Since the discovery of the topological states of light by Haldane and Raghu over a decade ago,1^,2 topological photonics has occupied the center stage of modern electromagnetics. This flurry of attention is motivated by the extraordinary properties of topological materials: they do not support light states in the bulk region, but when terminated with an opaque boundary, they may support unidirectional edge states. The edge states are immune to backscattering and their propagation is impervious to small defects and imperfections of the interface. The topologically protected transport of light paved the way for unique and exciting applications in photonics.1^–23

The net number of edge modes in a topological system can be determined using the bulk-edge correspondence principle.3^,24^–26 Typically, the topological classification of photonic materials is done through the calculation of the Chern invariants, which are determined from the photonic band structure of the material and the Bloch eigenstates.1^,2 The calculation of the invariants usually requires the system to be periodic, so that the wave vector space is a closed surface with no boundary.27 This is not the case of systems with a continuous translational symmetry, e.g., bulk materials modeled as continuous media, for which the wave vector lies in the unbounded Euclidean plane. Notably, some time ago it was shown that arbitrary bianisotropic materials or waveguides with a continuous translational symmetry can be topologically classified, provided the material response has a wave vector cutoff.27 The cutoff may already be embedded in the physical response of the materials due to the nonlocality arising from charge diffusion,28^–30 or other effects. Alternatively, a synthetic spatial cutoff can be enforced by inserting a small air gap in between the relevant materials.25^,27 Without a suitable wave vector cutoff, the topology of a system with a continuous translational symmetry is ill-defined, and thereby the bulk-edge correspondence is inapplicable.25^,31

It was theoretically shown in Ref. 22 that the violation of the bulk-edge correspondence creates the conditions to abruptly halt the edge mode propagation at a singular point. Thereby, ill-defined topologies create unique opportunities and may lead to unique physical phenomena, such as the concentration of an incoming wave at a single point with a massive field enhancement.22^,32^–39 Electromagnetic energy sinks are expected to be useful to enhance nonlinear effects40^–46 and for energy harvesting.47^,48

Here, we investigate the propagation of topologically protected edge modes at the interface between magnetized ferrite and a metallic wall. We numerically and experimentally verify that separating the ferrite and the metallic wall with a small air gap effectively imitates a spatial cutoff, and thereby regularizes the ferrite topology. In particular, for the waveguide with the air gap it is possible to correctly predict the emergence of a unidirectional edge state using the bulk-edge correspondence. On the other hand, without the air gap, the ferrite topology is ill-defined, and it turns out that when it is paired with the metal wall the system does not support edge states. Taking advantage of this peculiar property, we experimentally demonstrate the formation of an electromagnetic energy sink at the junction of materials with well-defined and ill-defined topologies. The time dependence of the electromagnetic fields is assumed to be of the type e−iωt, with ω being the oscillation frequency.

Here, we study the edge modes supported by a waveguide formed by a magnetically biased ferrite and a metal. The two materials are separated by a small air gap with thickness d, as shown in Fig. 1(a). For now, it is assumed that the structure is uniform along the z direction.

The magnetically biased ferrite is characterized by a relative permittivity εF=εF′(1+itanδ), with tanδ being the electric loss tangent of the ferrite, and a frequency-dependent relative permeability tensor of the form μ‾=(μ11μ120μ21μ220001), with μ11=μ22=1+ωg(ωl−iωα)(ωl−iωα)2−ω2 and μ12=μ21=−iωωg/((ωl−iωα)2−ω2), where α is a damping factor that determines the resonance linewidth.49 Here, ωl=γBz is the Larmor precession frequency that depends on the static magnetic bias B0=+Bzz^. The gyromagnetic ratio is given by γ=e/me=1.759×1011C/kg, with e being the elementary charge and me the electron mass. The Larmor frequency for the magnetization is given by ωg=μ0Msγ, where Ms is the saturation magnetization. We consider a soft ferrite50 with permittivity εF′=14.52 and electric loss tangent tanδ=8×10−5 that is magnetized on average by a static field Bz=0.16T. The corresponding Larmor frequencies are ωl/(2π)=4.43GHz and ωg/(2π)=5.18GHz. The damping factor is equal to α=0.002. The biasing field is created by a permanent bonded neodymium magnet.51

The transverse electric (TE) polarized ferrite bulk modes, with Ez≠0, Hz=0, and ∂/∂z=0, have a frequency dispersion determined by εFμef(ω/c)2=k2, where c is the speed of light in a vacuum, k2=k·k with k being the wave vector, and μef=(μ112+μ122)/μ11. The band diagram of the ferrite in the limit of vanishing loss is depicted in Fig. 1(b). As seen, there is a bandgap defined by the frequency interval ωL<ω<ωH, with ωL=ωlωg/ωl+1 (ωL=2π×6.53GHz) and ωH=ωl+ωg (ωH=2π×9.61GHz). On the other hand, the metal is taken as aluminum, which in the microwave range can be regarded as a perfect electric conductor (PEC). Thus, the metal shares the same bandgap as the ferrite.

When two inequivalent topological materials share a common bandgap, their interface supports unidirectional edge states topologically protected against backscattering. The net number of edge modes is determined by the difference of the topological charges in the material.3^,24^–26 As already discussed in the introduction, systems with a continuous translation symmetry have an ill-defined topology. The topology can be regularized with a high-frequency cutoff kmax that ensures that when k→∞ the material response becomes reciprocal.27^,52 It should be noted that the suppression of the material response for k→∞ is not a mere mathematical trick but is rather a consequence of the Kramers–Kronig relations for spatially dispersive media and of the Riemman–Lebesgue lemma.53 Hence, in principle the realistic response of any material (e.g., the ferrite) has in it some intrinsic physical cutoff that is not captured by the local model description. Thus, introducing by hand the cutoff kmax in the local model description is a simple remedy to reproduce the actual (unknown) physical cutoff.

Using the topological band theory,27^,52^,54 it may be shown that the gap Chern number of the ferrite is Cgap,Fe=+1 for a finite kmax. The Chern number is robust to the presence of significant material loss.55^,56 In contrast, without the cutoff the gap Chern number is Cgap,Fe|local=sgn(ωg)(1+|ωgωl|)−1/2≈0.68,27 which evidently is not an integer. Thus, the spatial cutoff is essential to regularize the topology of the material, which otherwise is ill-defined.

The regularization concept can be geometrically explained as follows. Figure 2(a) depicts a torus with a vanishing inner radius. Any cross section of the torus that contains the symmetry axis consists of two kissing circles. It is evident that the surface is not differentiable at the central point. Furthermore, the number of holes (genus) is ill-defined because the two kissing circles touch a single point. Thus, the considered surface has an ill-defined topology. This problem can be fixed with an infinitesimally small perturbation of the original object. For example, by increasing the inner radius of the object in Fig. 2(a) by an infinitesimal amount, the resulting surface becomes topologically equivalent to a torus [Fig. 2(b)] with genus g=1. Alternatively, the top and bottom caps can be separated by an infinitesimal distance near the singular point, yielding an object topologically equivalent to a sphere [Fig. 2(c)] with genus g=0. Thus, even infinitesimally small perturbations of the original system can lead to a well-defined surface topology. Similarly, the topology of the ferrite becomes well-defined for an arbitrarily small value of 1/kmax. Curiously, the described geometrical ideas also highlight that the regularized topology may depend on the regularization procedure and on the type of high-frequency cutoff. An example that illustrates this property for the case of a magnetized electric plasma photonic crystal can be found in Ref. 57.

The gap Chern number of the metal vanishes due to reciprocity and therefore the metal is topologically trivial. Hence, from the bulk-edge correspondence principle, it is expected that the metal-ferrite waveguide supports a single gapless unidirectional guided mode (δC=|Cgap,Fe− Cgap,Al|=1).

Figure 1(c) depicts the calculated edge mode dispersion for different values of kmax (dashed curves). The spatial dispersion arising due to the spatial cutoff is rigorously taken into account in the simulation using the same methods as in Ref. 25. As seen, in agreement with the bulk-edge correspondence principle, the material interface supports exactly a gapless unidirectional edge-state, for any finite value of kmax (without an air gap in between the two materials).

The idealized kmax-cutoff can be imitated with a “synthetic cutoff” that is implemented in practice by adjoining a small air layer to the relevant ferrite interface. The equivalent cutoff is related to the thickness d as kmax∼1/d.25 Thus, with the synthetic cut-off, the metal and the ferrite become separated by an air gap. The rationale to add the air gap is the following. For waves with a small k, the air gap has little impact on the wave propagation, and the structure behaves similarly to a metal plate adjacent to a ferrite. On the other hand, for large values of k the response of a magnetic gyrotropic material with a spatial wave vector cutoff is similar to that of “air” [μ¯(ω,k)≍1] and this property is captured by the presence of the air gap. Note that the air gap thickness simply controls the transition between the two regimes. As seen in Fig. 1(c) [solid lines, also repeated in Fig. 1(b)], the synthetic cutoff imitates fairly well the idealized kmax-cutoff, especially near the top edge of the bandgap. The dispersion equation of the edge modes with the synthetic cutoff is derived in the Appendix.

As seen in Fig. 1(b), for any nonzero d, the ferrite-metal waveguide supports a single unidirectional edge state in most of the gap, consistent with the bulk-edge correspondence. However, different from Fig. 1(c) (dashed lines), with the synthetic cutoff the edge mode is not strictly gapless. In fact, the dispersion of the edge state may not span the entire gap ωL<ω<ωH for d≠0, and when it does there is a second edge mode for frequencies near ωL [see blue curve in Fig. 1(b)]. Hence, even though the synthetic cutoff regularization reproduces the main features of the idealized kmax-cutoff, the analogy is not strictly perfect.

For small values of d, the group velocity of the edge mode is negative (vg,x=∂ωk/∂kx<0) in most of the bandgap, so that the energy flows along the −x direction. Furthermore, for a small d the edge mode in most of the gap is associated with a positive kx and thereby is typically a backward wave. Interestingly, as d→0 the group velocity approaches zero and the edge mode propagation is exactly suppressed when the gap is removed. This property may be attributed to the ill-defined topology of the ferrite without a spatial cutoff. Note that the same type of behavior occurs when 1/kmax→0 [see Fig. 1(c)].

The results of the previous section show that the synthetic cutoff restores (at least partially) the bulk-edge correspondence and ensures that the waveguide supports a unidirectional edge mode. Here, we report the experimental verification of that property. Our prototype consists of a hollow aluminum waveguide partially filled with a magnetically biased ferrite block, as shown in Fig. 3(a). The magnetic bias is created by a permanent magnet placed at the back of the waveguide [not visible in Fig. 3(a)]. The ferrite block has the dimensions Lx=150mm, Ly=50mm, and Lz=10mm. The synthetic cutoff is implemented with an air gap thickness d=5mm. In the first part of the experiment, the air gaps in between the lateral (left and right) walls of the metallic guide and the ferrite block are filled with an absorber (Eccosorb LS-26).

The edge mode is excited with small monopole antennas oriented along the z direction in the middle of the air gap of the bottom lateral wall. The monopoles are fed by SMA cables, denoted in Fig. 3(a) by ports 1 and 2. The top and bottom walls (z=const.) are metallic plates that exactly emulate the two-dimensional scenario studied in Sec. 2, with E=Ezz^ and ∂/∂z=0.

The theoretical dispersion of the edge mode is given by the blue curve in Fig. 1(b) for d=5mm. To verify the edge mode propagation, we measured the scattering parameters in the bandgap of the ferrite. The results are depicted in Fig. 3(b) and reveal a considerable asymmetry in the scattering parameters, with the transmission from port 1 to port 2 being significantly stronger than the transmission from port 2 to port 1. The reflection parameters S11 and S22 (not shown) are similar for both ports. Note that the transmission level is generally very low because of the large mismatch between the input impedance of the monopole antennas and the impedance of the feeding lines. As neither the ferrite nor the metal supports bulk states in the bandgap, the enhanced transmission level S21 in the bandgap must be due to the excitation of the edge mode that flows along the −x direction, which is the only possible radiation channel.

We measured the spatial distribution of the edge mode electric field using a near-field scanner. The near-field scanner is based on a robotic arm that controls the position of a small metallic probe oriented along the z direction connected to a vector network analyzer (R&S ZVB20). In order to measure the electric field distribution of the edge mode, the top cover of the waveguide was replaced by a metallic plate drilled with an array of subwavelength circular holes. The holes are of 2 mm diameter and separated from each other by 1.5 mm in each direction. The subwavelength-sized holes allow for some leakage of the edge mode energy, and thereby the external characterization of the fields, without interfering significantly with the propagation inside the waveguide. We considered two frequencies inside the bandgap, f1=7GHz and f2=7.25GHz. The measurement apparatus is shown in Fig. 3(c). The electric field distributions for port 1 and port 2 excitations are shown in Figs. 4(a) and 4(b) and Figs. 4(c) and 4(d), respectively.

Figures 4(a) and 4(b) show that when the waveguide is excited at port 1 the radiation flows along the air gap toward port 2 for both frequencies. In contrast, when the waveguide is excited at port 2 no field propagates towards port 1. Thus, the near-field scan confirms the nonreciprocal transmission in the waveguide that is in agreement with the scattering matrix results of Fig. 3(b). The distance between the monopole antenna and the nearest lateral wall (an absorber) is 3 mm in the experiment. For this reason, the field measured above the perforated metallic plate has a negligible amplitude in Figs. 4(c) and 4(d). Moreover, since the measured electric field for a port 1 excitation is mostly concentrated in the ferrite–air–metal junction and propagates toward the −x direction, it is evidently associated with the unidirectional edge mode [see blue curve of Fig. 1(b)]. The measurements are characterized by some high-frequency noise because the fields above the perforated metal plate are rather weak away from the air-gap channel. Moreover, the oscillations in the field amplitude along the propagation path may be attributed to two factors: (1) the static magnetic field created by the magnets is not homogeneous in the ferrite block (the inhomogeneous biasing disturbs the permeability of the ferrite and thus modifies the propagation characteristics of the edge mode) and (2) the field above the perforated metal plate (where the measurement is done) is far less confined than inside the waveguide. The latter point is illustrated in Fig. 5(a), which depicts a snapshot of the measured electric field for a port 1 excitation at the frequency f2=7.25GHz. Indeed, above the top plate the edge mode spreads along the surface due to diffraction effects.

We also simulated the response of the ferrite-metal “edge waveguide” using the commercial software CST Studio Suite58 [Figs. 5(b) and 5(c)]. As seen in Fig. 5(b), the simulation results follow closely the experimental results depicted in Fig. 5(a). The simulation of Fig. 5(c) shows a time snapshot of the electric field inside the waveguide. In this case, the fields are strongly confined to the air gap near the bottom metallic wall. Furthermore, the full-wave results confirm that there is a unique unidirectional edge mode propagating in the waveguide, in agreement with the bulk-edge correspondence. The edge mode wavelength estimated from the experimental results is λg≈49.5mm, whereas the value obtained from the full-wave simulations and from the modal Eq. (2) is λg≈55mm. The difference between the two values is attributed to the lack of homogeneity of the bias magnetic field in the experiment. For completeness, we note that in the scenario of Fig. 5 the edge mode is a forward wave: the phase and group velocities are both directed along −x.

As previously discussed, the synthetic spatial cutoff determined by the air gap is essential to guarantee that the system supports a unidirectional edge mode. The air layer adjoined to the ferrite interface effectively regularizes the ferrite topology, ensuring that the bulk-edge correspondence is applicable in the spectral range of interest. On the other hand, when the air gap is removed the topology of the ferrite (described by the local-material model) becomes ill-defined and the waveguide does not support edge modes. Here it is worth mentioning again that the ferrite may have some unknown intrinsic physical cutoff that is not captured by its local dispersive model. In such a case, the intrinsic kmax of the ferrite is expected to be rather large (as compared to the light free-space wavenumber in the bandgap), as it is determined by diffusion effects or by the intrinsic periodicity of its crystal lattice. Since the characteristic size of any edge mode emerging due to an intrinsic nonlocality of the ferrite response is on the order of 1/kmax, the presence of any (infinitesimal) amount of loss would in practice immediately dissipate the edge mode, making this situation indistinguishable from an ill-defined topology. Thus, the ferrite may be regarded for all practical purposes as a material with an ill-defined topology. As shown next, this peculiar behavior creates the conditions for the formation of a topological singularity (energy sink) where the wave propagation is abruptly halted.22^,32^–39

To demonstrate the formation of an energy sink in the considered structure, we removed the absorber in the left-hand side of the ferrite block. The air gap that now separates the ferrite from the metal at this lateral interface creates an edge waveguide with a well-defined topology supporting a unidirectional edge mode. Thus, when the metallic waveguide is excited by the monopole antenna, the unidirectional guided mode can propagate around the sharp corner of the ferrite block without any backscattering and flow along the lateral left wall. Importantly, at the top lateral wall there is no air separation between the ferrite and the metal. Without the synthetic cutoff, the ill-defined topology of the ferrite reveals itself. In fact, the absence of edge states at the top lateral wall and the existence of an edge state in the left lateral wall is a clear violation of the bulk-edge correspondence. The point where the ferrite block meets the top sidewall corresponds to the transition between the well-defined topology and the ill-defined topology. The edge mode comes to an immediate halt at that point leading to the formation of an energy sink that dissipates all the incoming energy.

To experimentally verify these ideas, the structure was excited at port 2 for two frequencies inside the common bandgap of the materials, f1=7GHz and f2=7.25GHz. The electric field was measured at the surface of the perforated metal plate.

Figure 6 depicts the squared electric field distribution in a truncated region of the waveguide close to port 2. The experimental results demonstrate that the excited edge mode propagates without back-reflections around the sharp corner of the ferrite block, coming to a halt at the top corner of the structure. This can be seen in Fig. 6, which shows a large field concentration near the top sidewall of the waveguide. All the energy coupled to the edge mode is ultimately absorbed at the energy sink, leading to a massive field enhancement.22 Note that the field intensity near the sink is much stronger than in any other point in the lateral channel, being comparable to the field measured near the monopole antenna. In principle, the contrast between the field at the hotspot and the field in the lateral channel is even more pronounced inside the waveguide.

Nonreciprocal photonic systems with a continuous translation symmetry typically have an ill-defined topology. The scattering of unidirectional modes in such systems is not constrained by the bulk-edge correspondence. In this work, we experimentally verified that the topology of a ferrite described by a local material model can be effectively regularized with a synthetic wave vector cutoff. The cutoff is implemented by adjoining a small air layer to the ferrite interface. Furthermore, it was experimentally demonstrated that by pairing two waveguides, one formed by materials with a well-defined topology and another with an ill-defined topology, it is possible to bring an edge mode to an immediate halt at a topological singularity where the electromagnetic fields are massively enhanced. Our findings suggest that photonic systems with ill-defined topologies are unique platforms for extreme wave phenomena.

Here, we derive the modal equation of the edge modes that propagate in the waveguide depicted in Fig. 1(a). To do so, we write the electric field of the TE-waves as a superposition of decaying exponentials along the y direction, so that Ez(x,y,ω)=E0eikxx{e−γF(y−d),y>dsinh(γ0y)sinh(γ0d),0<y<d,(1)with kx being the propagation constant of the edge mode (along x). Note that Ez is continuous and vanishes at the metal wall (y=0). Here, the attenuation constants in the ferrite and air are γF=kx2−μefεF(ω/c)2 and γ0=kx2−(ω/c)2, respectively.

The magnetic field is obtained from the Maxwell equations using H=1iωμ0μ¯¯−1·∇×E. Enforcing the continuity of Hx=1iωμ01μ112+μ122(μ11∂yEz+μ12∂xEz) at y=d, we obtain the desired dispersion equation γF+μefγ0coth(γ0d)−μ12μ11ikx=0.(2)

David E. Fernandes received his MSc degree in 2010 and his PhD in 2015 from the University of Coimbra. Since 2016, he has been a post-doc researcher at Instituto de Telecomunicações, based in the Department of Electrical Engineering, University of Coimbra, where he is also an invited assistant professor. His research interests are mainly focused on metamaterials, nonreciprocal effects, and topological phenomena.

Ricardo A. M. Pereira received his integrated master’s deree in physics engineering from the University of Coimbra (UC) and worked as a researcher on the topics of simulation, modelling and control of metamaterials. Currently, he is a PhD student at the University of Aveiro, studying how to increase the beam efficiency of radiative wireless power transfer systems by using quasioptics for focusing microwave beams.

Sylvain Lannebère received his PhD in metamaterials from the University of Bordeaux in 2011. Since 2013, he has been a researcher at the Instituto de Telecomunicações of Coimbra, Portugal. His research interests include electromagnetism, metamaterials, nanophotonics, and topological materials.

Tiago A. Morgado received his “Licenciado” and his PhD degrees in electrical and computer engineering (specialty of telecommunications) from the University of Coimbra in 2007 and 2012, respectively. He is currently an assistant researcher within the FCT CEEC Individual 2017 at Instituto de Telecomunicações – Coimbra and an invited assistant professor in the Department of Electrical and Computer Engineering of the University of Coimbra. His current research interests concern electromagnetic metamaterials and their applications, plasmonics, and nanophotonics.

Mário G. Silveirinha is a professor at the University of Lisbon, Portugal and a senior researcher at Instituto de Telecomunicações. He is an IEEE Fellow, an OSA Fellow, and an APS Fellow. His research interests include electromagnetism, plasmonics and metamaterials, quantum optics, and topological effects.