Topological Protection in Disordered Photonic Multilayers and Transmission Lines
TTopological Protection in Disordered Photonic Multilayers and Transmission Lines
D. M. Whittaker and R. Ellis
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK. (Dated: February 9, 2021)The Su-Schrieffer-Heeger (SSH) model is the simplest example of a lattice with non-trivial topol-ogy. It supports mid-gap topologically protected states, whose energies are unaffected by disorder.We show that photonic multilayer structures provide an exact implementation of an SSH lattice,provided each layer has the same propagation thickness. From this, it follows that the cavity modein a conventional semiconductor microcavity is a protected SSH mid-gap state. We demonstrate thisexperimentally using controlled disorder in a mathematically equivalent system, a radio frequencytransmission line made from sections of coaxial cable with high and low impedances. We also showtheoretically that transmission lines connected to form networks map onto topologically interestinglattices in in higher dimensions.
The Su-Schrieffer-Heeger (SSH) model[1] is well knownas the ‘toy’ system used to introduce the ideas of topol-ogy in solid state physics. It consists of a one dimensionalchain of sites coupled by alternating weak and strongbonds. The SSH model has chiral symmetry, meaningthat its spectrum is exactly symmetric when reflectedabout the middle of the band-gap. This symmetry isa prerequisite for non-trivial topology in one dimension.It provides topological protection for mid-gap states, sotheir energies are unaffected by variations in the param-eters of the system.Although the SSH model has its origins in poly-mers, there have been numerous attempts to constructphotonic[2–6], acoustic[7, 8] and cold-atom[9] systems inorder to implement the topological physics it describes.They all share the idea of creating a one-dimensionalchain of resonators with alternating strong and weak cou-plings. However, this is not in itself enough to ensure chi-ral symmetry; it can be broken by effects such as on-sitepotentials, second nearest neighbour hopping and cou-pling to states not included in the manifold described by
FIG. 1. (a) Schematic of part of an SSH lattice, showingthe intra- and inter-cell hopping amplitudes, v and w . Asingle period, of length d, consists of two sites. (b) Bandstructure for the case v = 1 / w = 3 /
4. The band structureis symmetric about the dashed line at ε = 0, the chiral point. the model. Such effects are unavoidable in these complexsystems, so they do not exhibit exact chiral symmetry,and thus do not have the topology of the SSH model.We show that, by contrast, there is a class of very simpleone-dimensional systems, described mathematically bytransfer matrices, which possess exact chiral symmetryand map perfectly onto SSH chains. We consider two ex-amples: multilayer photonic structures, where the layershave different refractive indicies, and transmission lineswith sections of different impedances. These structureshave chiral symmetry if the propagation length of eachlayer, or line section, is identical.A consequence of this mapping is that in semiconduc-tor microcavity structures[10, 11], the cavity mode is anSSH mid-gap state. Disorder can be introduced intothese structures, without breaking the chiral symmetry,by varying the refractive indicies and layer thicknesses ina way which leaves the propagation lengths unchanged.They thus provide an ideal experimental platform fordemonstrating the topological protection which is centralto the SSH model. For the photonic case, we show numer-ically that when disorder is introduced in this way, theenergy of the cavity mode is unaffected. We also demon-strate the protection experimentally in a mathematicallyanalogous system of coaxial cables[14–16], where an en-semble of disordered structures can be constructed veryeasily by swapping in cables of different impedances.It is important to distinguish the spectral chiral sym-metry considered here from the spatial inversion symme-try possessed by all periodic bilayer structures. Inver-sion symmetry has been demonstrated to have topolog-ical consequences[12], which have been discussed in thecontext of photonic multilayers by Xiao et al[13], whoshow that it can be used to determine the conditions forfinding a localised mode at an interface. However, this isnot the topology of the SSH model. That requires chiralsymmetry, which can also be found in multilayers, butonly if the layers have the same propagation lengths. Aswe show, chiral symmetry can be maintained in the pres-ence of disorder, while inversion symmetry is inevitablybroken, so is not associated with topological protection.The basic SSH model consists of a one dimensionalchain of sites, all with the same energy. A particle canhop between adjacent sites, with hopping amplitudes al-ternating along the chain, as in Fig.1(a). The system a r X i v : . [ phy s i c s . op ti c s ] F e b FIG. 2. Band structure of a Bragg stack. The schematicillustrates a section of the stack, which has a period of twolayers with refractive indicies n = 3 and n = 1. The layerthicknesses satisfy the Bragg condition n d = n d , givingchiral symmetry. The band structure is periodic in frequency,with period 2 πc/ ( n d + n d ). The dashed lines show thechiral points; the bandstructure is exactly symmetric whenreflected about these lines. has two energy bands, shown in Fig.1(b), and also twodistinct topologies, depending on which of the inter- andintra-cell hopping amplitudes is greater. An importantproperty of the model is that the spectrum possesseschiral symmetry; it is exactly symmetric when reflectedabout the middle of the band-gap, which we will callthe chiral point. The chiral symmetry is not restrictedto the simplest case of alternating hopping amplitudes;it is present in any chain where there are only nearestneighbour hoppings, and the on-site energies are all thesame. In such systems, energy states generally occur inpairs, which map onto each other when reflected aboutthe chiral point. However, there are also special states,found at the chiral point, which map onto themselves,and are thus unpaired. In the SSH model, these are mid-gap states. They can occur at the end of a chain, orat an interface between two domains of different topol-ogy, created by repeating one of the hoppings in adjacentlinks. When the system parameters are varied, perhapsdue to disorder, the energy of these states cannot change:if an unpaired state were to move from the chiral point, itwould violate the chiral symmetry. Such states are saidto be topologically protected. PHOTONIC MULTILAYERS
The photonic structures considered here consist of aone dimensional series of layers, characterised by a refrac-tive index n and thickness d . The simplest such struc-ture, the Bragg stack, is a periodically repeated bilayer,which has the bandstructure shown in Fig.2. By intro- ducing defects in the periodicity, such as repeating layers,localised states can be created in the band gaps.The simplest way to calculate the properties of such alayered structures is to use the transfer matrix formalism.The transfer matrix for a layer relates the amplitudes ofthe electric and magnetic fields, E and B , on either sideof the layer. It is given by (cid:18) E (cid:48) B (cid:48) (cid:19) = (cid:18) cos ( ωd/c ) in − sin (( ωd/c )) in sin (( ωd/c )) cos ( ωd/c ) (cid:19) (cid:18) EB (cid:19) (1)where ω is the angular frequency, d = nd is the propa-gation length of the layer and c is the speed of light invacuum. Multiplying matrices for individual layers givesthe transfer matrix for a structure, which can be used tocalculate spectral properties such as transmission. Thespectral symmetries of the transfer matrix follow directlyfrom the periodicity and symmetries of the trigonometricfunctions in it. Apart from trivial phases, it is periodic infrequency, with period πc/d , and symmetric about valuesof ω which are integer multiples of πc/ (2 d ). If d is thesame for every layer, the transfer matrix for the wholestructure will display these symmetries.We next demonstrate that such chiral multilayers formexact implementations of an SSH chain. Consider astructure where each layer has the same propagationlength, d . Defining the fields E i and B i at the i th in-terface, the transfer matrices can be used to eliminatethe B i , giving a relationship between the E i : n i − ,i E i − + n i,i +1 E i +1 = ( n i − ,i + n i,i +1 ) cos ( ωd/c ) E i (2)Here, n i,i +1 is the refractive index of the layer betweenthe i th and ( i + 1) th interface. If we now define an ‘en-ergy’ ε = cos ( ωd/c ) and scaled fields ˜ E i = ( n i − ,i + n i,i +1 ) / E i , this becomes a tight-binding model, t i − ,i ˜ E i − + t i,i +1 ˜ E i +1 = ε ˜ E i , (3)where the hopping amplitude is t i,i +1 = n i,i +1 [( n i − ,i + n i,i +1 )( n i,i +1 + n i +1 ,i +2 )] / . (4)This tight binding model has the form of an SSH chain:there are only nearest-neighbour hoppings and the on-siteenergies are all zero.When applied to the case of a conventional bilayerBragg stack, Eq.(4) gives two values for the hopping am-plitudes, n / ( n + n ) and n / ( n + n ), for the layerswith index n and n , respectively. The tight binding sys-tem is then identical to the simplest SSH model. If thelayer propagation lengths are commensurate rather thanidentical, it is necessary to divide the layers into sub-unitsof equal thickness. The tight binding model then hasmore than two layers per unit cell, corresponding to gen-eralised SSH systems, such as the SSH-4 model[17, 18].A conventional microcavity structure is made by join-ing two Bragg stacks with the same termination, givinga defect with twice the thickness of a normal layer. Thedoubled layer corresponds to two hops with the sameamplitude, so this is an exact experimental realisation ofthe interface between SSH domains of different topology.The cavity mode is found at the mid-gap chiral point,and corresponds to the topologically protected interfacestate. This means that its frequency should be unaffectedby disorder consisting of variations in the refractive indi-cies of the layers of the structure. However, chiral sym-metry needs to be maintained, so the propagation length, d = nd , of each layer must be kept constant, by adjustingits thickness when the refractive index is changed.In Fig.3, we show transmission spectra, calculated us-ing the transfer matrix method, which demonstrate topo-logical protection for a microcavity in which the refrac-tive index of each layer is varied randomly, adjusting thethickness to maintain the chiral condition. We considertwo structures, with and without a chiral symmetry pointin the lowest gap. The first (Fig.3(a)) is a conventionalcavity satisfying the Bragg condition n d = n d , giv-ing a chiral point in the middle of the gap, so the cavitymode is topologically protected. The second structure(Fig.3(b)) is more complicated. The transmission thick-ness of a period, n d + n d is constant, but the ratio ofhigh to low index material is reversed at the interface: onone side n d = 3 n d , while on the other n d = 3 n d .These mirrors have identical bandstructures and the in-terface supports a cavity mode in the middle of everygap. However, the structure maps onto an SSH-4 model,so there is no chiral point in the first gap, and no topolog-ical protection. The difference is evident in the spectra:for the chiral case, there is absolutely no change in theenergy of the cavity peak, while without the protection,each instance of the disorder gives a distinct peak. InFig.3(c), we plot the electic field profile of the mode forthe chiral case. There is a different profile for each in-stance of the disorder, but in all cases the field goes tozero at alternate interfaces. This relates to the form ofthe protected state in SSH systems, which has zero am-plitude on alternate sites.The main factor determining how closely exact chiralsymmetry can be approached is likely to be the accuracyof the fabrication, ensuring that the propagation lengthof each layer is the same. The description in terms ofa constant, real refractive index also needs to be consid-ered. The frequency dispersion of the index is not reallyan issue, because the topological protection only requiresthe matching to be exact at the chiral point. Absorption,corresponding to a complex refractive index n + iκ , canbreak the chiral symmetry. However, it produces frac-tional energy shifts ∼ ( κ/n ) , which can be made verysmall with appropriate material choices. Furthermore, amode of quality factor Q , without absorption, will alsobe weakened to the point where it becomes unobservablewhen κ/n ∼ /Q , so the maximum detectable shift dueto absorption will be of order of the linewidth divided by Q . Q values of thousands are achievable in semiconduc-tor microcavities, so this is a small effect. FIG. 3. Topological protection for a chiral multilayer photonicstructure. (a) and (b) show calculated transmission spectra,in the frequency range around the middle of the first band-gap, for a cavities created by joining two disordered Braggmirrors. Each mirror consists of five periods, terminated atthe interface with a high index layer. The structures are de-signed (see text) so that both have a mid-gap cavity mode; in(a) this corresponds to a chiral point, making the mode topo-logically protected, while in (b) the chiral point is in a highergap, so there is no protection. The colours correspond to teninstances of disorder, in which the refractive index in eachlayer is varied randomly by ± n = 3 and n = 1, while the layer thickness is adjustedto keep the propagation length d = nd constant. The blackcurves are reference spectra without disorder. (c) shows elec-tric field profiles for the protected mode in (a), plotted as afunction of the optical depth, so the layer interfaces, shownin the black and white bar, coincide for all structures. TRANSMISSION LINE STRUCTURES
The chiral and topological properties discussed abovefollow from the form of the transfer matrices which de-scribe the structures. Similar mathematics occurs inother situations where waves are one-dimensional, suchas acoustic waves in multilayers and waves on strings.Hence, it should be possible to find the physics of topo-logical protection in these systems. We shall considerthe propagation of radio-frequency signals in coaxial ca-ble transmission lines. For an ideal, lossless, cable, thetransfer matrix relates the voltage, V , and current, I ateither end, according to (cid:18) V (cid:48) I (cid:48) (cid:19) = (cid:18) cos ( ωd/v ) iZ sin ( ωd/v ) iZ − sin ( ωd/v ) cos ( ωd/v ) (cid:19) (cid:18) VI (cid:19) , (5)where the impedance Z = (cid:112) L/C , and the propagationvelocity v = 1 / √ LC , with L and C the inductance andcapacitance per unit length. This has the same form asEq.(1), so exact analogues of photonic multilayer struc- FIG. 4. Experimental transmittance spectra for cavity struc-tures made from coaxial cables. (a) spectra for structureswithout disorder, shown schematically in the inset. Each hastwo chiral mirrors, made from five repeats of a pair of cableswith impedances 50Ω (white) and 93Ω (black), surroundinga 50Ω section defining the cavity. The physical lengths of the50Ω and 93Ω cables in the mirrors are 1.0m and 1.24m re-spectively, giving the same propagation lengths. Three cavitylengths, d c , are used: for d c = 2 . d c = 0 .
5m and 1.5m, it is broken.The band-gap corresponds to the strong dip in transmission,between about 40 and 60 MHz, containing the peak due tothe cavity mode. (b)-(d) show normalised spectra, in theregion of this mode, for eight different structures with dis-order, where half of the 93Ω sections are selected randomlyand replaced with 75Ω cables. Identical mirrors are used witheach cavity, indicated by the line colours. In the chiral case,(c), the disorder has very little effect on the mode frequency,demonstrating that it is topologically protected tures can be constructed by joining sections of transms-sion line with different impedances[14–16].Transmission lines made from coaxial cable sectionswith the same propagation length d = dc/v , are a conve-nient system for demonstrating the topological protectiondiscussed above. It is straightforward to make an ensem-ble of instances of disorder by swapping in sections of ca-ble with different impedances. Transmittance spectra forcavities constructed in this way are shown in Fig.4. Weconsider three cavity structures: in each case, the mirrorsconsist of a repeated chiral bilayer, but in one the chiralsymmetry is maintained as the cavity is formed by re-peating one of the low impedance layers ( d c = 2 . d , of all the cables is the same, our result, Eq.(2),generalises to (cid:88) j Z − ij V j = (cid:88) j Z − ij cos ( ωd/c ) V i (6)where the sums are over the nearest neighbours of site i , to which it is directly connected. A similar resulthas been derived by Zhang and Sheng[19], for microwavewaveguide networks. With the equivalent rescaling andidentification of ε = cos ( ωd/c ), this again looks like atight binding model. Provided any loops have an evennumber of nodes, this network will have chiral symme-try, and so potentially non-trivial topology. In such anetwork the spatial positioning of sites is unimportant,the only relevant consideration being how they are con-nected. Hence lattices can be fabricated which are impos-sible under the physical constraints of three dimensionalspace[20, 21] and Euclidean geometry[22]. Transmissionline structures can also be seen as a bridge between pho-tonic systems and topological circuits[23, 24]. It is natu-ral to combine transmission lines with discrete electron-ics located at the nodes, providing, for example, gain andloss or nonlinearity. This would extend the range of mod-els which can be studied to include PT symmetric[25] andnonlinear systems.To conclude, we have shown that commensurate pho-tonic multilayers provide exact implementations of SSHchains, with perfect chiral symmetry. A Bragg stack,with equal propagation lengths for the layers, corre-sponds to the basic SSH model, and conventional mi-crocavity modes are the interface states between regionsof different topology. 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