Abstract
New structures with richer electromagnetic properties are in high demand for developing novel microwave and optic devices aimed at realizing fast lightbased information transfer and information processing. Here we show theoretically that a topological photonic state exists in a hexagonal LC circuit with shortrange textures in the inductance, which is induced by a band inversion between p and dlike electromagnetic modes carrying orbital angular momentum, and realize this state experimentally in planar microstrip arrays. Measuring both amplitude and phase of the outofplane electric field accurately using microwave nearfield techniques, we demonstrate directly that topological interfacial electromagnetic waves launched by a linearly polarized dipole source propagate in opposite directions according to the sign of the orbital angular momentum. The open planar structure adopted in the present approach leaves much room for including other elements useful for advanced information processing, such as electric/mechanical resonators, superconducting Josephson junctions and SQUIDs.
Introduction
To harness at will the propagation of electromagnetic (EM) waves constitutes the primary goal of photonics, the modern science and technology of light, expected to enable novel applications ranging from imaging and sensing well below the EM wavelength to advanced information processing and transformation. So far, systems with spatially varying permittivity and/or permeability, or arrays of resonators were explored, and EM properties unavailable in conventional uniform media have been achieved, such as negative refractive index, superlensing, cloaking and slow light^{1,2,3,4,5,6}, etc.
Inspired by the flourishing topological physics emerging in condensed matter^{7,8,9,10,11,12,13}, robust EM propagation at the edge of photonic topological insulators immune to backscattering from sharp corners and imperfections came into focus in the past decade. This is achieved by oneway edge EM modes in systems with broken timereversal symmetry (TRS)^{14,15,16,17,18,19,20,21,22,23}, and by pairs of counterpropagating edge EM modes carrying opposite pseudospins in systems respecting TRS^{24,25,26,27,28,29,30,31,32,33,34,35} (for a recent review, see ref. ^{36}). Topological photonic systems with TRS which avoid the need of application of external magnetic fields—albeit at the price of sacrificing partially absolute robustness—attract increasing interest since they are more compatible with semiconductorbased electronic and optical devices. In a twodimensional (2D) topological photonic crystal with C_{6v} symmetry proposed recently^{27}, p and dlike EM modes of opposite parities with respect to spatial inversion are tuned to generate a frequency band gap, and the sign of the orbital angular momentum (OAM) plays the role of an emergent pseudospin degree of freedom (for OAM and spin angular momentum, a related physical quantity, of EM modes in various circumstances, see previous works^{37,38,39,40}). While EM modes with pseudospin up and down are degenerate in bulk bands due to TRS, thus hard to manipulate, they are separated into two opposite directions in the topological interface EM propagation, which can be exploited for realizing novel EM functionality. However, up to now only field strengths along the interface between photonic crystals distinct in topology and transmission rates at output ports have been measured. Details of pseudospin states of the topological EM propagation remain unclear in topological photonic systems explored so far, which hampers their advanced applications (the valley degree of freedom and related OAM have been revealed in a photonic graphene by selectively exciting the two sublattices in terms of interfering probe beams^{41}).
In this work, we present the direct experimental observation on pseudospin states of unidirectional interface modes in topological photonic metamaterials. Based on the insight obtained by analyzing a lumped element circuit model with honeycombtype structure, we propose that topological EM propagation can be achieved experimentally in a planar microstrip array, a typical transmission line in the microwave frequency band^{42} constructed as a sandwich structure of bottom metallic substrate, middle dielectric film and top patterned metallic strips, which is common in various electronic devices. When the metallic strips form a perfect honeycomb pattern, linear frequencymomentum dispersions appear in the normal frequency EM modes, very similar to the Dirac cones in the electronic energymomentum dispersions seen in graphene. Introducing a C_{6v}symmetric texture with alternating wide and narrow metallic strips opens a frequency band gap. In addition, a band inversion between p and dlike EM modes arises when the interhexagon strips are wider than the intrahexagon ones, yielding a topological EM state mimicking the quantum spin Hall effect (QSHE) in electronic systems. Taking advantage of the planar and open structure of this metamaterial, we measure distributions of both amplitude and phase of the outofplane electric field along the interface between two topologically distinct microstrip regimes using nearfield techniques. EM waves are launched from a linearly polarized source located close to the interface. We resolve the weights of p and dlike EM components in the interface modes and clarify their dependence on the source frequency swept across the bulk frequency band gap. We further map out the circulating local Poynting vectors and reveal explicitly the pseudospin states locked to the propagating directions. The simple structure of the present topological microstrip device displaying local OAM in its EM modes enables easy fabrication and onchip integration, which is advantageous for harnessing EM transport inside the metamaterial, and potentially the system can be exploited for building novel microwave antennas which emit EM waves carrying OAM. Furthermore, the open 2D structure adopted in the present approach leaves much room for including other elements, such as electric/mechanical resonators, superconducting Josephson tunneling junctions, and SQUIDs, which are useful for advanced information processing.
Results
Topological phase transition in lumped element circuit
As a simplified model of our system (see Fig. 1a, b), we begin with a lumped element circuit shown schematically in Fig. 1c. Onnode capacitors with a uniform capacitance C establish shunts to a common ground plane. Link inductors with inductance L_{0} connect the nearest neighbor nodes within the honeycomb structure (drawn in red in Fig. 1c) and inductors with inductance L_{1} (shown in blue in Fig. 1c) connect to the next hexagonal cell. Topological LC circuits were proposed and realized previously^{34,35}, in which crosswirings with permutations were adopted to generate the nontrivial topology. In contrast, in the present approach, the nontrivial topology emerges purely from the symmetry of 2D honeycomb structure^{27,43}.
The voltage on a given node i with respect to the common ground is described by (see Supplementary Note 1 for details)
Taking the hexagonal unit cell shown in Fig. 1c, the normal frequency modes are governed by the following secular equation:
with \({\mathbf V} \!=\! {\mathbf V}_0{\mathrm{exp}}\left( {i{\mathbf{k}} \cdot {\mathbf{r}}  i\omega t} \right) \equiv [V_1V_2V_3V_4V_5V_6]^t{\mathrm{exp}}\)\(\left( {i{\mathbf{k}} \cdot {\mathbf{r}}  i\omega t} \right)\) for the voltages at the six nodes (see Supplementary Figure 1 for the numbering of nodes), where \(X = {\mathrm{exp}}(i{\mathbf{k}} \cdot {\mathbf{a}}_1)\), \(Y = {\mathrm{exp}}(i{\mathbf{k}} \cdot {\mathbf{a}}_2)\), \(\omega _0^2 = 1/L_0C,\) τ = L_{0}/L_{1}, and the asterisk means complex conjugating.
Figure 2 displays the frequency band structures for three typical values of τ. As shown in Fig. 2a for τ < 1, there is a global frequency band gap around 1.515 GHz, and the EM modes exhibit double degeneracy both below and above the frequency band gap. When τ = 1, the frequency band gap is closed at the Γ point, center of the Brillouin zone (BZ), and two sets of linear dispersions, known as photonic Dirac cones, appear with fourfold degeneracy at the Γ point as displayed in Fig. 2b (accidental Dirac cones were achieved before in square lattices and were used to manipulate EM transport with zero refractive index^{44}). For τ > 1, a global frequency gap reopens as shown in Fig. 2c. While the frequency band structures in Fig. 2a, c look similar to each other, the EM modes in the two cases are different which can be characterized by the eigenvalue of the twofold rotation operator C_{2}^{45,46}, or equivalently the parity with respect to the 2D spatial inversion, at the highsymmetry points of BZ (see Supplementary Figure 2 for details). For τ < 1 (Fig. 2a), the parities of the eigen EM modes below the frequency band gap are given by “+ − −” at both Γ and M points. In contrast, for τ > 1 (Fig. 2c), the parities are given by “+ + +” at the Γ point while “− + −” at the M point. With the parities of EM modes different at the Γ and M points, the case τ > 1 features a nontrivial topology. Therefore, the lumped element circuit exhibits a topological phase transition with the pristine honeycomb structure as the transition point.
The details of the phase windings at the Γ point as displayed in Fig. 2d–g reveal that the EM eigenmodes can be designated as p_{±} and d_{±} orbitals. A \(k \cdot p\) Hamiltonian can be formulated around the Γ point based on these four orbitals^{27}, where the 4 × 4 matrix is block diagonalized into two 2 × 2 matrices, associated with the EM modes with positive and negative OAM, respectively (see Supplementary Note 2 for details). This \(k \cdot p\) Hamiltonian takes the same form as the Bernevig–Hughes–Zhang model of QSHE proposed for HgTe quantum wells^{12}, where the two 2 × 2 blocks are associated with the electronic spinup and down states. Parallelizing these two Hamiltonians, it is clear that the sign of OAM of the eigen EM mode in the present topological microstrip arrays plays the same role as the spin in spin–orbit coupled electronic systems, indicating that the sign of the OAM behaves as an emergent pseudospin degree of freedom^{27,28,29,30,31,47,48,49}. In the case of Fig. 2c, band inversion between p and d orbitals takes place, which is thus topologically distinct from the case of Fig. 2a, in agreement with the conclusion derived from the analysis based on the eigenvalue of C_{2}.
While for simplicity and transparency, we describe the topological phase transition by a lumped element circuit with a twovalued inductance and uniform capacitance, the phenomenon is generic for circuits with textured capacitances and/or inductances. The system can also be reformulated in terms of propagating electric and magnetic fields with dielectric permittivity and magnetic permeability as relevant parameters. Therefore, the physics revealed here also applies to a broad class of planar networks of waveguides^{50} including coaxial cables and striplines.
Observing pseudospin and p–d orbital hybridization
We then implement experimentally the topological photonic state revealed above by designing the planar microstrip arrays as shown in Fig. 1b. Because the distributed capacitances of the metallic strips with respect to the ground plate estimated following the standard procedure^{42} are smaller than the lumped ones by one order of magnitude, the distributed capacitances can be incorporated to good approximation into the onnode capacitance, resulting in the lumped element circuit discussed above. The widths of the metallic strips in the trivial and topological designs are chosen in such a way that the two bulk systems give a common frequency band gap, taking into account the common lumped capacitance. In order to reveal explicitly the topological EM properties, we put these two microstrip arrays side by side as displayed in Fig. 1a, b. As shown in Fig. 3a, obtained by numerical calculations based on a supercell for the lumped element circuit, two frequency dispersions appear in the common bulk frequency gap due to the inclusion of the interface between the two halfspaces of distinct topology. It is interesting to note that these interface modes are characterized mainly by two degrees of freedom, namely pseudospin and parity as resolved experimentally below.
In order to detect these topological interface EM modes experimentally, we launch an EM wave from a linearly polarized dipole source located in the interface with a frequency within the common bulk frequency gap (see inset of Fig. 1b). It is noticed that injecting an EM wave into the system without disturbing the bulk frequency band is a feature inherent to the bosonic property of photons which is not available for electrons. As displayed in Fig. 3b, c for the distributions of the outofplane electric field E_{z} obtained by the fullwave simulations and experimental measurements (see Methods for details), respectively, the EM wave propagates only along the interface. Figure 3d shows the phase distribution of the outofplane electric field E_{z} obtained by the fullwave simulations, which exhibits clockwise/counterclockwise phase winding in the half of the sample to the left/right of the source, as is clearly revealed by the mirror symmetry with respect to a mirror line perpendicular to the interface and passing through the source (indicated by the two darkblue arrows in Fig. 3d). The two insets show the zoomedin views of the phase distributions in the two typical hexagonal unit cells close to the interface, with the left/right one accommodating the clockwise/counterclockwise phase winding, which specifies the down/up pseudospin state of EM modes. This demonstrates a clear pseudospinmomentum locking in the topological interface EM propagation, mimicking the helical edge states in QSHE.
Now we investigate the variation of phase distribution in the topological interface EM modes when the source frequency is swept across the bulk frequency band gap. The interface EM modes intersecting the frequency bands below and above the band gap are composed from both p and d orbitals, which can be resolved by analyzing the phase winding noticing that for a p/d orbital the phase winds 2π/4π over a hexagonal unit cell (see Supplementary Figure 3 and 4, and Supplementary Note 3 and 4 for details). As displayed in Fig. 3e–p obtained by the fullwave simulations and experimental measurements, at a frequency close to the lower band edge (Fig. 3e, h, k, n) the interface EM modes consist mainly of p orbitals, and at a frequency close to the upper band edge (Fig. 3g, j, m, p) the interface EM modes are predominately d orbitals, whereas p and d orbitals contribute equally at the center frequency of the band gap (Fig. 3f, i, l, o). Figure 3q displays the full frequency dependence of the weights of p_{+} and d_{+} orbitals evaluated in terms of the Fourier analysis on the phase distribution in the right zoomedin hexagon in Fig. 3d (same results are obtained for the left hexagon and p_{−} and d_{−} orbitals as assured by the mirror symmetry), with a systematic frequency shift of 0.03 GHz between the experimental results and the ones obtained by the fullwave simulations due to the tolerance of the material and structural parameters in the fabrication. Because the p and d orbitals correspond to the dipolar and quadrupolar EM modes, respectively, using a linearly polarized dipole source we can generate and guide EM waves with the desired sign of OAM by choosing the propagation direction, and desired relative weight of dipolar and quadrupolar EM modes by choosing the working frequency within the topological band gap. These properties may be exploited to design topologybased microwave antennas and receivers.
The OAM accommodated in the hexagonal unit cell of the present microstrip array is intimately related to the local Poynting vector through the Faraday relation (see Supplementary Note 5 for details). For a harmonic mode with frequency ω the local Poynting vector is given by
where \({\mathbf{E}} = E_z{\mathbf{z}} = \left {E_z} \righte^{i\varphi }{\mathbf{z}}\) (with x, y, and z being the unit vectors in the three spatial directions) and H are the outofplane electric field and the inplane magnetic field, respectively. For EM modes with fixed OAM such as \(p_ \pm\) and \(d_ \pm\) defined in the hexagonal unit cell, the local Poynting vectors circulate around the edges of the hexagon. It is obvious that \(p_ \) and \(d_ \) orbitals accommodate the Poynting vectors circulating clockwise whereas \(p_ +\) and \(d_ +\) orbitals accommodate those circulating counterclockwise, which correspond to the two pseudospin polarizations in the present system. One can evaluate explicitly the amount of angular momentum carried by the local Poynting vector given in Equation (4)
It can be shown (see Supplementary Note 5 for details) that for the EM mode \(E_z = \left {E_z} \right{\mathrm{exp}}\left( {i\varphi } \right) = \left {E_z} \right{\mathrm{exp}}\left( {il\theta } \right)\) where θ is the azimuthal angle and l = ±1 (for p_{±}) or l = ±2 (for d_{±}) one has \({\mathbf{L}} = \frac{{\left {E_z} \right^2}}{{2\mu \omega c^2}}l{\mathbf{z}}\). Therefore, one photon of energy ℏω carries a quantized OAM lℏ along the normal of microstrip plane.
The local Poynting vectors measured experimentally at 1.44 GHz are displayed in Fig. 4a for the two typical hexagonal unit cells close to the interface, and those obtained by the fullwave simulations at 1.47 GHz are displayed in Fig. 4b over a wide region along the interface (shown schematically in Fig. 4c). Good agreement is achieved between experiments and simulations. On the left/righthand side of the source, the local Poynting vectors circulate clockwise/counterclockwise in the hexagonal unit cells in both topological and trivial regimes. In hexagons above the interface (i.e., in the topological regime), the density of the local Poynting vectors is larger at the bottom edge (closer to the interface) than that on the top edge, which generates net energy flows in two directions along the interface. In hexagons below the interface (i.e., in the trivial regime), although the local Poynting vectors circulate in the same ways as those in the topological regime, the density of the local Poynting vectors is smaller at the top edge (closer to the interface) than that on the bottom edge, opposite to that in the topological regime, which therefore contributes the same net energy flows. This yields the winding phases, or equivalently the circulating local Poynting vectors, in hexagonal unit cells and the unidirectional energy flow along the interface. The distribution of the local Poynting vectors in Fig. 4 indicates explicitly that OAM of EM mode governs the topological interface EM propagation in the present system.
As seen above, the planar and open structure of the present system permits us to observe directly the pseudospin states, pseudospinmomentum locking, and furthermore the p–d orbital hybridization in the interface EM modes, which constitute the essence of the topological state preserving TRS. So far, the relevance of pseudospin in topological interface propagation has been inferred based on comparisons between experimental observations and theoretical analyses, and there are few experimental studies on the relative weight of orbitals with opposite parities in topological interface propagations.
Discussion
EM waves with OAM attract considerable current interest. It becomes clear that optical fields with OAM are ideal for many important applications such as communications, particle manipulation, and highresolution imaging^{51}. Even at microwave frequency, tunable OAM provides a new degree of freedom, which can be used for controlling onchip propagation and can be exploited to develop novel high information density radar and wireless communication protocols^{52,53}. The OAM explored in the present work is defined in unit cells and is oriented perpendicular to the propagating direction of the topological interface EM modes, different from OAM carried by light vortices in continuous media that is parallel to the propagating direction. To figure out a way to emit efficiently EM modes carrying OAM supported by the microstrip structure with C_{6v} symmetry into free space is one of the most intriguing future problems. The topological EM properties achieved in the planar circuit can not only be exploited for microwave photonics^{54} and plasmonics^{55}, but can also be extended up to the infrared frequency regime^{56}. In the photonic wire laser working in the terahertz band^{57,58}, a patterned, doublesided metal waveguide is used for confining and directing the emission from a quantumcascade laser, with the metal–semiconductor–metal structure essentially being the same as the microstrip array investigated in the present study. In terms of a honeycombpatterned network with typically micrometer stripwidths one can achieve a topological quantumcascade laser, where the emission and propagation of terahertz EM waves are governed by OAM. It is also worth noticing that the 2D structure of the present scheme makes the topological microwaveguiding compatible with various lithographically fabricated planar devices. Extension of the lumped element circuit discussed in the present work to a network including resonators of quantum features, such as quantum bits (qubit) based on SQUID structures^{59}, is of special interest.
Methods
Preparation of the sample and the fullwave simulations
To prepare the perimeter of the whole sample, we load lumped resistors between the metallic strips and the common ground plane (i.e., bottom metallic substrate), which corresponds to a perfect matching boundary condition. The values of lumped resistors are selected according to the characteristic impedances Z_{0} of microstrip lines, 115, 74, 97, and 66 Ω for microstrip lines with widths of 1.0, 2.6, 1.5, and 3.2 mm, respectively^{42}. In order to numerically simulate the system, we perform threedimensional fullwave finiteelement simulations using Computer Simulation Technology Microwave Studio software based on a finite integration method in the time domain. The dielectric loss tangent (tan δ) of the substrate and the conductivity of the metallic microstrip lines are set to be 0.0079 and 5.8 × 10^{7} S/m, respectively. The internal resistance in each lumped capacitor is taken as 1 Ω. An open boundary condition is applied for the whole sample including the terminal resistors. In order to demonstrate fully the unidirectional interface EM transport governed by pseudospin, we also prepare interfaces with sharp turns and stimulate the system by a source with signals overlapping exclusively with one of the two pseudospins of the interface modes (see Supplementary Figure 5 and Supplementary Note 6 for details).
Experimental setup
Signals generated from a vector network analyzer (Agilent PNA Network Analyzer N5222A) are transported into a port located in the sample, which works as the source for the system (see Supplementary Figure 6 and Supplementary Note 7 for details). A small homemade rod antenna of 2 mm length is employed to measure the outofplane electric field E_{z} at a constant height of 2 mm from the microstrip lines. We make sure by the fullwave simulations that the field distribution thus measured is almost the same as that at the very surface of microstrip lines. The antenna is mounted to a 2D translational stage to scan the field distribution over the whole system with a step of 2 mm. A finer step of 1 mm is taken in order to measure accurately the field distribution in several typical hexagonal unit cells, which reveals the pseudospin structure. The measured data are then sent to the vector network analyzer. By analyzing the recorded field values, we obtain the distributions of both amplitude and phase of the outofplane electric field E_{z}, which are used for analysis of detailed phase windings, weights of p and d orbitals, and local Poynting vectors in the topological interface states.
Code availability
All the computer codes that support the findings of this study are available from the corresponding authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
H. Chen and Y. Sun are supported by the National Key Research Program of China (No. 2016YFA0301101), the National Natural Science Foundation of China (Grant Nos. 11234010, 61621001, and 11674247), the Shanghai Science and Technology Committee (Nos. 18JC1410900 and 18ZR1442900), and the Fundamental Research Funds for the Central Universities. X. Hu is supported by GrantsinAid for Scientific Research No.17H02913, Japan Society of Promotion of Science.
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Y.L. prepared the sample and conducted experimental measurements and the fullwave simulations. Y.S., W.Z., Z.G., and J.J. helped with experiments. T.K. joined discussions on the model and theoretical analyses. H.C. and X.H. conceived the idea, supervised the project, and wrote the manuscript. All authors fully contribute to the research.
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