1. Introduction

Beam-splitters play a key role in many optical devices, in particular interferometers, with a consequently broad variety of applications ranging from astrophysics [1] to quantum computing [2] as well as to many areas of optical electronics such as optical modulators [3] for fibre optic telecommunications amongst much more. Indeed to achieve complete control over the flow of light, power division and redirection devices are required, of which beam-splitters are those most commonly utilised; a recurrent theme is the desire to have broadband lossless splitters, capable of multiple and tunable re-direction, and with dimensions that are sub-micron, or at most microns, in size.

Several different beam-splitting approaches have been successfully implemented ranging from coupled bent dielectric slab waveguides or ridge waveguides atop substrates, to photonic crystal/ grating devices using waveguide or self-collimation or more recently non-reciprocal media or using topological designs. The dielectric waveguide approach is exemplified for polarisation beam splitters, as used for Mach-Zehnder interferometers, by [3, 4] and recently with more compact designs [5]. Splitters based upon dielectric waveguides traditionally have disadvantage of scale, even for high-index dielectrics requiring them to be of the order of several wavelengths in length to minimise scattering and losses at corners or shallow bends. This scale limitation, and a desire to minimise radiation losses at the bend, motivated [6] to take advantage of photonic waveguides created by removing rows or columns of the crystal array; here we design the topological analogues. These photonic waveguide devices have been successfully applied to beam-splitters and branched waveguides with T-shaped and Y-shaped branches implemented by [7–9], amongst many others, and the insertion of additional, or modified, array elements in the neighbourhood of the corner [10] can improve transmission or broaden frequency range; this has spawned ever more elaborate designs to optimise for various scenarios [11, 12]. A related, but different, approach to creating splitters within photonic crystals is to take advantage of self-collimation [13] and self-guiding created by strong dynamic anisotropy [14]. Beam splitters, including three-way splitters [15], are created and applied to devices [16, 17] using lattice array alterations to create effective mirrors or partial mirrors.

More recently, ideas from topological insulators [18] have been transposed into photonics [19], as reviewed in [20], showing promise for robust one-way edge states protected against disorder by topology. This promise is tempered by the requirement for time reversal symmetry (TRS) to be broken, and the electron spin to be mimicked by pseudospin in continuum systems. A simpler, passive and time reversal symmetric, but less robust approach is to attempt to reproduce the valley-Hall effect and utilise ideas from the field of valleytronics [21], for instance [22] create dielectric photonic topological arrangements leading to reflectionless guiding and designs for optical delay lines. Topological designs have been recently implemented for telecommunication wavelengths [23] on a CMOS-compatible chip thus bringing these concepts closer to application. These valley-Hall devices are locally topologically nontrivial however globally trivial, and therefore cannot draw upon the full power of the analogy with topological insulators, but do have advantages in terms of simplicity of construction as one need only break spatial inversion or reflectional symmetry, together with suppressing backscatter. Given the emergence of topological guiding there is now interest in developing this for photonic circuits, [24], and a natural drive to explore the potential of these new ideas.

The vast majority of this valleytronics literature takes advantage of periodic hexagonal or honeycomb lattices, utilising ideas from graphene, in particular the symmetry properties of the hexagonal Brillouin zone and symmetry induced Dirac cones at the KK′ vertices. Perturbations of the structure, that break symmetries, then gap the Dirac points giving topologically nontrivial band-gaps and well-defined KK′ valleys. The valleys have opposite chirality and are related by parity and/or reflectional symmetry as well as TRS. A key point is that by engineering a large Fourier separation between the valleys one suppresses intervalley scattering and ultimately the valley is used as an information carrier [20]. One negative that emerges is geometrical: The hexagonal systems can only create two-way energy-splitters [25], and three-way topological splitting would require other geometries for which symmetry induced Dirac points do not occur–hence the recipe outlined above cannot be employed.

Fortunately, one can engineer accidental Dirac points [26–28] for square systems and, although they are no longer at the high symmetry points, they provide pragmatic advantages over their hexagonal counterparts. For instance, allowing three-way topological beam-splitting away from a well defined nodal point. Previously, the only means to partition energy, more than two-ways, along topologically protected interfaces is by utilising the tunneling phenomena, described in [25]; however a drawback of this approach is that the outgoing energy is dependent upon the decaying tail of the initial interface state; here we have no such issues. Interestingly, [25] did show a three-way non-topological energy-splitter however this has the inherent disadvantage of being less robust and hence more prone to backscattering.

Recently [29] showed beam splitting for a model system of waves on elastic plates, for a fourth order elastic plate equation and masses of infinitesimal radius, here we explore to what extent three-way beam splitters can be employed in electromagnetism. We do this via a combination of group and k·p theory [30] coupled with detailed numerical simulations to extract the interfacial edge states, zero-line modes (ZLMs), and further we take these idealised edge states and perform numerical simulations, using [31], showing transport around right-angled bends and three-way energy splitting. A typical splitter that we construct is shown in Fig. 1, it is constructed from dielectric inclusions arranged upon a square lattice, within each elementary cell there is an arrangement of inclusions and the choice of arrangement is critical for splitting; we illustrate this importance by contrasting two arrangements. A judicious choice of inclusion arrangements, and the connection of quadrants of material yield the passive splitter that, as in Fig. 1, takes an incoming wave and then splits it in three.

 

Fig. 1 A three-way splitter designed on a square lattice using accidental degeneracies and geometry (right panel) shows the junction region between four quadrants each containing a set of dielectric inclusions placed around a rotated square. The quadrants differ in only the rotation given to the inclusion arrangement within it. In the left panel, the wave energy incoming along the leftmost interface is split at the junction into three interface waves. In this simulation the normalized angular frequency ωd/c is 4.78 and we treat TM polarisation.

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We operate at telecommunication wavelengths using a photonic crystal (PC) consisting of Indium Phosphide (InP) dielectric pillars in air. Drawing upon the design and fabrication of a dielectric carpet in [32], we contrast two designs of PCs within a square lattice array; the first of which has 6 InP dielectric pillars of diameter 200 nm, height 2µm, and minimum center-to-center spacing of 250 nm arranged around an equilateral triangle, Fig. 2(a), and the second has 8 pillars with similar dimensions arranged around a square as in Fig. 2(b). The refractive index of InP is n = 3.16, and we consider primarily (TM) polarization whereby the electric field is parallel to the pillars’ axis and discuss the analogous transverse electric (TE) polarization, the magnetic field is parallel to the pillar axis, in the concluding remarks. These two designs exemplify the potential of topological designs for energy splitting, and in particular how to achieve three-way splitting, for intrachip communication devices.

 

Fig. 2 The (non-rotated) arrangement of inclusions taken in the elementary cell: (a) equilateral triangular and (b) square arrangements and (c) the first Brillouin zone relevant for periodic arrangements of these cells upon a square lattice. For a pitch d, the radius of inclusions is 0.05d and their center-to-center spacing is 0.2 in the triangular case (a) the radii of the two types of inclusions in the square case are 0.075d and 0.05d and their center-to-center spacing is 0.25d The high-symmetry points of the Brillouin zone we refer to are Γ = (0, 0), X = (π/d, 0), Y = (0,π/d) and M = (π/d,π/d).

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The Maxwell equations split naturally into p and s polarizations, with forcing created by an electric line source or magnetic current dipole at position r_s respectively [33] as TM

2. Engineering a non-symmetry repelled Dirac cone for a square lattice

To create three-way topological splitting we first need to engineer accidental degeneracies along the Brillouin zone as shown in Fig. 2(c), boundary, it is not sufficient to simply generate Dirac points, and the arrangement of inclusions in the interior of the physical elementary cell, and their symmetry, plays a key role. We utilise perturbation theory and symmetry arguments to design the structures we study; the inclusions are either in a triangular formation or are arranged along a square, see Fig. 2(a) and 2(b). As we shall see the interaction of the internal inclusion symmetries with the symmetries of the lattice lead to fundamentally different behaviours; ultimately we shall see that the triangular inclusion case is unable to create splitters for reasons uncovered by symmetry arguments. We begin by determining the criteria under which accidental Dirac points are created, then gap them using geometrical rotations that break symmetry to create band-gaps with protected edge states; these edge states form the building blocks of the splitters we design.

2.1. Perturbation theory and band interactions

We begin by considering infinite periodic media and interpret the dispersion diagrams of Fig. 3 that contrast the triangular and square cases. We take the inclusion arrangements shown in Figs. 2(a) and 2(b); when we rotate the inclusion arrangements, the rotation is taken around the centroid of the inclusions. The most notable difference between the non-rotated cases, Figs. 3(a) and 3(b), are the single Dirac point for bands 4 and 5 in (a) vis-a-vis two in (b); the most important bands are coloured red and correspond to n = 3, 4, 5, 6 in the index notation we adopt. As advertised in the introduction a symmetry breaking perturbation, in this case a rotation anti-clockwise, gaps Dirac points to create band-gaps as indicated in Figs. 4(a) and 4(b). A critical issue, of course, is where the accidental Dirac points arise, or indeed whether they arise at all [26], and this requires an analysis of the band structure.

 

Fig. 3 The band diagrams for (a) triangular and (b) square arrangement of inclusions; the red curves show the branches of most interest, bands 3-6, in red; the accidental degeneracy is clear along MX in (a) and along both MX and YM in (b). Also shown are the eigenstates for the bands of interest at X.

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A minor point is that, due to the precise position of the inclusions, spaced at 0.25d, the square case has an additional glide and reflectional symmetry and there is a Dirac point exactly at X for bands 5 and 6, but this has no influence on the analysis here.

To understand how the bands interact we consider the eigenfunctions ${ℰ}_{\mathbf{n}\mathbf{\kappa }}(\mathbf{x})$, with n the band index and κ the Bloch wavevector, i.e. the Bloch momentum vector in reciprocal space, in the first Brillouin zone such that ${\nabla }^2{ℰ}_{n\mathbf{\kappa }}(\mathbf{x})+a(\mathbf{x}){\omega }_{n\mathbf{\kappa }}^2{ℰ}_{n\mathbf{\kappa }}(\mathbf{x})=0$ with ωnκ the eigenfrequency. We consider Bloch waves and the eigenstates $|{ℰ}_{n\mathbf{\kappa }}〉$ relate to periodic eigenstates $|u_{n\mathbf{\kappa }}〉$, which form a complete basis, via

The completeness of the periodic eigenstate basis, means we can expand |u_nκ〉 in terms of the basis set $\{u_{j{\mathbf{\kappa }}_0}(\mathbf{x})\}$ where κ₀ is fixed, to deduce

This relationship connects the eigenstates at a given point in reciprocal space, κ₀, to the frequency, and hence determines the local dispersion relation, and can be used to investigate band repulsion or attraction. We concentrate upon the behaviour near the point X as both sets of inclusions share σ_v mirror symmetry; the triangular case does not have σ_h symmetry and this leads to the "missing" Dirac point along YM in Fig. 3(a).

Looking further at the band diagrams, say that of Fig. 3(a) one observes, at the wavenumber X, close to the Dirac point of interest, that the eigenstates of bands 3 − 6 have characteristic orbitals familiar from quantum mechanics [34]. The symmetries inherent in these eigenstates motivate the use of the mathematical language of symmetry, that is, group theory: The point group symmetry of the structure is G_Γ = C_2v ; this is also the point group symmetry at X, G_X = C_2v (Table 1). The C_2v point group arises from a combination of spatial (reflectional) and time-reversal symmetries; the latter relates κ →−κ. If we were to solely use the spatial symmetry then we would plot around half of the BZ; however this would be the incorrect IBZ and the resulting figure would contain redundant paths (see Figs. 5 and 7 in [35]). The irreducible representations (IRs) at X are one-dimensional hence there is no symmetry induced degeneracy; however, two of the IRs can be tuned such that an accidental degeneracy, that is not symmetry repelled, is created along MX (and also along YM for the square case).

Table 1. Character table showing the classes, irreducible representations (IRs) and basis functions for C_2v

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The four bands highlighted in red in Figs. 3(a) and 3(b) (bands numbered 3 − 6 inclusive) are associated with the eigenstates shown alongside the dispersion curves; these match the basis function symmetries of the C_2v group given in Table 1; hence this indicates that bands 3 − 6 are symmetry induced and the sequential ordering of them (lowest to highest) is deduced numerically, via the eigenstates, as: {B₂, A₁, B₁, A₂} and these are shown in Fig. 3. The rotated, symmetry-breaking, cases of Figs. 4(a) and 4(b) retain, at least broadly, the same eigenstate structure.

 

Fig. 4 The band diagrams for (a) triangular and (b) square rotated arrangement of inclusions; the red curves show the branches of most interest, bands 3-6, in red. The inclusion sets are rotated anti-clockwise through π/12 in (a) and π/18 in (b). The associated eigenstates, at X, show the orbitals and parity.

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Intuitively, we expect the two bands (band 4 and 5) forming the accidental degeneracy, A₁, B₁, to be strongly coupled with each other; the other two symmetry induced bands, B₂, A₂, will have limited influence on the local curvature or slope of the A₁, B₁ bands. The even more distant bands (bands 1 and 2 and those above 6) will have negligible effect on the A₁, B₁ bands and we quantify this by separating the bands into the the symmetry set eigenstates (SSE, bands 3 6) and the remainder that lie outside the SSE. Equation (5) becomes

Multiplying Eq. (6), by the conjugated states ${ℰ}_{p{\mathbf{\kappa }}_0}^{*}\left(\mathbf{x}\right)$ or ${ℰ}_{q{\mathbf{\kappa }}_0}^{*}\left(\mathbf{x}\right)$ (where p ∈ SSE, q ∉ SSE) and integrating over the primitive cell in physical space we obtain the following two equations,

Our main interest is in the neighbourhood of the Dirac point so we set n = p ∈ SSE, and perturb with κ = κ₀ + ∆κ, then

Therefore the second summation in Eq. (10), coupling states within the SSE to those outside, falls into second-order, i.e. $\mathcal{O}\left({\left|\mathrm{\Delta }\mathbf{\kappa }\right|}^2\right)$, hence the first-order equation is the 4 × 4 system,

2.2. Compatibility relations and creating accidental degeneracy along XM

Bands vary continuously, except possibly at accidental degeneracies where mode inversion may occur, which in turn leads to a discontinuity of the intersecting surfaces. Hence when moving along a continuous band of simple eigenvalues the eigenstates continuously transform; departing from the high symmetry point X, the associated IRs describing the transformation properties of the eigenstates smoothly transition into IRs that belong to the point groups along XΓ or XM.

In physical space both cellular structures we consider in Fig. 2 have σ_v spatial symmetry, this is equivalent to σ_h symmetry in Fourier space. Note that the relevant symmetry, that is responsible for gapping the Dirac cone, is the point group symmetry of the Dirac cone in κ, not the spatial symmetries of the cellular structure. From the definition of a point group symmetry, i.e. any symmetry operator $\widehat{R}\in G_{\Gamma }$ that satisfies, $\widehat{R}\mathbf{\kappa }=\mathbf{\kappa }$ mod G , where G is a reciprocal lattice basis vector, implies that κ ∈ XM solely has the mirror symmetry operator, σ_h within its point group. Similarly, for a κ ∈ ΓX, only the vertical mirror symmetry operator, σ_v satisfies the point group criterion. The symmetries of the eigenstates, for a κ belonging to either of the paths, XM and XΓ, are shown within the basis functions column of Table 2.

Table 2. Character table showing the classes, irreducible representations (IRs) and basis functions for reflections σ_v,h.\

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We now simplify the 4 × 4 system of Eq. (12) by assuming it is dominated by the two strongly coupled bands, bands 4 and 5 with IRs A₁ and B₁, which will result in a 2 2 system. As we move away from X, the A₁, B₁ IRs belonging to the C_2v, see Table 1, from× continuity of the bands, transform into the IRs of the σ_v,h, see Table 2, and reveal compatibility relations of the IRs. For the symmetry σ_h the eigenstates at X and along XM satisfy the following,

The bands (A₁, B₁) at X are therefore compatible with (A, A) along XΓ which implies band repulsion and, as we observe, an inability to have a band crossing along XΓ.

Importantly, note that, in deriving Eq. (12) we have only assumed that κ₀ belongs to a particular symmetry set band (surfaces 3 − 6) (the band at κ₀ must be continuously connected to the same band at X). Therefore, the compatibility relations allow us to choose any expansion point along the the path ΓXM where the eigenfunction basis set, Eq. (5), transforms accordingly i.e. $|{ℰ}_{A_1}〉\to |{ℰ}_A〉$.

In order to solve the 2-band eigenvalue problem, Eq. (12), we compute the determinant of the truncated matrix,

Note that the Dirac cone occurs along the spatial symmetry path, σ_h , of the structure due to the opposite parities of the A, B bands; band repulsion occurs along the XΓ path [27] thereby resulting in a partial band gap along XΓ. If ${\omega }_{B_1}>{\omega }_{A_1}$, then the partial gap along XΓ isolates the Dirac cone along a portion of the IBZ path, ΓXM. Equation 16 solely demonstrates that the curves cross linearly along the high-symmetry line. For a pictorial proof of the locally linear behaviour, in all directions away from the Dirac cone, please see Figs. 1-3 in [36].

We have designed situations where either two, or four, (for triangular and square cases respectively) pairs of Dirac points are created by an accidental degeneracy and we have shown how they are created and given a prescription for their occurence. These Dirac points have been gapped by a symmetry breaking perturbation and band gaps have been created.

3. Results and discussion

3.1. Edge states: Zero line modes

We aim to use our knowledge of the designed band-gaps to create situations whereby an interface will support edge states. We begin by taking a half-space of one medium and place it above another; the only difference between the two media being that the symmetry of the cells in the upper and lower media is broken by clockwise and counter-clockwise rotations respectively (see Figs. 5(a) and 5(b) for the triangular case). This has the effect, in Fourier space, of interchanging X and Y, the Berry curvatures [37] are opposite in sign at X and Y, and hence at the interface between such media can create valley Hall edge states; these are named zero line modes (ZLMs) due to their origin as arising from opposite Berry curvature in the adjoining media.

 

Fig. 5 The interfaces created when regions of perturbed media neighbour its counter-rotated twin. Panels (a) and (b) show, for the triangular case, that there are two distinct interfaces where the triangles in opposing media are either point-to-point or face-to-face. The square arrangement of inclusions is shown in (c) where the top left medium is oppositely orientated to the others (used for propagation around the simple bend) and (d) where the top left and bottom right media are oppositely orientated to the other quadrants (used for propagation in a splitter). Red dots show interfaces with nonzero Berry curvatures, whilst purple spirals show an overlap between regions with opposite Berry curvature where ZLMs can reside.

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We compute ZLMs by considering finite tall ribbons one cell thick with Bloch conditions applied on the sides and with periodic conditions at the very top and bottom; the ribbons in the simulations are 40 cells in length which is sufficient to ensure that the exponential decay of the edge states does not interact with the precise boundary conditions at the top and bottom of the ribbon. We extract decaying edge states, even and odd, around the interface where, recalling the polarisation chosen, the even modes are the physically relevant ones. The eigenstates and dispersion curves are shown in in Fig. 6 and the parity of the ribbon eigenstates is inherited from the bulk eigenstates of Figs. 3 and 4. Notably the triangular and square cases have a crucial difference: ordering of the media matters, that is, stacking the clockwise above the counter-clockwise or vice-versa leads to two different interface types for the triangular case whilst for the square case it is irrelevant which ordering is taken (see Fig. 5) as it only has a single distinctive interface. Therefore, in the triangular case, the even and odd-parity modes exist along different interfaces whilst for the square case the different parity modes are hosted on the same interface; the orthogonality of the opposite parity modes inhibits coupling between them. In reality, there should only be 2 edge states in Fig. 6(f); these pertain to the even and odd-modes. The presence of another 2 (almost overlapping) states is a numerical artifact due to periodic conditions applied at the top/bottom of the ribbon. The differences in the interfaces in the triangular versus square cases form a key distinction that impacts upon energy transport around corners and splitting.

 

Fig. 6 Edge states (ZLMs): For the interface between oppositely tilted triangles (a-c), and squares (d-f). The even and odd modes are shown in (a,d) and (b,e) respectively at normalised frequencies given by the circles, (5.77, 4.69), and crosses, (5.74, 4.62), in the dispersion curves of (c,f).

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Full scattering simulations performed using the commercial finite element package COMSOL [31] are shown in Fig. 7, these are for the even ZLM relevant for TM polarisation and for the triangular case; very similar ZLMs are found for the square case (not shown). The excitation is a line source just outside the leftmost edge of interface and a very clear ZLM is excited that can be identified, on each ribbon, with the eigenstate from the ribbon. One feature that is also evident is the long-scale wave envelope with a wavelength that alters with frequency; this can be described via an effective medium approach [38] as applied to edge states [29].

 

Fig. 7 ZLMs along the face-to-face interface between media composed of oppositely tilted triangles at normalised frequencies 5.62, 5.65, 5.67 illustrating the long-scale envelope and concentration of field to the interface. The source position is shown circled and the direction of the ZLM propagation is indicated by arrow.

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3.2. Energy transport around a sharp bend and energy splitting

We now consider two related, yet distinct, problems: redirecting energy around a sharp bend using just topology, see Fig. 8(a), which is the topological alternative to the photonic crystal waveguides pioneered in [6], and a three-way energy splitter (see Fig. 8(b)). Both panels in Fig. 8 are for inclusions placed around a square; the triangular case, despite creating a clear ZLM as in Fig. 7, is incapable of supporting a ZLM along the vertical interface as those interfaces, c.f. Fig. 5, do not have non-zero Berry curvature. The square arrangement of inclusions has the very useful property that the vertical interfaces are exactly the same as the horizontal ones; the added benefit of having two reflectional symmetries in the square inclusion case is now evident there is now non-zero Berry curvature along both interfaces and both support the same ZLMs. It is this insight that allows for the design of the splitter and allows transport around the bend.

 

Fig. 8 (a) Propagation around a π/2 bend at the interface between a quadrant of squares oppositely orientated to the others, c.f. Fig. 5(c). (b) The three-way splitter, where the quadrants alternate in their relative rotation, c.f. Fig. 5(d). In both panels the excitation is a monopole source, with position shown circled, located at the leftmost edge of the crystal on the interface between the media. The directions of the incoming ZLM and outgoing ZLMs are shown using arrows.

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To get a measure of the efficiency of the energy transport we calculate the electric intensity along each interface, i.e. $I={\displaystyle {\int }_S|E{|}^{2}dS}$ for equal areas and measure the efficiency as the ratio of the outgoing intensities to that of the incoming intensity. For Fig. 8(a) the outgoing interface gives 0.954 and for Fig. 8(b) the right, upper and lower interfaces give 0.286, 0.253 and 0.329 respectively.

We now want to optimise the transport properties. First, we may wish to minimise backscatter from the junction. To do so we note that the Fourier separation between the Dirac point of the unperturbed bulk dispersion curves and high-symmetry point is highly relevant for the transmission properties of the topological guide [25, 39]; transmission improves for short wavelength, as opposed to long wavelength, envelopes, hence, for transmission post the junction, it is desirable to increase the distance between the Dirac cone and the point X. The latter holds due to the connection between the bulk and projected bandstructures [40]; the Brillouin zone reduces to one-dimension because the only relevant wavevector component is along the interface. All wavevectors are projected onto the ΓX line in Fourier space, hence if the distance between X and the Dirac cone is increased then the Fourier separation between oppositely propagating modes, along the topological guide, is increased. A mechanism to do this is to alter the system parameters; Eq. (16) demonstrates that the slopes of the A and B bands can be increased or decreased by the system parameters thereby altering the position of the band intersection.

Second, we return to envelope wavelength and note that the distance from the source to the junction will play a role. If we took a finite length slab then fitting an integer, or half integer, numbers of envelope wavelengths along the lead interface gives a Fabry-Perot resonance with perfect transmission or perfect reflection from the far edge. We can use this knowledge to tune the system and the sharp bend is optimised by having a node of the long-scale envelope at the junction and so the energy is smoothly transported around the corner. Whilst for the splitter, the perfect reflection scenario concentrates energy at the junction for subsequent redistribution to the exit leads.

This is the first example of a three-way splitter passively created due to its topology and as a result of the inherited protection should be less prone to backscatter and therefore forms the prime candidate for three-way splitting in time reversal systems.

4. Concluding remarks

We have constructed the first-ever three-way topological beam-splitter for optics. Although we have concentrated upon the TM polarisation, it is clear that TE polarisation will also generate splitters using the geometrical designs we present; the main change being that the odd (and not even) modes, now excited by dipoles, would be the physically relevant fields. We briefly illustrate this in Fig. 9 for a square C_4v case; here the inclusions of Fig. 2(b) are augmented by a central inclusion of radius 0.15d. The situation is almost identical to TM, except that the bandgap is smaller (the action of the central inclusion is to help create the bandgap) and the decay of the edgestate, shown in Fig. 9(c), is slower.

 

Fig. 9 The magnetic, TE, case. (a) shows a square inclusion case, (b) rotated by π/15, and (c) the odd ZLM eigenstate (at normalised frequency 5.43).

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One crucial difference from the majority of the valleytronics literature; is that we have chosen to operate on a square, and not hexagonal, lattice; the hexagon arrangement has several advantages -the Dirac points are symmetry induced, and the band-gaps obtained by gapping them are broad. However, due to the two distinct zigzag interfaces, present within hexagonal structures, and the conservation of chirality and phase, only two-way topological energy-splitting is possible, [41–45]. For the square geometry, we have two interfaces, however due to the presence of TRS, they are effectively identical; therefore, the conservation of chirality and phase no longer restricts propagation along a third outgoing interface because the incoming ZLM needs only to couple to itself along all three outgoing leads.