Observation of a higher-order topological bound state in the continuum
Alexander Cerjan, Marius Jürgensen, Wladimir A. Benalcazar, Sebabrata Mukherjee, Mikael C. Rechtsman
OObservation of a higher-order topological bound state in the continuum
Alexander Cerjan, ∗ Marius J¨urgensen, Wladimir A. Benalcazar, Sebabrata Mukherjee, and Mikael C. Rechtsman † Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Dated: June 12, 2020)Higher-order topological insulators are a recently discovered class of materials that can possesszero-dimensional localized states regardless of the dimension of the lattice. Here, we experimentallydemonstrate that the topological corner-localized modes of higher-order topological insulators canbe symmetry protected bound states in the continuum; these states do not hybridize with thesurrounding bulk states of the lattice even in the absence of a bulk bandgap. As such, this classof structures has potential applications in confining and controlling light in systems that do notsupport a complete photonic bandgap.
Topological materials have garnered significant inter-est for their ability to support boundary-localized statesthat manifest exotic phenomena, such as the backscatter-free chiral edge states found in Quantum Hall systems[1–14], and edge-localized states found in systems withquantized dipole moments [15–17]. Recently, it was dis-covered that crystalline symmetries can give rise to a newclass of materials with topological phases that can pro-tect zero-dimensional corner-localized states in two di-mensions, or more generally d − n dimensional states atthe boundaries of d dimensional lattices, with n ≥ ∗ [email protected] † [email protected] topological phases have now been demonstrated in awide range of different physical platforms, including mi-crowaves [45], photonics [46–48], acoustics [49–54], elec-tric circuits [55, 56], and atomic systems [57], all of theseprevious studies have been limited to insulator-like sys-tems, and exhibit their corner-localized states spectrallyisolated from their surrounding bulk bands.Here, we experimentally realize a higher-order topo-logical bound state in the continuum using a two-dimensional waveguide array comprised of evanescently-coupled waveguides [58, 59]. To show that our waveguidearray possesses a BIC, we perform three separate exper-iments. First, by injecting light into the corner of thearray, we prove that the lattice exhibits a corner-boundmode when the lattice is in its topological phase, andthat this mode disappears across the topological phasetransition. Second, by using an auxiliary waveguide tocouple into the array, which fixes the effective energy ofthe initial excitation, we show that this corner-localizedmode appears at zero energy, and is degenerate with bulkstates of the lattice. For consistency with previous stud-ies, we refer to ‘zero energy’ as the propagation constant /energy of a single waveguide, which for chiral-symmetriclattices is at the center of the spectrum. Finally, we showthat our bound state transforms into a resonance whenwe break chiral symmetry by detuning the index of refrac-tion of the members of one sublattice. Together, theseexperiments prove that the corner-localized state of ourhigher-order topological waveguide array is a symmetry-protected BIC, and does not hybridize with the bulkbands so long as the necessary symmetries remain intact.Our experimental array consists of a square lattice inwhich each unit cell contains four waveguides and is C v symmetric, as shown in Fig. 1a,b [28, 41, 60, 61]. Aseach waveguide within the lattice only supports a sin-gle bound mode for the wavelengths we consider, andthe coupling between waveguides decreases exponentiallywith increasing separation, our waveguide array can beapproximated using a tight-binding model with onlynearest-neighbor couplings, such that the lattice is chiral(sublattice) symmetric. The diffraction of light throughthe structure is governed by i∂ z | ψ ( z, λ ) (cid:105) = ˆ H ( λ ) | ψ ( z, λ ) (cid:105) . (1)Here, | ψ ( z, λ ) (cid:105) is the envelope of the electric field on each a r X i v : . [ phy s i c s . op ti c s ] J un inter (cid:1) intra (a) (c)(b) (d) (f) (cid:2) L o c a l d e n s i t y o f s t a t e s T o p o l og i c a l T r i v i a l L o c a l d e n s i t y o f s t a t e s (cid:2) (cid:3) /(e (cid:2) D e n s i t y o f s t a t e s D e n s i t y o f s t a t e s FIG. 1. Higher-order topological insulator in a waveguide array. (a),(b) Schematic of a higher-order topological insulator inits topological (a) or trivial (b) phase. The unit cell of each phase is indicated in the dashed-black square. The distancesbetween adjacent waveguides within a unit cell, and between neighboring unit cells, are shown as l intra and l inter , respectively.Schematics are not to scale. (c),(d) Density of states (top panel) and associated local density of states (bottom panel) foreach band for the topological and trivial phases of the higher-order topological insulator in finite geometries, respectively. Thedensity of states is calculated using the tight-binding approximation. (e) Bulk band structure for the higher-order topologicalinsulator, which is identical for both the topological and trivial phases, calculated using full-wave numerical simulations for λ = 850 nm, l intra = 17 µ m, and l inter = 13 µ m. (f) White light transmission micrograph of the output facet of a waveguidearray with l intra = 13 µ m and l inter = 11 µ m. An auxiliary waveguide into which light can be injected, 20 µ m away from thearray, is indicated with a black arrow. of the waveguides at propagation distance z and wave-length λ . The coupling coefficients, t intra and t inter , inˆ H are determined by the spacings between neighboringwaveguides within the same unit cell, l intra , and betweenadjacent unit cells, l inter .For an array which is infinite in the transverse plane,the Bloch Hamiltonian of the lattice can be written as h ( k x , k y ) = (cid:18) QQ † (cid:19) , (2) Q = (cid:18) t intra + t inter e ik x a t intra + t inter e ik y a t intra + t inter e − ik y a t intra + t inter e − ik x a (cid:19) , (3)in which a is the lattice constant. To assist with com-parisons with the topological literature, we will refer tothe eigenvalues of the waveguide array, ˆ H , as energies, E ,while noting that physically these values correspond toshifts in momentum, β = − E = k z − k , of | ψ (cid:105) along the z axis. Here, k = ωn /c , where n is the index of refrac-tion of the borosilicate glass into which the waveguidesare fabricated and ω is the frequency of the injected light.As all of these modes are bound modes of the waveguides,‘zero energy’ refers to the energy at the middle of thisspectrum.The presence of C v symmetry permits two distincttopological phases depending on the ratio of the relativespacings between neighboring waveguides within and be-tween adjacent unit cells. When the lattice is in its topo- logical phase, with l intra /l inter >
1, the bands possessdifferent representations of C v ( C v ) at the correspond-ing high-symmetry points in the Brillouin Zone, M ( X and Y ), than at Γ . However, when the lattice is in itstrivial phase, with l intra /l inter <
1, the bands possessthe same symmetry representation at all of the high-symmetry points. The topological phase transition oc-curs at l intra /l inter = 1, when the bulk bandgap closesat the high-symmetry points, allowing for the exchangeof their representations of these crystalline symmetries.In a finite lattice, these two phases can be distinguishedby their density of states, as well as the associated lo-cal density of states of each band, shown in Fig. 1c,d.In its topological phase, this lattice exhibits both edge-localized states in its bulk bandgaps protected by C symmetry, as well as a corner-induced filling anomalyalso protected by C [28, 41]. In Fig. 1c, the presenceof these extra corner-localized states can be observed inthe local density of states of the central bulk band of thelattice. Note that when l intra is interchanged with l inter ,both the topological and trivial phases of the array havethe same bulk band structure consisting of four bands, asdisplayed in Fig. 1e. An example of a facet of a waveguidearray is shown in Fig. 1f.In the presence of C v and chiral symmetries, the lat-tice will always have gapless bulk bands at zero energy,regardless of its topological phase. These same two sym- intra ( μ m) ( (cid:3) s − (cid:3) e ) / ( (cid:3) s + (cid:3) e ) (cid:5) s10-1 10 12 14 1664 2 E x p e r i m e n t Trivial latticeTopological lattice (cid:1) intra = (cid:1) inter (cid:1) intra ( μ m) ( (cid:3) s − (cid:3) e ) / ( (cid:3) s + (cid:3) e ) (cid:5) s10-1 10 12 14 1664 2 S i m u l a t i o n (b)(c) (d)(f) (e)(g)(a)subsystem environment I n t e n s i t y ( a . u . ) FIG. 2. Bound state in a higher-order topological insulator. (a) Schematic of a waveguide array in the topological phasewith the boundary of the ‘subsystem’ for n s = 3 indicated. (b),(c) Experimentally observed (b) and numerically simulated (c)fractional power, as a function of the size of the subsystem, n s , and the spacing between adjacent waveguides within the sameunit cell, l intra . Spacing between adjacent waveguides in neighboring unit cells is fixed at l inter = 13 µ m, and the wavelengthof the light is λ = 850 nm. The maximum propagation distance of the array is L = 7 . l intra = l inter = 13 µ m. (d) Experimentally observed intensity at the output facetfor l intra = 17 µ m. Light is injected into the left-most waveguide at the corner of the array, marked with a white arrow. (e)Experimentally observed intensity at the output facet for l intra = 9 µ m. (f)-(g) Same as (d)-(e), except for full wave numericalsimulations of the waveguide array. metries also pin the corner-localized states to zero en-ergy, guaranteeing that the states will always be degen-erate with the bulk bands of the array, while simulta-neously protecting the corner-localized modes from hy-bridizing with the surrounding bulk states [41]. This pro-tection comes in two parts. First, two combinations ofthe four corner states have incompatible symmetry rep-resentations with those of the surrounding bulk bandsat zero energy, and thus cannot hybridize with them.Then, the two remaining combinations of corner local-ized states must be both rotationally symmetric partners,with the same energy, and chiral symmetric partners,with opposite energies, forcing their energies to remainpinned at zero. This prevents these two corner-localizedstates from hybridizing with the degenerate bulk states tochange their energies or modal profiles, and as such anyhybridization of the corner states with the surroundingbulk states is simply a change in basis that does not altertheir underlying spatially-localized nature. This meansthat all four corner states in our lattice are topologicallyguaranteed to be zero-dimensional symmetry-protectedbound states in the continuum.In our experiment, it is not possible to completelyremove the next-nearest-neighbor couplings which existbetween the waveguides in the array, which means ourlattice is not perfectly chiral symmetric. However, thedecay length of the corner state due to this slight sym-metry breaking ( ∼
25 m) is significantly longer than thepropagation length in our experiments, L = 7 . H ( r ) = ˆ H x ( x ) + ˆ H y ( y ) [34].However, despite the fact that Eqs. (2)-(3) are separa-ble, separability is not what protects the higher-ordertopological BICs we consider here. Analytically, one stillobserves higher order topological BICs when additionalterms have been added to the lattice’s Hamiltonian whichobey C v and chiral symmetries but break separability[41], see Supplementary Information.To experimentally prove that our waveguide array con-tains a higher-order topological BIC, we first inject lightinto the corner of the array, and observe whether mostof the light remains confined to this corner or diffractsinto the bulk. To assess the localization of the light atthe output facet of the array, we divide the array intotwo regions, the ‘subsystem’ which represents the squareof unit cells with side length n s closest to the corner,as indicated in Fig. 2a, while the remaining waveguidescomprise the ‘environment.’ This terminology is cho-sen for consistency with previous studies of bound statesin the continuum, in which the subsystem (where lightis confined) and its surrounding radiative environmentare typically physically distinct regions containing dif-ferent types of structures. We then compare the totaloutput power observed in the subsystem, P s , with thatobserved in the environment, P e , using the figure of merit( P s − P e ) / ( P s + P e ). For this ‘fractional power,’ valuesnear +1 correspond to all of the output power being lo-calized in the subsystem, while values of − a) (b) Trivial latticeTopological lattice FIG. 3. Observation of the BIC and surrounding continuum ina higher-order topological insulator. (a) Experimentally ob-served intensity at the output facet for a topological waveg-uide array, with l intra = 13 µ m and l inter = 11 µ m. Lightis injected into the array at λ = 900 nm using an auxiliarywaveguide placed 20 µ m away from the corner of the lattice(marked with a white arrow). The total length of the arrayis L = 7 . l intra = 11 µ mand l inter = 13 µ m. function of the topological phase of the array, l intra /l inter ,as well as the the size of the subsystem, n s . Here, we canclearly see that the light remains localized to the sub-system, regardless of its size, until the arrays approachthe topological phase transition, at l intra /l inter = 1. Theobserved intensity at the output facet is shown for an ex-ample of both the topological and trivial arrays in Figs.2d-g. Note that the increase seen in the fractional powerfor large subsystem sizes for some topologically trivialarrays is due to spurious reflections off of some of thewaveguides at the top and bottom of the array, as wellas back-reflections off of the far side of the array. Never-theless, it is clear from Fig. 2e that these arrays do notpossess a bound state. Thus, these results indicate thatthe topological waveguide array possesses a bound statethat is connected to the topological phase of the lattice,but does not yet prove that the bound state is degeneratewith the surrounding bulk bands.To prove that this topological bound state is a BIC,we use a waveguide array with an auxiliary waveguideweakly coupled to the lattice and placed near one of thecorners. Since this waveguide is identical to all othersin the lattice, it effectively acts as a fixed zero-energysource. All light injected into it can only excite statesnear zero-energy in the array. As can be seen in Fig. 3a,when the waveguide array is in its topological phase andthe auxiliary waveguide is placed near a corner of the lat-tice, the dominant excited mode of the waveguide arrayis the topological corner-localized mode. However, when l intra and l inter are reversed, the bulk of the lattice re-mains completely unchanged, but the array is now in thetrivial phase. Upon excitation using an auxiliary waveg-uide, we see that bulk states of the lattice are excited,and there is no corner-localized mode. Since the latticebulk is identical in both cases, we can conclude that thereare zero-energy bulk states that are degenerate with the corner-localized mode in the topological case. This ex-perimentally proves that the corner-localized topologicalbound states in this array are BICs.Finally, to demonstrate that this higher-order topolog-ical BIC is protected by chiral symmetry, we purposefullybreak chiral symmetry by increasing the refractive indexon two of the four waveguides in the unit cell, as indicatedin Fig. 4a, which has the effect of decreasing the effectiveon-site energy of these two lattice sites. This has severaleffects on the array. First, this opens a bulk bandgapin the center of the spectrum, in which one of the twocentral bulk bands remains at ‘zero-energy’ (which is nolonger at the middle of the spectrum), while the other’senergy decreases, as shown in Fig. 4b. Second, as themodal profile of each corner-localized state is only sup-ported on two of the four waveguides in the unit cell(diagonally across from one another), this change alsobreaks the four-fold degeneracy of the corner-localizedstates. Instead, the pair of corner-localized states whosemodal profiles overlap with the perturbation decreasetheir energy, remaining degenerate with the higher of thetwo central bulk bands, while the other pair of corner-localized states remain degenerate with the bulk band at‘zero energy’ (which is now not the center of the spec-trum). This can be seen in the local density of statesfor each band of the array, shown in Fig. 4c. However,now that chiral symmetry has been broken, the corner-localized modes are allowed to hybridize with states fromtheir respective bulk bands, transforming from BICs intoresonances of the lattice. We can observe this transi-tion of one of the BICs into a resonance by incremen-tally increasing the strength of the sublattice symmetrybreaking, and coupling into the lattice using an auxiliarywaveguide, which remains at zero energy, as shown inFigs. 4d-h. As chiral symmetry is lost, as in Figs. 4e-h,the wavefunction within the array begins to disassociatefrom the corner, and the maximum of this wavefunctiontravels into the bulk of the array and along the edges,signifying that all of the states that are being excited bythe auxiliary waveguide have significant spatial overlapwith the other modes of the lattice. In other words, thecorner-localized state has become a resonance and is nolonger a BIC. This is in clear contrast to what is seen inFig. 4d, where chiral symmetry is intact and the wave-function in the lattice remains localized to the corner,indicating the presence of a BIC.In conclusion, we have experimentally observed ahigher-order topological bound state in the continuumin a waveguide array. This BIC is protected by C v andchiral symmetries, and is topologically guaranteed to ex-ist at zero energy in the lattice. Moreover, as these statesare able to confine light to a zero-dimensional mode in theabsence of a bulk bandgap, which is a greater reductionin dimensionality than is found in other systems support-ing BICs [34], they are a promising candidate for creatingcavities in low-index photonic platforms in the absence ofa complete photonic bandgap. Current designs for pho-tonic crystals that support band gaps require refractive (cid:1) (cid:2) (cid:3) / (cid:4) -1 10 (cid:1) (cid:5) (cid:3) / (cid:4) (cid:6) (b)(a) AuxiliarywaveguideUnit cell20 μ m 13 μ m11 μ mConstant refractive indexIncreased refractiveindex in (e)-(h) (d) (e) (f) (g) (h)Increasing chiral symmetry breakingChiral symmetric 0 0 (cid:6) L o c a l d e n s i t y o f s t a t e s (c) D e n s i t y o f s t a t e s FIG. 4. Observation of a BIC turning into a resonance as chiral symmetry is broken by detuning the sublattices of the lattice.(a) Schematic of a higher-order topological waveguide array with broken chiral symmetry. The auxiliary waveguide wherelight is initially injected is 20 µ m away from the waveguide array. The waveguides colored in green have been fabricated usingslower writing speeds, resulting in a larger refractive index, and thus a decreased on-site energy. (b) Bulk band structure for thehigher-order topological insulator with broken chiral symmetry, calculated using full-wave numerical simulations for λ = 900 nm, l intra = 13 µ m, l inter = 11 µ m, and in which two of the waveguides have an increased refractive index, ∆ n = 3 . · − , asopposed to ∆ n = 2 . · − . These parameters correspond to the experimental results shown in (f), below. (c) Density of states(top panel) and associated local density of states for each band (bottom panel) for a tight-binding lattice with broken chiralsymmetry. Zero energy (center of the spectrum) of the chiral symmetric (unperturbed) array is marked. (d) Experimentallyobserved intensity at the output facet of the symmetric waveguide array, with the same refractive index on all of the waveguides,with l intra = 13 µ m and l inter = 11 µ m, for an incident wavelength of λ = 900 nm. The length of the array in the z direction is L = 7 . indices of at least n = 2 . n = 1 . n ∼ .
5. We expectthat zero-dimensional bound states in the continuum ofthe kind described here will lead to an expanded range ofdevices in which cavity and defect modes, for enhancinglight-matter coupling, can be found.
METHODS
We fabricated our waveguide arrays with a Yb-dopedfiber laser (Menlo BlueCut) system emitting circularlypolarized sub-picosecond (260 fs) pulse trains at 1030 nmwith a repetition rate of 500 kHz. The light was focusedinside a borosilicate glass (Corning Eagle XG) sampleusing an aspheric lens. The borosilicate glass samplewas mounted on high-precision x - y - z translation stages(Aerotech). Each individual waveguide was written bytranslating the glass sample once through the focus ofthe laser at a speed of 10 mm / s. This process results in waveguides which only support the fundamental mode,which has an elliptical profile. However, by then orient-ing the lattice to be ‘diagonal’ with respect to the sur-face of the glass slide (i.e. the surface of the glass slidelies along the [1 1] direction of the lattice), the couplingconstants between neighboring waveguides in both the[1 0] and [0 1] directions are the same, up to fabricationimperfections. By measuring two-waveguide couplers, wedetermined that our waveguides can be modeled in ourfull-wave numerical simulations using a Gaussian profile,∆ n ( r ) = ∆ n e − r /σ r (4)with ∆ n = 2 . · − and σ r = 4 µ m. For a separation of13 µ m at λ = 850 nm, this yields a tight-binding couplingcoefficient of t = 1 .
62 cm − . For the waveguides used inbreaking the chiral symmetry of the lattice as highlightedin Fig. 4, the writing speed was incrementally decreasedto be [8 , , ,
2] mm / s in Fig. 4e-h, respectively. Thesereduced writing speeds can be modeled numerically asshifts of the refractive index of the waveguides of ∆ n =[3 . , . , . , . · − .The waveguide arrays were measured using a commer-cial supercontinuum source (NKT SuperK COMPACT)with a filter to select the desired wavelength (SuperKSELECT). The beam was focused into the samplesing an aspheric lens with an NA of 0 .
15 (ThorLabsC280TMD-B) and imaged onto the camera with anachromatic doublet (ThorLabs AC064-015-B-ML). Theimages were taken using a CMOS camera (ThorLabsDCC1545M).
ACKNOWLEDGMENTS
This work was supported by the US Office of Naval Re-search (ONR) Multidisciplinary University Research Ini-tiative (MURI) grant N00014-20-1-2325 on Robust Pho-tonic Materials with High-Order Topological Protection,the ONR Young Investigator Award under grant numberN00014-18-1-2595 as well as the Packard Foundation un-der fellowship number 2017-66821. W.A.B. acknowledgesthe support of the Eberly Postdoctoral Fellowship at thePennsylvania State University. M.J. acknowledges thesupport of the Verne M. Willaman Distinguished Grad-uate Fellowship at the Pennsylvania State University.
SUPPLEMENTAL INFORMATION FOR: OBSERVATION OF A HIGHER-ORDER TOPOLOGICALBOUND STATE IN THE CONTINUUMI. HIGHER ORDER BOUND STATE IN THE CONTINUUM WITHOUT SEPARABILITY
One known mechanism for creating bound states in the continuum (BICs) is through separability [34], in which theHamiltonian of the system can be divided into two (or more) parts,ˆ H ( r ) = ˆ H x ( x ) + ˆ H y ( y ) (S1)that only depend on a single spatial coordinate. Then, by finding a localized bound state of each individual portion,for example, H x ( x ) ψ n ( x ) = E ( x ) n ψ n ( x ) and H y φ m ( y ) = E ( y ) m φ m ( y ), the combined state, ψ n φ m , is a bound state ofˆ H ( r ) with energy E ( x ) n + E ( y ) m , which may reside within the continuum ˆ H ( r ). Furthermore, one may suspect that thisis the origin of the BIC we report in the main text, as the tight-binding model corresponding to our waveguide arraysis separable. -0.02 -0.04 I m [ (cid:1) ] Re[ (cid:1) ] -3 -2 -1 -10 -2 -10 -4 -10 -6 -10 -8 -10 -10 Subsystem size, (cid:2) s I m [ (cid:1) ] (b) (c) (d)(a) (cid:2) s ⋯⋮ ⋱ FIG. S1. (a) Schematic depicting the tight-binding model used as an example of a non-separable higher-order topological BIC.Dashed lines indicate couplings of t intra = 0 .
25, solid black lines indicate couplings of t inter = 1, and solid red lines indicatechiral-preserving non-separable couplings, t = 0 . γ = 0 .
05. Only a single corner of the latticeis shown, but all four corners are part of the subsystem for these simulations, and do not have any on-site loss. (b) Eigenvaluesfor this lattice shown in the complex plane for n s = 3, i.e. the total number of unit cells in each corner which constitute the‘subsystem’ is 3 ×
3. The lattice has n l = 16 unit cells along its sides, for a total lattice size of 16 ×
16. (c) The probabilitydensities of the four eigenstates corresponding to the BICs from (b). (d) Plot of the decay rate of the eigenvalues of the latticeas n s is increased. However, this is not the origin of the protection of the BICs that we observe in our waveguide arrays [41]. Todemonstrate this explicitly, in Fig. S1 we show numerical tight-binding calculations of a higher-order topologicalinsulator that obeys both of the necessary protecting symmetries of the BIC, C v and chiral symmetry, but breaksseparability. This lattice is schematically shown in Fig. S1a. We then divide this finite lattice into two regions, the‘subsystem,’ a region of n s unit cells next to each of the four corners of the lattice, and the ‘environment,’ which are ec a y r a t e , I m [ (cid:1) ] ( c m − ) Environmental loss(a) (b) Subsystem size, (cid:3) s D ec a y r a t e , I m [ (cid:1) ] ( c m − ) FIG. S2. (a) Plot of the decay rates of the 30 least lossy eigenvalues of the lattice as a function of the added loss, γ , tothe waveguides in the environment. The plateau seen in the decay rate of the corner-localized mode corresponds to its trueradiative rate in an infinite lattice. Here, n l = 32 (i.e. the total lattice size is 32 × n s = 4. (b) Plot of the decay ratesof the 30 least lossy eigenvalues of the full lattice as a function of the edge length of the device region, n s for γ = 0 .
02 and n l = 32. In both (a) and (b), the decay rate of the corner-localized mode is reaching a nearly constant value at approximatelyIm[ E ] = − · − cm − . the remainder of the unit cells in the lattice. A small, but non-zero, amount of loss is then added to the environmentto simulate radiation loss, 0 < γ (cid:28)
1. As can be seen in Fig. S1b, this lattice still possesses four essentially realeigenvalues, whose corresponding eigenstates are exponentially localized to the corners of the lattice, see Fig. S1c.Finally, by changing the size of the device region, we can confirm that the energies of these corner-localized statesexponentially converge to be real, confirming that these are BICs, not resonances, as shown in Fig. S1d.Thus, even though the waveguide array that we study in the main text is separable, this separability is not whatprotects the BICs that we observe in our array. Even in the absence of separability, our arrays would still exhibitBICs, so long as the two necessary protecting symmetries, C v and chiral, were preserved. II. EFFECT OF NEXT-NEAREST-NEIGHBOR COUPLINGS IN THE WAVEGUIDE ARRAY
As discussed in the main text, in our waveguide arrays it is impossible to completely remove next-nearest-neighborcouplings between waveguides as the coupling strength is an exponentially decaying function of the spatial separationbetween the waveguides, t ( λ ) = e − α ( λ ) l , in which α ( λ ) is a wavelength-dependent constant. In particular, this meansthat the couplings between diagonally adjacent waveguides in our arrays will be non-vanishing, and break chiralsymmetry, placing a practical limit on the decay length of the corner modes. However, in practice, the couplingcoefficients corresponding to this process are small relative to the dominant energy scale in the array, i.e. the largerof t intra or t inter depending on whether the lattice is in the topological phase. For example, for the topological latticeshown in Fig. 2d,f of the main text, t diag /t inter ∼ .
08, where t diag is the next-neighbor coupling strength betweenwaveguides across the diagonal in the unit cell.Moreover, we can calculate the effect of this coupling constant on the now-finite decay length of the corner-localizedmode using a tight-binding model with a small amount of non-Hermitian loss added to an ‘environment’ region.Mathematically, this corresponds to ˆ H tot = ˆ H − (cid:88) j ∈ env. wgs. iγ | j (cid:105)(cid:104) j | , (S2)in which ˆ H is given by Eq. 1 of the main text, and in which the sum runs over those waveguides in the environment.As can be seen in Fig. S2a, the decay length of the corner-localized mode quickly saturates as a function of the addedloss to the environment, and this plateau corresponds to the radiative loss of the corner-localized resonance in aninfinite lattice without added loss by the Limiting Absorption Principle [65–68]. Moreover, we can confirm this decaylength of the corner-localized mode by fixing the added environmental loss and varying the size of the device region,shown in Fig. S2b. In both cases, the decay length converges to approximately Im[ E ] = − · − cm − As stated inthe Main Text, for the topological lattice shown in Fig. 2d,f, the decay length is L decay ∼
25 m, which is significantlylonger than the waveguide arrays in our experiment ( L = 7 . [1] Klitzing, K. v., Dorda, G. & Pepper, M. New Method forHigh-Accuracy Determination of the Fine-Structure Con- stant Based on Quantized Hall Resistance. Phys. Rev.ett. , 494–497 (1980). URL https://link.aps.org/doi/10.1103/PhysRevLett.45.494 .[2] Halperin, B. Quantized Hall Conductance, Current-Carrying Edge States, and the Existence of ExtendedStates in a Two-Dimensional Disordered Potential. Phys.Rev. B , 2185–2190 (1982).[3] Thouless, D. J., Kohmoto, M., Nightingale, M. P. &den Nijs, M. Quantized Hall Conductance in a Two-Dimensional Periodic Potential. Phys. Rev. Lett. ,405–408 (1982). URL https://link.aps.org/doi/10.1103/PhysRevLett.49.405 .[4] B¨uttiker, M. Absence of backscattering in the quan-tum Hall effect in multiprobe conductors. Phys. Rev.B , 9375–9389 (1988). URL https://link.aps.org/doi/10.1103/PhysRevB.38.9375 .[5] Haldane, F. D. M. Model for a Quantum Hall Effectwithout Landau Levels: Condensed-Matter Realizationof the ”Parity Anomaly”. Phys. Rev. Lett. , 2015–2018 (1988). URL http://link.aps.org/doi/10.1103/PhysRevLett.61.2015 .[6] Haldane, F. D. M. & Raghu, S. Possible Realizationof Directional Optical Waveguides in Photonic Crystalswith Broken Time-Reversal Symmetry. Phys. Rev. Lett. , 013904 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.100.013904 .[7] Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljaˇci´c,M. Observation of unidirectional backscattering-immunetopological electromagnetic states.
Nature , 772–775 (2009). URL .[8] Umucalılar, R. O. & Carusotto, I. Artificial gauge fieldfor photons in coupled cavity arrays.
Phys. Rev. A ,043804 (2011). URL https://link.aps.org/doi/10.1103/PhysRevA.84.043804 .[9] Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M.Robust optical delay lines with topological protection. Nat. Phys. , 907–912 (2011).[10] Fang, K., Yu, Z. & Fan, S. Realizing effective magneticfield for photons by controlling the phase of dynamicmodulation. Nat. Photon. , 782–787 (2012).[11] Kitagawa, T. et al. Observation of topologically pro-tected bound states in photonic quantum walks.
Nat.Commun. , 1872 (2012). URL .[12] Rechtsman, M. C. et al. Photonic Floquet topo-logical insulators.
Nature , 196–200 (2013).URL .[13] Khanikaev, A. B. et al.
Photonic topological in-sulators.
Nat. Mater. , 233–239 (2013). URL .[14] Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor,J. M. Imaging topological edge states in silicon photonics. Nat. Photon. , 1001–1005 (2013). URL .[15] Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons inPolyacetylene. Phys. Rev. Lett. , 1698–1701 (1979).[16] King-Smith, R. & Vanderbilt, D. Theory of polarizationof crystalline solids. Physical Review B , 1651 (1993).[17] Zak, J. Berrys phase for energy bands in solids. Physicalreview letters , 2747 (1989).[18] Benalcazar, W. A., Teo, J. C. Y. & Hughes, T. L. Clas-sification of two-dimensional topological crystalline su- perconductors and Majorana bound states at disclina-tions. Phys. Rev. B , 224503 (2014). URL https://link.aps.org/doi/10.1103/PhysRevB.89.224503 .[19] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L.Quantized electric multipole insulators. Science ,61–66 (2017). URL http://science.sciencemag.org/content/357/6346/61 .[20] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L.Electric multipole moments, topological multipole mo-ment pumping, and chiral hinge states in crystalline in-sulators.
Phys. Rev. B , 245115 (2017). URL https://link.aps.org/doi/10.1103/PhysRevB.96.245115 .[21] Song, Z., Fang, Z. & Fang, C. ( d − Phys. Rev. Lett. , 246402(2017). URL https://link.aps.org/doi/10.1103/PhysRevLett.119.246402 .[22] Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F.& Brouwer, P. W. Reflection-Symmetric Second-OrderTopological Insulators and Superconductors.
Phys. Rev.Lett. , 246401 (2017). URL https://link.aps.org/doi/10.1103/PhysRevLett.119.246401 . Publisher:American Physical Society.[23] Schindler, F. et al.
Higher-order topological in-sulators.
Science Advances , eaat0346 (2018).URL https://advances.sciencemag.org/content/4/6/eaat0346 . Publisher: American Association for theAdvancement of Science Section: Research Article.[24] Zilberberg, O. et al. Photonic topological boundarypumping as a probe of 4d quantum hall physics.
Nature , 59–62 (2018).[25] Wieder, B. J. & Bernevig, B. A. The axion insulator asa pump of fragile topology. arXiv:1810.02373 (2018).[26] van Miert, G. & Ortix, C. Higher-order topological in-sulators protected by inversion and rotoinversion sym-metries.
Phys. Rev. B , 081110 (2018). URL https://link.aps.org/doi/10.1103/PhysRevB.98.081110 .[27] Ezawa, M. Minimal models for wannier-type higher-ordertopological insulators and phosphorene. Phys. Rev. B , 045125 (2018). URL https://link.aps.org/doi/10.1103/PhysRevB.98.045125 .[28] Benalcazar, W. A., Li, T. & Hughes, T. L. Quantiza-tion of fractional corner charge in C n -symmetric higher-order topological crystalline insulators. Phys. Rev. B , 245151 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.99.245151 .[29] Lee, E., Kim, R., Ahn, J. & Yang, B.-J. Higher-order band topology and corner charges in monolayergraphdiyne. arXiv preprint arXiv:1904.11452 (2019).[30] Sheng, X.-L. et al. Two-dimensional second-ordertopological insulator in graphdiyne. arXiv preprintarXiv:1904.09985 (2019).[31] Schindler, F. et al.
Fractional corner charges in spin-orbit coupled crystals. arXiv preprint arXiv:1907.10607 (2019).[32] Petrides, I. & Zilberberg, O. Higher-order topolog-ical insulators, topological pumps and the quantumHall effect in high dimensions.
Phys. Rev. Research ,022049 (2020). URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.022049 . Publisher: AmericanPhysical Society.[33] von Neumann, J. & Wigner, E. ¨Uber merkw¨urdigediskrete eigenwerte. Phys. Z. , 465 (1929).[34] Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. Soljaˇci´c, M. Bound states in the continuum. Nat. Rev.Mater. , 16048 (2016). URL .[35] Friedrich, H. & Wintgen, D. Interfering resonancesand bound states in the continuum. Phys. Rev. A ,3231–3242 (1985). URL https://link.aps.org/doi/10.1103/PhysRevA.32.3231 .[36] Plotnik, Y. et al. Experimental Observation of Opti-cal Bound States in the Continuum.
Phys. Rev. Lett. , 183901 (2011). URL https://link.aps.org/doi/10.1103/PhysRevLett.107.183901 .[37] Weimann, S. et al.
Compact Surface Fano States Em-bedded in the Continuum of Waveguide Arrays.
Phys.Rev. Lett. , 240403 (2013). URL https://link.aps.org/doi/10.1103/PhysRevLett.111.240403 .[38] Hsu, C. W. et al.
Observation of trapped lightwithin the radiation continuum.
Nature , 188–191(2013). URL .[39] Zhou, H. et al.
Perfect single-sided radiation andabsorption without mirrors.
Optica , 1079–1086(2016). URL .[40] Cerjan, A., Hsu, C. W. & Rechtsman, M. C. BoundStates in the Continuum through Environmental Design. Phys. Rev. Lett. , 023902 (2019). URL https://link.aps.org/doi/10.1103/PhysRevLett.123.023902 .[41] Benalcazar, W. A. & Cerjan, A. Bound states in thecontinuum of higher-order topological insulators.
Phys.Rev. B , 161116 (2020). URL https://link.aps.org/doi/10.1103/PhysRevB.101.161116 .[42] Joannopoulos, J. D., Johnson, S. G., Winn, J. N. &Meade, R. D.
Photonic Crystals: Molding the Flowof Light (Second Edition) (Princeton University Press,2011).[43] Men, H., Lee, K. Y. K., Freund, R. M., Peraire,J. & Johnson, S. G. Robust topology opti-mization of three-dimensional photonic-crystal band-gap structures.
Opt. Express , 22632 (2014).URL .[44] Cerjan, A. & Fan, S. Complete photonic band gapsin supercell photonic crystals. Phys. Rev. A ,051802 (2017). URL https://link.aps.org/doi/10.1103/PhysRevA.96.051802 .[45] Peterson, C. W., Benalcazar, W. A., Hughes, T. L. &Bahl, G. A quantized microwave quadrupole insulatorwith topologically protected corner states. Nature ,346 EP – (2018). URL http://dx.doi.org/10.1038/nature25777 .[46] Noh, J. et al.
Topological protection of photonic mid-gapdefect modes.
Nature Photonics (2018). URL https://doi.org/10.1038/s41566-018-0179-3 .[47] Mittal, S. et al.
Photonic quadrupole topological phases.
Nature Photonics , 692–696 (2019). URL . Num-ber: 10 Publisher: Nature Publishing Group.[48] Li, M. et al. Higher-order topological states in pho-tonic kagome crystals with long-range interactions.
Na-ture Photonics , 89–94 (2020). URL . Number: 2Publisher: Nature Publishing Group.[49] Serra-Garcia, M. et al. Observation of a phononicquadrupole topological insulator.
Nature , 342–345 (2018).[50] Ni, X., Weiner, M., Al`u, A. & Khanikaev, A. B. Observa-tion of higher-order topological acoustic states protectedby generalized chiral symmetry.
Nature Materials ,113 (2019). URL .[51] Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang,B. Acoustic higher-order topological insulator ona kagome lattice. Nature Materials , 108–112(2019). URL . Number: 2 Publisher: Nature Pub-lishing Group.[52] Xue, H. et al. Realization of an Acoustic Third-Order Topological Insulator.
Phys. Rev. Lett. ,244301 (2019). URL https://link.aps.org/doi/10.1103/PhysRevLett.122.244301 . Publisher: AmericanPhysical Society.[53] Ni, X., Li, M., Weiner, M., Al`u, A. & Khanikaev,A. B. Demonstration of a quantized acoustic oc-tupole topological insulator. arXiv:1911.06469 [cond-mat, physics:physics] (2019). URL http://arxiv.org/abs/1911.06469 . ArXiv: 1911.06469.[54] Xue, H. et al.
Observation of an acoustic octupoletopological insulator.
Nature Communications ,2442 (2020). URL . Number: 1 Publisher: NaturePublishing Group.[55] Imhof, S. et al. Topolectrical-circuit realization oftopological corner modes.
Nature Physics , 925–929 (2018). URL .[56] Bao, J. et al. Topoelectrical circuit octupole insulatorwith topologically protected corner states.
Phys. Rev.B , 201406 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.100.201406 . Publisher: Ameri-can Physical Society.[57] Kempkes, S. N. et al.
Robust zero-energy modes inan electronic higher-order topological insulator.
Na-ture Materials , 1292–1297 (2019). URL . Num-ber: 12 Publisher: Nature Publishing Group.[58] Davis, K. M., Miura, K., Sugimoto, N. & Hi-rao, K. Writing waveguides in glass with a fem-tosecond laser. Opt. Lett. , 1729–1731 (1996).URL . Publisher: Optical Society ofAmerica.[59] Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B: At. Mol.Opt. Phys. , 163001 (2010). URL http://stacks.iop.org/0953-4075/43/i=16/a=163001 .[60] Liu, F. & Wakabayashi, K. Novel Topological Phasewith a Zero Berry Curvature. Phys. Rev. Lett. ,076803 (2017). URL http://link.aps.org/doi/10.1103/PhysRevLett.118.076803 .[61] Chen, Z.-G., Xu, C., Al Jahdali, R., Mei, J. & Wu, Y.Corner states in a second-order acoustic topological in-sulator as bound states in the continuum.
Phys. Rev. B , 075120 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.100.075120 .[62] Freymann, G. v. et al.
Three-Dimensional Nanostruc-tures for Photonics.
Advanced Functional Materials , 1038–1052 (2010). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/adfm.200901838 .63] B¨uckmann, T. et al. Tailored 3D Mechanical Meta-materials Made by Dip-in Direct-Laser-Writing OpticalLithography.
Advanced Materials , 2710–2714 (2012).URL https://onlinelibrary.wiley.com/doi/abs/10.1002/adma.201200584 .[64] Norris, D. J., Arlinghaus, E. G., Meng, L., Heiny, R.& Scriven, L. E. Opaline Photonic Crystals: How DoesSelf-Assembly Work? Advanced Materials , 1393–1399(2004). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/adma.200400455 .[65] v. Ignatowsky, W. Reflexion elektromagnetisches Wellenan einem Draht. Ann. Phys. , 495–522 (1905). URL http://onlinelibrary.wiley.com/doi/10.1002/andp.19053231305/abstract .[66] `Eidus, D. M. On the principle of limiting absorption.
Mat. Sb. (N.S.) , 13–44 (1962).[67] Schulenberger, J. R. & Wilcox, C. H. The limiting ab-sorption principle and spectral theory for steady-statewave propagation in inhomogeneous anisotropic media.
Arch. Rational Mech. Anal. , 46–65 (1971). URL http://link.springer.com/article/10.1007/BF00250177 .[68] Cerjan, A. & Stone, A. D. Why the laser linewidth isso narrow: a modern perspective. Phys. Scr. , 013003(2016). URL http://stacks.iop.org/1402-4896/91/i=1/a=013003http://stacks.iop.org/1402-4896/91/i=1/a=013003