Abstract
Topological insulators have unconventional gapless edge states where disorderinduced backscattering is suppressed. In photonics, such edge states lead to unidirectional waveguides which are useful for integrated photonic circuitry. Cavity modes, another type of fundamental component in photonic chips, however, are not protected by band topology because of their lower dimensions. Here we demonstrate that concurrent wavevector space and realspace topology, dubbed as dualtopology, can lead to lighttrapping in lower dimensions. The resultant photonicbound state emerges as a Jackiw–Rebbi soliton mode localized on a dislocation in a twodimensional photonic crystal, as proposed theoretically and discovered experimentally. Such a strongly confined cavity mode is found to be robust against perturbations. Our study unveils a mechanism for topological lighttrapping in lower dimensions, which is invaluable for fundamental physics and various applications in photonics.
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Introduction
Photonic crystals (PhCs) are periodic structures of electromagnetic materials which offer versatile tailoring of photonic spectrum and wave dynamics^{1,2,3}. Recently, photonic quantum anomalous Hall effects^{4,5,6,7,8,9,10,11}, photonic Floquet topological insulators^{12,13,14}, photonic quantum spin Hall insulators^{15,16,17,18,19,20,21,22}, topological photonic quasicrystals^{23,24,25,26}, and photonic Zak phases^{27,28,29} are realized or proposed using various PhCs. Topology^{30,31,32,33,34,35,36} is revealed as a mechanism for lighttrapping on the edges of PhCs^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29}, leading to topological surface states which are much more robust than conventional surface states^{3}. However, until now, topological lighttrapping at (sub)wavelength scale is achieved only for edges which have one less dimension than the bulk. Lowerdimensional lighttrapping protected by topological mechanism has not yet been discovered in photonics. In twodimensional (2D) photonic systems, such lowerdimensional wave localization, once realized, can lead to robust cavity modes (zerodimensional (0D) photonic states), which are demanded in photonics and hybrid quantum systems^{37}.
Here, we predict theoretically and observe experimentally robust lighttrapping into a 0D cavity mode in a 2D PhC, as induced and protected by the dualtopology mechanism. The characteristic localization length l_{loc} of the cavity mode is close to the wavelength in vacuum λ. The frequency of the topological cavity mode is found to be more robust than conventional PhC cavities. Strong and robust lighttrapping is ubiquitously useful in the stateofart photonics such as miniature photonic devices, integrated photonic/quantum chips, and cavity quantum electrodynamics. We emphasize that although the dualtopology mechanism was first proposed in electronic systems^{38,39,40}, it is much more feasible to observe and utilize dualtopology to induce wave localization in photonics than in electronics. In fact, observation of such d2 dimensional wave localization in a ddimensional electronic system has not yet been achieved.
Results
Dualtopology mechanism
The proposed structure is a rectangularlattice PhC with spatial periodicities a_{ x } and a_{ y } along the x and y directions, respectively. The photonic band gap (PBG) has a Chern number \({\it{{\cal C}}} = 1\) and the Zak phase^{33} along the Brillouin zone (BZ) boundary line XM is θ_{XM} = π. These two special properties constitute the nontrivial topology in wavevector space. On the other hand, a dislocation is an object with realspace topology: any closeloop of lattice translation including the dislocation has a mismatch between the starting and ending points. The vector connecting these points, the Burgers vector, serves as the realspace topological invariant (see Fig. 1a).
To elucidate the lighttrapping mechanism through the dualtopology, we employ the cutandglue technique^{38} (see Figs. 1b and c) which consists of two steps: first, a chunk of PhC with trivial topology is inserted into the dislocation structure which is then split into two halves. This step introduces two oneway edge channels at the opposite boundaries of the chunk, due to the wavevector space topology. The dispersions of these two edge channels must intersect at a timereversal invariant wavevector. The nontrivial Zak phase along the XM line ensures such an intersection to be at \(K_x = \frac{\pi }{{a_x}}\) ^{27,38}. For a finitewidth chunk, the tunnel coupling between the opposite edges opens a gap at the intersection (Dirac) point. These two coupled edges can be described by the 1D massive Dirac Hamiltonian, \({\it{{\cal H}}} = vq_x\sigma _z + m\sigma _x\), where v is the group velocity of the edge states, σ_{ z } = ±1 represents the two edge channels and q_{ x } ≡ k_{ x } − K_{ x } is the wavevector relative to the Dirac point. The Dirac mass m (m is real, see Supplementary Note 1) depends on the interedge coupling. In the weak coupling regime, the Dirac mass is determined by the overlapping integral of the electromagnetic fields of the two edge states^{3}, i.e., \(m \propto {\int} {\mathrm{d}\,{\mathbf{r}}\,\left( {{\mathbf{E}}_ + \cdot \widehat \varepsilon \cdot {\mathbf{E}}_  ^ \ast + {\mathbf{H}}_ + \cdot \widehat \mu \cdot {\mathbf{H}}_  ^ \ast + {\mathrm{c}}.{\mathrm{c}}.} \right)}\) where the subscripts ± represent the edge states at the upper and lower boundaries, respectively. Due to the dislocation, the Dirac mass becomes position dependent, since the upper and lower edges experience different numbers of lattice translations (see Fig. 1b). If the Burgers vector is B = (a_{ x }, 0), the phase difference between the two edge states will experience a πphase elapse across the dislocation, since K_{ x }a_{ x } = π. This πphase elapse leads to a signchange in the Dirac mass, forming a mass domainwall at the dislocation. According to the Jackiw–Rebbi theory^{30}, there will be a photonicbound state localized at the dislocation center.
The second step in the cutandglue procedure is to glue the two halves together by reducing the width of the chunk gradually to zero. As the width of the chunk reduces, the interedge coupling becomes stronger and stronger. The magnitude of the Dirac mass increases and the edge states are gradually moved into the bulk bands. However, because the Dirac mass domainwall persists as it is guaranteed by the real and wavevectorspace topology, the Jackiw–Rebbi soliton mode always exists in the PBG, even when the chunk of topologically trivial PhC is removed.
The above scenario of lighttrapping by the dualtopology mechanism can be mathematically summarized as^{38,39,40,41,42}
where \({\mathbf{\Theta }} = \left( {\frac{{\theta _{{\mathrm{XM}}}}}{{a_x}},\frac{{\theta _{{\mathrm{YM}}}}}{{a_y}}} \right)\) with θ_{XM} and θ_{YM} being the Zak phases along the XM and YM lines with contributions from all bands below the gap, respectively, and N_{loc} = 0 or 1 is the number of localized photonic modes. The existence of the mass domainwall and the soliton mode has a Z_{2} nature, reflecting whether the Dirac mass switches sign or not (see Supplementary Note 2). We emphasize that the Chern number, though not appeared in the above equation, provides the ground for the edge states and the Jackiw–Rebbi soliton^{41,42,43,44}. This crucial requirement differs our mechanism from those in refs. ^{45,46}, beside the strong wave localization observed in this work.
Design and characterization of the PhC
To realize lighttrapping due to the dualtopology mechanism, we design a rectangularlattice PhC with two yttrium iron garnet (YIG) cylindrical pillars in each unit cell. With the metallic cladding above and below, the bulk photonic bands of interest here are the 2D transversemagnetic (TM) harmonic modes. Magnetized by a magnetic field of 900 Oe along the z direction, a PBG (denoted as PBG II) between the third and fourth bands with Chern number \({\it{{\cal C}}} = 1\) is developed for a_{ y } = 2a_{ x } = 24 mm, R = 2 mm, and d = 17 mm. This topological PBG is realized by the gyromagnetic effect which gaps out the two Dirac points (indicated by the red arrow) on the YM line (see Fig. 2a). The dispersions along the BZ boundary lines YM and XM after the gap opening are shown in Fig. 2b together with the Zak phases along these lines. The Zak phases and the Chern number are calculated numerically using the Wilsonloop approach (see Supplementary Note 3). The electric field profiles at high symmetry points (Fig. 2b) indicate connection and consistency between our diatomic photonic unit cell and the Su–Schrieffer–Heeger model^{31}: the parity inversion between the X and M points leads to θ_{XM} = π, while the absence of parity inversion between the Y and M points gives θ_{YM} = 0^{27,33}. We obtain from numerical calculations that the Zak phases are θ_{YM} = 0 and θ_{XM} = π for the first three bands (Fig. 2b).
The edge states at opposite boundaries indeed intersect at \(K_x = \frac{\pi }{{a_x}}\), as verified by the finiteelement simulation method (see Fig. 2c), which is consistent with the nontrivial Zak phase along the XM line, θ_{XM} = π ^{27,38}. The edge states dispersion measured in experiments using the Fouriertransformed field scan method (see Methods) agrees well with the dispersion from the finiteelement simulation, as shown in Fig. 2c. The nonreciprocal photon flow along the edge channel, characterized by the difference between the forward and backward transmission, is presented in Fig. 2d. Pronounced nonreciprocal photon flow exists in the frequency window of 12.21–12.84 GHz (indicated by the black dashed box in Fig. 2d), while relatively weak nonreciprocal photon transport exists for lower frequencies as well. The bulk band gap is between 12.05 and 12.60 GHz (see Supplementary Note 4). The edge states spectrum spans a much larger frequency range of about 10.7–13.2 GHz. Pronounced nonreciprocal transmission may appear only in a fraction of this range, which is possibly caused by the impedance mismatch between the feed probe and the edge/bulk states for the finitesize sample studied in our experiments. A higher frequency window of nonreciprocal transmission is also visible, which is likely due to the nonreciprocal edge states in the higher PBG (the shallow red and blue curves in Fig. 2c) and irrelevant to the study in this work.
Photonic realization of the dualtopology mechanism
In our design, the dislocation is formed by first taking half column of unit cells (indicated by the reddashed rectangular region in Fig. 3a) away from the perfect PhC. We then compress the rest of the PhC to the vacancy left to restore the lattice order. This can be done by continuous deformation of the whole PhC, leaving only the dislocation center as a structure defect. The final appearance of the sample measured in experiments (including an inserted chunk of PhC with trivial band topology) is given in Fig. 3b. The topologically trivial PhC is designed using the same YIG pillars but with different lattice constants \(\left( {a_x \!\! \prime = a_x/2} \right),\left( {a_y \!\! \prime = a_y/2} \right)\) and the distance between the pillars (d′ = 5 mm) (see Supplementary Information Secs. 5 and 6).
To reveal the signchange feature in the edge states, we use a twopoints pumping scheme. The feed source from a vector network analyzer Agilent E8363A is divided into two branches by power divider and connected to the two feed probes (indicated by the red asterisks in Fig. 3b) through a pair of Gore phase stable cables, which guarantees that the two feed probes have nearly the same signal (including magnitude and phase). The signchange picture in Fig. 1b is confirmed by the finiteelement simulation of the edge states for the dislocation structure with an inserted chunk of topologically trivial PhC (see Fig. 3c): on the left (right) side of the dislocation the two edge waves are of opposite (the same) phases. Such a signchange feature, being a smokinggun signature of the Dirac mass domainwall, is confirmed in our experiments using the twopoints pumping scheme (Fig. 3d). Although each feed probe pumps only one of the edge states, the relative phase of the two edge states clearly indicates the π phase elapse induced by the dislocation.
To further verify the cutandglue picture, we study the spectrum and field profiles of the photonic states within the PBG II during the gluing processes using finiteelement simulations. To employ periodic boundary conditions in the finiteelement simulation, we use a supercell with two chunks of trivial PhCs and four dislocations (indicated by the green boxes in Fig. 4) to calculate the eigenspectrum and the field profiles. The Burgers vectors of the four dislocations are summed up to zero which is crucial for the periodic boundary condition. The supercell has a length of L_{ x } along the x direction, while the length along the y direction depends on the thickness of the two chunks of topologically trivial PhCs L_{trivial}. We start from the situation with \(L_{{\mathrm{trivial}}} = 4a_y \!\! \prime\) where the photonic spectrum in Fig. 4a indicates merely bandfolding of the edge states on all the boundaries. The field profiles in Fig. 4d and e show that the two opposite edge channels are nearly independent and uncoupled. When the thickness is reduced to \(L_{{\mathrm{trivial}}} = a_y \!\! \prime\), the coupling between opposite edge channels become considerable. As shown in Fig. 4b, the spectrum of edge states no longer spans the whole PBG, a small gap is developed due to such coupling. The magnitude of this gap is equal to 2m where m is the Dirac mass for the edge states. Within this edge gap there are photonic states which are weakly localized on the dislocations, i.e., the Jackiw–Rebbi modes. The localization lengths are comparable with L_{ x }/2, as indicated in the electric field distributions in Fig. 4f and g. Hence, the spectrum of the Jackiw–Rebbi states is still dispersive, due to considerable coupling between nearby Jackiw–Rebbi modes. The field profiles in Fig. 4f and g indicate strong mixing between opposite edge channels as well as weak field localization on the dislocations. Finally, for L_{trivial} = 0, the ingap photonic spectrum becomes nearly flat (Fig. 4c), indicating strongly localized states. Indeed, the electromagnetic fields are entirely localized on the dislocations, as shown in Fig. 4h and i. In this case, all the edge states are moved into the bulk bands, since the two chunks are entirely removed from the structure. In this way, the cutandglue picture is completely manifested in electromagnetism through finiteelement simulation of the Maxwell equations.
Experimental observation of the topological cavity mode
The experimental setup for the dislocation structure and the measurement scheme is shown in Fig. 5a. The electromagnetic wave is excited through the feed probe near the bottom cladding and detected by the detectprobe near the top cladding. The feed probe is fixed at a position near the dislocation, while the detection position is changeable. Both the amplitude and the phase of the local electromagnetic fields are scanned using a 2D mapping system in a frequencyresolved manner (see Methods).
Our finiteelement simulation study indicates that there is no localized state in the topologically trivial PBG between the second and the third bands (denoted as PBG I, 8.12–10.17 GHz), whereas there is one cavity mode in PBG II (topologically nontrivial PBG, 12.05–12.60 GHz) which is localized on the dislocation (see Fig. 5b). This observation confirms that the emergence of the midgap cavity mode is solely due to the dualtopology mechanism: without the nontrivial topology in wavevector space, the dislocation alone cannot induce 0D lighttrapping. The electromagnetic field profile of the cavity mode indicates strong lighttrapping on the dislocation with a localization length l_{loc} = 1.0λ (estimated from \(l_{{\mathrm {loc}}} = \sqrt {A/\pi }\) where A is the Gaussian modalarea). Experimentally, we measure the transmission between two points which are located at different sides of the dislocation (inset of Fig. 5c). The transmission has only one peak at 12.37 GHz in the PBG II (the shaded regions in Fig. 5c–f), indicating only one localized midgap mode. The mode profile measured at the peak frequency in experiments is comparable with the field profile of the topological cavity mode from the finiteelement simulation (see the insets of Fig. 5c).
We now study the transmission and the field profiles when the dislocation structure is perturbed. For instance, we replace one of the YIG pillar close to the dislocation by a metallic pillar of the same size. As shown for the cases in Fig. 5d–f, the topological cavity mode is found to be resilient against such perturbations. For all these cases, there is only one peak in the transmission spectrum in the PBG II, which reflects the robustness of the cavity mode and the topological lighttrapping mechanism (for simulations comparable with the experimental results, see Supplementary Note 7). From the experimental observations, we notice that the frequency of the topological cavity mode is also stable against perturbations. The peak frequencies in Fig. 5d–f are 12.43, 12.50, and 12.43 GHz, respectively (corresponding to a relative change of 0.5%, 1%, and 0.5% compared to the unperturbed one, separately). A comparative study with conventional photonic cavities formed by defect modes in a PhC with a comparable PBG (but topologically trivial) shows that the frequency of the topological cavity mode is more stable than the conventional PhC cavities (see Supplementary Note 8). For the latter system, the number of cavity modes can also be modified when a nearby dielectric pillar is replaced by a metallic pillar of the same size, whereas there is always one topological cavity mode due to the dualtopology mechanism in the PBG. This observation is another evidence that the topological cavity mode is more stable than conventional PhC cavities.
Discussion
Achieving topological lighttrapping at d2 dimensions in a ddimensional photonic system opens up the possibility of realizing many new physical effects and applications. One possible application in photonic circuitry, a waveguidecoupler which couples two chiral edge channels through the topological cavity mode, is demonstrated in the Supplementary Note 9. The lowerdimensional topological lighttrapping unveiled in this work gives rise to several important open questions: whether dualtopology can be exploited to induce wave localization in nonHermitian or nonlinear systems where topological lasing^{28,47,48,49}, paritytime symmetry, and other important effects can emerge^{50}; what are the consequences when such topological cavity modes are coupled with quantum dots or other singlephoton emitters; how to extend to higher dimensions, such as a topological waveguide induced by a 1D dislocation line in a 3D PhC; can the dualtopology mechanism be extended to quasiperiodic photonic systems^{23,24,25,26} where higherdimensional physics can be simulated in lowdimensions? All these questions can lead to interesting physics and applications in the future. Our work paves the way toward lowerdimensional topological wave localization through dualtopology.
Methods
Materials and sample fabrication
All the samples in this paper are fabricated by the low loss commercial YIG ferrite pillars. The saturation magnetization is measured as 4πM_{s} = 1884 G by a vibrating sample magnetometer and relative permittivity is retrieved as 15.26–0.003i by the transmission/reflection method which can be treated as a constant at the microwave frequencies of interest (i.e., the PBG II). The fired ferrite is machined into pillars with a radius of R = 2 mm and height h = 10 mm. The topologically trivial PhC is realized by reducing the lattice constants to \(a_x \!\! \prime = a_x/2\) and \(a_y \!\!\prime = a_y/2\), as well as d → d′ = 5 mm, while keeping the radius, height, and material of the pillars unchanged.
Experimental setup and measurement
The measurement setup for the topological dislocation, as schematically shown in Fig. 5a, consists of a vector network analyzer Agilent E8363A, a 2D mapping system, in a structure with top and bottom metallic cladding using aluminum plates. The upper (fixed) metallic plate has an area of 1 × 1 m^{2}. The lower metallic plate (movable) has an area of 0.5 × 0.5 m^{2}. The mapping area can be as large as 0.5 × 0.5 m^{2} when a single detectprobe is used. An array consisting of 364 permanent magnet NdFeB pillars are embedded into a 3mmthick aluminum plate (the lower metallic plate) in the sample with the dislocation. This plate works simultaneously as the metallic cladding as well as the external magnetic field source. Each NdFeB pillar is of radius 1.5 mm and height 3 mm. It can induce maximally 2800 Oe surface magnetic field. These NdFeB pillars apply onetoone external magnetic field to the YIG pillars set between the magnet plate and top aluminum plate. Since the NdFeB pillars are outside of the metallic cladding, they do not affect the electromagnetic waves inside the cladding, except providing the magnetic fields. On average, the NdFeB pillars provide an external magnetic field of about 900 Oe. The whole photonic structure is surrounded by microwave absorbers. The field profiles are measured by scanning the electromagnetic fields through changing the position of the detectprobe.
Band structure and simulation
The band structure and all the simulations were calculated by using the commercial software COMSOL MULTIPHYSICS with the RF module.
Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
F.F.L., Q.L., P.C., R.X.W., and Y.P. thank the support of National Natural Science Foundation of China (NSFC Grant Nos. 61671232, 61771237). Y.P. thanks partial support of Project Supported by Fundamental Research Funds for the Central Universities and Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. H.X.W., Z.X., and J.H.J. acknowledge supports from the National Natural Science Foundation of China (NSFC Grant No. 11675116) and the Soochow University. S.J. acknowledges support from Natural Sciences and Engineering Research Council of Canada.
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J.H.J. initiated the project and led the collaboration. J.H.J., Y.P., R.X.W., and S.J. guided the research. H.X.W., Z.X., and J.H.J. designed the photonic architecture, and performed theoretical analysis and calculations. F.F.L., Q.L., P.C., R.X.W., and Y.P. designed and conducted the experiments. All authors contributed to the analysis of results and the underlying mechanisms. J.H.J. wrote the manuscript.
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Li, FF., Wang, HX., Xiong, Z. et al. Topological lighttrapping on a dislocation. Nat Commun 9, 2462 (2018). https://doi.org/10.1038/s4146701804861x
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DOI: https://doi.org/10.1038/s4146701804861x
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