cid:18) Ω L Ω L (cid:19) = − (cid:18) M s M s (cid:19) . (2)If the incidence is taken to be zero and if one imposes fi-nite scattering signals, as occurs in lasing, one must thusensure that the determinant of the matrix in the LHSof Eq. (2) is zero, which yields M s M s − M s M s =∆ M s = 0, which is markedly different from the simplecondition M s = 0 for acoustic or optical systems, forexample. For the transmitted signals, it is easy to ob-tain their expression, by taking the first and third linein Eq. (22) of the SM [37]. This gives for exampleΩ R = M s Ω L + M s Ω L . This complexity stems thus fromthe interplay between propagating and evanescent wavesthat cannot be ignored for flexural systems, as clearlydemonstrated by the lasing equation that mixes ampli-tudes of both kinds of waves. The variation of the param- FIG. 3: Contourplot of the variation of the parameter ∆ M s = M s M s − M s M s versus the frequency and landscape of theimaginary part of the Young’s modulus (in GPa), in logarith-mic scale. The dark regions correspond to the lasing regime(i.e., ∆ M s ≈ eter ∆ M s responsible for infinite outgoing amplitudes(i.e., lasing) is depicted in Fig. 3 versus the frequency ofthe flexural wave and the imaginary part of the youngmodulus (in GPa). The dark regions correspond to thelasing regime (i.e., ∆ M s ≈ (cid:61) ( E ) can be defined for each frequency, belowwhich no lasing can occur. From this landscape of theimaginary part of Young’s modulus it can also be seenthat the lowest threshold occurs for a frequency of 541Hz, as was discussed previously.A similar reasoning can be made for the CPA effect.In fact, for perfect absorption to occur, one must cancelthe outgoing waves for finite incoming waves. The sameanalysis as before shows that for CPA to take place, onemust ensure that M = 0 and M s = Ω R / Ω L , over-looking again evanescent waves. However, in the generalcase, if also an evanescent wave is allowed to be inci-dent, one must ensure that M s Ω L + M s Ω L = 0 and M s Ω L + M s Ω L = Ω R . Yet, in the CPA case, Ω L = 0as it corresponds to an exponentially growing field when x → −∞ . Thus for the CPA operation, evanescent wavesare not directly present in the condition on the ampli-tudes. However, their indirect effect in the M -matrix isstill present. Therefore, one needs to launch two wavesincoming from opposite directions, and by changing their(complex) amplitude ratio, one will be able to selec-tively excite the lasing or CPA mode, as can be seenin Fig. 2(d). If their amplitude ratio M s = Ω R / Ω L , onewill achieve the CPA operation mode. In the rest of thecases, i.e., M s (cid:54) = Ω R / Ω L one will have the FLASERmode. This is, however, a very narrowband effect forboth CPA and lasing.To summarize, an effect reminiscent of lasing is dis-covered in this Letter by making use of PT -symmetry ina specific frequency and gain/loss regimes. This mecha-nism takes roots in the coherent perfect absorber laser ef-fect. This CPAL device can be used as an ultra-sensitiveflexural sensor to detect sub-micrometer displacementsor as a perfect absorber of flexural energy. However,in stark contrast to Maxwell’s equations, the flexuralwave equation assumes displacements that are small incomparison with the thickness of the plate, so our de-vice could be used as a source, since by applying verysmall displacements, these will be amplified and coher-ently transmitted to the outside. On the detection side,if some strong signal/noise impinges on the device, it canbe hugely amplified and can lead to its dislocation. 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