Far-field phonon coupling in valley metamaterial circuits

Domain: physics
Published: 2025-12-11
Editorial outcome: minor_revision

Abstract
On-chip whispering-gallery-mode cavities enable versatile bosonic wave manipulation but typically rely on near-field evanescent coupling. Here we experimentally demonstrate broadband far-field phonon coupling in a valley metamaterial cavity integrated with a Dirac-cone waveguide—termed a “Dirac strip”. The far-field coupling is confirmed by transmission spectroscopy and spatiotemporal field mapping over distances up to approximately five wavelengths, enabling multiplexed, distance-robust coupling pathways that overcome near-field limitations. By combining near-field and far-field cavities on the same substrate, we achieve amplification and control of non-Hermitian dynamics through loss and distance modulation of sympathetic resonances, directly resolved via piezo-laser interferometry. This work establishes a scalable phononic platform for far-field coupling and paves the way for parallel topological wave processors.

Introduction
Whispering gallery modes (WGMs), characterized by reflections along curved cavity interfaces1, have become foundational elements in waveguide-cavity coupling systems with broad utility across acoustics2, elastodynamics3, optics4, and electronics5. These modes enable key functionalities including low-threshold lasing6, enhanced sensing7, and critical coupling–where dissipative loss (Γ) equals radiative loss (γ)8. The pursuit of advanced on-chip coupling schemes has garnered considerable interest in scalable information processing3–5,8.

Recent advances in enhancing waveguide-cavity coupling have exploited non-Hermitian physics in waveguide-cavity systems through deliberate modulation of external loss and gain. This approach enables phenomena such as exceptional points9,10 that yield nonlinear dynamics11, nonreciprocal transmission12, and spectral flows13,14. Non-Hermitian systems are non-conservative and characterized by energy exchange15,16, encompassing three situations: loss only, gain only, or both. Concurrently, topological metamaterials have transformed continuum waveguide and cavity design, giving rise to topological edge states (TESs)17,18, topological whispering gallery modes (TWGMs)19–21, and corner states22,23. These bosonic wave counterparts encode topological protection and synthesized chirality24,25, liberating cavity designs from perfect circular geometries26,27. Valley-phononic TWGMs can exhibit Hermitian20 or non-Hermitian behaviour through unit cell-level modulation of gain and loss26.

Despite these advances, achieving far-field coupling remains a fundamental challenge in waveguide-cavity systems28. All approaches for coupling in such systems to date rely on evanescent near-fields, which intrinsically confine energy transfer to sub-wavelength distances. Far-field coupling schemes, by contrast, if achievable would enable unprecedented spatial flexibility for on-chip integration, full utilization of wave degrees of freedom, and direct observation of inter-cavity dynamics in multi-resonator architectures. Yet their construction has not been possible owing to the absence of broadband, robust, and distance-insensitive coupling mechanisms.

In this work, we experimentally demonstrate broadband, robust waveguide-cavity far-field coupling. This is achieved through topological phonon interactions in on-chip valley waveguide-cavity systems using a Dirac-cone waveguide strip (a ‘Dirac strip’), which enables long-distance coupling beyond conventional near-field limits and markedly improves coupling efficiency. This topological channel mediates enhanced waveguide-cavity coupling through the broadband and robust features of valley metamaterials. By co-locating near- and far-field cavities, we observe sympathetic resonances—simultaneous resonance in both cavities at a single frequency—and reveal non-Hermitian dynamics modulated by material loss and inter-cavity distance.

Results
Valleymetamaterialcircuits:threeconfigurations
Three valley metamaterial circuits are investigated in this work (Fig. 1a). Sample 1 is a far-field cavity (FFC) waveguide circuit composed of triangular pillars, with side length s l=500μm and height h p=292μm(seeFig.1b).Samples2and3correspondtothenear-fieldcavity(NFC)waveguidecircuitanddual-cavitywaveguidecircuit,respectively,withs l=526μmandh p=289μm.Allsamplessharethesamelatticeconstanta=641μmandwaferthicknesse t=525μm,withgeometrical parameters detailed in Supplementary Table 1. The rhombus-shapedcavitieshavesidelength7a,andthewaveguidesspan38a horizontally. Rightward terminal boundaries are oblique to suppressreflections.TheDiracstripsmaintaina3awidthhorizontally,extendingvertically7layersforFFCstructuresand3layersforNFCstructures.Indual-cavityconfigurations,theDiracstripsofNFCandFFCarepositionedwitha7a(center-to-center)horizontalseparation.

We fabricated valley phononic crystal (PnC) plates by etching arraysoftriangularpillarsinahoneycomblattice(seeMethods).DiracconesappearattheKpointoftheBrillouinzoneforθ=0°(Fig.1c,d,upperpanels),whilematerialanisotropyintroducesaminorbandgap(insets).Rotatingthepillarstoθ=20°(phaseA)andθ=–20°(phaseB)generatesbroadbandgapsforantisymmetricplatewaves(Fig.1c,d,lowerpanels),withslightfrequencyshiftsoftheDiracconesbetweenSample1andSamples2–3.

TheunitcelloftriangularpillarspossessesC 3vsymmetrywhenθequalszero,andtopologicalphasetransitionsareinducedbychangingtherotationangleθtobreakthissymmetry.ThesevalleyPnCssupportthetopologicalstatesreportedpreviously24,29. We adoptthis classic paradigmandfabricateallthreesamplesonsiliconwaferstoexplorenovelmechanismsofwavepropagation.

Singlefar-fieldwaveguide-cavityphononsystem
Our first on-chip waveguide-cavity phonon system integrates an FFC coupledtoastraightwaveguideviaaDiracstrip(Fig.2a),toenabletheobservation of far-field coupling. PnCs A and B exhibit opposite substratethicknesse s=e t-h p.Topologicalphasesarecontrolledbytuningtherotationanglesθofthetriangularpillars.c,dBandstructuresoftheunitcellforθ=0°(toppanels)andθ=±20°(bottompanels)areshownforSample1in(c)andSamples2and3in(d).ThecolorscaleindicatRestheratiRoof|u z|tothetotaldisplacement|u t|intheunitcell,definedbyη z= ju zjdV= ju tjdV.Insetsshowunitcellschemesandazoom-inofthedispersionforθ=0°.

ThecavityeigenfrequenciesresultfromembeddingPnC-BwithinaPnC-Adomain(Fig.2b).Withintheantisymmetricplatewavebandgap,fourTWGMs(TWGM-TWGM )areidentified(blacksolidlines),exhibitingadecreaseinfre 1 quency 4 withincreasingh p.Fortheexperi- m m e o n d t e a s l , s e a v m id p e le nc (h ed p= 2 9 2 μ m ) , t l h e e ak T a W g G e M be 1 y a o t n 1. 79 8 M H z c a t he c a vi c t o y. u I p n le c s o t n o t b ra u s l t k , t a o n p ti o s l y o m g m ica e l tr p ic ro b t u e l c k t - i w o a n ve an b d ran b c ro h a e d s b e a m n e d rg p e ro w p it e h rt w ie a s v 3 e 0,3 e 1. n T er h g e y m co o n d fi e n le e d d totheDiracstrip,whereastheirfrequencyrangesoccupytheeigenfrequenciesofTWGM 1 ,TWGM 3 andTWGM 4 (SupplementaryFig.8).Theseguidedbulk-wavebranchesaremodulatedbystrippillarrotation.Forexample,atstrippillarangleθ=10°,thesebulk-wavebran- chesoccupyexclusivelytheTWGM 4 frequency,directlyshapingtheD 1 -D 4 andP 1 -P 4 |u z|versusθprofiles(SupplementaryFig.8).

In situ time-resolved observations are performed by spatially mappingu zoverthewaveguide-cavitycircuit(seeMethods).Normal- ized |u z| is measured at representative positions: Pure PnC (i.e., unmodifiedPnC),FFC,Inlet,andOutlet(Fig.2a).AsshowninFig.3a, theFFC|u z|(bluecurve)exhibitspeaksP 1 -P 4 at1.808,1.867,1.937and1.976MHz, corresponding to the calculated frequencies of TWGM- 1 TWGM .ComparedtothePurePnCcase(graybackground),theoutlet 3 4 |u z|(red curve) shows distinct minima at D 3 ≈0.32 (1.937MHz) and D ≈0.30(1.976MHz). 4

The |u z| fieldmaps confirm incident wave coupling from wave- guidetocavity(Fig.3b).Minimalenergyreachesthetruncatedoutlet at frequencies corresponding to D /D owing to TWGM /TWGM 3 4 3 4 excitation, whereas partial wave leakage at D prevents outlet dip 2 observation.Themapof|u z|atD 1 isshowninSupplementaryFig.9, comparedtotheFFCsystem,D /P persistinexperiment/simulation 3 3 (Supplementary Fig. 11) regardless of cavity position. TWGM fre- quenciesshiftwithchangeins landh p,buttherobustnessofD 3 con- firmstheexistenceofstablecoupling.Crucially,NFCexhibitsa10kHz frequencyshiftfromD 3 (1.955MHz)toP 3 (1.965MHz)—unobservedin theFFCsystems.

Dual-cavitywaveguidesystem
Our final waveguide-cavity phonon system integrates both FFC and NFC,bridgedtoasharedwaveguideviatwoDiracstrips(Fig.4a).The cavitiesarepositionedonoppositesidestopreventdirectcavity-to- cavity coupling. With parameters matching the single NFC circuit parameters, we measure u z to obtain |u z| at key positions (Fig. 4a). FocusingonTWGM whichgeneratestransmissiondipsinindividual 3 cavities,thetoppanelofFig.4bshowspeakvaluesin|u z|of0.66(NFC) and0.34(FFC)at1.956MHz(theFFCTWGM 3 frequency),andcorre- spondingvaluesof0.53(NFC)and0.47(FFC)at1.973MHz(theNFC TWGM frequency).Thisconfirmssympatheticresonances33ateach 3 TWGM frequency—i.e.,bothcavitiesareresonatingsimultaneously— validatedbythe|u z|fieldmaps(middle/bottompanelsinFig.4b).Wave energyisthusexchangedbetweencavities.Thetime-resolvedu zmap (SupplementaryMovieS2)revealsclockwisecirculationintheupper cavityandanticlockwisecirculationinthelowercavity.

In the top panel of Fig. 4b, the outlet |u z| shows a dip D F3 at 1.956MHz.IsolatedFFCmeasurementsconfirmthattheD dipori- F3 ginatesfromtheresonanceoftheTWGM intheFFC,whereasreso- 3 nancehybridizationinthedual-cavitysystemadditionallycouplesD F3 totheTWGM intheNFC.ThisoverlapexplainswhyD ≈0.29here— 3 F3 slightlylower than D ≈0.32 in the singleFFC systems (simulation: F3 0.22vs.0.26).EnergyconcentratesprimarilyintheNFC(Fig.4bbot- tom panel), which exhibits broad-spectrum |u z| profiles in Supple- mentaryFig.12.Thecomputedfieldsat1.955MHz(D ,toppanelin F3 Fig.4c)revealastrongFFCu z (bottompanel),differingslightlyfrom Fig. 4b. Introducing material loss α (Fig. 4d top panel), peaks P F3 (1.955MHz) and P (1.966MHz) persist (Fig. 4d middle/bottom). N3 Supplementary Figs. 13–14 show displacements versus α, which displaypeaksin|u z|(P N1 -P N4 /P F1 -P F4 )decreasingwithα,whereasdips likeD /D exhibitV-shapedprofileswithdistinctminima. F2 F4

Space-controlledsympatheticresonances
We now modulate inter-cavity coupling by adjusting the horizontal distance L from −7a to 16a, probing how sympathetic resonance affects transmission dips. The simulated normalized |u z| at three positions is displayedinFig.5a:waveguideOutlet(left),NFC(middle), andFFC(right),withoutmaterialloss.FourbrightbarslabeledP -P N1 N4 emergeintheNFCpanel,whereasfourbarsP -P appearintheFFC F1 F4 panel.FrequenciesforP -P andP -P matchtheirisolatedcavity N1 N4 F1 F4 counterparts.

Energy exchange between cavities is evidenced by weak bright bars in the NFC at the frequencies corresponding to P -P (e.g., F1 F4 dashed lines at P /P ) and in the FFC at the frequencies corre- F2 F3 spondingtoP -P .Theoutletpanelshowsdarkbars(D -D ,D ) N1 N4 F1 F4 N4 correspondingtoresonances(P -P ,P ). F1 F4 N4

We now analyze the dual-cavity energy exchange mechanism usinggoverningequationsofmotion.LetP Fj 0denoteFFCpeaksand D Fj 0denoteoutletdipsforthesinglecavitycircuitandP Fj 00andD Fj 00 denotetheanalogousquantitiesforthedual-cavitycircuit(j=1,2,3,4). Withoutmateriallosses,theou(cid:2)tletdips(cid:3)inthedual-cavitycircuitare given by D Fj 00=D Fj 0×P Fj 0=P Fj 00 1+P FNj 00 (see theory). According to thisformula,theoreticallypredictedresults(squaremarkers)forD - F2 D areconsistentwithdirectsimulations(solidlines)fromthedual- F4 cavitymodel(Fig.5b).RelevantresultsforD appearinSupplemen- F1 taryFig.15.

Methods
Experimentalmethods
Thephononiccrystalsampleswerefabricatedon525μm-thicksiliconwafers using lithography and dry etching techniques. During lithography,a6μmphotoresistlayerwasappliedasanetchingmasktoprotectunpatternedregions,followedbysiliconetchinginanLPX-ICPsystem using a BOSCH process to achieve vertical sidewalls. The BOSCH process alternated between passivation and etching steps, withparametersdetailedinSupplementaryTable2.Pillarheightcontrol was achieved by timing the etching duration based on pre-calibrated silicon etch rates, and final geometries were optically verified.

Forallsamples(Fig.1),thewavepropagationwascharacterized using the piezo-laser ultrasonics for in-situ time-resolved measurement of out-of-plane displacement (Supplementary Fig. 1). The displacement u z was monitored across a 41a×29a rectangular region encompassingcavitiesandwaveguides,withspatialsamplingintervalsof641μm(a)alongxand555μm(0.866a)alongy.Time-resolveddataenabledderivationofaverageamplitudeattheInlet,Outlet,FFC,NFC,and Pure PnC regions (Figs. 3a and 4b, Supplementary Figs.11aand12a).Inaddition,wecarriedoutspatialfieldmappingof|u z| at the resonant frequencies (Figs. 3b and 4b, Supplementary Figs.9d,11band12b).

AsshowninSupplementaryFig.1,theexperimentalsetupcomprised a Polytec OFV 2570 laser Doppler vibrometer (LDV), a high-speedcamera,aRIGOLDG1032zsignalgenerator,apoweramplifier, and a Tektronix DPO4102B-L oscilloscope. Silicon samples were mountedonasupportingplatewithaspatialpositioningprecisionof~5μm.A5mm×1mmPZTdiskwasbondedtothebacksurfaceofthewaveguideinletusingconductiveadhesive.Aseven-cyclesinusoidalburst signal centered at frequency f c was generated, amplified, and deliveredtothePZTtransducer.Thenumberofcyclesintheburstwaslimited to facilitate identification of the incident wave packet as it propagated through the valley metamaterial phononic circuits. The LDV measured the out-of-plane displacement (u z) within target regions,whilethehigh-speedcamerarecordedthelaserbeamposition andenabledspatialpointselection.

Toenhanceexperimentalaccuracy,severalmeasuresweretaken. The input signal power was amplified by ~37dB to generate strong ultrasonicpulseswithoutdamagingthesiliconwafer.Ateachspatialpoint,512scanswereaveragedtoreducewhitenoise.Thesamplingfrequencywassettof s=100MHz,morethan47timesthetargetfrequencyf c(1.7–2.1MHz).Ateachspatialpoint,signalswererecordedwith 105 data points, corresponding to >1700 periods of 1/f c. This allowed fine spectral filtering near f c with a frequencyresolution of ~0.001MHz.Byrepeatingtheseprocedures,weobtainedtheamplitude of the out-of-plane displacement across the target region, enabling separation of neighboring TWGMs and dips. These proceduresalsoallowedustoreconstructanimationsandspatialsegmentsofwavedisplacementwithintheregionofinterest.

Numericalcalculations
We adopt the finite-element method to calculate dispersion in our silicon phononic crystal plates. Elastic constants are taken as C₁₁=165.6GPa, C₁₂=63.9GPa, C₄₄=79.5GPa, and mass density 2.331g/cm³38, with silicon assumed to exhibit near-isotropic behaviour.Unitcelldispersioniscomputedforrotationanglesθ=0°and±20°(Fig.1c,d).

Fortopologicaledgestates(TESs),asandwichedsupercell(Supplementary Fig. 5a) comprising 12-layer PnC-B (top), 12-layer PnC-A (middle), and 12-layer PnC-B (bottom) is constructed. Periodic boundary conditions are applied to the supercell B-A-B along the x-direction, whereas continuity conditions are applied along the y-direction. The results for disRpersion (RSupplementary Fig. 5b, c) includethedisplacementratio ju z jdV= ju t jdV ateachfrequency, ushinerg e a o u p r z h o a o l n o m gi b u c u a t l d c e w a n h v o i i t s t e y pe s o r u u i p n t- e g o r f c g - e p a l l l l a le ( n s r e i y d a e m n l d e o n t d o g e t t s a h l ( 7 d T a i s ) p p l o a c si m e s t ) m io a e n r n e e d s a . b n e a t l w yz e e e d n T= (cid:5) (cid:5) (cid:5) (cid:5)S S (cid:2) (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5) S + + S i pffi γ ffiffi a~ c (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5)1+ iðω(cid:2)ω (cid:2)γ Þ+Γ+γ (cid:5) (cid:5) (cid:5) (cid:5): ð6Þ + + 0 the inner PnC-B and outer PnC-A (Supplementary Fig. 6a). Periodic boundary conditions are applied to threepairsofparallelsidesofthehexagonalsupercell,yieldingthecorrespondingeigenfrequenciesanddisplacementfields(Fig.2b;SupplementaryFig.6b,c).

Wave propagation simulations for NFC-waveguide, FFC-waveguide (Fig. 2a), and dual-cavity structures (Fig. 4a) replicate experimental dimensions (Fig. 1a). Perfectly matched layers (PMLs) absorbboundaryreflections.Acircularsourceappliesunitz-axisforce over a 2.5mm radius at the waveguide inlet. Transmission spectra (Figs.3cand4c,d,SupplementaryFigs.9a,11c,12cand13)arebasedonthe average |u z| over 5a segments at Inlet, Outlet, FFC, NFC, and unpatternedPnCregions.

Insinglecavity-waveguidemodels(Fig.2a),theDiracstriplength D varies from 1 to 12 layers in 1-layer increments. Frequency- and distance-resolved|u z|maps(Fig.2c)aregeneratedfromaveragevaluesover5asegmentsattheOutletandFFC.ControlsimulationsreplacingDirac strips with PnC-A appear in Fig. 2d. For dual-cavity models (Fig.4a),theinter-cavitydistanceLsweepsfrom−7ato16ainstepsof a, producing frequency-distance |u z| maps at the Outlet, NFC, and FFC(Fig.5a).

Theoryofthewaveguide-cavitydynamics
Single-cavitywaveguidesystem.Thewaveguide-cavitycouplingcan beunderstoodfromthetheoreticalmodelshowninSupplementary fig. 2. The cavity includes dissipative loss Γ and radiative loss γ, respectively. The governing equation can be expressed as40–42 d d a~ t c =iωa~ c : ð3Þ SubstitutingEq.(3)intoEq.(1)yields The relationship between the input wave S + and the output wave S - canbewrittenas S (cid:2) =S + +i pffi γ ffiffi a~ c : ð5Þ Combining Eqs. (4) and (5), the transmission coefficient of the systemcanbeexpressedintheform T= (cid:5) (cid:5) (cid:5) (cid:5)S S (cid:2) (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5) S + + S i pffi γ ffiffi a~ c (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5)1+ iðω(cid:2)ω (cid:2)γ Þ+Γ+γ (cid:5) (cid:5) (cid:5) (cid:5): ð6Þ + + 0 AccordingtoEq.(6),oneobtains Δω=2ðΓ+γÞ T = Γ ð7Þ min Γ+γ whereT istheminimumvaluearoundω ,andΔωrepresentsthe min 0 angularfrequencycorrespondingtothefullwidthathalfofT .These min twoparameterscanbeobtainedfromthenumericallycalculated spectrum.ThesubstitutionofT andΔωintoEq.(7)yieldsthedis- min sipativelossΓandradiationlossγ.Then,byuseofEq.(6),onecanplot the fitted transmission profile around ω . Critical coupling 0 occurswhenΓ=γ,whereasΓ<γ(orΓ>γ)indicatesover-(orunder-) coupling.

Dual-cavity waveguide system. The theoretical model ofthe dual-cavitywaveguidesystemisillustratedinSupplementaryFig.3.Each cavityincludesdissipativelossΓ jandradiativelossγ j(j=F,N).Since thetwocavitiesareinsertedonseparatesidesofthewaveguide,direct cavity-to-cavitycouplingisprohibited.Thequantityφistheaccumulatedphasedifferencealongthewaveguidepath. ForanincidentwaveS +,thegoverningequationforthetwo-cavity system(a~ N=a N eiωt anda~ F=a F eiωt)canbewrittenas43,44 d d a~ N =ðiω N (cid:2)Γ N (cid:2)γ N Þa~ N+i pffi γ ffiffi N ffiffi ðS + +i pffi γ ffiffi F ffiffi e(cid:2)iφa~ F Þ, ð8Þ d d a~ t F =ðiω F (cid:2)Γ F (cid:2)γ F Þa~ F+i pffi γ ffiffi F ffiffi ðS + e(cid:2)iφ+i pffi γ ffiffi N ffiffi e(cid:2)iφa~ N Þ, ð9Þ w fie h l e d re ca ω vi N ti a e n s, d re ω s F p a e r c e tiv th e e ly. re D s i o re n c a t n c c a e v f i r t e y q -t u o e -c n a c v ie it s y o c f ou th p e lin n g ea i r s - i a g n n d or f e a d r- . Thetransmissioncoefficientforthesystemisgivenby T= (cid:5) (cid:5) (cid:5) (cid:5)S S (cid:2) (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5) ðS + +i pffi γ ffiffi N ffiffi a~ N S Þe(cid:2)iφ+i pffi γ ffiffi F ffiffi a~ F (cid:5) (cid:5) (cid:5) (cid:5) ð10Þ = (cid:5) (cid:5)e(cid:2) + iφ+iðe(cid:2)iφ pffi γ ffiffi N ffiffi a~ N =S + + + pffi γ ffiffi F ffiffi a~ F =S + Þ (cid:5) (cid:5): Byfittingthesimulatedtransmissionprofile,wecanobtainparametersincludingΓ jandγ j(j=F,N).SubstitutingthemintoEqs.(8)and (9)yieldsa~ Nanda~ F.Onecanthenplotthetheoreticallyfittedprofileof transmission W m h is e s n io ω n = fr ω o F m , Eq.(10). T= (cid:5) (cid:5)e(cid:2)iφ+iðe(cid:2)iφ pffi γ ffiffi N ffiffi a N =S + + pffi γ ffiffi F ffiffi a F =S + Þ (cid:5) (cid:5) =D Fj 00: ð11Þ Specifically, D Fj 00= (cid:5) (cid:5) (cid:5) (cid:5) eiφ e ð 2 ½i i ð φ ω ½ið (cid:2) ω ω (cid:2) F ω Þ+ N Þ Γ + F+ Γ N γ F + (cid:4) γ (cid:2) N (cid:4) γ ½i F ð Þ ω ð½i (cid:2) ðω ω (cid:2) F Þ ω + N Γ Þ F + + Γ γ N F + (cid:4)(cid:2) γ N γ (cid:4) F (cid:2) γ N γ N Þ (cid:5) (cid:5) (cid:5) (cid:5), ð12Þ wheretheD Fj ”istheoutletdipinthedual-cavitycircuit. CombiningEqs.(8)and(9),theamplitudesfortheNFCandFFC aregivenby (cid:5) (cid:5) (cid:5) (cid:5)S a~ N + (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5) e2iφ½iðω i p (cid:2) ffi γ ffi ω ffi N ffi N ffiffi f Þ e + 2i Γ φ N ½ið ω γ (cid:2) N (cid:4)½ ω ið F ω Þ+ (cid:2) Γ ω F F + Þ γ + F Γ (cid:4) F (cid:2) + γ γ F F g (cid:4)(cid:2)γ F γ N (cid:5) (cid:5) (cid:5) (cid:5), ð13Þ (cid:5) (cid:5) (cid:5) (cid:5)S a~ + F (cid:5) (cid:5) (cid:5) (cid:5)= (cid:5) (cid:5) (cid:5) (cid:5) e2iφ½iðω i p (cid:2) ffi γ ffi ω ffi F ffiffi f N e Þ iφ + ½ Γ ið N ω + (cid:2) γ N ω (cid:4) F ½i Þ ð + ω Γ (cid:2) F+ ω F γ Þ F + (cid:4)(cid:2) Γ F e + iφ γ γ F N (cid:4) g (cid:2)γ F γ N (cid:5) (cid:5) (cid:5) (cid:5)=P0 F 0 j, ð14Þ whereP Fj ”referstotheamplitudepeakfortheFFC(ω=ω F )inthedual-cavitycircuit.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents the topological and non-Hermitian physics in on-chip waveguide-cavity systems, addressing challenges in far-field phonon coupling and multi-cavity dynamics. The authors introduce broadband and robust \"Dirac strip\" to conquer far-field coupling in valley topological systems. This is experimentally validated through observations of displacement peaks or dips via pump-probe interferometry, demonstrating robust phonon coupling in single-cavity system. The dual-cavity system further reveals non-Hermitian behaviors, where material loss and inter-cavity spacing modulate synchronizing oscillations—findings supported by a theory of dual-cavity energy exchange. The experimental technique is quite smart, while the theory underlines well the influences of observed synchronizing oscillations.\nGiven the rigorous experiments, theoretical insights, and broad interdisciplinary significance (phonon, photon, and topological metamaterials), I find this work quite suitable for publication in Nature Communications. I strongly recommend its publication after addressing the following issues:\n1. The triangular pillar serves as the unit cell in this work. However, the motivation of this selected specific geometry was not elucidated in the main text or supplementary information. Why choose this type of unit cell as the fundamental configuration?\n2. In the last paragraph of the Introduction, the second sentence is ambiguous. What does the authors want to express about \"in range and efficiency\"? I suggest modifying this sentence to make it clearer.\n3. In the first paragraph of Results Section, the authors notify the Supplementary Figs. 5-7 in one place. This is not clear. The explicit linking between Supplementary Figs. 5-7 to relevant sentences in this paragraph shall be given.\n4. After Fig. 1, clear description is needed about the part \"The modeled antisymmetric bulk wave…… shaping the D1-D4 and P1-P4 |uz| profiles\". Additionally, is it possible to add a displacement field at strip pillar angle θ = 10° as an example in Supplementary Figure 10?\n5. While the key achievements are highlighted in the Discussion, more descriptions on detailed findings are needed to provide crucial guidance for further research, e.g., the feasible layers of Dirac strip, conditions for stronger critical coupling, etc.\n6. The authors enable the far-field phonon coupling and present the non-Hermitian features such as synchronizing oscillations. Why these two are important or essential needs highlight.\n7. The authors should present numerical methods to calculate the eigenfrequencies of supercells for TESs and TWGMs, e.g., B-A-B elongated and B-inner A-outer hexagonal supercells. Additionally, it should be clarified whether periodic boundary conditions are applied.\n8. How the authors distinguish the transmission peaks or dips in experiment? As we can see the frequency difference between the two peak frequencies in Fig. 3b-d is only about 0.01 MHz versus the central frequency 1.96 MHz.", "reviewer_2": "The noteworthy result presented in the manuscript is the demonstration of 'far-field' coupling between a waveguide and cavities in a valley phononic plate crystal, through a 'Dirac strip'. Among the claims, I was not convinced by what the authors call 'synchronized oscillations': in the understanding of this reviewer there is no oscillator in the experiments presented, but resonators excited by a single-frequency driving source (the term 'oscillator' has a clear meaning in physics and engineering and appears to be misused). The discussion of non-Hermitian circuits also sounds off for the same reason: there is dissipative (material,structure) loss and radiation loss, for sure, but there is no gain. The Methods section says that 'direct cavity-to-cavity coupling is prohibited', then why does the main text advertises 'synchronized oscillations in the resonators', whereas clearly the same external source drives both of them?\nStrangely, the supplemental information is much clearer and informative than the main manuscript. It is written in simple engineering terms and the figures are clear and not composed of too many panels. What the main manuscript calls 'pump-probe' is actually a piezoelectric source and a laser vibrometer, there is no laser pump (while first reading the text I thought laser ultrasound was being used in the experiment but this not the case at all).\nGlobally the methodology is consistent, but most of the useful information in given in the SI, such as geometric parameters, band gap frequencies, Dirac cone deformation with angle theta - I personally tend to dislike this way of writing papers, aiming at impact rather than accuracy. I recommend a reorganization of the manuscript with important figures of the SI included in the main text (SI figures 1, 2, 5 typically). The revision should be made in order to clarify the methodology and to reconsider the claims: there is no reason to oversell what has been achieved, the experiment is rather nice and seems well conducted. As a final note, the experimental results in figures 2 and especially figure 3 are not really successful in demonstrating localized fields established inside the cavities; whatever the experimental reasons it seems the conclusions can only rely on numerical simulations - this should be acknowledged."}, "author_response": "Dear Editor,\nWe would like to thank you and the reviewers for the quick response. We also thank the\nReviewers for the recognition of this work and for their valuable suggestions. We greatly\nappreciative the opportunity to revise this work. All concerns and recommendations have been\ncarefully addressed to enhance the quality of our manuscript. The point-by-point responses are\nhighlighted by the red text in the manuscript. Our responses are given below.\n---------------------------------------------------------------------------\nResponse to reviewer #1\nReviewer #1: This manuscript presents the topological and non-Hermitian physics in on-chip\nwaveguide-cavity systems, addressing challenges in far-field phonon coupling and multi-cavity\ndynamics. The authors introduce broadband and robust \"Dirac strip\" to conquer far-field\ncoupling in valley topological systems. This is experimentally validated through observations\nof displacement peaks or dips via pump-probe interferometry, demonstrating robust phonon\ncoupling in single-cavity system. The dual-cavity system further reveals non-Hermitian\nbehaviors, where material loss and inter-cavity spacing modulate synchronizing oscillations—\nfindings supported by a theory of dual-cavity energy exchange. The experimental technique is\nquite smart, while the theory underlines well the influences of observed synchronizing\noscillations.\nGiven the rigorous experiments, theoretical insights, and broad interdisciplinary significance\n(phonon, photon, and topological metamaterials), I find this work quite suitable for publication\nin Nature Communications. I strongly recommend its publication after addressing the following\nissues:\nAuthors: We thank the referee for the positive comments. We have carefully revised both\nthe main text and supplementary information, highlighting modifications in red.\nReviewer #1: 1. The triangular pillar serves as the unit cell in this work. However, the\nmotivation of this selected specific geometry was not elucidated in the main text or\nsupplementary information. Why choose this type of unit cell as the fundamental configuration?\nAuthors: Thank you for noting the choice of the triangular pillar as the unit cell. The\nmotivation for selecting this specific geometry is rooted in its symmetry properties (C\n3v\nsymmetry when θ equals zero) and the potential to induce topological phase transitions. We\nadopted this classic valley PnC that support topological states, at the aim of exploring novel\nmechanisms of wave propagation.\nRelevant revisions are made as follows:\nPage 2 in the main text, “The unit cell of triangular pillars possesses C symmetry when θ\n3v\nequals zero, and topological phase transitions are induced by changing rotation angle θ to break\nthis symmetry. These valley PnCs support the topological states reported previously24,29. We\nadopt this classic paradigm and fabricate all three samples on silicon wafers to explore novel\nmechanisms of wave propagation.”\nReviewer #1: 2. In the last paragraph of the Introduction, the second sentence is ambiguous.\nWhat does the authors want to express about \"in range and efficiency\"? I suggest modifying\nthis sentence to make it clearer.\nAuthors: Thank you for this valuable suggestion. For the word \"range\", we mainly focus\non the far-field phonon coupling over long distances by using the Dirac strip. For the word\n\"efficient\", we underline that the Dirac strip enhances the coupling efficient between the\nwaveguide and the cavity.\nRelevant revisions are made as follows:\nPage 2 in the main text, “This is achieved through topological phonon interactions in on-chip\nvalley waveguide-cavity systems using a Dirac-cone waveguide strip (a 'Dirac strip'), which\nenables long-distance coupling beyond conventional near-field limits and significantly\nimproves coupling efficiency.”\nReviewer #1: 3. In the first paragraph of Results Section, the authors notify the Supplementary\nFigs. 5-7 in one place. This is not clear. The explicit linking between Supplementary Figs. 5-7\nto relevant sentences in this paragraph shall be given.\nAuthors: Thanks for your kind reminder. We have addressed this part you mentioned to\nexplicitly link each relevant sentence to the corresponding Supplementary Figs. 5-7 (now Figs.\n1b-1d and Supplementary Figs. 3-4), in the main text (Pages 2-3).\nRelevant revisions are made as follows:\nPage 2 in the main text, “Figure 1a shows the three valley metamaterial circuits investigated in\nthis work. Sample 1 is a far-field cavity (FFC) waveguide circuit composed of triangular pillars,\nwith side length s = 500 μm and height h = 292 μm (see Fig. 1b).”\nl p\nPage 2 in the main text, “Dirac cones appear at the K point of the Brillouin zone for θ = 0°\n(Figs. 1c–d, upper panels), while material anisotropy introduces a minor band gap (insets).”\nPage 3 in the main text, “PnCs A and B exhibit opposite topological phases (Supplementary\nFig. 3), and their interface supports topological edge states (TESs, Supplementary Fig. 4) and\ntopological whispering gallery modes (TWGMs, Fig. 2b and Supplementary Fig. 5).”\nReviewer #1: 4. After Fig. 1, clear description is needed about the part \"The modeled\nantisymmetric bulk wave…… shaping the D1-D4 and P1-P4 |uz| profiles\". Additionally, is it\npossible to add a displacement field at strip pillar angle θ = 10° as an example in Supplementary\nFigure 10?\nAuthors: Based on your first point, we have reorganized this part and provided a more\ndetailed description, in the main text (Page 5). Besides, we note the influence of the bulk branch\nof the Dirac strip on cavity-waveguide coupling, in the Supplement (Page 7).\nRegarding the second question, we have added a displacement field at strip pillar angle θ\n= 10° for TWGM in Supplementary Figure 10 (now Supplementary Figure 7a), in the\nRelevant revisions are made as follows:\nPage 5 in the main text, “The modeled antisymmetric bulk-wave branches emerge with wave\nenergy confined to the Dirac strip, whereas their frequency ranges occupy the eigenfrequencies\nof TWGM , TWGM and TWGM (Supplementary Fig. 7). These guided bulk-wave branches\n1 3 4\nare modulated by strip pillar rotation. For example, at strip pillar angle θ = 10°, these bulk-\nwave branches occupy exclusively the TWGM frequency, directly shaping the D -D and P -\n4 1 4 1\nP |u| versus θ profiles (Supplementary Fig. 7).”\n4 z\nPage 6 in the Supplement, “\nSupplementary Figure 7. The influence of pillar rotation in the PnC-C strip (based on the sample\nin Fig. 2a). a The FFC-waveguide structure and the u distribution when θ = 10° for TWGM (upper\nz 4\npanel). Calculated normalized |u| retrieved at the outlet (dip value, orange circle) and at the lower edge\nz\nof the cavity (peak value, blue circle), versus the pillar rotation of the PnC-C in the model (upper panel).\nDashed lines are fitted curves. PnC-C bridges the waveguide and the cavity, and becomes the Dirac strip\nwhen θ = 0. b Supercell (upper panel) used to calculate the dispersion (lower left panel) when θ = 0o.\nThe color scale indicates the ratio of u to the total displacement u in the supercell. The middle panel\nz t\nshows a map of u at 1.93 MHz when θ = 0o. Evolution of the eigenfrequencies (lower right panel) versus\nz\nthe pillar rotation for PnC-C when k = 0.”\ny\nPage 7 in the Supplement, “Thanks to the overlap between the bulk branch of the Dirac strip\nand the resonant frequency of the cavity for TWGM , highly-efficient matched coupling\n3\nbecomes possible between the horizontal waveguide and the far-field cavity.”\nReviewer #1: 5. While the key achievements are highlighted in the Discussion, more\ndescriptions on detailed findings are needed to provide crucial guidance for further research,\ne.g., the feasible layers of Dirac strip, conditions for stronger critical coupling, etc.\nAuthors: Thanks for your advice about more detailed findings in the Discussion for\nfurther research. So, we have added related contens to enrich the findings in the main text (Page\n9), i.e., in single-cavity systems, adjusting Dirac strip length optimizes far-field coupling. in\ndual-cavity systems, simultaneous cavity excitation creates a deeper outlet dip, and tuning inter-\ncavity distance modulates energy exchange.\nRelevant revisions are made as follows:\nPage 9 in the main text, “In the single-cavity waveguide system, adjusting the length of the\nDirac strip optimizes the far-field phonon coupling. In the dual-cavity waveguide system, the\nsimultaneous excitation of both cavities produces a pronounced outlet dip compared to the\nsingle-cavity system. Furthermore, tuning the inter-cavity distance allows modulation of energy\nexchange between the dual cavities.”\nReviewer #1: 6. The authors enable the far-field phonon coupling and present the non-\nHermitian features such as synchronizing oscillations. Why these two are important or essential\nneeds highlight.\nAuthors: Thanks for pointing out the need to highlight the importance of far-field phonon\ncoupling and non-Hermitian feature.\nFirst, the far-field phonon coupling is vital since it enables long-distance cavity energy\nlocalization. This is crucial for exploring functionalities in complex systems like dual cavity\nsystems. We have added relevant contents in the main text (Page 6).\nSecondly, the non-Hermitian behavior, featuring sympathetic resonance resulting from the\nenergy exchange between dual cavities, is the other significant innovation. It contributes to\napplications such as high-performance sensors and information processing. We have heighted\nits importance in the main text (Page 9).\nRelevant revisions are made as follows:\nPage 6 in the main text, “Far-field phonon coupling enables long-distance cavity energy\nlocalization and unlocks functionalities in complex systems, such as dual-cavity configurations.”\nPage 9 in the main text, “The discovery of energy exchange between dual cavities is not only\ncrucial for the emergence of non-Hermitian phenomena, but also underpins sympathetic\nresonance at shared frequencies. Such non-Hermitian behavior is vital for high-performance\nsensing and information processing.”\nReviewer #1: 7. The authors should present numerical methods to calculate the\neigenfrequencies of supercells for TESs and TWGMs, e.g., B-A-B elongated and B-inner A-\nouter hexagonal supercells. Additionally, it should be clarified whether periodic boundary\nconditions are applied.\nAuthors: Thank you for this comment on numerical methods. For both TESs and TWGMs,\nperiodic boundary conditions are applied along the x-direction and continuity conditions along\nthe y-direction for the B-A-B supercell, and periodic boundary conditions are also applied to\nthree pairs of parallel sides for the hexagonal supercell. We have added related details in the\nMethods of main text (Page 11).\nRelevant revisions are made as follows:\nPage 11 in the main text, “Periodic boundary conditions are applied to the supercell B-A-B\nalong the x-direction, whereas continuity conditions are applied along the y-direction.”\nPage 11 in the main text, “Topological whispering gallery modes (TWGMs) are analyzed using\na rhombus cavity supercell (side length 7a) positioned between the inner PnC-B and outer PnC-\nA (Supplementary Fig. 5a). Periodic boundary conditions are applied to three pairs of parallel\nsides of the hexagonal supercell, yielding the corresponding eigenfrequencies and displacement\nfields (Fig. 2b; Supplementary Figs. 5b and 5c).”\nReviewer #1: 8. How the authors distinguish the transmission peaks or dips in experiment?\nAs we can see the frequency difference between the two peak frequencies in Fig. 3b-d is only about\n0.01 MHz versus the central frequency 1.96 MHz.\nAuthors: Thank you for raising this experimental issue. We were able distinguish\ntransmission peaks or dips with a frequency difference of only 0.01 MHz in the experiment,\nmainly relying on the following measures:\nFirstly, the input signal power was amplified by ~37 dB to enable the generation of large\nultrasonic pulses and ensure undamaged silicon wafer. At each space point, 512 times of scans\nwas used to reduce the amplitude of white noise.\nSecondly, we set the sampling frequency f = 100 MHz, being over 47 times of target\ns\nfrequency f (1.7 MHz < f < 2.1 MHz). At each space point, we recorded signal by 105 data\nc c\npoints, being over 1700 times of target frequency period (1/f ). Therefore, it became possible to\nc\nanalyze wave pulses with fine filtering width towards target frequency f , with step variation\nc\nevery 0.001 MHz.\nBy repeating the above procedures, we can obtain the amplitude of out-of-plane\ndisplacement within the target space region. These data were further processed to separate\nneighboring |u| peaks (at cavity edge) or dips (at waveguide outlet). Besides, these procedures\nz\nalso allowed us to get the animation or segment of wave displacement within target space region.\nWe have added relevant experimental methods before Fig. 6 in the main text (Pages 10-11).\nRelevant revisions are made as follows:\nPages 10-11 in the main text, “As shown in Fig. 6, the experimental setup comprised a\nPolytec OFV 2570 laser Doppler vibrometer (LDV), a high-speed camera, a RIGOL DG1032z\nsignal generator, a power amplifier, and a Tektronix DPO4102B-L oscilloscope. Silicon\nsamples were mounted on a supporting plate with a spatial positioning precision of ~5 μm. A 5\nmm × 1 mm PZT disk was bonded to the back surface of the waveguide inlet using conductive\nadhesive. A seven-cycle sinusoidal burst signal centered at frequency f was generated,\nc\namplified, and delivered to the PZT transducer. The number of cycles in the burst was limited\nto facilitate identification of the incident wave packet as it propagated through the valley\nmetamaterial phononic circuits. The LDV measured the out-of-plane displacement (u) within\nz\ntarget regions, while the high-speed camera recorded the laser beam position and enabled\nspatial point selection.\nTo enhance experimental accuracy, several measures were taken. The input signal power\nwas amplified by ~37 dB to generate strong ultrasonic pulses without damaging the silicon\nwafer. At each spatial point, 512 scans were averaged to reduce white noise. The sampling\nfrequency was set to f = 100 MHz, more than 47 times the target frequency f (1.7–2.1 MHz).\ns c\nAt each spatial point, signals were recorded with 105 data points, corresponding to more than\n1700 periods of 1/f . This allowed fine spectral filtering near f with a frequency resolution of\nc c\n~0.001 MHz. By repeating these procedures, we obtained the amplitude of the out-of-plane\ndisplacement across the target region, enabling separation of neighboring TWGMs and dips.\nThese procedures also allowed us to reconstruct animations and spatial segments of wave\ndisplacement within the region of interest.”\nResponse to reviewer #2\nReviewer #2: The noteworthy result presented in the manuscript is the demonstration of 'far-\nfield' coupling between a waveguide and cavities in a valley phononic plate crystal, through a\n'Dirac strip'. Among the claims, I was not convinced by what the authors call 'synchronized\noscillations': in the understanding of this reviewer there is no oscillator in the experiments\npresented, but resonators excited by a single-frequency driving source (the term 'oscillator' has\na clear meaning in physics and engineering and appears to be misused).\nThe Methods section says that 'direct cavity-to-cavity coupling is prohibited', then why does\nthe main text advertise 'synchronized oscillations in the resonators', whereas clearly the same\nexternal source drives both of them?\nAuthors: Let us first express our thanks for the positive comment on ‘far-field coupling’.\nWe also appreciate the insightful comment on 'synchronized oscillations'. We agree that\nthe term 'synchronized oscillations' does not accurately reflect the response behavior between\nthe two cavities.\nAs you have stated pointed, in our experiment, the two cavities were not independent\n'oscillators', but were resonators simultaneously excited at the same frequency under a driving\nsource. We have therefore replaced the term 'synchronized oscillations' with 'sympathetic\nresonances' throughout the main text (Pages 1-2, 6-9).\nAs for the logical contradiction in your proposal between 'direct coupling prohibited' and\n'synchronous oscillation', it stems from our misused 'synchronized oscillations'. The 'direct\ncavity-to-cavity coupling is prohibited' refers to the absence of short-range direct interaction\nbetween the two cavities. The observed dual-cavity resonances are mediated by a long-range\ncoupling pathway through the Dirac strips, which enables simultaneous excitation of TWGMs\nin both cavities by the same external driving source. Therefore, there is no longer a\ncontradiction between 'direct cavity-to-cavity coupling is prohibited' and 'sympathetic\nresonances'.\nRelevant revisions are made as follows:\nPage 1 in the main text, “We also combine near- and far-field cavities on the same substrate,\nand thereby amplify and control non-Hermitian dynamics through loss- and distance-modulated\nsympathetic resonances, directly resolved via piezo-laser interferometry.”\nPage 2 in the main text, “By co-locating near- and far-field cavities, we observe sympathetic\nresonances—simultaneous resonance in both cavities at a single frequency—and reveal non-\nHermitian dynamics modulated by material loss and inter-cavity distance.”\nPage 6 in the main text, “This confirms sympathetic resonances33 at each TWGM frequency—\n3\ni.e., both cavities are resonating simultaneously—validated by the |u| field maps\nz\n(middle/bottom panels in Fig. 4b).”\nPage 7 in the main text, “Enhanced coupling and sympathetic resonances in the dual-cavity\nwaveguide system.”\nPage 8 in the main text, “Space-controlled coupling and sympathetic resonances in the dual-\ncavity waveguide system.”\nPage 8 in the main text, “We now modulate inter-cavity coupling by adjusting the horizontal\ndistance L from -7a to 16a, probing how sympathetic resonance affects transmission dips.”\nPage 9 in the main text, “In addition, by integrating dual cavities in both the near and far fields\nwe establish deterministic control over non-Hermitian dynamics through material loss and\nspatial separation, experimentally observing sympathetic resonances.”\nReviewer #2: The discussion of non-Hermitian circuits also sounds off for the same reason:\nthere is dissipative (material, structure) loss and radiation loss, for sure, but there is no gain.\nAuthors: Thank you for the comment on non-Hermitian system. Truly, there is no gain in\nour system. We use the concept of \"non-Hermitian\" for several reasons:\nFirst, the non-Hermitian system is non-conservative, requiring only energy exchange. This\ncan occur with only loss, only gain, or both present simultaneously [Rev. Mod. Phys. 93, 01505\n(2021); Nat. Phys. 14, 11 (2018)]. We follow this more general conclusion of \"non-Hermitian\"\nsystem, in the main text and included relevant references (Page 1).\nSecondly, although our system lacks gain, it does experience energy loss. Our dual-cavity\nsystem not only includes the material loss, but also energy loss to the external environment\ngenerated by the dual-cavity system itself, making it intrinsically a non-conservative system.\nThirdly, this work studies the non-Hermitian phenomena based on the energy exchange\nbetween two cavities. From this perspective, we derive the theory that considers dual-cavity\nenergy exchange based on the sympathetic resonance. We have added related contents in the\nmain text (Page 9).\nRelevant revisions are made as follows:\nPage 1 in the main text, “Non-Hermitian systems are non-conservative and characterized by\nenergy exchange15,16, encompassing three situations: loss only, gain only, as well as the\ncoexistence of both.”\nPage 9 in the main text, “The discovery of energy exchange between dual cavities is not only\ncrucial for the emergence of non-Hermitian phenomena, but also underpins sympathetic\nresonance at shared frequencies.”\nReviewer #2: What the main manuscript calls 'pump-probe' is actually a piezoelectric source\nand a laser vibrometer, there is no laser pump (while first reading the text I thought laser\nultrasound was being used in the experiment but this not the case at all).\nAuthors: Thank you for pointing out the term 'pump-probe'. The experiments used a\npiezoelectric excitation source and a laser vibrometer, without involving laser pumping. Our\nexperimental method, more precisely, is a piezo-laser method. To address your concern, we\nhave revised the 'pump probe' to 'piezo-laser' in the main text (Page 1).\nRelevant revisions are made as follows:\nPage 1 in the main text, “We also combine near- and far-field cavities on the same substrate,\nand thereby amplify and control non-Hermitian dynamics through loss- and distance-modulated\nsympathetic resonances, directly resolved via piezo-laser interferometry.”\nReviewer #2: Strangely, the supplemental information is much clearer and informative than the\nmain manuscript. It is written in simple engineering terms and the figures are clear and not\ncomposed of too many panels. Globally the methodology is consistent, but most of the useful\ninformation in given in the SI, such as geometric parameters, band gap frequencies, Dirac cone\ndeformation with angle theta - I personally tend to dislike this way of writing papers, aiming at\nimpact rather than accuracy.\nI recommend a reorganization of the manuscript with important figures of the SI included in\nthe main text (SI figures 1, 2, 5 typically). The revision should be made in order to clarify the\nmethodology and to reconsider the claims: there is no reason to oversell what has been achieved,\nthe experiment is rather nice and seems well conducted.\nAuthors: Thank you for your detailed comments on the content balance between the main\ntext and SI. We agree with your that key information should be prioritized in the main text to\nensure accuracy. In response to you specific suggestion, we have systematically reorganized\nour manuscript.\nFirst, we have combined the original SI Figures 1 and 5 into a new Figure 1 in the main\ntext. At the same time, we have moved the original descriptions of these two figures to the\ncorresponding positions in the main text, and we have removed duplicate sentences. We have\ntaken these contents as the first section of the main text to present the useful information such\nas sample structures, geometric parameters, and frequency dispersion of the unit cell (Pages 2-\n3).\nSecondly, as the original SI Figure 2 is directly related to the experimental method, we\nhave moved the original SI Figure 2 (now Fig. 6) and related contents (including Tables 1-2 )\nto the Experimental methods section of the main text (Pages 10-11).\nIn summary, we have reexamined and reorganized the entire manuscript, and improved its\ncompleteness and accuracy.\nRelevant revisions are made as follows:\nPages 2-3 in the main text, “\nValley metamaterial circuits: three configurations\nFigure 1a shows the three valley metamaterial circuits investigated in this work. Sample 1\nis a far-field cavity (FFC) waveguide circuit composed of triangular pillars, with side length s\nl\n= 500 μm and height h = 292 μm (see Fig. 1b). Samples 2 and 3 correspond to the near-field\np\ncavity (NFC) waveguide circuit and dual-cavity circuit, respectively, with s = 526 μm and h =\nl p\n289 μm. All samples share the same lattice constant a = 641 μm and wafer thickness e = 525\nt\nμm, with geometrical parameters detailed in Table 1 (see Methods). The rhombus cavities have\nside length 7a, and the waveguides span 38a horizontally. Rightward terminal boundaries are\noblique to suppress reflections. The Dirac strips maintain a 3a width horizontally, extending\nvertically 7 layers for FFC structures and 3 layers for NFC structures. In dual-cavity\nconfigurations, the NFC and FFC are positioned 7a apart horizontally. Simplified models of the\nthree samples are shown in Supplementary Figs. 1 and 2.\nWe fabricated valley phononic crystal (PnC) plates by etching arrays of triangular pillars\nin a honeycomb lattice (see Methods). Dirac cones appear at the K point of the Brillouin zone\nfor θ = 0° (Figs. 1c–d, upper panels), while material anisotropy introduces a minor band gap\n(insets). Rotating the pillars to θ = 20° (phase A) and θ = –20° (phase B) generates broad band\ngaps for antisymmetric plate waves (Figs. 1c-d, lower panels), with slight frequency shifts of\nthe Dirac cones between Sample 1 and Samples 2–3.\nThe unit cell of triangular pillars possesses C symmetry when θ equals zero, and\n3v\ntopological phase transitions are induced by changing rotation angle θ to break this symmetry.\nThese valley PnCs support the topological states reported previously 24,29. We adopt this classic\nparadigm and fabricate all three samples on silicon wafers to explore novel mechanisms of\nwave propagation.\nFig. 1 | Samples and dispersion relations for the valley phononic plates. a Experimental samples\nfor the FFC-waveguide, NFC-waveguide, and dual-cavity waveguide circuits. b PnC plate, unit cell, and\nBrillouin Zone. Geometrical parameters include lattice constant a, pillar side length s, pillar height h ,\nl p\nsilicon wafer thickness e, and substrate thickness e = e - h . The topological phases are controlled by\nt s t p\ntuning the rotation angles θ of the triangular pillars. c, d Band structures of the unit cell for θ = 0° (top\npanels) and θ = ±20° (bottom panels) are shown for Sample 1 in (c) and Samples 2 and 3 in (d). The\ncolor scale indicates the ratio of |u| to the total displacement |u| in the unit cell. Insets show the scheme\nz t\nof unit cells and a zoom-in of the dispersion for θ = 0°.”\nPage 10 in the main text, “\nTable 1. Geometrical parameters of three samples\nSample 1 Sample 2 Sample 3\nSingle far-field Single near-field Dual\ncavity cavity cavity\nLattice constant a (μm) 641\nWafer thickness e (μm) 525\nt\nPillar height h (μm) 292 289\np\nPillar side length s (μm) 500 526\nl\nTable 2. The recipe for the BOSCH process\nPlaten power Pressure SF6 C4F8 Time\nProcess\n(Watt) (mTorr) (sccm) (sccm) (s)\nPassivation 0 19 0 85 7\nEtching 20 37 175 0 12\n”\nPages 10-11 in the main text, “\nAs shown in Fig. 6, the experimental setup comprised a Polytec OFV 2570 laser Doppler\nvibrometer (LDV), a high-speed camera, a RIGOL DG1032z signal generator, a power\namplifier, and a Tektronix DPO4102B-L oscilloscope. Silicon samples were mounted on a\nsupporting plate with a spatial positioning precision of ~5 μm. A 5 mm × 1 mm PZT disk was\nbonded to the back surface of the waveguide inlet using conductive adhesive. A seven-cycle\nsinusoidal burst signal centered at frequency f was generated, amplified, and delivered to the\nc\nPZT transducer. The number of cycles in the burst was limited to facilitate identification of the\nincident wave packet as it propagated through the valley metamaterial phononic circuits. The\nLDV measured the out-of-plane displacement (u) within target regions, while the high-speed\nz\ncamera recorded the laser beam position and enabled spatial point selection.\nTo enhance experimental accuracy, several measures were taken. The input signal power\nwas amplified by ~37 dB to generate strong ultrasonic pulses without damaging the silicon\nwafer. At each spatial point, 512 scans were averaged to reduce white noise. The sampling\nfrequency was set to f = 100 MHz, more than 47 times the target frequency f (1.7–2.1 MHz).\ns c\nAt each spatial point, signals were recorded with 105 data points, corresponding to more than\n1700 periods of 1/f . This allowed fine spectral filtering near f with a frequency resolution of\nc c\n~0.001 MHz. By repeating these procedures, we obtained the amplitude of the out-of-plane\ndisplacement across the target region, enabling separation of neighboring TWGMs and dips.\nThese procedures also allowed us to reconstruct animations and spatial segments of wave\ndisplacement within the region of interest.\nFig. 6 | Experimental piezo-laser setup for measuring ultrasonic wave fields in the cavity-\nwaveguide circuit.”\nReviewer #2: As a final note, the experimental results in figures 2 and especially figure 3 are\nnot really successful in demonstrating localized fields established inside the cavities; whatever\nthe experimental reasons it seems the conclusions can only rely on numerical simulations - this\nshould be acknowledged.\nAuthors: Thank you for your careful review on the experimental results of original\nFigures 2 and 3 (now Figs. 3 and 4).\nFirst, for FFC-waveguide system, we analyzed the reason why the experimental wave\nfields in the cavity are not as uniform as the numerical results. This is due to the backscattering\ncaused by the corners of the rhombus cavity, a similar phenomenon is observed in references\n[Nat. Photon. 17, 386 (2023); Nano Lett. 24, 5570 (2024)]. In addition, there exists boundary\neffects from the metamaterial to the pure silicon wafer, and the limitations in the number and\npower of incident pulses. These factors prevent waves from fully propagating along the\nrhombus path as simulations, making it difficult to clearly capture the local field inside the\ncavity. Therefore, we acknowledge the full display of the wave field depends on the simulations.\nNevertheless, experimental results are still essential to comprehensively demonstrate the\nperformance of the cavity-waveguide system. We have added relevant contents in the main text\n(Page 6).\nThen, similar situation is for experimental wave fields of the dual-cavity waveguide\nsystem. In addition to material loss and bulk waves, the experimental characterization of the\nfull wave field along the rhombus path is difficult because of the limited circle and power of\nthe ultrasonic pulses. The ideal display of the local field inside the cavity still relies on\nsimulations, while experimental results are indispensable for further analysis. We have added\nrelevant contents in the main text (Page 7).\nRelevant revisions are made as follows:\nPage 6 in the main text, “Supplementary Fig. 8 shows TWGM generation at P /D perturbed\n1 1 1\nby bulk waves in the Pure PnC, and verifies the experimental perturbation by bulk waves at\nP /D .”\n1 1\nPage 6 in the main text, “Besides material loss and the influence of bulk waves, the experiment\nalso encounters the effects of backward scattering32 (arising from cavity corners20) and\nboundary effects at the metamaterial–silicon interface, whereas the input waves are limited in\nnumber and pulse power (see Methods). These factors can prevent waves from fully circulating\nalong the ring path as ideally as in simulations. While the full demonstration of wave fields\nrelies on simulations, experiments remain essential to reveal the functional capabilities of the\nwaveguide circuits, serving as a foundation for future applications. Far-field phonon coupling\nenables long-distance cavity energy localization and unlocks functionalities in complex systems,\nsuch as dual-cavity configurations.”\nPage 7 in the main text, “Besides material loss, bulk waves are visible beyond the cavities in\nthe wave field of Fig. 4b. As with the FFC circuit, experimental characterization of the full\nwave field along the ring paths is limited by ultrasonic pulses of finite duration and power.\nSimulations capture the ideal wave-field behavior, while experiments remain essential to reveal\nthe main factors influencing wave propagation.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In the respond letter, authors point to point response to my comments about the details of this study. After thorough-going analysis and clear description the details of the study, authors reasonably show the significance of this work. In a word, this paper is a careful study and the findings are of considerable interest. This paper is worth to be published.", "reviewer_2": "The authors have conducted a very convincing revision. I am highly satisfied with all answers to my comments and suggestions. The correction of terminology, I hope, makes the manuscript clearer regarding the precise achievements. The redistribution of information between main text and supplemental text is now adequate.\nOverall this is a very well written manuscript (I did not spot any typo), concise and precise in the methodology. The conclusions are well supported by the results. I believe the manuscript is ready for publication and I recommend acceptance."}, "author_response": "Author:\nThank you again for your positive comment and recommendation. We are delighted that you\nrecognize our study as a meticulous work with significant findings.\nAuthor:\nWe once again thank you for your careful review. Your professional comments and suggestions\nhave greatly helped us to improve the quality of the present paper."}]

Decision letter