1. Introduction

Recently, topological insulators have received considerable attentions because of intriguing properties that topologically protected surface states are immune to backscattering, even at the presence of large impurities [1,2]. In analogy to electronic systems [3], optical quantum Hall effect was proposed theoretically using photonic crystals (PCs) [4] and unidirectional electromagnetic propagations were observed in microwave experiments [5]. These efforts opened the research area of topological photonics. PCs, consisting of periodic arrangement of low loss dielectrics, possess the ability to control the propagation of electromagnetic waves for a wide range of frequencies [6–9] which are a perfect platform to realize topological photonic properties. Optical and acoustic analogy to quantum spin Hall effect [10,11] were soon proposed and realized [12–18], where spin-polarized one-way edge states exist at the interfaces of photonic topological insulators. Using dielectric materials only is crucial that these unidirectional edge states can be extended to optical frequencies contributing to future optical communications. By engineering C_6v lattice symmetry in dielectric PCs [14] at microwave frequencies, we succeeded in demonstrating unidirectional photonic channels [17].

Whispering gallery mode (WGM) exists due to the total internal reflection on concave surfaces. As light wave of certain frequencies can be confined and allowed given a certain geometry, even a single molecular biosensing is possible using WGMs [19–22]. Various structures have been explored to sustain WGMs including microspheres, micro-rings and optical fibers [23–26]. By combining optical topological insulators and microcavity geometries, novel design principles of lasers were proposed [27,28] and other microcavities were also been designed for different applications [29]. Finite topological PC particles with discrete interface states were also theoretically discussed [30].

In this paper, we experimentally investigate the WGMs existing in two-dimensional (2D) PC cavities consisting of dielectric materials. Different from previously studied infinite system, we observe discrete WGMs propagating unidirectionally along the topological PC cavity circumference. In contrast to previous WGMs that have a strong dependence to cavity geometry, topological WGMs remain unchanged if we keep the same circumferences but allow different shapes in topological PC cavities. As pure dielectric PCs with reasonable permittivities are used in our microwave experiments, the structure can be scaled down to optical frequencies and a topological optical coupler is suggested.

2. Topological photonic crystal cavity

The schematic to construct WGM topological cavity using 2D PCs is depicted in Fig. 1(a). Six alumina cylinders in air constitute one unit cell of the triangular lattice of PC₁ and PC₂, whose lattice constant is a, relative permittivity of alumina ε = 7.5 and their diameters d = 0.24a. As enlarged in the inset, for such a composite artificial molecule, two geometrical parameters determine its photonic band properties: the intra (inter) couple distance h₁ (h₂) defined as the distance between neighboring (nearest) cylinders, where 2h₁ + h₂ = a. The intra (inter) couple distance is h₁ = 0.36a (h₂ = 0.28a) for PC₁, while h₁ = 0.30a (h₂ = 0.4a) for PC₂. As discussed in [17], the change between h₁ and h₂ induces a topological phase transition of the photonic band properties of the corresponding PCs and a topological (with h₁>h₂) PC₁ and a trivial (with h₁<h₂) PC₂ can be constructed, as shown in red and blue cylinders respectively. In [17], at the interface between semi-infinite PC₁ and PC₂, topological interface states with in-plane orbital angular momenta were observed and their unidirectional propagations were visualized.

 

Fig. 1 (a) Schematic picture of 2D finite topological PC cavity with the lattice constant a, the diameter of dielectric cylinders d = 0.24a and relative permittivity ε = 7.5. The radius of hexagonal topological PC₁ cavity (in red) is R = 3a surrounding by trivial PC₂ (in blue). (b) Discrete topological interface states (red dots) of PCs labeled by numbers calculated with R = 3a (left panel), R = 2a (middle panel) and R = 6a (right panel) respectively. (c) Nearly degenerate E_z eigenfield distributions of discrete WGMs of R = 3a cavity. The red and blue lines along the circumference of the topological cavity indicate “+” and “-” eigen-field distributions respectively. Two enlarged views next to the first-order mode distribution show phase differences at the topological cavity circumference (white dashed lines).

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Now we extend to a finite situation, where finite cavities made of topological PC₁ are considered. We start with a hexagon-shape cavity first. The side length of the topological PC cavity is R = 3a surrounding by trivial PC₂. The surrounding trivial PC₂ layers are set larger than the topological PC cavity side length to avoid unwanted coupling between topological cavities when periodic boundary conditions are applied. We consider Transverse Magnetic (TM, with $E_z$ polarization) polarization only in this work. Though this is a finite system, we still can obtain its photonic modes by doing eigenmode analyses of a periodic supercell as in Fig. 1(a) using finite element numerical solver COMSOL Multiphysics when setting the wave vector k = 0. The calculated modes are shown in the left panel of Fig. 1(b). The modes obtained within the common bulk photonic band gap frequencies between PC₁ and PC₂ correlate to the interface between topological and trivial PCs, as indicated by red dots where modes corresponding to either bulk PC₁ or PC₂ bands are shown in black dots. Five sets of discrete topological interface states are found numerically for the R = 3a cavity, as labeled next to the dots. The reason that WGMs exist in a microcavity is that constructive interference occurs when the interface states wind around the microcavity:

For traditional WGMs, Eq. (1) is satisfied only in the situation that an adiabatic change on the circumference occurs. When there is a sharp bending, the propagating wave number $k_{//}(\omega )$will change dramatically and thus WGM frequencies are very sensitive to the shapes of cavities. However, the interface state between a topological PC and a trivial one can even propagate at the existence of sharp corners [17]. In other words, these sharp bending and corners will not alter its propagation properties and WGM frequencies shall be insensitive to the shape of topological cavities. When we modify the shape of topological cavity from a hexagon to a rhombus, the number of topological WGMs and their eigenfield distributions are unchanged. Figure 2(a) depicts schematics of four rhombic cavities (left panels) and their eigenfield distributions of the zeroth-order modes (right panels). Even though the shapes of rhombic cavities are different, we maintain their circumferences to be the same as 18a, which is also the circumference of the hexagonal cavity in Fig. 1. The bottom edges of four rhombic topological cavities are 5a, 4a, 3a and 2a in turn. In all scenarios, five sets of WGMs are observed, which is a direct consequence of Eq. (1) with the same circumference L, and these modes frequencies and corresponding eigenfield distributions can be found in Figs. 2(b) and 2(a) respectively. This unique mode property of topological PC cavity owes to the unidirectional propagation of interface states allowed on the cavity circumference. Even if there are defects or fabrication imperfections on the PC cavities, as no backscattering is allowed, the WGMs are not perturbed. We acquire similar results for cavities with rectangle or triangle shapes in numerical simulations.

 

Fig. 2 (a) Eigenfield distributions of topological interface states of PCs at different shapes with identical normalized frequency and responding topological discrete states shown in (b).

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3. Experimental results

We carry out microwave experiments to verify and detect topological WGMs in PC cavity. The photo of experimental setup can be found in Fig. 3(a). Alumina cylinders with height h = 8 mm, relative permittivity ε = 7.5 and diameter d = 6 mm are used to assemble into PC₁ and PC₂ structures as shown in Fig. 1(a). The intra (inter) couple distance is h₁ = 9 mm (h₂ = 7 mm) for topological PC₁ highlighted by orange hexagonal region, while h₁ = 7.5 mm (h₂ = 10 mm) for trivial PC₂. The lattice constant is a = 25 mm and the hexagon cavity size is R = 3a = 75 mm. A chiral excited source indicated by a dashed circle locates at the center of one unit cell of topological PC₁. The pseudospin up topological interfacial state (with positive in-plane angular momentum, as denoted by the black circular arrow) is excited and propagates anti-clockwise along the cavity. The chiral source is constructed by four antennas with an anticlockwise π/2 phase decrease between neighbors [31,32]. Absorbing materials (in blue) are surrounding PCs to avoid unwanted scattering. As we are interested in TM polarization, the sample and source are assembled inside a parallel-plate microwave field mapper system where the time-harmonic $E_z$ field distributions are measured by a probing antenna connected to a vector network analyzer (VNA, Agilent E5071C) through the top plate (not shown here). The distance between the two parallel plates is 8.5 mm, where an extra 0.5 mm air layer between alumina cylinders and top plate is allowed to guarantee a smooth movement of the top plate in order to automatically measure field distributions.

 

Fig. 3 (a) Photo of experimental setup to observe topological WGMs in PC cavity by alumina cylinders with relative permittivity ε = 7.5, height h = 8 mm, diameter d = 6 mm and lattice constant a = 25 mm. The intra (inter) couple distance is h₁ = 9 mm (h₂ = 7 mm) for topological PC₁ (orange region), while h₁ = 7.5 mm (h₂ = 10 mm) for trivial PC₂. A chiral source shown in the inset is used to excite WGMs. (b) Total electric field intensity calculated and discrete modes consistent to eigenmode analysis are found. (c) Measured total electric field intensity spectrum by an integration of measured electric field amplitudes of all frequencies interested. (d) Measured electric field distributions of topological WGMs at 7.531 (upper panel) and 7.709 GHz (lower panel).

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In experiments, we inject electromagnetic waves at different frequencies through the antenna-array and the corresponding E_z field distributions are obtained. We then integrate amplitudes of electric fields of the whole sample region including both topological PC₁ and trivial PC₂. The frequencies are inside the common photonic band gap between PC₁ and PC₂. If a WGM existing along the cavity circumference is excited, we shall expect a largely enhanced integrated field. Figure 3(c) depicts our experimental results that the fields are enhanced at discrete frequencies. For comparison, we performed numerical simulations by COMSOL Multiphysics of electric field distributions of WGMs in topological cavity, whose parameters used are the same as those in experiments. We calculate electric field intensity of the simulated topological cavity at different frequencies shown in Fig. 3(b). Only discrete modes with very high fidelity are obtained and the frequencies are consistent with the discrete WGMs obtained from eigenmode analyses (Fig. 1, by setting a = 25 mm) and we acquire similar mode distributions. The corresponding mode indices are labeled. Due to the existence of air gap in our microwave field mapper system, the corresponding frequencies slightly shift from numerical simulations and the inevitable leakage of electromagnetic wave through the air gap lowers the mode fidelity of WGMs as in experiments. However, good consistence between the number of modes and corresponding mode indices can be found between experiments and simulations. We observe discrete WGMs existing on a topological PC cavity. The excited WGM distributions at the microwave frequencies 7.531 and 7.709 GHz are shown in the upper and lower panels of Fig. 3(d) respectively, corresponding to the upward and downward triangles in Fig. 3(c). We observe good confinement of the WGMs on the interface between PC₁ and PC₂ and the field distribution reflects the corresponding index m of WGMs. Moreover, by replacing the chiral source with a monopole antenna with no preferred in-plane angular momentum, identical mode spectrum as in Fig. 3(b) is obtained. However, the WGM will wind around the topological PC cavity clockwise/anti-clockwise if pseudospin excitation is used. This leads to the directional coupler design as follows.

4. Coupler

Our topological PC cavities are constructed with low loss dielectrics and the permittivity required is possible at optical and infrared frequencies. Thus their physical properties demonstrated in our microwave experiments remain unchanged if the sample is scaled to micron size. In numerical simulations, we propose a topological optical coupler device using silicon dielectric cylinders with lattice constant a = 0.25 μm, relative permittivity ε = 12.5 and diameter d = 0.06 μm. The intra (inter) couple distance is h₁ = 0.09 μm (h₂ = 0.07 μm) for topological PC₁, while h₁ = 0.075 μm (h₂ = 0.1 μm) for trivial PC₂. The configuration of coupler is shown in the left panel of Fig. 4(a). The topological PC₁ (orange regions) and trivial PC₂ (gray region) constitute a topological cavity and a waveguide where blue regions represent absorbing layers around PC structures. A topological interface state excited by a chiral source with pseudospin up state (indicated by a black circular arrow) at 561 THz propagates unidirectionally along the topological waveguide and couples evanescently to the WGM in topological cavity. Light can thus be trapped inside the topological PC cavity.

 

Fig. 4 The simulated results of an optical directional coupler. (a) At 561 THz, left-incident wave can couple to the topological cavity. The schematic diagram is shown in the left panel. PCs are composed of silicon dielectric cylinders with lattice constant a = 0.25 μm, relative permittivity ε = 12.5 and diameter d = 0.06 μm. (b) The WGM can couple to the right output if pseudospin up mode is excited at 539 THz.

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Moreover, WGMs in topological PC cavity can also directionally couple to external waveguide. The pseudospin up (down) mode will wind clockwise (anticlockwise) and propagate to left (right) of the external waveguide. Thus by selecting the proper excitation of different pseudospin states, a directional coupler is achieved. As shown in Fig. 4(b), at 539 THz, as pseudospin up source is used (indicated by the black circular arrow), WGM winding clockwise around the cavity can only couple to the rightward propagating mode in the external mode and its emission is to the right output of the waveguide. If we change the excited source to be pseudospin down with reversed in-plane angular momentum, anticlockwise winding along the cavity is expected and the emission is to the left output.

5. Conclusion

In conclusion, topological WGMs in 2D finite dielectric PC cavity are demonstrated in microwave experiments. Fruitful physics occurs when considering a finite geometry. The original continuous topological interface state dispersion on a semi-infinite structure becomes discrete and its physical origin is discussed. The number of modes is solely dependent to the circumference of the topological cavity. Different to traditional cavities, as the topological WGMs are backscattering immune, the detailed geometry of a topological PC cavity does not affect the number of WGMs. Using low loss dielectrics only, our topological PC cavity scheme shall be compatible to modern semiconductor fabrication process and a directional coupler working at optical frequencies is also proposed.