Observation of spatiotemporal dynamics for topological surface states with on-demand dispersion

Zhanyuan Zhang; Zhihao Lan; Pengbai Xu; Leiming Wu; Menglin L. N. Chen; Wei E. I. Sha; Yi Xu; Yuwen Qin

doi:10.1364/PRJ.530245

1. INTRODUCTION

Surface states are an interesting wave phenomenon that exists in various kinds of wave systems [1], such as Shockley electronic states at terminated semiconductors [2], elastic waves on the surfaces of solids [3], and surface plasmon polaritons [4,5]. They possess distinct physical properties that are absent in bulk media, facilitating numerous fundamental discoveries and important applications leveraged by their unique surface nature, including but not limited to topological physics [6 –8], photonic time crystals [9], communications [10,11], three-dimensional waveguiding [12,13], unidirectional propagation [14,15], lasing [16], and sensing [17], just to name a few.

Topological physics provides a new twist in manipulating guided waves at an interface, where unidirectional transmission immune to certain types of defects can be achieved [18,19]. As a result, topological photonic crystals (PhCs), especially PhC slabs supporting topological edge states, have attracted great attention because of their capability in manipulating light at chip scale. To date, several kinds of quantum-Hall-related topological electronic states in condensed matter physics have been extended to photonic systems [20 –37], facilitating various on-chip applications [34,38 –48]. In particular, dispersion management of topological PhCs can also provide an effective approach to realize precise manipulation of light. The dispersion management of either bulk bands or topological edge states enables new possibilities to realize higher-order topology [49,50], topological lasers [51], topological slow light [32,33,52,53], and topological rainbow trapping [54 –56]. However, interfacing two PhCs with opposite topological properties is indispensable to sustain topological edge states, which substantially limits the on-demand dispersion manipulation of these edge states. Topological surface states provide an effective means to overcome this limitation, which offer more degrees of freedom to modulate the dispersion of surface states [57 –60]. Although dispersion management of surface waves is crucial for achieving versatile functionalities [6 –9], on-demand manipulation of dispersion properties of the topological surface waves remains a promising but unsolved challenge. More importantly, the observation of spatiotemporal dynamics for these surface waves with on-demand dispersion, which can pinpoint handcrafted dispersion, is also absent, which prevents the advanced manipulation of electromagnetic (EM) surface waves in spatiotemporal dimensions.

In this paper, we theoretically propose and experimentally demonstrate a concept of topological surface states with on-demand dispersion, as shown in Fig. 1(a). The proposed topological surface states, where the electromagnetic field localizes at the interface between the PhC slab with modulated boundary cylinders and the air, are supported by a truncated valley PhC with surface modulation, which provides more degrees of freedom for achieving on-demand dispersion management of the surface states. Both the direction and magnitude of group velocity ( $v_{g} = d ω / d k_{x}$ , where $ω$ is the angular eigenfrequency) of the surface states can be tailored substantially. Microwave experiments for the observation of dynamical spatial trapping of EM surface waves with flat dispersion in the time domain and topological surface waves with opposite group velocities serve as two typical examples to validate the proposed concept. Our results could bring new insight in designing surface wave based integrated devices with functional applications, such as optical buffer, high- $Q$ cavity, and backward beam routing.

Figure 1.Topological surface states with on-demand dispersion. (a) Schematic of the topological surface states with on-demand dispersion. The dispersion of the topological surface states can be effectively manipulated by modifying the structural parameter $d_{s}$ . (b) The unit cell of the 2D kagome PhC supporting topological surface states. The lattice constant is $a$ , and the 2D unit cell is formed by three dielectric cylinders with a radius $r = 0.15 a$ . The relative permittivity $ε_{r}$ is 9.6. (c) Band diagrams of kagome PhCs with (blue, $d = 0.5 a$ ) and without (red, $d = 0.56 a$ ) inversion symmetry. The valley gap generated at the $K$ point by breaking the inverse symmetry of the unit cell is indicated by the gray region. (d) Phase distributions of $E_{z}$ at the $K$ point for the first and second bands. (e) Berry curvature for the first band of the unit cell without inversion symmetry shown in (c). (f) The super cell of the 2D PhC supporting topological surface states. The radius of the boundary cylinder is $r_{e}$ , and the distance between the boundary cylinder and the PhC is $d_{s}$ . (g) The eigenmode dispersion of the super cell shown in (f) with different structural parameters, where a typical electric field distribution $Re (E_{z})$ of the surface state is also provided in the inset. (h), (i) Schematics for the dynamical spatial trapping of the EM pulse and backward beam routing mediated by topological surface states with on-demand dispersion, respectively.

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2. RESULTS

A. Physical Properties of the Two-Dimensional PhC

The two-dimensional (2D) PhC is considered first, which will be generalized to a three-dimensional (3D) case, i.e., a PhC slab, in the next section. The 2D unit cell of the kagome PhC with lattice constant $a$ approximately equal to 26.7 mm is shown in Fig. 1(b), which consists of three dielectric cylinders in air with a radius of $r = 0.15 a$ and relative permittivity of $ε_{r} = 9.6$ . The height of the cylinders is infinite in the 2D case. The inversion symmetry of the unit cell can be tuned by adjusting the structural parameter $d$ . The band structures for the transverse magnetic (TM) modes ( $E_{z}$ polarized) of the kagome PhC with ( $d = 0.5 a$ ) and without ( $d = 0.56 a$ ) inversion symmetry are shown in Fig. 1(c). A Dirac point is formed between the first and second bands at the $K / K^{'}$ point when the inversion symmetry of the unit cell is maintained. Moreover, when the inversion symmetry of the PhC is broken (for example, $d = 0.56 a$ ), the Dirac point will be gapped out, giving rise to a valley gap marked by the gray shadow, as shown in Fig. 1(c). The phase distributions of the eigenmodes ( $E_{z}$ ) at the $K$ point for the bands below and above the valley gap are shown in Fig. 1(d), from which one can see that the phase windings of the two bands are opposite, indicating their distinct topological properties. The non-trivial topology in momentum space can be characterized by the Berry curvature, $F (k) = \nabla_{k} \times A_{n} (k),$ (1)where $A_{n} (k) = ⟨ u_{n k} | i \nabla_{k} | u_{n k} ⟩$ (2)is the Berry connection and $| u_{n k} ⟩$ is the spatially periodic part of the Bloch function for the $n$ th band. The Berry curvature distribution of the first band below the valley gap is shown in Fig. 1(e), which exhibits opposite signs at the $K$ and $K^{'}$ points, indicating that gapping out the Dirac points can lead to non-trivial valley Chern numbers at $K / K^{'}$ (note that the second band above the valley gap has the opposite feature to the first band and is thus not shown here).

By truncating the non-trivial PhC with a valley gap using boundary cylinders with a radius $r_{e}$ in the 2D super cell shown in Fig. 1(f), topological surface states with on-demand dispersion can be obtained, where Bloch and perfectly matched layer (PML) boundary conditions used in the simulations are also marked in the figure. Because of the surface state nature, one has more degrees of freedom to manipulate the dispersion properties of the topological surface states by modifying the structural parameters of the boundary cylinders. The radius $r_{e}$ of the boundary dielectric cylinders and the distance $d_{s}$ between the boundary cylinders and the bulk PhC are the key parameters to achieve on-demand dispersion of the topological surface states. The calculated eigenmode dispersions for seven typical cases in terms of $d_{s}$ and $r_{e}$ indicated in Fig. 1(f) are shown in Fig. 1(g) (see Appendix A). It can be seen that the dispersion of the surface states is changed from concave to convex by simply tuning $d_{s}$ . It means that the direction and magnitude of the corresponding $v_{g}$ for the surface states can be substantially modified, leading to reversed dispersion properties (see Appendix C). Because the dispersion of the surface states experiences continuous shape change, a flat dispersion can be achieved in the intermediate stage, as shown in Fig. 1(g), facilitating the dynamical spatial trapping of an EM pulse, as shown in Fig. 1(h). As can be seen from Fig. 1(g), changing $r_{e}$ will mainly shift the dispersion of a given shape, indicating the opportunity for achieving on-demand dispersion management at a given frequency (see Appendix D). At the same time, leveraged by the guided modes with reversed group velocities, on-chip backward beam routing could be achieved, which is presented in Fig. 1(i).

B. Physical Properties of the PhC Slab with Finite Height

As can be seen from the above analysis, the dispersion of the topological surface states in this 2D PhC can be manipulated precisely. Guiding by the total internal reflections at boundaries parallel to the $X - Y$ and $X - Z$ planes, we can generalize the topological surface states from 2D to 3D using a PhC slab with finite height. The schematic unit cell of the 3D kagome PhC is shown in the inset of Fig. 2(a), where the radius of the dielectric cylinders and the lattice constant are the same as those of the 2D PhC, whereas the height of the unit cell is $h = 1.3483 a$ . In order to obtain a topological bandgap similar to the 2D case, the inversion symmetry of the unit cell is broken ( $d = 0.6 a$ ). The band structures for the quasi-TM modes of the kagome PhC slab with (blue line) and without (red line) inversion symmetry are shown in Fig. 2(a), where the results are similar to the 2D cases (see Appendix E). This non-trivial unit cell with a topological bandgap is used to design a super cell that is similar to the 2D one, where the eigenmode dispersions of the surface states with different $d_{s}$ (see Appendix E) are shown in Fig. 2(b) (see Appendix A). The corresponding group indices ( $n_{g} = c / v_{g}$ , where $c$ is the speed of light in vacuum) are also provided in Fig. 2(c). Similar to the 2D case, the dispersion of the topological surface states also experiences the shape change from concave to convex with an increase of $d_{s}$ , and the group index approaches infinity when $d_{s} = 0.8038 a$ . It means that the $v_{g}$ of this topological surface state is nearly zero. This interesting phenomenon can be used to achieve dynamical light trapping based on topological surface states. It should be noted that the direction of $v_{g}$ is reversed for the case of $d_{s} = 1.0 a$ and a partial direction of $v_{g}$ is reversed for the case of $d_{s} = 0.7 a$ , which can also be seen from the band structures shown in Fig. 2(b). Additionally, the mapping relationship among $v_{g}$ , $d_{s}$ , and $r_{e}$ is too complex to be retrieved, which is out of the scope of this paper. The data-driven deep learning method could be a solution [61]. Figures 2(d) and 2(e) show the near-field distributions of the electric field [ $Re (E_{z})$ ], Poynting vectors, and its phase distributions $Arg (E_{z})$ for the topological surface states when $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ at $k_{x} = 1.1 π / a$ , respectively. The corresponding Poynting vectors form a clockwise or anticlockwise rotation around the phase vortex center. Therefore, electromagnetic energy flows from one supercell to its unilateral supercell, forming an effective wave-guiding channel. The near-field distributions for the super cell with other $d_{s}$ are provided in Appendix E. As can be seen from these figures, a phase vortex center can be clearly identified near the boundary, and the electric field ( $E_{z}$ ) is also strongly localized at the truncated surface without any leakage to the air, validating the surface state nature.

Figure 2.Eigen- and excitation solutions of topological surface states in a PhC slab. (a) The band diagrams of the kagome PhC with (blue, $d = 0.5 a$ ) and without (red, $d = 0.6 a$ ) inversion symmetry, where the 3D unit cell is shown by the inset. The height of the dielectric cylinders is $h = 1.3483 a$ . The light and dark gray areas indicate the light cone and the bandgap, respectively. (b) The eigenmode dispersion of the super cell for different structural parameters $d_{s}$ . (c) The corresponding calculated group indices of the topological surface states for different structural parameters shown in (b). (d), (e) Real parts of $E_{z}$ , Poynting vectors, and phase distributions of $E_{z}$ for the cases of $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ , respectively. (f) A finite PhC slab consists of 23 periods along the propagation direction. The red star indicates the location of the CPMD sources, where the energy fluxes $P_{L}$ and $P_{R}$ are calculated at two planes indicated by two blue dashed boxes ( $1.34 a \times 3.75 a$ ) labeled by $L$ and $R$ . Both planes are $3.5 a$ away from the boundary of the waveguide along the $Y$ -axis, where parts of them are inside the PhC ( $2.2 a$ in length) to take the whole surface state into account. The background material is air. (g)–(j) The calculated normalized transmission spectra for different structural parameters $d_{s}$ , which are obtained by integrating the power in the blue dashed boxes (see Appendix A). The value “1” (“ $- 1$ ”) represents the energy flux exclusively to the left (right) side. The red and blue curves represent the transmission spectra under the excitation of pseudospin-down and pseudospin-up CPMD sources, respectively. (k) The corresponding distributions of electric field $| E |$ under the excitation of the same CPMD source (pseudospin-down) shown in (g)–(j), where the red stars indicate the locations of the CPMD sources.

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To study the propagation properties of the topological surface states with on-demand dispersion, we consider a finite PhC slab (23 periods along the propagation direction) under the excitation of circularly polarized magnetic dipole (CPMD) sources (see Appendix A), as shown in Fig. 2(f). The normalized transmission spectra at different $d_{s}$ excited by CPMD sources with opposite pseudospin states (see Appendix A) are shown in Figs. 2(g)–2(j), where pseudospin dependent unidirectional propagation of the surface states can be clearly observed. It should be noted that for cases of $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ , the CPMD sources with the same pseudospin state are placed at the same $Y$ position in the waveguide, but their transmission spectra show completely opposite characteristics because of the reversed dispersion properties controlled by $d_{s}$ . The near-field distributions of the electric field $| E |$ for reversed dispersion properties under the excitation of the pseudospin-down CPMD source are shown in Fig. 2(k), which can intuitively validate the pseudospin-dependent unidirectional transmission characteristic (see Appendix F) and indicate the topological properties of the proposed surface states. In particular, for the case of ultra-flat dispersion ( $d_{s} = 0.8038 a$ ) with nearly zero $v_{g}$ , the electric field $| E |$ is mainly localized close to the CPMD source, and the dynamical EM wave trapping in the surface waveguide will be demonstrated in the following.

C. Experimental Demonstrations

Microwave experiments are performed to validate the theoretical results, where the experimental setup is shown in Fig. 3(a). A finite topological PhC slab composed of 23 periods along the $Y$ direction is fabricated using alumina ceramic cylinders ( $ε_{r} = 9.6$ , $\tan δ \approx 4 \times 10^{- 4}$ , $r = 4 mm$ , and $h = 36 mm$ ). The boundary cylinders ( $r_{e} = r$ , $h = 36 mm$ ) are introduced at the surface. Because a CPMD source at the considered frequency range is difficult to implement, a magnetic dipole is used to excite the topological surface states in the microwave experiment. A home-made loop antenna is placed inside the finite PhC waveguide, which is presented in the inset of Fig. 3(a) (see Appendix B), where it mimics a magnetic dipole with a dipole moment along the $X$ direction. The normalized spectra $| S_{21} |$ of an electric monopole antenna at the blue star marked in Fig. 3(a) indicate the transmission characteristics of surface states in the topological PhC slab, where different cases of $d_{s}$ are shown in Figs. 3(b)–3(e) (see Appendix B). The corresponding simulation results calculated by the 3D finite difference time domain (FDTD) method are also provided (see Appendix F). As can be seen from these figures, good agreement between the simulation and experimental results can be achieved, which confirms that the proposed topological surface states can be excited effectively by using the loop antenna. Furthermore, the distributions of $| S_{21} |$ measured at the area marked by the red dashed rectangle in Fig. 3(a) for different $d_{s}$ are provided in Figs. 3(f)–3(i). As can be seen from these experimental results, the EM fields are mainly localized at the surface of the PhC waveguide. Furthermore, the electric field shown in Fig. 3(h) is mainly trapped at the location close to the source, which agrees well with the simulation results shown in Fig. 2(k). Moreover, in order to verify the reversed dispersion properties for the cases of $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ in experiment, Fourier transform is used to analyze the near-field distribution acquired in experiment, where the corresponding retrieved band structures are shown in Figs. 3(j)–3(m). The corresponding simulation results of eigenmode dispersion (white dashed lines) are also provided, where they qualitatively agree with each other except for small frequency shift caused by the permittivity uncertainty of the cylinders. By increasing $d_{s}$ , the shape of the dispersion changes continuously, from a concave type to a flat type and to a convex type at last, enabling the realization of on-demand dispersion management for topological surface states. The flattest dispersion is obtained at $d_{s} = 0.8038 a$ , which can facilitate the experimental observation of dynamical spatial trapping of EM waves in the time domain.

Figure 3.Steady results of microwave experiments. (a) The experimental setup and fabricated dielectric PhC slab to demonstrate the topological surface states with on-demand dispersion. (b)–(e) Experimentally acquired $| S_{21} |$ (red points) and calculated normalized electric field amplitude $| E_{z} |$ (blue solid lines) for different structural parameters $d_{s}$ acquired at the blue star marked in (a) under the excitation of a loop antenna. (f)–(i) The distributions of $| S_{21} |$ for different $d_{s}$ measured by a monopole antenna at the red dashed box outlined in (a). (j)–(m) Experimentally retrieved band structures calculated by the Fourier transform of the steady states and their corresponding simulation results (white dotted line).

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Then we consider the propagation dynamics of the topological surface states with on-demand dispersion. The simulated spatiotemporal results are calculated by using 3D FDTD method (see Appendix A), where they are presented in the right column of Figs. 4(a)–4(c). In simulation, the central frequencies of the time domain Gaussian pulse are set to 4.361 GHz, 4.465 GHz, and 4.511 GHz for three different $d_{s}$ , while the width and offset of the Gaussian pulses are set to 40 ns and 100 ns, respectively. As can be seen from Fig. 4(b), the EM pulse is trapped at the position of $Y \approx 11.2 a$ for the case of $d_{s} = 0.8038 a$ , while for other cases ( $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ ) shown in Figs. 4(a) and 4(c), the EM pulses can transmit through the PhC along the $Y$ direction. To verify the theoretical results, inverse Fourier transform of the experimentally acquired results in frequency domain is leveraged to validate the dynamical spatial trapping of EM waves mediated by the topological surface states (see Appendix G). The retrieved results of spatiotemporal dynamics in experiment are provided in the left columns of Figs. 4(a)–4(c), whose central frequency, width, and offset of the Gaussian pulses are the same as those used in the simulation. It can be seen that the experimental results agree reasonably well with the simulation ones. For the case with the smallest $v_{g}$ [ $d_{s} = 0.8038 a$ shown in Fig. 4(b)], the electric field is trapped at the center of the waveguide rather than propagating along the waveguide. Meanwhile, the trapping position of the electric field in the waveguide can be effectively manipulated (see Appendix G). This interesting phenomenon may further facilitate the rainbow trapping based on topological surface states [54 –56]. Both the observations of different dispersion relationships shown in Figs. 3(j)–3(m) and spatiotemporal dynamics with distinct transmission properties shown in Figs. 4(a)–4(c) consolidate the possibility of flexible dispersion manipulation for topological surface states [62]. We further analyze the time dependent evolutions of surface states in momentum space, where the experimental and simulation results are shown in Figs. 4(d)–4(f). There is a long-life trapping mode for the case of $d_{s} = 0.8038 a$ with the smallest $v_{g}$ , where this mode exists in the range of $160 ns < t < 220 ns$ with an almost constant wave number. The lifetime of topological surface states in simulation is longer than the experimental one, which is because of the material losses in the experiment. It also should be noted that this long-life mode still decays because of the finite size effect and material absorption of the PhC slab. Meanwhile, there exists no EM wave for the cases of $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ that possess higher $v_{g}$ in this range. This interesting phenomenon indicates that the topological surface states with on-demand dispersion can achieve nearly zero $v_{g}$ , enabling the observation of dynamical spatial trapping of EM waves in the time domain.

Figure 4.Spatiotemporal results of microwave experiments. (a)–(c) Evolutions of experimentally retrieved $| Re (S_{21}) |$ in spatial and time domains, which are acquired along the blue line (Fig. 13 in Appendix G), and the corresponding simulation results are also provided. The red arrows marked in (b) indicate the dynamical positions of the Gaussian pulse. (d)–(f) Fourier transform of experimental results at every moment. The zoom-in views for I, II, and III marked by the white dashed boxes are provided. The simulation results are also provided.

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To further provide an intuitive picture, $| S_{21} |$ proportional to the electric field for three different $d_{s}$ at typical moments are presented in Figs. 5(a)–5(c). The full videos are available in Visualization 4, Visualization 7, and Visualization 8 (MP4). For the topological surface state with the largest $v_{g}$ shown in Fig. 5(a), it passes through the waveguide rapidly. With the increase of $d_{s}$ (see Appendix G), the topological surface state still exists in the waveguide at $t = 141.0484 ns$ , which indicates that the $v_{g}$ of topological surface states is significantly reduced. The case of a flat dispersion ( $d_{s} = 0.8038 a$ ) with the smallest $v_{g}$ is shown in Fig. 5(b), from which one can see that the topological surface state is effectively trapped at a certain position of the waveguide. Further increasing $d_{s}$ to $1.0 a$ , the dispersion property of the surface states reverses, and the corresponding $v_{g}$ increases, where it can also pass through the waveguide rapidly. According to the time domain data of the experiment, the corresponding $v_{g}$ for $d_{s} = 0.3 a$ , $d_{s} = 0.5 a$ , and $d_{s} = 1.0 a$ is $0.16 c$ , $0.08 c$ , and $0.13 c$ , respectively. These results agree well with the experimental results shown in Figs. 3(j), 3(k), and 3(m). It should be noted that the $v_{g}$ associated with the topological surface state at $d_{s} = 0.8038 a$ is difficult to quantitatively evaluate due to the spatial trapping of the EM wave. However, the corresponding $v_{g}$ can be approximately evaluated according to the retrieved band structure shown in Fig. 3(l), which is about $0.001 c$ . This $v_{g}$ is smaller than the one ( ${> 10}^{- 6} c$ ) calculated by eigenmodes because of the finite size effect.

Figure 5.Snapshots of spatiotemporal results in experiment. (a)–(c) The near-field distributions of $| S_{21} |$ for different $d_{s}$ at four typical moments.

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In order to further demonstrate the potential application of the topological surface states with on-demand dispersion, we design a triangular PhC waveguide structure for achieving beam routing, which is presented in Fig. 6(a) (see Appendix H). A dielectric waveguide with a square cross section (side length is 17.6 mm) and a dielectric resonator with a radius of 8 mm as well as a height of 11 mm are used to mimic point sources with different pseudo-spin states [34]. By modifying the input direction of the waveguide, the pseudospin states in the dielectric resonator can be changed effectively to realize unidirectional transmission. The normalized electric field amplitudes as a function of the frequency for $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ at the blue and red stars in the transmission direction are shown in Figs. 6(b) and 6(c), respectively (see Appendix B). The simulation results qualitatively agree with the experimental results, where the frequency shift between the simulation and experimental results may come from the difference of permittivity or size used between experiments and simulations. The corresponding near-field distributions $| S_{21} |$ of the steady states are provided in Figs. 6(d) and 6(e), and the dynamical results are provided in Figs. 6(f) and 6(g). The unidirectional transmission with very small backward scattering in both steady and dynamical results further consolidates that the proposed surface states possess topological properties. Meanwhile, these experimental results can also indicate that reversed dispersion properties can be achieved by modifying the structural parameter $d_{s}$ . Red arrows and blue triangles indicate the propagation directions of the EM pulse and the location of the PhC slab structure, respectively. It can be seen that reversed dispersion properties can be used to route the EM pulse along different directions, where negative propagation direction with respect to the input one can also be achieved as shown in Figs. 6(d) and 6(f). As demonstrated by all the simulation and experimental results, the proposed topological surface states with on-demand dispersion provide promising opportunities to precisely tailor the magnitude and direction of $v_{g}$ on the surface of photonic devices. Furthermore, our proposed PhC slab can effectively reduce the size of topological optical devices, facilitating compact optical integration of topological photonic devices.

Figure 6.Experimental results of backward beam routing. (a) The experimental setup and fabricated dielectric PhC slab to demonstrate the beam routing effect. (b), (c) Experimentally acquired $| S_{21} |$ (red points) and simulated electric field amplitude $| E_{z} |$ (blue solid lines) for different structural parameters $d_{s}$ obtained at the positions marked by the blue and red stars in (a), respectively, where the excitation of the loop antenna is placed on the dielectric waveguide. (d), (e) The acquired distribution of $| S_{21} |$ for $d_{s} = 0.3 a$ and $d_{s} = 1.0 a$ measured by a monopole antenna. Red arrows and blue areas indicate the direction of transmission and the location of the PhC slab, respectively. (f), (g) The typical near-field distributions of $| S_{21} |$ at a specified moment ( $t = 114.009 ns$ ). The central frequencies of the reconstructed pulses are set to 4.362 GHz and 4.507 GHz, respectively. And the corresponding pulse width and offset are set to 40 ns and 100 ns, respectively.

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3. DISCUSSION AND CONCLUSION

In summary, we theoretically propose and experimentally demonstrate a concept of topological surface states with on-demand dispersion. The dispersion properties of the surface guided modes can be feasibly tailored by changing the structural parameters of the boundary cylinders. The ultimate management of dispersion for the topological surface states enables the direct experimental observation of dynamical EM pulse trapping. Our experimental results validate the existence of topological surface states, and their EM dynamics further confirms the on-demand dispersion management. The unprecedented dispersion management of topological surface wave states might provide a paradigm shift for dispersion engineering in topological PhCs, which not only enables the on-demand manipulation of topological surface guided waves, but also effectively reduces the size of topological waveguides compared with conventional PhC waveguides based on topological edge states between two different PhCs. It is anticipated that our findings could pave new ways for the ultimate management of dispersion for the topological surface waves and shed new light on designing topological microcavities with higher $Q$ factor, surface state based rainbow trapping, and functional optical integration of topological PhC devices.

Category: Nanophotonics and Photonic Crystals

Received: May. 17, 2024

Accepted: Oct. 8, 2024

Published Online: Nov. 29, 2024

The Author Email: Yi Xu (yixu@gdut.edu.cn), Yuwen Qin (qinyw@gdut.edu.cn)

DOI:10.1364/PRJ.530245

CSTR:32188.14.PRJ.530245