1. Introduction

Lots of the macroscopic properties of bulk media can be understood through the interplay between atoms and waves. When dealing with optical waves, this microscopic interaction allows to explain their transparent or opaque nature for example. The latter is actually driven by two different mechanisms: the chemical composition of the medium and/or the spatial arrangement of the atoms. The interplay between the structure and the composition can obviously lead to more complex macroscopic properties. Understanding all of these microscopic phenomena could provide rich features, and it is interesting to mimic them by building analogs at different scales.

The first class of artificial media which permit to mimic the structural patterning of bulk media was introduced in 1987 [1,2], and they are known as photonic crystals [3]. These systems are the three-dimensional generalization of Bragg mirrors and generally consist in a periodic mixture of two different dielectric materials. Applying Floquet-Bloch theorem [4,5], one can retrieve the propagating properties of electromagnetic waves. Such an approach permits to build a parallel between the propagation of photons within the photonic crystals and the propagation of electrons in metals as it is performed in solid state physics [6]. Notably the emergence of non-propagating bands, or band-gaps, is analogous to the insulating nature of some bulk materials. This property occurs for frequencies where the free space wavelength roughly scales with the lattice period of the medium.

The second class of artificial media mostly appeared after the pioneering work of J. Pendry [7–11] and are known as metamaterials [12–14]. As for them, they mimic the interaction regarding the composition of the medium. They are made of subwavelength unit cells, each of them interacting with the incoming wave. The macroscopic response of the medium is described in terms of effective parameters which result from all the independent interactions. Among many other interesting features, negative permittivity [8], negative permeability [7,9] or even negative refractive index [11,15] have been reported. These media inherit their macroscopic properties from their subwavelength building blocks and are therefore inherently patterned at scales far smaller than the free space wavelength.

Throughout this review, we will make a bridge between those two classes of materials and play with both the structure and the composition. We therefore build what we call crystalline metamaterials in order to manipulate the waves. This strategy gives more degrees of freedom than each sub-category of materials in order to mimic topological phenomena for electromagnetic waves at a macroscopic scale.

In a first part, we will carefully describe the uniaxial wire medium which operates in the microwave frequency range and is made of parallel metallic wires. We will therefore describe as precisely as possible the experimental advantage of such a medium. After describing its intrinsic propagating properties, we will show the different degrees of freedom we have in order to tune the propagation properties.

In a second part, we will focus on a specific frequency range for which this medium inhibits the propagation. This is known as a hybridization bandgap. We will show how one can take advantage of this absence of propagating modes in order to induce near-field hoping between defects placed within it. By doing so, we will experimentally mimic the Hamiltonian which rules the propagation of electrons in graphene. We will end this part by showing perspectives and notably how it allows to experimentally explore more complex Hamiltonians.

In a third part, we will play directly on the properties of the propagating band. In this section we will demonstrate the possibility to build an analog of a Semenoff insulator [16]. We will take advantage of our experimental platform to map the Berry curvature. Eventually, we will exploit the bulk properties to create chiral edge states which enable routing capabilities.

The fourth and last part will be dedicated to an analog of a $\mathbb {Z}_2$ topological insulators based on the strategy proposed by Wu and Hu [17]. Typically, we will explain how one can create a pseudo-spin in our medium and therefore build topologically protected helical edge states.

2. Wire medium

In this part, we focus on the propagation medium which will be at the basis of all the experimental works presented in this review: the so-called wire medium. Our first use of such a medium goes back to 2010 [18] and detailed calculations can be found in [19].

2.1 Polariton-like dispersion relation

The typical medium under study is represented in Fig. 1(a). It consists of a parallel arrangement of metallic wires, with a lattice constant $a$ which is small compared to the free-space wavelength. If we considered a medium with infinite wires, we would observe very interesting propagating properties such as hyperbolic dispersion [20–22], but we will focus on wires with finite lengths, thus building a two-dimensional medium of parallel finite-length identical wires. This length is chosen to roughly match the half of the operating wavelength. In such a medium the plane cutting normally the wires at half their length is a symmetry plane. For some practical reasons and without the lack of generality, the half-wavelength-long wires can be replaced by quarter-wavelength wires soldiered to a metallic ground plane.

 

Fig. 1. a) The wire medium consists on a dense collection ($a\ll \lambda$) of half-wavelength-long parallel wires (If the wires are on a ground plane the length is reduced to $\lambda /4$). b) Dispersion relation of waves propagating in (a) following the model of the polariton, an avoiding crossing between the light line and the local resonance (red). c) The effective permittivity which also allows to describe the dispersion in the wiremedium.

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To describe the propagation of electromagnetic waves within such a medium one could start from the medium made of infinitely-long wires and add some boundaries as it was done step by step in [19]. But, a more intuitive way to understand what happens in the finite-length medium is to start by a single wire. Indeed, the latter acts as a resonant scatterer for electromagnetic waves with an electric field polarized along the wire axis. In quantum physics, the coupling between the continuum of propagating waves, ie. the photons, and a localized resonant state gives rise to a quasi-particle which is called a polariton [23]. More classically, this interaction is described as an avoiding crossing [24] between the light line and the resonance frequency, as sketched in Fig. 1(b). The resulting dispersion relation (black line) features all of the interesting characteristics needed in this review:

This toy-model, based on the analogy with the polariton, can be more rigorously caught by the techniques of effective parameters retrieval developed in the community of metamaterials [12–14]. Here, the finite-length wire medium is well described by an effective permittivity which exhibits a Lorentzian shape near the resonance frequency of a single wire $f_0$ as shown in Fig. 1(c). The three previously discussed frequency regions now correspond to a high effective permittivity (blue), a negative permittivity (red) and a low permittivity (green). The high effective permittivity is typically what we exploited in order to beat the diffraction limit with time reversal from the far-field [18,27].

At this stage, we would like to mention that such a polariton-like behaviour is not limited to the resonant wires as long as one can find a subwavelength unit cell, which acts as a meta-atom. Similar behaviours have been observed for split-ring resonators interacting with the magnetic field [28], for dielectric Mie resonators [29] or even plasmonic nanoparticles at higher frequencies [30]. Also, we have performed experiments with Helmhotlz resonators [31,32] and quarter-wavelength-pipe resonators [33] in airborne acoustics, and with bubbles for underwater acoustics [34]. Eventually, the case of pillars stuck to a plate or holes within a membrane play a similar role for respectively the guided elastic waves within the plate [35] or the hydroelastic waves at the surface of water [36]. In this review, we will focus on the wire-medium and give some technical experimental details specific to this example, but one can certainly be inspired to build a similar experimental platform with another unit cell and with another type of wave.

2.2 Experimental platform

2.2.1 Sample preparation

Our first experimental realization of a wire medium [18,37] was the half-wavelength design shown in the top of Fig. 2(a). It was made of $20 \times 20$ copper rods of length 40 mm and diameter 3 mm. We operated at frequencies near 375 MHz, for which the lattice constant of 12 mm is deeply subwavelength. The rods were maintained parallel with a structure made of Teflon, which is a low-loss dielectric material at these frequencies. The overall transverse dimensions of the sample remain below the freespace wavelength. We then miniaturized the distances at play (the rod’s length and diameter and the lattice constant) as in [38], with the downsize of the manual threading of the rods which revealed really time consuming. A more efficient strategy, described in [39], consists in 3D printing a plastic sample made of Acrylonitrile butadiene styrene (ABS), and then adding a metallic layer which is largely thicker than the skin depth at this frequency. The corresponding samples have been based on the quarter-wavelength design with a ground plane as shown in the bottom of Fig. 2(a). The rods are rigidly connected to the ground plane and no holder is needed any more. The typical rod’s height is between one centimeter to a few ones thus operating in the GigaHertz range, the lattice constant remaining below the centimeter.

 

Fig. 2. a) Photographs of two samples (half wavelength-long wires on the top and quarter-wavelength-long wires on a ground plane on the bottom). Typical experiments consist in measuring the transmission between a small source and a probe both placed in the near-field of the sample. b) Transmission spectrum measured for the sample of (a) displaying resonance peaks associated to subwavelength modes (blue region) and a a dip in transmission corresponding to the hybridization bandgap (red region) (reproduced from [37]). c) When scanning the probe position we obtain the complex transmission maps that we can place on frequency versus wavenumber graph. Repeating this procedure gives the following dispersion relation which follows precisely the polaritonic model (reproduced from [18]).

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2.2.2 Typical measurements

As everything occurs on scales smaller than the freespace wavelength, we need to probe the near-field of the sample. Home-made near-field probes were designed. They simply consist on a core cable from a coaxial cable sticking out of a SMA connector (Fig. 2(a)). For the quarter wavelength resonators design, the magnetic field is maximum near the latter so the near-field probe could also consists in a small loop (visible in the figure) stick near the ground plane.

A typical experiment consists in measuring the transmission between the source and the probe both placed at two different positions within the sample. For the half wavelength design, we place one probe on each side of the experiment, while for the quarter wavelength design the loop probe is near the ground plane and the second one is positioned few millimeters away from the top of the rods. The complex monochromatic transmissions are measured with a high signal-to-noise ratio thanks to a vector network analyser, on a frequency range which spans all of the interesting parts of the polariton dispersion relation discussed in the previous part. The absolute value of the transmission measured in the sample of Fig. 2(a) [37] is reproduced in Fig. 2(b). Below $f_0$, it displays many resonant peaks which correspond to the subwavelength modes we mentioned, and above $f_0$ the transmission drastically drops as expected in the absence of propagation.

We then repeat the same transmission measurement while the probe scans the entire face of the wire medium. Given the macroscopic scale of our samples we have no difficulty for scanning with a step thinner than the lattice constant. We therefore reconstruct the complex transmission field maps frequency by frequency with a good spatial resolution. The real part of four of them is represented in Fig. 2(c). One advantage of this complex measurement is that we can perform a 2D spatial Fourier transform of the maps and obtain the wavenumbers associated to each of them. A systematic procedure gives the experimental dispersion relation shown in the same figure. This is in great agreement with the polariton picture previously discussed.

We also would like to emphasize the fact that the wavenumbers at play are larger than the free space one. As a consequence, these subwavelength modes are mostly trapped within the sample giving rise to a stationary-like pattern due to reflection at the edges. On a spectral point of view, this manifests as the presence of many resonance peaks in the transmission spectrum.

2.3 Degree of freedom of the system

Let us now review what are the degrees of freedom available in the medium to modify the behaviour of waves within such a medium.

2.3.1 Shortened wires create a resonant cavity or a waveguide

The general polariton picture results from an interaction of the wave with the collection of resonators. Thus, contrary to photonic crystals, removing one wire does not create a cavity, the dimensions at play being too small. However, a resonant mode can be introduced by directly putting a resonator with a resonance frequency $f_1$ which falls within the hybridization bandgap. Interestingly, the design of such a resonator solely consists in shortening one of the wires. The shortened wire, which cannot radiate energy toward the far-field because it is embedded within a medium which does not support wave, therefore acts as a resonant cavity. Figure 3(a) shows the transmission map measured on top of the wires while the source is below the shorter wire: a resonant defect at the resonant frequency of the shorter wire is evidenced. The map also reveals the confinement of the field on distances far below the freespace wavelength [37,38]. Thanks to the inhibition of radiation as well as the strong confinement, these defects can exhibit extremely high Purcell factors [41]. Furthermore, the attenuation length of the hybridization bandgap varies with two parameters: the resonator density and the operating frequency. It is thus possible to control the spatial spreading of the trapped mode by increasing or decreasing the number of resonators around the defect. Or, this can be controlled by adjusting the resonant frequency $f_1$ of the defect: the larger the detuning, the less confined is the field around the defect [37,38].

 

Fig. 3. a) (Left) A shorter wire within the medium creates a defect. By mapping the transmission on top of the wires (right) while the source is below the shorter wire one retrieves the existence of a trapped mode which creates a subwavelength cavity (reproduced from [38]). b) A complex path of shorten wires is created within a quarter-wavelength wire medium. The mapped electic field on top of the wires while the two extreme wires are connected reveal the existence of a waveguide (reproduced from [39]) c) Modifying the unit cell of the wire medium, has allowed to observe negative refraction as shown in the map of the right, in a honeycomb lattice of identical wires (left) or in a lattice which contains two wires resonating at different frequencies (middle). This was reproduced from [40].

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Although the energy is strongly confined to the site of the defect, the latter exhibits an evanescent decrease. As a consequence, if a second defect is placed on its near vicinity the energy can "tunnel" from one site to its neighbor. By extending this argument, one can create a waveguide by building a path of shortened wires [37,39]. Note that the path does not have to be straight as shown in Fig. 3(b). Such a waveguide presents a transverse confinement which is again far smaller than the freespace wavelength as well as a group index which can be order of magnitudes higher than classical waveguides [39]. Indeed, the propagating mechanism is the equivalent of the one in so-called Coupled Resonator Optical Waveguides (CROWs) [42–44]: the energy hops from site to site with a nearest neighbor coupling. The physics is therefore well caught by a tight-binding Hamiltonian [6].

2.3.2 Modifying the unit cell: crystalline metamaterial

The second strategy for modifying the propagation within the wire medium consists in creating global changes instead of local defects. To do so, we modify the unit cell itself. And, as stated in the introduction of this article we have two levers: the structure or the composition. Playing on the structure of the unit cell comes down to keeping all the wires identical and patterning them differently, while playing on the composition keeps the wires at the same position but changes the resonance frequency of some of them. Despite the small distances at play the polariton-like modes propagating in the wire medium can feel its spatial patterning because the effective wavelength is small. In other words, multiple scattering also occurs at scales far smaller than the freespace wavelength [45].

We first experimentally explored this strategy in acoustics with soda cans [45] and then reproduced similar results with the wire medium [40]. The basic idea beyond those experiments was to generate a macroscopic effective property, namely a negative refraction, by creating a unit cell which exhibits a superposition of a monopolar-like resonance (as the single wire does) and a dipolar one. The simplest strategy consists in creating a unit cell with two resonators. By playing on either the structure or the composition, we ended on creating the two unit cells of Fig. 3(c): a honeycomb lattice of wires or a square lattice of wires with two slightly detuned resonators per unit cell. In both cases, a resonance where the two resonators are out-of-phase spectrally overlaps with the monopolar resonance where the two resonators are in phase. From a macroscopic point of view this dipolar resonance is responsible for the creation of a so-called optical branch within the hybridization band gap created by the monopolar one. It thus results in a propagating band with a negative slope, which overall corresponds to negative refraction as demonstrated experimentally in [40,45] (the map on Fig. 3(c)). It is also interesting to note that the long distance correlations does not play a crucial role for obtaining such results and a medium made of pairs of resonators also presents this negative refraction even if they are randomly positioned [46].

As a summary, we have seen in this section that a half-wavelength-long metallic wire (or its quarter-wavelength-long version on a ground plane) behaves like a meta-atom in the microwave range for electromagnetic waves. When we create a medium from this unit cell, it gives rise to a polariton-like dispersion relation. Experimentally we can probe the complex amplitude of the field over a relatively broad spectral range with a resolution well below the lattice constant thanks to the macroscopic scale of the sample. Also, thanks to the subwavelength nature of the propagating modes we can monitor several wavelengths inside the sample without building giant samples. Eventually, we have seen that we can modify locally (by creating defects) or globally, by creating meta-molecules, the medium and induce new properties such as a cavity, a waveguide or negative refraction. The next three sections will be dedicated to the introduction of topological features from this very simple experimental platform.

3. Hybridization band gap allows to mimic tight-binding Hamiltonians

In this section, we focus on the hybridization band gap and the opportunities that it offers in order to mimic tight-binding Hamiltonians. Indeed, as we have seen in the previous section if one create several defects (ie. slightly detuned resonators) they only couple to the nearest neigbours due to the absence of propagating solutions in this spectral range. This section first follows the results of Ref. [47] before we propose new perspectives based on this work.

3.1 Honeycomb lattice of tight-binded resonators

In order to reproduce the Hamiltonian of electrons propagating in graphene [48], we propose the protocol presented in Fig. 4, whose starting point is free space, in which waves can propagate at all frequencies. We set up a first sub-network of quarter-wavelength-long wires whose resonant frequency is $f_0$ organised on a triangular lattice (Fig. 4(a)). This creates the previously described polariton-like dispersion relationship in the system. Consequently, the waves are no longer authorized to propagate in the medium for frequencies slightly greater than $f_0$, corresponding to the hybridization band gap. We then use this forbidden band to introduce defects into the system. As explained previously, the latter are resonators whose resonance frequency $f_1$ is greater than $f_0$. These defects are arranged according to a second sub-network, with a honeycomb lattice this time, around the first triangular sub-network (Fig. 4(b)). The bandgap, by confining the field onto the sites of the honeycomb array of defects, enforces the energy to "tunnel" from a given site to its nearest neighbor and prevents any coupling with more distant resonators. It indeed induces a tight-binding type of coupling in the honeycomb network: the two ingredients of the Hamiltonian of electrons in graphene are present.

 

Fig. 4. a) A triangular sub-lattice of long wires (black disks) creates a polariton with a hybridization bandgap above $f_0$ (grey region). b) Adding a honeycomb sub-lattice of shorter wires (black circles) generates two new defects bands within the bandgap (orange lines), precisely described by the tight-binding Hamiltonian of graphene (reproduced from [47]).

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We calculate the dispersion relation of the infinite crystal composed of these two sub-networks using COMSOL Multiphysics software. This amounts to applying periodic boundary conditions to the walls of a unit cell, ie. implementing Floquet/Bloch’s theorem [4,5]. These dispersion relations, projected on the main directions of the network ($\Gamma \textrm {M}$, $\Gamma \textrm {K}$, $\textrm {KM}$) are represented in Figs. 4. As planned in the protocole, before introducing the shortest wires, the dispersion relation is indeed well described by the model of the polariton with the opening of a forbidden band above $f_0$ (shaded gray area in the figure). After adding the defects, two new bands appear in the previous forbidden band. The latter are similar to the calculated bands with nearest neighbor tight-binding Hamiltonian of graphene. Indeed, they present a Dirac cone at the K-point of the first Brillouin zone.

3.2 Experimental results

We designed experimentally the corresponding sample shown in Fig. 5(a). The latter corresponds to a quarter wavelength design. Its global shape corresponds to a triangle in order to respect the symmetries of the different sub-networks. The sides of the triangle are of a dimension comparable to the freespace wavelength ($\approx 10~\textrm {cm}$). Applying the same typical experimental procedure, we map the electric field for several excitation frequencies. The average spectrum (not shown here) reveals the existence of resonant modes around 5 GHz within the bandgap, in good agreement with the numerical results of the previous paragraph.

 

Fig. 5. a) Picture of the sample. b) (top) Experimental field maps at the frequencies 4.88 GHz (Left), 4.92 GHz (center), 5.05 GHz (right) and (bottom) their corresponding spatial Fourier transform after applying the symmetries of the lattice. c) Experimental band structure extracted from the maps on the reciprocal space which shows Dirac cones at the corner of the Brillouin zone (inset). d) Experimental mapping of the relative phase between the two sites of the honeycomb lattice for the low (left) and high (right) frequency bands. Different vortices appear at the corners of the Brillouin zone (insets). (figure reproduced from [47])

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We are now interested in the experimental field maps corresponding to some of these resonances (Fig. 5(b)). All of them show that the field is rather carried by the shortest wires consistently with the fact that a lattice of defects have been created within the hybridization band gap of the metamaterial. However, this is not yet sufficient to claim that this crystal is an electromagnetic analogue of graphene. With this in mind, we pass into reciprocal space in order to relate the frequency of these modes with their Bloch wavevector. For this we calculate the two-dimensional spatial Fourier transforms of each of these modes. Since our whole experimental protocol is linear, we can perform some modifications on these Fourier transforms. Consequently, we add the absolute values of the initial transform with those modified according to certain symmetries authorized by the hexagonal shape of the first Brillouin zone. In this case, the latter are the rotation symmetry $\textrm {C}_{6}$ (angle of 60 degrees), two mirrors as well as the periodic repetition of this first Brillouin zone imposed by the Bloch theorem. Thanks to this procedure, we obtain the maps in reciprocal space ($k_x$, $k_y$) represented below the maps in the real space in Fig. 5(b). By keeping only their maxima, we extract the isofrequency contours. The shape of these contours varies greatly with frequency, proving the dispersive nature of the medium. Some of them are circles centered on the origin of the Fourier plane which is related to an isotropic propagation in the metamaterial. However, for some frequencies the contours become circles centered around the corners $\textrm {K}$ of the first Brillouin zone. This strong anisotropy testifies the crystalline nature of the modes.

In order to demonstrate these crystalline properties, we carry out the previous procedure for several frequencies in order to reconstruct the band structure corresponding to these defect modes. We superimpose the different isofrequency contours and display them in three dimension (Fig. 5(c)). This representation of the results clearly shows that we succeeded in obtaining experimentally a discretized version of the graphene band structure within the wire medium. Notably, this is now possible to distinguish the two different bands of the dispersion relation which are conically connected at the $\textrm {K}$ points of the first Brillouin zone. Although this band structure does not present a perfectly symmetrical profile with respect to the Dirac frequency unlike that calculated with the nearest neighbour Hamiltonian, one can find a better match by adding next-nearest neighbour terms. A more systematic study where we played with the attenuation distance within the bandgap was made in acoustics [49].

Last but not least, our measurements combine the simultaneous knowledge of the complex maps in the real space and the position of the mode in the first Brillouin zone in the reciprocal space. This allows to project the data onto the Pauli basis [48]. This is directly obtained for one mode, corresponding to one wavenumber, by taking the phase shift between the two sites of the unit cell of the honeycomb lattice of defects. As we have numerous cells within our sample, we can average over them and impose a higher weight on the sites which have more energy. We then carry out this procedure for all the measured maps and for both the defect bands. Ideally, for each point in the reciprocal space we should obtain a phase difference that we will represent as an arrow. The experimental results are presented on Fig. 5(d). The resulting arrows undeniably exhibit vortices at the corners of the Brillouin zone, whose features depend on the corner (inset), and are inverted between the two bands. This quantity is not gauge invariant (one of the two sites of the honeycomb cell is chosen as the reference site) but these peculiar windings of the phase around the Dirac cones are closely related to their topological properties. Namely quantized Berry phases equal to $\pm \pi$ according to the type of vortex can be obtained when following a contour around each $\textrm {K}$ point.

3.3 Perspectives: more complex Hamiltonians

After having obtained the band structure of a hexagonal network governed by a tight-binding model, we are now interested in more complex periodic structures.

3.3.1 Dice lattice (or T3)

The first structure that we consider here is the T3 network [50]. The latter is commonly called dice lattice, because it gives the illusion of being a stack of cubes represented in perspective (Fig. 6(a)). It is a triangular network to which some inter-site links have been removed. For the triangular network, each site is equivalent and has three links. In the case of dice, three links are removed in order to distinguish two types of sites within the network: two peripheral sites (3 links) and a central site (6 links). The corresponding unit cell is therefore composed of three sites linked differently to each other. In reciprocal space, the Brillouin zone remains an hexagon similar to that of the hexagonal lattice of the previous part.

 

Fig. 6. a) The $T3$ lattice or equivalently called dice lattice, a triangular lattice with anisotropic links. b) Implementation of (a) with a wire medium made of three sub-lattices of wires with different heights. c) Numerical band structure of (b), exhibiting Dirac points at the corner of the Brillouin Zone with with a flat band at the same frequency. d) The brick-wall lattice a square lattice with anisotropic links. e) Implementation of (d) within a wire medium made of two sub-lattices of wires with different heights. f) Tuning the height of the wires of one sub-lattice leads to the merging and disappearance of two Dirac cones within the band structure.

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We propose here a simple protocol in order to reproduce the anisotropic coupling such as in the dice lattice. We start from a triangular network of sub-wavelength resonators of resonance frequency $f_1$. The next step consists in adding a second sub-network of resonators whose sites correspond to the links that we want to remove (black sites on the lattice of Fig. 6(b)). These resonant inclusions have a resonance frequency $f_0$ lower than $f_1$. They therefore create a local band gap within the medium and the resonators of the first triangular sub-network behave like defects within the latter. In this way, an evanescent coupling is established between them. Unlike the case of graphene, this evanescent coupling is made anisotropic because of the position of the low-frequency resonators.

Also, to exactly mimic the dice Hamiltonian preliminary simulations showed that we need to slightly detune the central site which has 6 links compared to the peripheral ones which only have 3 links. Again, the strategy is very simple and one only needs to change the height of this central wire. After a numerical parametric study we managed to obtained the lattice of Fig. 6(b). Its two dimensional dispersion relation calculated by applying Bloch periodic conditions is shown in Fig. 6(c). On top of exhibiting Dirac cones similarly to the honeycomb lattice, this band structure features a flat band at the Dirac frequency. It has been shown that the eigenmodes corresponding to this triple degeneracy can be described as pseudo-spin 1 particles. These particles are responsible for exotic properties such as Klein’s super tunneling [51]. Moreover, the presence of the flat band is in itself interesting. Indeed, the latter is tightly related to wave localization, topological phenomena and non-linear effects for example [52,53]. This work opens the door to further experimental investigations of these transport phenomena.

3.3.2 Brick-wall lattice

The second structure that we study is the so-called brick-wall network displayed in Fig. 6(d). Again, this network starts from a very simple lattice, a square lattice, on which we remove some links. We therefore end on a square lattice with two atoms per unit cell with anisotropic links. Here again, we exploit the possibility to generate this anisotropy in the link between sites resonating at frequency $f_1$ by adding the longer resonators, resonating at $f_0$ and responsible for the hybridization bandgap, on specific positions. Typically we place them solely on the positions where a link has to be removed and we end on the square unit cell of Fig. 6(e) with 3 wires per unit cell.

A first calculation on this lattice demonstrates that it is possible to obtain two Dirac cones in the band structure of such a square lattice (first panel of Fig. 6(f)). It somehow proves that the Dirac cones are not specific to the hexagonal symmetry. The second very interesting feature that needs to be explored is the possibility to change the relative strength of the two links. The attenuation distance of a bandgap being strongly frequency dependent this can easily be done by changing the height $h_0$ of the wires responsible for it. These preliminary simulations demonstrate a feature that has been predicted theoretically [54] and demonstrated experimentally in many different tight-binding systems [55–59] as we increase $h_0$ the two cones move toward the $\Gamma$ point where they merge, and then disappear (Fig. 6(f)). These results show that this platform potentially allows to experimentally measure a merging of cones at $\Gamma$ in an open system.

4. Mimicking the quantum valley-Hall effect

After having played with the hybridization bandgap in the previous section, we will now modify the polaritonic propagating band itself. We no longer exploit the inhibition of propagation but will play on the lattice itself in order to induce new topological wave transport. In this section, we concentrate on building an analogue of a so-called Semenoff insulator [16]. Saying differently, we will create pseudo-spin–valley at the origin of the propagation of waves which carry a circular polarization associated to the propagation direction, as described in [60].

4.1 Breaking the degeneracy between $\textrm {K}$ and $\textrm {K'}$

The starting point of this study is the honeycomb lattice of identical wires. The latter was previously introduced in this review in section 2.3.2 to demonstrate that playing on structure and building a biperiodic lattice can induce negative refraction [40,45]. Actually, a more complete calculation of the band structure is shown in Fig. 7(a). This is still in agreement with the previous statement since for most of the frequencies the isofrequency contours are circles centered on the $\Gamma$ point, but this description fails to highlight the existence of Dirac cones at the $\textrm {K}$ points. These Dirac cones are slightly different from the ones studied in paragraph 3.1 since it takes into account the interaction with many more neighbors, but it also results from the geometrical properties of the honeycomb pattern.

 

Fig. 7. (Reproduced from [60] except c) a) Honeycomb lattice of identical wires (top) and its band structure which exhibits Dirac cones at the 6 corners of the first Brillouin ZOne. b) (top) Slightly shortening one of the two wires of the unit cell (black circles) and elongating the other one (black disks) breaks the Dirac degeneracy (bottom). The insets show the corresponding numerical electric field maps which exhibit a crystalline angular orbital momentum on the bulk meta-molecules whose sense of rotation depends on the band. c) Photograph of the experimental sample (top) and experimental extraction of the Berry curvature for the lower (middle) and upper (bottom) bands, showing the breaking between the valleys $\textrm {K}$ and $\textrm {K'}$.

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We are now playing with the composition of the unit cell of the previous network. Instead of having two identical resonators, we slightly modify the resonance frequency of each of them (see details in Fig. 7(b)). The unit cell remains a diamond with two resonators and the corresponding Brillouin zone is the same hexagon as before. This lattice is therefore both biperiodic and bidisperse. The full dispersion relation calculated numerically in COMSOL Multiphysics corresponding to this crystalline metamaterial is shown in Fig. 7(b). It has a lot of similarities with the honeycomb case. Indeed the two bands are present, the high frequency one exhibiting a negative slope, which proves that we only slightly disturbed the previous system. However, this change was enough to break the Dirac degeneracy at the corners of the Brillouin zone.

In order to understand this gap opening, we go back to the real space and look at the modes corresponding to the $\textrm {K}$ point for the lower and upper bands. To make things clearer, in Fig. 7 we define a hexagon with a black outline on the various field maps. The latter is composed of two triplets of resonators with different resonance frequencies. These trimers are represented by triangles which have either their base below (green) or above (yellow) of the corresponding Bloch wavevector (inset of the field maps). For the lower band the electric field is concentrated on the wires which resonate at lower frequency (black discs), and null on those which resonate at higher frequency (black circles). Moreover, for this wavenumber, the electric field oscillates with a phase delay equal to $2\pi /3$ between the three black wires. This means that if the field is maximum on one of the vertices of the triangle, then it is minimum on another and zero on the last. The phase of these modes is then scrolled in order to reproduce their propagation within the metamaterial in the $\Gamma \textrm {K}$ direction. This highlights a counter-clockwise rotation of the electric field within the triangle of long wires, as shown by the green arrows on the green triangle [61]. On the contrary, the upper band at the $\textrm {K}$ point presents a field which concentrates on the shorter wires (circles), which indeed have a higher resonance frequency, and exhibit oppositely clockwise rotation of the field on the yellow triangle. A more complete study [60] reveals that the fields at the $\textrm {K'}$ point rotate in the opposite direction. The fact of having slightly modified the composition of the honeycomb unit cell therefore allowed us to introduce a well-defined circular polarization within the medium. This rotation which accompanies the wave during its propagation is linked to the direction and the frequency of the latter.

We would like to conclude this part by highlighting the usefulness of our experimental platform which provides the complex maps. A systematic analysis of the measured relative phases between neighboring resonators of the lattice for all measured transmission maps allows to extract the Berry curvature, a quantity that has been previously measured experimentally with cold atoms for example [63]. The procedure is very similar to the one previously introduced in section 3.2 for extracting the phase between the two resonators of the unit cell, except that the Pauli’s matrix $\sigma _z$ has now a non-zero contribution due to the opening of a bandgap near the Dirac frequency. After a change in the basis compared to the previous maps of arrows according to [64], we obtained the two maps of the Berry curvature in the reciprocal space of Fig. 7(c). They present sharp peaks centered at each corner of the first Brillouin zone with a change of sign between the two valleys $\textrm {K}$ and $\textrm {K'}$. Therefore the valleys carry their own topological charge $\pm 1/2$. Nevertheless, the total Chern invariant remains zero because of the time reversal symmetry of the system. Such a medium is therefore equivalent to a Semenoff insulator [16], a trivial insulator. But, it still possesses topological valley charges at each Dirac point and we will aim at exploit them in the next paragraph.

4.2 Valley edge states

To exploit this topological property, we need to build a domain wall between two media with opposite valley charges. It consists in setting up an interface in such a way that valleys of opposite charges face each other. This interface supports guiding mode for frequencies within the forbidden band of each crystal on the two sides of the interface [65]. These guided modes have an additional degree of freedom, a valley polarization, which limits reflections and allows information to be selectively routed, but are not inherently topologically protected. This was done with a device involving bi-layer graphene [66,67]. Similar promising characteristics have also been demonstrated in microwaves with photonic crystals [68,69] or spoof-plasmons [70], and at optical wavelengths with a network of evanescently coupled waveguides [71].

An easy way to setting up such an interface is to add a mirror symmetry along the $\textrm {KK'}$ axis. This transforms $\textrm {K}$ into $\textrm {K'}$ on both sides of this mirror. In real space, this protocol corresponds to the two types of interfaces between a crystal and its symmetrical one represented in Fig. 8. The first interface, represented in red, is obtained by putting the two long wires (black discs) facing each other. The second one (blue interface) does the same but with short wires (circles).

 

Fig. 8. a) (Left) Band structure of the red interface made by putting two long wires in front of each other showing a new band in the bandgap (red).(Right) Corresponding edge modes propagating along the interface, whose direction of propagation is locked to a crystalline orbital angular momentum on the interface metamolecule (inset). b) Same as (a) for the blue interface made by putting two short wires in front of each other. (reproduced from [60])

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We then numerically compute the dispersion relations corresponding to each of these interfaces. In both cases, they exhibit the existence of a new propagation band which crosses the bandgap common to the two metamaterials on both sides (Fig. 8). The field pattern associated to this band is mainly concentrated at the interface, thus being defined as a edge guided mode. Moreover, depending on the type of interface, the slope of the dispersion relation is different: red is positive when blue is negative.

In order to understand deeper the link between the angular momentum of the crystalline modes in the bulk medium and the emergence of these interface modes we come back to the real space and look at the field maps. Note here that we study the modes propagating in the direction of increasing $x$. These are therefore the points of the dispersion relation where the slope, or the group velocity, is positive. We first focus on the case of the red interface (Fig. 8(a)). In this case, the electric field is mainly concentrated on a limited number of resonators in the vicinity of the interface. To follow the polarization at this interface, we define a fixed hexagon (black outline) along the interface. The latter includes six resonators and it is again possible to define two triangles with different orientations with respect to the direction of propagation. Here again, green has its base below the interface and it is the reverse for yellow. Although this interface meta-molecule is similar to that of the bulk crystal, its composition is different. Indeed the two triangles are no longer formed by a single type of resonator (circles or black discs) but each of them has two long wires (black discs) and one short wire (circles). The electric field is arranged on these two hybrid triangles at the same time. Then, we apply the same procedure as for the bulk polarization by scrolling the phase in order to reproduce the propagation of the wave along the interface. This gives no longer one but two helices, each defined on one of the triangles, which turn in opposite directions: the yellow triangle is counter-clockwise while the green one is clockwise. This superposition gives rise to an anti-symmetrical mode at the interface.

If we now focus on the blue interface (Fig. 8(b)), similar conclusions can be made. The mode corresponding to a positive group velocity is this time located in the $k_x<0$ part. Its profile, symmetrical with respect to the interface, presents dipolar variations in the direction of propagation, which is in agreement with the negative slope of the dispersion relation. Again, the field is mainly carried by a modified meta-molecule at the interface: the long (black discs) and short (circles) wires have reversed their position compared to the red one. By scrolling the phase as before, we show once again that the propagation of the wave in the direction $k_x$ is accompanied by a combination of two opposite rotations. But, these two polarizations rotate in the opposite direction compared to those of the red interface.

After having verified experimentally that a single edge supports a guided mode within the bandgap created near the Dirac frequency, we focused in more complex cases which can exploit this polarization. We designed one crossroad made of six red interface branches (Fig. 9(a)) and one crossroad with four branches which mixes the two types of interfaces (Fig. 9(b)). Due to the limits in the size of the samples which create reflections at their ends, we prefer to show results in the time domain [62]. We generate a $20\textrm {-ns-long-pulse}$ at the input (branch number 1) and show a snapshot of the intensity 40 ns later. In both cases, these crossraods demonstrate a specific routing of the waves in certain directions. This can be intuitively explained in the real space. Indeed, the propagation direction is linked to the sense of rotation of the field on the meta-molecule at the interface. Arriving at the intersection, the specific sense of rotation injected at the initial position is conserved, and determines the directions in which the wave is authorized to continue. This is sketched for both experiments with only the green triangles but one has to keep in mind the presence of the yellow ones.

 

Fig. 9. a)(Left) Sample creating a crossroads made of six red interfaces. (Center) Snapshot of the pulse propagation. b) Same as (a) in the case of a crossroads with two red interfaces and two blue ones. The routing is governed by the polarization of the edge states as explained by the sketch on the right. (reproduced from [60]).

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To conclude this section, we have several key aspects that we would like to emphasize. First, this study on a bidisperse/biperiodic crystalline metamaterial has permitted to experimentally measure a topological quantity, namely the Berry curvature in a honeycomb lattice, hence giving it a more practical meaning. Second, thanks to our experimental platform which give phase sensitive-measurements in the real space, we have been able to explain the specific chiral properties of the edge modes by visualizing the polarization on the meta-molecule at the interface. Third, as a consequence of the previous statement, we specifically built waveguides (not shown here but described in [60]) which only contains the metamolecules of the interface and are embedded in a hybridization bandgap. The latter has more confining properties and we believe it could open more perspectives in terms of applications. Four, we recently demonstrated in acoustics [72] (but one can easily design a wire medium based experiment) that an equivalence between the change in the composition made here and a change in the structure of a Kagome lattice leads to very similar chiral edge states. Eventually, we have only studied here the zigzag edges of each semi-infinite crystals building the interface. We believe that there is still to explore with these analogues of a Semenoff insulator.

5. Mimicking the quantum spin-Hall effect

This last section focuses on our unique realization of the analogue of a true topological insulator. Indeed, as we work with the wave equation in non-magnetic stationary media, all our solutions must respect a time reversal symmetry. And in two-dimensions the unique non-trivial topological phase which preserves this symmetry is known as $\mathbb {Z}_2$ topological insulator. In electronics it is the consequence of a spin-orbit coupling which typically manifests in the quantum spin-Hall effect [73,74]. The first experimental photonic analogue of this electronic effect was proposed in 2013 [75] based on a design with split ring resonators. In 2015, Wu and Hu proposed [17] a general recipe to emulate a pseudo-spin in an hexagonal lattice of resonators exhibiting a $C6$ rotational symmetry. In this section, we will show this strategy applied to the wire medium and mostly present the results of the Ref. [76]. Similar realizations of the same scenario have been made in photonics [77–79], but also in acoustics [80–83] and with elastic waves [84,85].

5.1 Creating the band inversion

Here again we directly play on the polaritonic band. We first consider the regular honeycomb lattice initially presented in section 2.3.2 with identical wires in each site. The latter has a unit cell which has the shape of a diamond and which has two wires resonating at the same frequency $f_0$. The corresponding dispersion relation is therefore the one presented in the previous section in Fig. 7(a) which presents Dirac cones at $\textrm {K}$ points.

The next step consists in reproducing the double spin degeneracy at the origin of $\mathbb {Z}_2$ topological insulator. This amounts to superimpose a second Dirac cone on top of the already existing ones. Here, rather than finding it in an extra wave state as in [75], we directly use the ones which are already present in the dispersion relation. Indeed, the main idea of the Ref. [17] is to make coincide the Dirac cone at $\textrm {K}$ point and the one at $\textrm {K'}$. Thus, by considering the honeycomb lattice with a non-primitive unit cell which comprises six resonators arranged in a hexagon, we fold the dispersion relation of Fig. 7(a) onto the dispersion relation of Fig. 10(b). This way, we bring the two cones to the center of the first Brillouin zone where they superimpose (see inset). We have thus created a quadruple conical degeneracy at the $\Gamma$ point.

 

Fig. 10. Lattice and numerical dispersion relations for a a triangular lattice of an hexagon of wires. In (a) (respectively in (c)) the hexagons are squeezed (respectively stretched) compared to (b) which actually corresponds to the honeycomb lattice (see the black primitive unit cell). The two transformations of the unit cell opens a gap at the Dirac cone of b) (Note that this is a doubly degenerated Dirac cone since it comes from the folding of the cones at $K$ and $K'$ as sketched in the inset). Experimentally (field maps in the bottom), a band inversion is observed in the shape of the field in the meta-molecule at the edges of the gaps. (reproduced from [76])

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At this stage, the double Dirac cone that we have obtained is the result of a purely mathematical folding of the bands. In order to give it a physical meaning, the super-cell at the origin of the folding must become the primitive cell. To do so, we induce two types of structural modifications within the latter, by squeezing or expanding the hexagon within the super-cell. This brings us to the two metamaterials of respectively Fig. 10(a) and (c). Applying Bloch periodic boundary conditions in COMSOL Multiphysics gives the dispersion diagrams represented in the same figure. They both exhibit the opening of a bandgap (blue shaded area) where we originally had the double Dirac cone.

In order to highlight, the difference in the topology of the two gaps we need to look at the complex electric field for the modes at the edges of this bandgap. The experimental maps obtained on two triangular samples containing several unit cells are shown at the bottom of Fig. 10. When we look at the field on a single unit cell (ie. a squeezed or and expanded hexagon) the modes at the edges of this bandgap are of a dipolar or quadrupolar symmetry (see the circular insets in the maps). These symmetries correspond respectively to the molecular orbits of type $p$ or $d$. Naturally, the $p$ orbits arrive at lower frequency since the wavenumber along the curvilinear path of the hexagon is smaller. Therefore, the squeezed crystal of Fig. 10(a) which exhibits $p$ modes below the bandgap and $d$ modes above it appear as a natural crystal. Inversely, the expanded crystal of Fig. 10(c) exhibits a band inversion since the $d$ modes appear below the gap while the $p$ modes are above. The transition between these two inverted band structures occurs at the $\Gamma$ point under the form of a Dirac cone, known to be the transition point between two topological phases.

5.2 Edge state

The experimental protocol to explore the edge states at the interface between two media with different topologies is very simple. It is sufficient to take the two samples from the previous paragraph and put them facing each other while respecting the symmetries of the network. This is shown in Fig. 11(a). We then carry out the same type of transmission measurement by scanning the sample while exciting from a source antenna at the bottom of the edge.

 

Fig. 11. a) Interface made by putting two topologically different samples in front of each other. b) Corresponding band structure of the edge state within the band gap of the bulk. c) Experimental electric field map showing the topological interface mode at 4.93 GHz.

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The dispersion relation of the corresponding interface shows the appearance of two propagating bands within the forbidden band of each of the media (Fig. 11(b)). The structural modifications corresponding to the latter being quite significant, we can clearly see the opening of a second forbidden band between these bands at the point $k_y=0$. In a sense we do not have a complete analogue of $\mathbb {Z}_2$ topological insulator which should not show this gap. Indeed near the edge the local symmetry is broken and the two pseudo-spins are coupled: it results in the opening of this very small gap at $\Gamma$ point as explained in [52].

The averaged spectrum over the interface (not shown in this review article but presented in [76]) exhibits two transmission peaks within the frequency range corresponding to the common bandgap of both crystals. The field map corresponding to the first peak at 4.93 GHz shows a confined field along this interface. It slightly more penetrates on the right side of the interface because the band gap of the topological sample on the right is less efficient.

In this part we have demonstrated numerically and experimentally the existence of modes propagating at the interface between two media with different topologies. Although the latter do not have topological protection strictly speaking, since the time reversal partner exists, their symmetry properties allow them to propagate in certain well-defined directions but also to acquire a certain robustness with regard to reflections on steep turns.

To conclude this part we would like to say again that spatially patterning the wire medium has led to new wave transport. Typically, we succeeded to adapt the protocol proposed by Wu and Hu [17] in order to create a macroscopic analogue of a $\mathbb {Z}_2$ topological insulator. This study, which we have also reproduced numerically for the acoustical case [49], once again demonstrates the relevance of this approach, which goes beyond the usual effective medium description. Moreover, it provides access to experimental data such as the local symmetry of the field at the band edges which are quite rare in the community of macroscopic topological insulators.

6. Conclusion

All along this review we have highlighted the role of composition and structure in a locally resonant metamaterial. This approach goes beyond the usual description in terms of effective parameters at play in the metamaterial community. It also brings them closer to the photonic crystals since we describe them through their band structure.

In the first section, we introduced our typical medium: the wire medium. We started from the simplest case where the unit cell is built from a single resonator, which we called meta-atom. The corresponding network is regular and its dispersion relation exhibits a polariton-like behaviour: a subwavelength propagating band followed by a hybridization band gap. We also reviewed in this section the versatility of our experimental platform. Not only it is very easy to build experimental samples, but we also have experimentally access to phase sensitive electric field maps. Also, given the fact that the lattice constant can be completely independent of the freespace wavelength owing to the subwavelength nature of the modes at play, the experimental samples have reasonable dimensions while being macroscopic in the same time. We ended this section by providing the different degrees of freedom that we have to modify the general polariton behaviour. We showed that local or delocalized changes in the composition (changing the length of the wires) or the structure (changing the spatial pattern) of such a medium could lead to interesting features.

The next three sections were dedicated to different realizations of topological photonics based on this very simple experimental platform. We first exploited the existence of the hybridization bandgap. The latter cancels the propagation of waves and therefore creates a tunneling effect between resonant defects placed in the medium. This has allowed us to mimic the tight-binding model of graphene but also to envision more complex anisotropic Hamiltonians. In the next two sections, we directly played on the propagating band of the polariton which features subwavelength modes. We respectively built an analogue of a Semenoff insulator and a $\mathbb {Z}_2$ topological insulator. Both experimental realizations had allowed to map the complex field at the scale of the medium, which overall permitted a better understanding of the chirality of the waves at play in all these realizations.

This work still opens up several perspectives, notably in the topological physics domain. One first strategy is to transpose these approaches to three dimensions by finding the adequate unit cell. The study of 3D topological insulators or Weyl materials for example is an exciting perspective [86]. Another path is to combine these structural designs with time reversal symmetry breaking by introducing gyromagnetic materials [87,88] in the unit cell of the wire medium. The great diversity among topological photonics phases combined with the benefits of locally resonant metamaterials provides many opportunities for fundamental research at the macroscopic scale.