Reconfigurable topological valley-Hall interfaces: Asymptotics of arrays of Dirichlet and Neumann inclusions for multiple scattering in metamaterials

Recongurable top ological v alley-Hall in terfaces: Asymptotics of arra ys of Diric hlet and Neumann inclusions for m ultiple scattering in metamaterials Ric hard Wiltsha w ∗ 1,2 , Henry J. Putley 3 , Christelle Bou Dagher 4 , and Mehul P . Makwana 1 1 Departmen t of Mathematics, Imp erial College London, London, SW7 2AZ, United Kingdom 2 Graduate School of Engineering, The Universit y of Osaka, Suita, Osaka 565-0871, Japan 3 Sc ho ol of Physics and Astronomy , Universit y of Birmingham, Birmingham, B15 2TT, United Kingdom 4 Sc ho ol of Physics and Astronomy , Universit y of Southampton, Southampton, SO17 1BJ, United Kingdom Abstract W e study t wo-dimensional p erio dic metamaterials in which idealised cylindrical inclusions are mo delled b y b oundary conditions. In the scalar time-harmonic setting, the bac kground eld satises the Helmholtz equation, and high-contrast inclusion limits reduce to Dirichlet or Neumann conditions, with direct analogues in dielectric and acoustic media. By switc hing the condition assigned to selected inclusions, we break p oin t-group symmetries of the primitive cell and thereby lift symmetry-induced degeneracies in the Flo quet–Bloch sp ectrum of hexagonal and square lattices, opening v alley-t yp e band gaps with Berry curv ature lo calised near opposite v alleys. T o analyse innite and nite structures within a unied framew ork, we derive matched- asymptotic p oin t-scatterer approximations for mixed Dirichlet–Neumann arrays. F or doubly p eriodic systems, this yields a nite-dimensional generalised eigen v alue problem for the Floquet– Blo c h spectrum; for nite arrays, it yields a generalised F oldy m ultiple-scattering system. In b oth hexagonal and square lattices, geometrically identical crystals can realise distinct v alley- Hall phases solely through b oundary-condition assignment while retaining an ov erlapping bulk gap. Spatially v arying this assignment therefore creates and relocates in ternal interfaces without altering the underlying geometry , enabling the asso ciated v alley-Hall interfacial modes to b e rep ositioned within the same crystal. Keyw ords: v alley-Hall in terfaces; recongurable top ological metamaterials; top ological photonics; Flo quet–Bloch sp ectrum; multiple scattering; singular p erturbations 1 In tro duction T op ological photonics [ 1 – 5 ] concerns wa v e-propagation phenomena controlled by sp ectral top ology in structured electromagnetic media, including top ological-insulator phases and the emergence of in terface-lo calised mo des that lie in bulk sp ectral gaps and propagate along material b oundaries or in ternal domains. In innitely p erio dic settings, these mo des are naturally framed via the Flo quet– Blo c h sp ectrum and its symmetry- and top ology-driven degeneracies. Moreov er, nite top ological w av eguides b eha v e as F abry–Pérot resonators [ 6 ] and can b e understo o d in terms of leaky cavit y mo des, oering a p o werful framew ork for designing nite top ological devices [ 7 , 8 ]. A canonical route to v alley-Hall guiding in photonic and phononic crystals is to b egin with a lattice whose band structure possesses Dirac-type or symmetry-induced degeneracies and then to break in version [ 7 , 9 – 11 ] or reection symmetry of the primitive cell. This lifts the degeneracy and op ens a bulk band gap [ 9 ], thereby engineering v alley-Hall-type insulators that preserv e time-reversal symmetry (TRS) [ 11 – 30 ]. The resulting gap is accompanied b y Berry-curv ature localisation [ 31 ] (equiv alen tly , a non- zero v alley Chern num ber [ 12 , 32 ]) near distinct v alleys and by non-trivial v alley indices for the adjacen t bands [ 33 ]. V alley-Hall insulators of this kind hav e b een designed, b oth theoretically and exp erimen tally , for guiding wa v es in photonic [ 10 – 12 , 34 – 37 ], acoustic [ 38 – 40 ], elastic [ 4 , 14 , 41 – 43 ], and water-w a ve [ 44 ] systems. When t wo bulk media with o verlapping gapp ed sp ectra and opp osite v alley indices are joined, their interface supp orts conned mo des, often termed zero-line mo des (ZLMs) [ 14 ]. In practice, ∗ Corresp onding author: r.wiltshaw17@imperial.ac.uk, wiltshaw.ric hard.w82@osaka-u.ac.jp 1 v alley-Hall interfaces are commonly created b y joining tw o ge ometric al ly distinct c hiral unit cells, for example mirror-related arrangemen ts of inclusions. The central design question addressed here is dieren t: we retain a xed geometric pattern of inclusions and instead control the b oundary c onditions imp ose d on the inclusions . This provides an active mechanism by which the bulk phase can b e switched lo cally , so that the lo cation of the interface–and hence the spatial p osition of the ZLMs–can b e mo ved within the same underlying crystal. T op ological metamaterials provide a route to structurally robust wa v e transp ort, most notably through b oundary , in terface, or domain-w all mo des residing within bulk band gaps. Ho wev er, for practical devices, a dominan t limitation is that the top ological phase and microstructure are xed during fabrication; as a result, the propagation path, op erating frequency band, and functionality are dicult to alter post ho c without sacricing the robustness one seeks. Ov er roughly the last decade, the eld has therefore shifted from static top ological insulators and crystals tow ards re- congurable top ological platforms[ 45 ] capable of switching phases, op ening and closing gaps, and dynamically rerouting edge or interface c hannels across photonic, elastic, and phononic settings. These capabilities ha ve b een realised using mechanisms ranging from physically rotating or trans- lating symmetry-brok en unit cells [ 46 , 47 ] to electronically programming unit-cell states [ 48 ] or switc hing material phases for phononic biosensing [ 49 ]. In photonics and electromagnetic metama- terials, this trend is exemplied by digitally reprogrammable v alley-Hall-type platforms [ 50 ], in whic h the top ological phase and domain w alls are up dated through fast electronic switching at the unit-cell level. Related developmen ts include on-chip v alley photonic crystals [ 51 ], where electrical con trol, for example via thermo-optic [ 52 ] or thermo-acoustic [ 53 ] tuning, mo dulates the transmis- sion of kink or edge channels. In parallel, phase-change approac hes [ 54 , 55 ] exploit large, reversible material contrasts, often without geometric motion, to toggle the existence and sp ectral p osition of top ological edge states, thereby motiv ating non-volatile “top ological memory” [ 56 ] concepts. Recongurable top ological metamaterials hav e rapidly evolv ed from demonstrations of static robustness into activ ely switc hable, routable, and programmable w av e platforms spanning electro- magnetics and photonics, elasticity , and acoustics/phononics, with switc hing sp eeds of nanoseconds in digitally reprogrammed metasurfaces [ 50 ] or thermally driv en [ 52 ] soft-matter transitions. A cross these platforms, top ological switching can b e organised into a small set of physically distinct con- trol mechanisms: geometry- or symmetry-based switc hing; material-prop erty switc hing, including phase-c hange and thermoresp onsive materials; and active or electronic mo dulation of eective coup- lings or b oundary conditions. Eac h approach carries characteristic trade-os in sp eed, reversibilit y , robustness, and manufacturabilit y . In this manuscript, we develop (i) a mixed Diric hlet–Neumann p oin t-scatterer mo del for small inclusions, (ii) a Flo quet–Blo c h eigenv alue formulation for innite arra ys, (iii) a generalised F oldy m ultiple-scattering formulation for nite collections, and (iv) applications to hexagonal and square lattices that illustrate symmetry breaking, v alley-t yp e gaps, ribb on ZLMs, and recongurable in- terface relo cation via b oundary-condition switching. The semi-analytical solutions are v alidated against full numerical Finite Element Metho d (FEM) simulations, and are shown to b e accurate. 2 Mo del F orm ulation W e consider the dynamic electric eld propagating in an isotropic dielectric structure. W e consider a tw o-dimensional scalar time-harmonic w av eeld in the plane x = x e x + y e y , obtained from a structured medium that is translationally inv ariant along the cylinder axis e z . The medium consists of a homogeneous bac kground in which cylindrical inclusions are embedded. In the cross-section, the inclusions are discs cen tred at points x = X I J and of radii η i , arranged p eriodically in primitive cells. The dimensionless equation go verning the electric eld p oten tial ϕ , in the plane, is given by [ 57 , 58 ]  ∇ 2 + Ω 2  ϕ ( x ) = 0 . (1) W e en umerate primitive cells by I = 1 , . . . , N and inclusions within a cell by J indices; later we will distinguish Diric hlet-type and Neumann-type idealised inclusions. The b oundary conditions along the surface b et ween tw o isotropic dielectrics are presented as [ 59 ] ϕ    r I J = η i = ϕ i    r I J = η i , ϵ n · ∇ ϕ    r I J = η i = ϵ i n · ∇ ϕ i    r I J = η i . (2) In ( 2 ), ϵ denotes the (piecewise constan t) dielectric p ermittivity , n is the unit normal at the inclusion b oundary , and subscript i denotes quantities within the inclusions. W e fo cus on t wo singular-con trast idealisations for the inclusion material: ϵ i → 0 and ϵ i → ∞ . In these limits, the con tinuit y conditions in ( 2 ) reduce to Neumann or Dirichlet constraints on the exterior eld at the inclusion b oundary 2 ∂ ϕ ∂ r I J    r I J = η i = 0 as ϵ i → 0 , (3) ϕ    r I J = η i = 0 as ϵ i → ∞ . (4) Although we phrase the discussion for photonic crystals, the same scalar Helmholtz mo del with Diric hlet/Neumann inclusions is also the standard idealisation of sound-soft/sound-hard scatterers in acoustics; the subsequent sp ectral and multiple-scattering framework therefore apply iden tically to the scalar phononic analogue. 3 The mixed Diric hlet-Neumann Inclusion problem W e consider small inclusions whose radii satisfy η i ≪ 1 in the non-dimensionalisation implicit in ( 1 ). In this regime, Dirichlet inclusions are asymptotically represented by monop oles [ 60 ], whereas Neumann inclusions require a combination of monop oles and dip oles [ 38 ] - both of whic h can b e view ed as a singular p erturbation to the wa v eeld approaching the cen tre of the inclusions. This is most naturally deriv ed via the metho d of matc hed asymptotic expansions [ 38 , 60 ]: an inner solution is matched to an outer solution of the Helmholtz equation, and the singular asymptotics naturally cancel during the matching pro cedure and enable the appropriate boundary conditions to b e asymptotically satised throughout the pro cedure. F or mixtures of Diric hlet and Neumann inclusions, the singular asymptotics exists tending tow ards the centre of each inclusion, and is therefore inherited from the corresp onding pure problems (refer to [ 38 , 60 ] for the deriv ations of the singular asymptotics). T o supp ort b oth of the computational settings used later, we treat: (i) an innite p eriodic arrangemen t (leading to a Flo quet–Blo c h eigenv alue problem for band structure); and (ii) a nite collection of inclusions (leading to a multiple-scattering problem with radiation conditions). Both follo w from the same mixed p oin t-scatterer appro ximation. 3.1 Singular p erturbations to appro ximate small Diric hlet and Neumann inclu- sions Assuming eac h inclusion radius is small and inclusions remain well separated within the primitiv e unit cell, the mixed b oundary-v alue problem gov erned b y ( 1 ) with Dirichlet conditions on a subset of inclusions and Neumann conditions on the remainder can b e reduced to a Helmholtz equation with distributional source terms supp orted at the inclusion centres. Dirichlet inclusions contribute monop ole sources; Neumann inclusions contribute a monop ole and a dip ole term - as follows {∇ 2 + Ω 2 } ϕ = 4 i ( N X I =1 P X J =1 a I J δ ( x − X I J )+ ) + ( N X I =1 Q X K =1 η 2 I K [ b I K δ ( x − X I K ) − c I K · ∇ δ ( x − X I K )] ) . (5) In ( 5 ), the indices ( I , J ) en umerate Dirichlet inclusions and ( I , K ) en umerate Neumann in- clusions within the I th primitiv e cells. Each primitive cell con tains P Diric hlet and Q Neumann inclusions resp ectiv ely . The co ecients a I J (Diric hlet) and ( b I K , c I K ) (Neumann) are determined b y asymptotically matching the inner and outer solutions so that the required b oundary conditions are approximately satised. The representation ( 5 ) is in terpreted in the distributional sense in the outer domain (the plane with the inclusion centres remo ved) and is the starting p oin t for the p eriodic and nite formulations that follow. 4 An eigenv alue problem for Flo quet–Blo c h w a v es W e now consider an innite doubly perio dic arrangemen t of inclusions for the mixed p oin t-scatterer form ulation in the xy -plane, as illustrated in Fig. 1 . In ( 5 ) we tak e N → ∞ and partition free space in to primitive cells translated by the tw o-dimensional Brav ais lattice R = n α 1 + m α 2 , n, m ∈ Z . (6) The recipro cal basis vectors β 1 and β 2 are dened by α i · β j = 2 πδ ij , i, j = 1 , 2 , (7) 3 and the corresp onding recipro cal Bra v ais lattice is G = n β 1 + m β 2 , n, m ∈ Z . (8) α 1 α 2 . . . . . . . . . . . . ( a ) β 1 β 2 . . . . . . . . . . . . ( b ) α 1 α 2 . . . . . . . . . . . . ( c ) β 1 β 2 . . . . . . . . . . . . ( d ) e x e y e z Figure 1: T wo-dimensional square (a) and hexagonal (c) Bra v ais lattices in physical space, gen- erated by α 1 and α 2 . P anels (b) and (d) show the corresp onding recipro cal lattices, generated by β 1 and β 2 . In all panels, Bra v ais lattice p oin ts are indicated by ◦ . Because the structure is p erio dic, it is sucient to w ork with a reference cell, whic h we lab el b y I = 1 ; the source co ecien ts in all other cells are then determined by the Blo c h phase. Let A denote the area of a primitive cell. F or wa v es propagating in an innite p erio dic medium, Blo ch’s theorem [ 61 , 62 ] implies that the eld may b e written as ϕ ( x ) = Φ( x ) exp( i κ · x ) . (9) Here Φ( x + R ) = Φ( x ) and κ is the Bloch w av ev ector. Once Φ is kno wn in a single primitiv e cell, the eld throughout the lattice follows from the Blo c h phase factor. It is therefore natural to expand the p erio dic part in recipro cal-lattice harmonics: Φ( x ) = X G Φ G exp( i G · x ) , (10) where the sum is tak en ov er the recipro cal lattice ( 8 ). W riting K G = κ + G , 4 and substituting ( 9 ) and ( 10 ) into ( 5 ), then multiplying b y exp[ − i ( κ + G ′ ) · x ] , integrating o ver the reference cell, and applying orthogonalit y , yields A  K G · K G − Ω 2  Φ G = P X J =1 4 a 1 J exp( − i K G · X 1 J ) i + Q X K =1 4 η 2 1 K [ b 1 K − i c 1 K · K G ] exp( − i K G · X 1 K ) i . (11) Hence the eld admits the series representation ϕ ( x ) = 4 i A X G ( P X J =1 a 1 J exp( i K G · r 1 J ) K G · K G − Ω 2 + Q X K =1 η 2 1 K [ b 1 K − i c 1 K · K G ] exp( i K G · r 1 K ) K G · K G − Ω 2 ) , (12) where r 1 J = x − X 1 J , r 1 K = x − X 1 K . W e denote r 1 J = | r 1 J | and r 1 K = | r 1 K | . As is standard for lattice sums of this type, the repres- en tation ( 12 ) is only conditionally conv ergen t. Moreov er, it b ecomes singular as the observ ation p oin t approaches an inclusion centre: the eld diverges as r 1 J → 0 at a Dirichlet inclusion and as r 1 K → 0 at a Neumann inclusion. Resolving these singular contributions is therefore the crucial step in obtaining a practical nite-dimensional eigenv alue problem. 4.1 Double sum asymptotics of the inner limit of the outer solution The conditional conv ergence of the Blo ch series creates the main computational diculty: in order to ev aluate the eld near an inclusion cen tre, one must isolate the singular contribution from the regular part of the sum. T o this end, we introduce a recipro cal-space cut-o R and decomp ose the series into truncated and residual parts: lim r 1 J → 0 ϕ = lim r 1 J → 0      ϕ tr z }| { X | G | R Φ G exp( i K G · x )      , lim r 1 K → 0 ϕ = lim r 1 K → 0      ϕ tr z }| { X | G | R Φ G exp( i K G · x )      . (13) Here, ϕ tr denotes the truncated sum and ϕ res the remainder b ey ond the cut-o. F or an absolutely con vergen t series, the residual term v anishes as R → ∞ , so truncation in tro duces negligible error. In the presen t problem, how ev er, the series diverges as r 1 J → 0 and r 1 K → 0 , and the residual part therefore contains the singular asymptotics required for the matc hed asymptotic analysis. F or a large but nite cut-o satisfying R ≫ 1 , Rr 1 J ≪ 1 , and R r 1 K ≪ 1 , the singular asymptot- ics of ϕ res can b e extracted analytically using the Euler–Maclaurin formula [ 63 ]. F or the Dirichlet case, [ 60 ] gives ϕ res ∼ 2 a 1 J iπ  log 2 r 1 J R − γ E  + . . . , as r 1 J → 0 and R → ∞ . (14) Here γ E denotes the Euler–Mascheroni constan t. Similarly , from the Neumann analysis in [ 38 ], one obtains ϕ res ∼ η 2 1 K exp ( i κ · r 1 K ) iπ n 2 c 1 K · e r 1 K r 1 K J 0 ( Rr 1 K ) + J i 0 ( Rr 1 K ) h ir 1 K ( i c 1 K Ω 2 + 2 b 1 K κ ) · e r 1 K − 2 b 1 K i − − J 1 ( R ′ r ) R ′ r h 2 i  c 1 K · κ cos(2 φ 1 K − 2 θ κ ) − ( c 1 K × κ ) · e z sin(2 φ 1 K − 2 θ κ )  + + ir 1 K ( i c 1 K Ω 2 + 2 b 1 K κ ) · e r 1 K i − − 2 κ h c 1 K · κ cos(3 θ κ − 3 φ 1 K ) + ( c 1 K × κ ) · e z sin(3 θ κ − 3 φ 1 K ) i J 2 ( R ′ r 1 K ) R ′ 2 r 1 K o + . . . , as r 1 K → 0 and R → ∞ . (15) Here θ κ and φ 1 K are the argumen ts of κ and x − X 1 K , resp ectiv ely . In addition, e r 1 K denotes the radial base vector centred on the 1 K th Neumann inclusion, J i 0 denotes V an der Pol’s zero-order Bessel-in tegral function, whereas J 0 , J 1 , and J 2 denote Bessel functions. Their small-argumen t 5 b eha viour is standard [ 64 ]; in particular, Humbert [ 65 ] sho wed that J i 0 ( x ) = C i ( x ) − log 2 = log x 2 + γ E − x 2 4 + O ( x 4 ) , as x → 0 , (16) where C i ( x ) is the cosine-integral function. 4.2 The outer limit of the inner problem The matching conditions require the outer limit of the corresp onding inner solutions near the Diric hlet and Neumann inclusions. These are inherited from the pure Dirichlet and pure Neumann problems. F or a Diric hlet inclusion, [ 60 ] gives lim r 1 J → 0 ϕ = 2 a 1 J iπ log η 1 J r 1 J . (17) F or a Neumann inclusion, we [ 38 ] obtained lim r 1 K → 0 ϕ ∼ 4 i π b 1 K Ω 2  1 − Ω 2 r 2 1 K 4  + 4 iπ c 1 K · e r 1 K Ω  r 1 K Ω 2 − Ω 3 r 3 1 K 16  + η 2 1 K  2 ib 1 K π  log r 1 K η 1 K + 3 4  + c 1 K · e r 1 K iπ  2 r 1 K − Ω 2 r 1 K  log r 1 K η 1 K − 7 4  . (18) 4.3 A generalised eigenv alue problem from the metho d of matched asymptotic expansions Assume η 1 J ≪ 1 and η 1 K ≪ 1 . In the vicinit y of a given inclusion, the lo cal singular b eha viour is asymptotically dominated by that inclusion alone. Since the F ourier series ( 12 ) div erges as r 1 J → 0 and r 1 K → 0 , we introduce a large but nite recipro cal-space cut-o satisfying R ≫ 1 , Rr 1 J ≪ 1 , and Rr 1 K ≪ 1 . Equations ( 14 ) and ( 15 ) quan tify the error introduced b y truncating the conditionally conv ergen t series. Com bining ( 13 ), ( 14 ), and ( 17 ) yields the Dirichlet matching condition X | G |