Journals Articles Conferences News About CLP
Submit a paper Email Alert
Search
Search
Articles
Journals
News
Advanced Search
Top Searches
2D MaterialsTransformation opticsQuantum PhotonicsSmall lasersSemiconductor UV PhotonicsSupersymmetric microring laser arrays

Photonics Research, Volume. 13, Issue 8, 2159(2025)

Continuously tunable topological negative refraction via a tailorable Bloch wavevector in momentum space Editors' Pick

Yidong Zheng1、†, Jianfeng Chen2,4、†, Zitao Ji1, and Zhi-Yuan Li1,3、*
Author Affiliations
  • 1School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
  • 2Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore
  • 3State Key Laboratory of Luminescent Materials and Devices, South China University of Technology, Guangzhou 510640, China
  • 4e-mail: jfchen@nus.edu.sg
    • show less
    Full TextGet PDF
    Figures&Tables (15)
    Equations (6)
    References (38)
    Tools
    Paper Information
    Topics

    Topological photonics provides a strategy that makes light transmission immune to structural-defects-induced backward scattering. Leveraging this, topological negative refraction enables robust, reflectionless light deflection, but directly controlling the refraction direction remains challenging. We demonstrate continuously tunable topological negative refraction at the interface between a one-way waveguide state and a free-space beam, overcoming the limitations of fixed refraction angles in conventional systems. The key insight is the ability to adjust the wavevector of the incident one-way waveguide state. Through manipulating the Bloch wavevector of the waveguide states in momentum space, we achieve a transition from negative to positive refraction. The unidirectional nature of these states prevents backscattering from defects, ensuring immunity to imperfections. As a prototypical demonstration, we achieve dynamic steering of refraction beams from -38° to +12° through active magnetic bias control. Our findings provide an exotic pathway for photon manipulation and a promising route toward topological photonics applications.

    1. INTRODUCTION

    Negative refraction is a well-known counterintuitive transmission phenomenon that exhibits the extraordinary ability of artificial materials and structures to manipulate electromagnetic waves and light. Negative refraction was first demonstrated in electromagnetic metamaterials [1] and later realized in photonic crystals [2] and artificial hyperbolic materials [3]. More recently, natural hyperbolic materials have also exhibited negative refraction at sub-wavelength scales [4–6]. The key to achieving negative refraction lies in constructing left-handed materials with specific energy flux-momentum relationships, where S·k<0. These materials and structures produce a negative in-plane momentum k||, resulting in a negative refraction angle θ according to Snell’s law k∥=k0 sin θ. More intriguingly, when the momentum k of the incident light is dynamically controlled over a wide range, tunable negative refraction—and even a transition to positive refraction—can be achieved [7–10]. Such tunable negative refraction holds significant potential for applications in beam steering [11], focusing [12–14], and imaging [15,16]. Notwithstanding such development, electromagnetic waves and light transmitted through these negative refraction materials and structures experience strong dispersion and large wavevectors, making them highly susceptible to inevitable reflections from imperfections.

    Recently, the introduction of topological protection has endowed negative refraction with robustness and exceptional features, such as being reflectionless. Several instances of topological negative refraction have been demonstrated in photonic valley-Hall [17–19] and Weyl systems [20]. Similar topological refraction has also been observed in acoustic systems [21–24]. However, these systems are constrained by limited Bloch wavevectors of the incident beam, typically concentrated near the valleys. This causes the refracted beam to be emitted at relatively fixed angles, limiting flexibility for practical applications. Additionally, the confined Bloch wavevector might lead to the energy being transmitted along the interface, rather than radiated into free space, thereby reducing the outcoupling efficiency of the refracted beam (see Appendix F). Hence, achieving a tunable topological negative refraction remains elusive.

    In this paper, we are the first to demonstrate tunable topological refraction across wide spectral and spatial ranges. The key insight is achieving a one-way state with a broad, one-way Bloch wavevector in momentum space. Specifically, changes in the incident k induce variations in the in-plane momentum k∥, resulting in a significant change in the refraction angle θ [Fig. 1(a)]. We achieve this by constructing a channel using two antichiral gyromagnetic photonic crystals with identical magnetization, separated by a distance. This configuration supports a one-way waveguide state with a monotonic dispersion, enabling continuous tuning of the Bloch wavevector by varying the excitation frequency [Fig. 1(b)]. The tuning of the Bloch wavevector along the high-symmetry line K′−M−K [Fig. 1(c)] allows control over the emission angle of the refracted beam, enabling steering from negative to critical to positive refraction [Fig. 1(d)]. Simultaneously, the one-way nature of a waveguide state ensures immunity to the imperfection on the transport path and avoids backscattering losses. Furthermore, we fabricated a prototype device for actively tuning the refraction angle through an external magnetic bias, achieving topological beam steering over a wide range from −38° to +12° at a specific frequency.

    Illustrations of tunable topological negative refraction. (a) Schematic illustration. Through tailoring the Bloch wavevector k of incident one-way topological states, the variation of in-plane wavevector k∥ can turn the refracted wave from negative to positive refraction, according to Snell’s law k∥=k0 sin θ. The gray dashed line represents the normal line of refraction. (b) Band structure of the topological photonic crystal that supports a one-way state. As the frequency increases, the Bloch wavevector of the one-way state moves along the high-symmetry lines K′−M−K. (c) Tailoring the Bloch wavevector of topological states in two-dimensional k-space. (d) Designed gyromagnetic photonic crystal with oppositely magnetized sublattices along the z-axis. Tailoring the Bloch wavevectors of topological states in k-space drives a transition from negative (θ1<0°) to critical (θ2=0°) to positive (θ3>0°) refraction.

    Figure 1.Illustrations of tunable topological negative refraction. (a) Schematic illustration. Through tailoring the Bloch wavevector k of incident one-way topological states, the variation of in-plane wavevector k∥ can turn the refracted wave from negative to positive refraction, according to Snell’s law k∥=k0 sin θ. The gray dashed line represents the normal line of refraction. (b) Band structure of the topological photonic crystal that supports a one-way state. As the frequency increases, the Bloch wavevector of the one-way state moves along the high-symmetry lines K′−M−K. (c) Tailoring the Bloch wavevector of topological states in two-dimensional k-space. (d) Designed gyromagnetic photonic crystal with oppositely magnetized sublattices along the z-axis. Tailoring the Bloch wavevectors of topological states in k-space drives a transition from negative (θ1<0°) to critical (θ2=0°) to positive (θ3>0°) refraction.

    Download full size
    View all figures

    2. TAILORABLE BLOCH WAVEVECTOR OF THE ONE-WAY WAVEGUIDE STATE

    We fabricate a sample consisting of two separated antichiral photonic crystals [25–27], with the right side exposed to free space [Fig. 2(a)]. The antichiral system is protected by chiral symmetry [25,27] (see Appendix C for its topological properties), which enables its edge states to connect two degenerate Dirac points, spanning a considerable wide range of wavevectors. The photonic crystal consists of yttrium-iron-garnet (YIG) gyromagnetic cylinders with a diameter of d=3  mm, arranged in a honeycomb lattice with a lattice constant of a=10  mm along the x direction (zigzag edge); see Appendix A for material properties. Opposite magnetic biases of μ0H0=±1500 Gauss (G) are applied to the two sublattices of each unit cell. The photonic crystal is separated by an additional distance of 0.2a in the middle, creating a channel along the x-axis. Each antichiral photonic crystal supports a pair of right-propagating edge states along the parallel edges. The channel formed by the two separated crystals supports two guided states, with their eigenmodal fields exhibiting odd and even symmetry. However, as the even mode emerges with the bulk states, we primarily focus on the one-way odd mode, which remains more far away from the bulk states in the momentum space [Fig. 2(b)]. This band structure demonstrates high stability across various waveguide widths (see Appendix C).

    Topological refraction via tunable Bloch wavevectors of one-way waveguide states. (a) Experimental sample. The constant of the photonic crystals along the x direction (zigzag edge) is a=10 mm and the width of the channel is w=w0+0.2a, where w0=a/3. Red and blue stars are two dipole sources with opposite phases. Blue arrow represents the transport of the waveguide state, while green arrow indicates the refracted beam at the interface (white dotted line). The supercell (white dotted rectangle) is used to obtain the dispersions in simulation. (b) Calculated and measured dispersions. The colored lines and background maps denote the simulated values and measured data, respectively. The blue curve indicates the one-way waveguide state. (c) Eigenmodal fields of the waveguide states. (d)–(h) Simulated Fourier spectra of the waveguide states at different frequencies: (d) f=9.51 GHz, (e) f=9.62 GHz, (f) f=9.80 GHz, (g) f=10.00 GHz, (h) f=10.11 GHz.

    Figure 2.Topological refraction via tunable Bloch wavevectors of one-way waveguide states. (a) Experimental sample. The constant of the photonic crystals along the x direction (zigzag edge) is a=10  mm and the width of the channel is w=w0+0.2a, where w0=a/3. Red and blue stars are two dipole sources with opposite phases. Blue arrow represents the transport of the waveguide state, while green arrow indicates the refracted beam at the interface (white dotted line). The supercell (white dotted rectangle) is used to obtain the dispersions in simulation. (b) Calculated and measured dispersions. The colored lines and background maps denote the simulated values and measured data, respectively. The blue curve indicates the one-way waveguide state. (c) Eigenmodal fields of the waveguide states. (d)–(h) Simulated Fourier spectra of the waveguide states at different frequencies: (d) f=9.51  GHz, (e) f=9.62  GHz, (f) f=9.80  GHz, (g) f=10.00  GHz, (h) f=10.11  GHz.

    Download full size
    View all figures

    The one-way waveguide state impinges upon the interface at a fixed incident angle of 30° and is refracted into the air side [Fig. 2(a)]. In simulations, the dispersion of the one-way waveguide state is calculated using a supercell (white dashed rectangle) in one column along the y direction. In experiments, the dispersion is measured by applying a Fourier transform to the complex electric field measured along the channel (see Appendix B for measurement details). The projected band structure of the photonic crystal is shown in Fig. 2(b). The colored lines represent simulated values, while the background maps denote the measured dispersion data. The measured dispersions are consistent with the simulation results. The waveguide state (blue band) connects the degenerate Dirac points at the K and K′ positions and exhibits positive group velocity (slope of the band vg=dω/dk), indicating its one-way transport property [26,27]. In the experiment, a pair of dipole sources with opposite phases (red and blue stars) is used to excite the topological state [Fig. 2(a)] due to its odd-symmetric eigenmodal field [Fig. 2(c)]. Although there is no complete bandage, the one-way topological state can still be efficiently excited, as it is well separated from other states in momentum space (see Appendix C for detailed analysis).

    To reveal the tunability of the Bloch wavevector of the one-way waveguide state, we apply Fourier transforms on its extended eigenmodal fields at various frequencies (see Appendix D for calculation details). The results, shown in Figs. 2(d)–2(h), indicate that as the frequency increases from 9.51 to 10.11 GHz, the waveguide states move along the high-symmetry lines K′−M−K in k-space [bright spots on the Brillouin zone boundaries in Figs. 2(d)–2(h), extended in the ky direction due to the localization in real-space y direction]. This clearly demonstrates that the channel supports a one-way waveguide state with distinct Bloch wavevectors at different frequencies. Given the broad distribution of topological states in momentum space, we can introduce a new degree of freedom—Bloch wavevector freedom—which can be independently and intuitively manipulated across a wide range, leading to the concept of tunable topological refraction.

    3. DEMONSTRATION OF TOPOLOGICAL REFRACTION

    We observe the transport behaviors of incident one-way states as they transmit into free space at the terminal. The waveguide state (f=9.68  GHz) propagates unidirectionally toward the interface and enters the air side with significant negative refraction [θ=−32°, Fig. 3(a1)]. The experimental observations agree well with the simulation results. Benefiting from its one-way properties, the waveguide state efficiently emits into the air without backscattering, despite slight bulk refraction caused by the lack of a complete bandgap (see Appendix F for a quantitative efficiency calculation). Additionally, the strong wavevector mismatch between the guided state and the edge states at the interface ensures that most of the energy is coupled into the air rather than confined to the right edge of the photonic crystals, preserving efficiency across a broad frequency range, and this is in distinct contrast with previous topological systems (see Appendix C and Appendix F). We also conducted k-space analysis [Fig. 3(a2)], which reveals the direction of the refracted beam. The projection lines of the two k-components of the topological state intersect the air light cone, and the negative k∥ results in a negative refraction angle (see Appendix E for the theoretical calculation). The background Fourier spectrum, obtained by applying a Fourier transform to the simulated electric field in Fig. 3(a1), further supports our theoretical analysis. The theoretical refraction angle shows an acceptable deviation of about 7° from the numerical and experimental values. This discrepancy arises because the theoretical refraction angle is calculated using bulk dispersions, which neglect the contribution of the interface geometry to refraction, while the numerical and experimental values take this contribution into account.

    Demonstration of topological refraction. (a) Negative refraction with θ=−32° at f=9.68 GHz. (b) Critical refraction with θ=−5° at f=9.80 GHz. (c) Positive refraction with θ=+14° at f=9.92 GHz. (a1), (b1), (c1) Simulated and measured electric field distribution in real space through near-field mapping. (a2), (b2), (c2) Fourier spectra of the simulated electric field (background) and theoretical k-space analysis (dots, lines, and arrows) of refraction process. The blue dots are k-components of incident waveguide states obtained from projected bands, the hexagon and semicircle represent the first Brillouin zone of the photonic crystal and the light cone of air, respectively, and the green arrows are theoretically calculated refraction angles. (d), (e) Immunity to metallic obstacles. (d) Simulated electric field distribution while inducing metallic obstacles in the channel at 9.68 GHz, together with the partial view of experimental sample. (e1), (e2), (e3) Experimental measurements of field distributions in the air side at (e1) f=9.68 GHz, (e2) f=9.80 GHz, and (e3) f=9.92 GHz. The metallic obstacles in the transport channel do not affect the final emergence and refraction angles of the negative refraction waves.

    Figure 3.Demonstration of topological refraction. (a) Negative refraction with θ=−32° at f=9.68  GHz. (b) Critical refraction with θ=−5° at f=9.80  GHz. (c) Positive refraction with θ=+14° at f=9.92  GHz. (a1), (b1), (c1) Simulated and measured electric field distribution in real space through near-field mapping. (a2), (b2), (c2) Fourier spectra of the simulated electric field (background) and theoretical k-space analysis (dots, lines, and arrows) of refraction process. The blue dots are k-components of incident waveguide states obtained from projected bands, the hexagon and semicircle represent the first Brillouin zone of the photonic crystal and the light cone of air, respectively, and the green arrows are theoretically calculated refraction angles. (d), (e) Immunity to metallic obstacles. (d) Simulated electric field distribution while inducing metallic obstacles in the channel at 9.68 GHz, together with the partial view of experimental sample. (e1), (e2), (e3) Experimental measurements of field distributions in the air side at (e1) f=9.68  GHz, (e2) f=9.80  GHz, and (e3) f=9.92  GHz. The metallic obstacles in the transport channel do not affect the final emergence and refraction angles of the negative refraction waves.

    Download full size
    View all figures

    As the frequency increases from 9.68 to 9.92 GHz [Figs. 3(a)–3(c)], the Bloch wavevectors of the topological states move along the high-symmetry lines K′−M−K in momentum space [Figs. 3(a2)–3(c2)], causing their momentum projection k∥ along the interface to evolve from negative to zero to positive values. Thus, the refracted beam deflects counterclockwise with increasing frequency, leading the refraction to transition from negative refraction [θ=−32°, Fig. 3(a)] to critical refraction [θ=−5° in experiment, Fig. 3(d)] to positive refraction [θ=+14°, Fig. 3(g)]. The effective operational frequency range for negative refraction is narrower than the theoretically predicted full K′−M−K path, as the waveguide mode becomes more delocalized and harder to excite near the K′ or K points, leading to increased mixing with bulk states (see Appendices C and D).

    We proceed to demonstrate the robustness of this negative refraction system by inserting a pair of metallic obstacles in the transport channel [Fig. 3(d)]. Due to the topological protection of the one-way waveguide states, the wave bypasses the metallic obstacles without backscattering, continuing to propagate forward and be refracted at the same angle as if the obstacles are not present. The measured near-field patterns [Figs. 3(e1)–3(e3)] confirm that the metallic obstacles do not alter the refraction angles. The inherent topological nature of the system ensures the absence of backscattering, distinguishing it from reciprocal topological systems [17,28], which may be subject to backscattering due to valley/pseudospin flipping [29–33] (see Appendix F).

    4. ACTIVE CONTROL OF TOPOLOGICAL REFRACTION

    We proceed to develop a prototype device capable of tuning the external magnetic biases to demonstrate the active control of topological refraction [Fig. 4(a)]. The critical operation involves manipulating the gap width (g) between the magnet array layers and photonic crystal through lifting the screw jacks and inserting the wedges [Fig. 4(b)]. Specifically, increasing the gap width (Δg) can reduce the external magnetic biases of photonic crystal [ΔH0, Fig. 4(c)], causing the Bloch wavevectors of one-way topological states at a certain frequency to move along the high-symmetry lines K′−M−K in momentum space [Δk, Fig. 4(d)]. This process essentially leverages the magnetic field to dynamically shift the frequency band of the one-way waveguide mode [Δf, red lines in Fig. 4(d)]. This operation further enables the transition of the output electromagnetic beams from negative to positive refraction [Δθ, Fig. 4(e)].

    Active control of topological refraction. (a) Experimental sample. (b) Partially enlarged view of (a) to show the interlayer structures. (c)–(e) Physical mechanism. (f) Measured refraction angles θ varying with the gap width g at 9.9, 10.0, and 11.1 GHz. (g)–(n) Active control of topological refraction realized by tuning the gap width g at 10 GHz. (g), (k) g=0 mm, μ0H0=1850G, θ=−38°. (h), (l) g=0.5 mm, μ0H0=1650G, θ=−18°. (i), (m) g=1.0 mm, μ0H0=1550G, θ=−5°. (j), (n) g=2.0 mm, μ0H0=1475G, θ=+12°. (g)–(j) Simulated results. (k)–(n) Measured results.

    Figure 4.Active control of topological refraction. (a) Experimental sample. (b) Partially enlarged view of (a) to show the interlayer structures. (c)–(e) Physical mechanism. (f) Measured refraction angles θ varying with the gap width g at 9.9, 10.0, and 11.1 GHz. (g)–(n) Active control of topological refraction realized by tuning the gap width g at 10 GHz. (g), (k) g=0  mm, μ0H0=1850G, θ=−38°. (h), (l) g=0.5  mm, μ0H0=1650G, θ=−18°. (i), (m) g=1.0  mm, μ0H0=1550G, θ=−5°. (j), (n) g=2.0  mm, μ0H0=1475G, θ=+12°. (g)–(j) Simulated results. (k)–(n) Measured results.

    Download full size
    View all figures

    Figure 4(f) shows how the refraction angle varies with the gap width g, each corresponding to a distinct external magnetic bias H0. As shown in Fig. 4(d), the excitation frequency determines the accessible Bloch wavevector of the incident waveguide mode, which in turn governs the tunable range of refraction. The experimentally accessible bandwidth is approximately 0.2 GHz, which is slightly narrower than the full bandwidth of the waveguide mode, due to limited wavevector accessibility near the spectral edges. We simulate and measure the near-field patterns on the air side at 10.0 GHz under gap widths of g=0  mm, 0.5 mm, 1.0 mm, and 2.0 mm [Figs. 4(g)–4(n)]. The simulated results [Figs. 4(g)–4(j)] agree well with the measured results [Figs. 4(k)–4(n)]. As the gap width g increases from 0 to 2.0 mm, the magnetic bias μ0H0 decreases from 1850 G to 1475 G, leading to a change in the Bloch wavevectors of the waveguide states in k-space. As a result, the refracted beams deflect counterclockwise and transition from negative refraction (−38°) to positive refraction (+12°). Although the measured near-field profiles are slightly broader than simulations—due to magnetic field non-uniformity and fabricated imperfection-induced mode leakage and spatial broadening—the key features are well preserved, and the associated deviations remain within an acceptable range. Our prototype device achieves beam steering through manual magnetic field adjustments in about a few seconds. To improve this, an electromechanical structure could reduce the response time to under 1 s, while the frequency-sweeping method typically operates in the millisecond range, making it promising for practical applications.

    5. CONCLUSION

    In summary, we have theoretically, numerically, and experimentally demonstrated continuously tunable topological negative refraction via a tailorable Bloch wavevector of one-way waveguide states. By creating a one-way photonic state with large, tunable wavevectors and manipulating it in momentum space, we enable continuous and wide-angle steering of the refracted wave, ranging from negative to positive refraction. Active control of the external magnetic bias allows refracted beam steering from −38° to +12° at a specific frequency. Benefiting from the unidirectional features of the waveguide states, the output waves are immune to structural defects. Our findings provide promising approaches for efficient electromagnetic beam manipulation and offer inspiration for the development of topological photonic devices for applications in microwave and optical antennas, telecommunication, radar, LiDAR, and more.

    APPENDIX A: MATERIAL PARAMETERS OF PHOTONIC CRYSTAL

    The gyromagnetic photonic crystals are composed of commercial yttrium iron garnet (YIG) cylinders and the surrounding air environment. The relative permittivity and permeability of air are ε=1 and μ=1, respectively. The relative permittivity of YIG is 14.5, with a dielectric loss tangent less than 0.5×10−4. Under sufficient magnetization of the external magnetic bias H0 along +z direction, YIG reaches its saturation magnetization μ0Ms=1950G. Due to the magneto-optics effect of YIG, its permeability becomes a tensor [34]: μ↔=(μriμk0−iμkμr0001),where μr=1+(ω0+iαω)ωm(ω0+iαω)2−ωm2 and μk=ωωm(ω0+iαω)2−ω2, with parameters ω0=γμ0H0, ωm=γμ0Ms, gyromagnetic ratio γ=1.76×107  rad s−1 G−1, and damping coefficient α=0.001. When flipping the direction of magnetic bias H0 into −z direction, both H0 and Ms become negative values and μk changes its sign.

    APPENDIX B: SAMPLE FABRICATION AND MEASUREMENT

    The structure of the experimental sample can be divided into three layers: the middle photonic crystal and the upper and lower magnet layers [Figs. 5(a) and 5(b)]. The photonic crystal layer consists of YIG cylinders with diameter of 3 mm and height of 5 mm, arranged in hexagonal honeycomb lattices. These cylinders are fixed on a 1 mm thick metal plate at their bottom and covered with a 1 mm thick metal plate on top, forming a closed cavity. The magnet layers are formed by embedding 3 mm diameter, 2 mm high NdFeB cylinder magnets into the 2 mm thick metal plates. The magnetization direction of the magnets is designed to provide opposite magnetization patterns for the two sublattices of the gyromagnetic photonic crystal, generating antichiral topological photonic states [26,27]. The design of separating the photonic crystal and magnet layers allows us to adjust the strength of the magnetic bias by varying the gap between the layers, without affecting the geometric structure of the photonic crystal.

    Sample fabrication and experiment measurement. (a) Exploded view of the experimental sample. (b) Structural details of a unit cell of antichiral gyromagnetic photonic crystal (PC). (c) Measurement of internal electromagnetic wave in sample. (d) 180° 3 dB bridge for creating a pair of dipole sources of opposite phases. (e) Schematic for measurement system. (f) Transmission spectra of one-way topological state.

    Figure 5.Sample fabrication and experiment measurement. (a) Exploded view of the experimental sample. (b) Structural details of a unit cell of antichiral gyromagnetic photonic crystal (PC). (c) Measurement of internal electromagnetic wave in sample. (d) 180° 3 dB bridge for creating a pair of dipole sources of opposite phases. (e) Schematic for measurement system. (f) Transmission spectra of one-way topological state.

    Download full size
    View all figures

    Subwavelength holes with a diameter of 1.5 mm are reserved in the top two metal plates, covering the air region and distributed along the waveguide [Fig. 5(c)]. These holes provide access for field probes to detect the internal electromagnetic field within the photonic crystal. As their size is much smaller than the operating wavelength, the resulting field leakage is negligible. We use a vector network analyzer (Keysight P9374A) as the microwave source and receiver [Fig. 5(e)], which supports electromagnetic waves measurement with frequency ranging from 300 kHz to 20 GHz. Two dipole antennas are connected to a 180° 3 dB bridge (Talent Microwave TBG-10180-3S-180), generating broadband (1–18 GHz) output signals of opposite phases serving as the excitation sources [Figs. 5(d) and 5(e)]. The transmission spectra measured within the waveguide region show a maximum forward-backward transmission contrast of approximately 20 dB in the 9.5–10.1 GHz range, confirming the effective excitation of the one-way topological state [Fig. 5(f)]. The dispersion measurement is obtained by performing a one-dimensional Fourier transform on the spectral data collected along the waveguide. The near-field distribution of the refracted beam is obtained by directly mapping point-to-point measurements in the air region.

    APPENDIX C: NUMERICAL SIMULATION AND EIGENMODES IN TOPOLOGICAL PHOTONIC CRYSTAL

    The finite element method (FEM) is applied in numerically calculating the eigenmodes of the photonic crystal, by using the commercial software COMSOL Multiphysics with the RF module. The detailed simulation setup of a supercell is depicted in Fig. 6(a). Each unit cell has two sublattices of opposite magnetization directions. Scattering boundary conditions are applied on the upper and bottom boundaries to absorb the superfluous electromagnetic waves. The lateral period boundaries are set to simulate transmission over an infinite distance along the x-axis.

    Eigenmodes in gyromagnetic photonic crystal. (a) Simulation setup for energy bands and eigenmodal field calculation. (b), (c) Projected energy bands and eigenmodal field distributions for the photonic crystal excitation analysis, respectively.

    Figure 6.Eigenmodes in gyromagnetic photonic crystal. (a) Simulation setup for energy bands and eigenmodal field calculation. (b), (c) Projected energy bands and eigenmodal field distributions for the photonic crystal excitation analysis, respectively.

    Download full size
    View all figures

    Projected band structures with varying distances Δw. (a) Schematic diagram. (b) Δw=0.1a. (c) Δw=0.2a. (d) Δw=0.3a.

    Figure 7.Projected band structures with varying distances Δw. (a) Schematic diagram. (b) Δw=0.1a. (c) Δw=0.2a. (d) Δw=0.3a.

    Download full size
    View all figures

    Topological negative refraction excited by distinct types of sources. (a) A pair of odd-symmetric dipole sources. (b) Single dipole source. (c) Electric field distributions at the blue dashed line positions in (a) and (b).

    Figure 8.Topological negative refraction excited by distinct types of sources. (a) A pair of odd-symmetric dipole sources. (b) Single dipole source. (c) Electric field distributions at the blue dashed line positions in (a) and (b).

    Download full size
    View all figures

    Scattering-based evaluation of topological properties in an antichiral photonic crystal. (a) Schematic of the simulation setup. Twisted boundary conditions are applied along the lateral edges, mapping the right boundary field to the left with an added twisting phase term eiΦx. An incident wave is launched through a waveguide formed by perfect magnetic conductors (PMCs). (b) Reflection phase φr as a function of the twisting angle Φx, demonstrating a full 2π winding at 9.8 GHz. This phase winding confirms the nontrivial topological nature of the antichiral edge state.

    Figure 9.Scattering-based evaluation of topological properties in an antichiral photonic crystal. (a) Schematic of the simulation setup. Twisted boundary conditions are applied along the lateral edges, mapping the right boundary field to the left with an added twisting phase term eiΦx. An incident wave is launched through a waveguide formed by perfect magnetic conductors (PMCs). (b) Reflection phase φr as a function of the twisting angle Φx, demonstrating a full 2π winding at 9.8 GHz. This phase winding confirms the nontrivial topological nature of the antichiral edge state.

    Download full size
    View all figures

    APPENDIX D: PERIODIC EXTENSION AND FOURIER SPECTRA OF THE EIGENMODAL FIELDS

    Although the projected band structure of the strip supercell provides the kx range of the waveguide mode, it does not fully determine its position in the two-dimensional k-space, as the ky component remains implicitly encoded in the eigenfield profile. To reveal the momentum distribution of the waveguide state in k-space, we perform a two-dimensional Fourier transform on its eigenmodal field. However, directly performing a Fourier transform to the eigenmodal field of a single supercell results in very low resolution in the kx direction, as a single supercell describes only a small portion of the entire wave packet. To address this, we periodically extend the field of the waveguide mode to obtain the complete wave packet distribution. After this periodic extension along the x direction, we then perform a two-dimensional Fourier transform, as illustrated in Figs. 10(a)–10(c). The detailed calculations are as follows.

    Fourier transform of the eigenmodal field of the topological state. (a) Evolution of the topological state’s eigenmodal field along the x direction. (b) Periodic extension of the topological state’s eigenmodal field to obtain complete wave packets. (c) Fourier spectra of the topological state at kx=(5/12) (2π/a). (d)–(h) Simulated Fourier spectra under point-source excitation in finite-sized photonic crystals, used to verify consistency with the eigenfield-based method.

    Figure 10.Fourier transform of the eigenmodal field of the topological state. (a) Evolution of the topological state’s eigenmodal field along the x direction. (b) Periodic extension of the topological state’s eigenmodal field to obtain complete wave packets. (c) Fourier spectra of the topological state at kx=(5/12) (2π/a). (d)–(h) Simulated Fourier spectra under point-source excitation in finite-sized photonic crystals, used to verify consistency with the eigenfield-based method.

    Download full size
    View all figures

    For a given kx, the eigenmodal field of the waveguide state, Ez,0(x,y), is obtained through the FEM numerical calculations (Appendix C). As shown in Fig. 10(a), while the propagating eigenmodes in the photonic crystal take the form of Bloch waves, the field in the next period can be obtained by multiplying the field of the previous period by a phase propagation factor, eikxa: Ez,j+1(x,y)=Ez,j(x,y)eikxa.

    By continuously replicating the eigenmodal field with this phase factor, we achieve a periodic extension of the topological state [Fig. 10(b)]. The extended field is derived from the original unit cell: Ez(x0+n·a)=Ez,0(x0)eikx(n·a).

    We extend the eigenmodal field along the x direction to a length of 14a to match its length in the y direction [Fig. 10(b)]. Both the periodic extension and the subsequent two-dimensional Fourier transform are performed using the numerical computation software MATLAB. After performing the two-dimensional Fourier transform, the Fourier spectra of the topological state are obtained [Fig. 10(c)]. The light spots indicate that the waveguide state is located on the boundaries of the first Brillouin zone of the photonic crystal, moving along the K′−M−K lines in k-space. Due to the waveguide state’s localization in the y direction of real space, it correspondingly becomes diffuse in the ky direction of momentum space, according to the principles of Fourier optics. By periodically extending and performing Fourier transforms on the eigenmodal fields at different kx values, we obtain the evolution of the waveguide states in momentum space [Figs. 2(d)–2(h)]. We also simulate the Fourier spectra of a finite-size waveguide structure excited by a pair of out-of-phase point sources for comparison [Figs. 10(d)–10(h)]. Despite the presence of partial bulk states, the waveguide mode still exhibits a clear trajectory along the K′−M−K path, consistent with the results from the eigenfield extension approach.

    APPENDIX E: THEORETICAL CALCULATION OF REFRACTION ANGLES

    The theoretical calculation of refraction angles relies on the momentum matching of the incident waveguide state and the refraction beam, following Snell’s law (k∥=k0 sin θ). For a given frequency f, the wavevector of a refraction beam in air is k0=ω/c, where ω=2πf and c is the speed of light in vacuum.

    For the frequency f, the x-component of the Bloch wavevector kx of the one-way waveguide state can be obtained from the projected band structure calculations. Its coordinate in two-dimensional k-space k(kx,ky) can be calculated from the mapping relationship between the projected band structure and the Brillouin zones [Fig. 11(a)]. For the waveguide state with known kx, where 13(2π/a)<kx<23(2π/a), there are four k(kx,ky) components in the k-space [marked with blue dots in Fig. 11(b)]: k1=(kx,−3(kx−232πa)),k2=(kx,3(kx−232πa)),k3=(kx−2πa,−3(kx−132πa)),k4=(kx−2πa,3(kx−132πa)).

    Theoretical calculation of refraction angle through wavevectors matching. (a) Mapping relationship between two-dimensional k-space and projected bands. (b) Wavevectors matching between incident waveguide states and refraction beam on the interface. (c) Theoretically calculated refracted angle under various magnetic biases.

    Figure 11.Theoretical calculation of refraction angle through wavevectors matching. (a) Mapping relationship between two-dimensional k-space and projected bands. (b) Wavevectors matching between incident waveguide states and refraction beam on the interface. (c) Theoretically calculated refracted angle under various magnetic biases.

    Download full size
    View all figures

    Simulated Fourier spectra (background) and theoretical wavevector projections (symbols and arrows) at 10 GHz under (a) μ0H0=1900 G, (b) μ0H0=1700 G, and (c) μ0H0=1500 G.

    Figure 12.Simulated Fourier spectra (background) and theoretical wavevector projections (symbols and arrows) at 10 GHz under (a) μ0H0=1900  G, (b) μ0H0=1700  G, and (c) μ0H0=1500  G.

    Download full size
    View all figures

    APPENDIX F: COMPARISON WITH THE VALLEY-HALL PHOTONIC CRYSTAL SYSTEM

    Here, we also provide a comparison with the valley-state negative refraction system proposed in previous work [17–19]. A valley photonic crystal consists of non-magnetized YIG cylinders, with a lattice constant of 10 mm and sublattice diameters of 2.75 and 2.55 mm, designed to operate within the same operating frequency range as the antichiral system. The valley-dependent topological states are locked at the K′ and K valleys within the bandgap, with a small Δk [solid magenta line in Fig. 13(b)]. The edge states along the interface tend to connect the valleys at the K′ and K points (dashed green and yellow lines). The high-frequency portion of the edge states is cut off by the air light cones; a frequency window for negative refraction is still retained. We use a right-handed chirality source to excite the valley-dependent waveguide states. At an excitation frequency of 9.6 GHz, nearly all the energy is transmitted along the upper boundary. As the frequency increases to 9.8 GHz, part of the energy is radiated into the air as negative refraction, though a significant portion still travels along the upper boundary. When the frequency reaches 9.9 GHz, the wavevector moves away from the valleys, preventing the chirality source from exciting the valley states that propagate to the right. Due to the small Δk and the presence of edge states, the valley-Hall system can only support a nearly constant fixed refraction angle within a narrow bandwidth.

    Negative refraction in valley photonic crystals. (a) Schematic diagram of valley photonic crystals. The channel is composed of an interface between two photonic crystals with opposite valley Chern numbers. (b), (c) Projected band structures and eigenmodal fields. The channel supports valley-dependent topological states, locked in the K′ and K valleys. The right edge of the photonic crystals supports trivial edge states. (d)–(g) Negative refraction excited by an RCP source at varying frequencies. Due to a small Δk, the refracted beam covers only a narrow angular range. The presence of edge states significantly reduces refraction efficiency, resulting in substantial energy leakage.

    Figure 13.Negative refraction in valley photonic crystals. (a) Schematic diagram of valley photonic crystals. The channel is composed of an interface between two photonic crystals with opposite valley Chern numbers. (b), (c) Projected band structures and eigenmodal fields. The channel supports valley-dependent topological states, locked in the K′ and K valleys. The right edge of the photonic crystals supports trivial edge states. (d)–(g) Negative refraction excited by an RCP source at varying frequencies. Due to a small Δk, the refracted beam covers only a narrow angular range. The presence of edge states significantly reduces refraction efficiency, resulting in substantial energy leakage.

    Download full size
    View all figures

    Efficiency of topological refraction in valley and antichiral photonic crystals. (a) Simulation setup for evaluating the transmission efficiency of topological refraction. (b) Electromagnetic wave transmission channels in photonic crystals. (c)–(f) Transmission and efficiency in (c), (d) antichiral photonic crystals and (e), (f) valley-Hall photonic crystals.

    Figure 14.Efficiency of topological refraction in valley and antichiral photonic crystals. (a) Simulation setup for evaluating the transmission efficiency of topological refraction. (b) Electromagnetic wave transmission channels in photonic crystals. (c)–(f) Transmission and efficiency in (c), (d) antichiral photonic crystals and (e), (f) valley-Hall photonic crystals.

    Download full size
    View all figures

    APPENDIX G: REFRACTION IN CHIRAL PHOTONIC CRYSTALS

    Momentum space manipulation of topological states is essential for achieving tunable negative refraction. However, simply introducing magneto-optical effects or unidirectionality to the system is insufficient to produce the same effect. To demonstrate this, we present the negative refraction in a Chern photonic crystal system, using two oppositely magnetized chiral crystals (see Fig. 15). The mirror symmetry confines the two waveguide states near the K and K′ valleys, making it impossible to achieve a wide Bloch wavevector. Simultaneously, edge states tend to connect high-symmetry points, bringing their bands very close to the incident beam. The well-matched wavevector between the incident beam and the edge state along the interface leads to most of the energy coupling to the edge rather than radiating into the air. Thus, although chiral crystals provide the strongest unidirectionality and robustness, they offer no advantage in beam manipulation.

    Refraction in a chiral system. (a) Schematic diagram of the chiral system. (b), (c) Projected band structures and eigenmodal fields. The channel supports odd and even modes locked between the K′ and K valleys, respectively. The interface of the system also supports one-way edge states. (d)–(g) Negative refraction under the excitation of a pair of odd-symmetric dipole sources. (h)–(k) Positive refraction under the excitation of a pair of even-symmetric dipole sources. The refraction angle shows only a small range of variation, and the efficiency is significantly reduced due to the presence of edge states at the interface.

    Figure 15.Refraction in a chiral system. (a) Schematic diagram of the chiral system. (b), (c) Projected band structures and eigenmodal fields. The channel supports odd and even modes locked between the K′ and K valleys, respectively. The interface of the system also supports one-way edge states. (d)–(g) Negative refraction under the excitation of a pair of odd-symmetric dipole sources. (h)–(k) Positive refraction under the excitation of a pair of even-symmetric dipole sources. The refraction angle shows only a small range of variation, and the efficiency is significantly reduced due to the presence of edge states at the interface.

    Download full size
    View all figures

    [1] R. A. Shelby, D. R. Smith, S. Schultz. Experimental verification of a negative index of refraction. Science, 292, 77-79(2001).

    [2] E. Cubukcu, K. Aydin, E. Ozbay. Negative refraction by photonic crystals. Nature, 423, 604-605(2003).

    [3] J. Yao, Z. Liu, Y. Liu. Optical negative refraction in bulk metamaterials of nanowires. Science, 321, 930(2008).

    [4] X. Lin, Y. Yang, N. Rivera. All-angle negative refraction of highly squeezed plasmon and phonon polaritons in graphene–boron nitride heterostructures. Proc. Natl. Acad. Sci. USA, 114, 6717-6721(2017).

    [5] A. J. Sternbach, S. L. Moore, A. Rikhter. Negative refraction in hyperbolic hetero-bicrystals. Science, 379, 555-557(2023).

    [6] Y. Gelkop, F. Di Mei, S. Frishman. Hyperbolic optics and superlensing in room-temperature KTN from self-induced k-space topological transitions. Nat. Commun., 12, 7241(2021).

    [7] A. A. High, R. C. Devlin, A. Dibos. Visible-frequency hyperbolic metasurface. Nature, 522, 192-196(2015).

    [8] A. Piccardi, A. Alberucci, N. Kravets. Power-controlled transition from standard to negative refraction in reorientational soft matter. Nat. Commun., 5, 5533(2014).

    [9] H. Hu, N. Chen, H. Teng. Gate-tunable negative refraction of mid-infrared polaritons. Science, 379, 558-561(2023).

    [10] H. Hu, N. Chen, H. Teng. Doping-driven topological polaritons in graphene/α-MoO3 heterostructures. Nat. Nanotechnol., 17, 940-946(2022).

    [11] B. Stein, J.-Y. Laluet, E. Devaux. Surface plasmon mode steering and negative refraction. Phys. Rev. Lett., 105, 266804(2010).

    [12] J. B. Pendry, S. A. Ramakrishna. Focusing light using negative refraction. J. Phys. Condens. Matter, 15, 6345-6364(2003).

    [13] R. Fleury, D. L. Sounas, A. Alù. Negative refraction and planar focusing based on parity-time symmetric metasurfaces. Phys. Rev. Lett., 113, 023903(2014).

    [14] A. Fang, T. Koschny, C. M. Soukoulis. Optical anisotropic metamaterials: negative refraction and focusing. Phys. Rev. B, 79, 245127(2009).

    [15] J. B. Pendry. Negative refraction makes a perfect lens. Phys. Rev. Lett., 85, 3966-3969(2000).

    [16] Z.-Y. Li, L.-L. Lin. Evaluation of lensing in photonic crystal slabs exhibiting negative refraction. Phys. Rev. B, 68, 245110(2003).

    [17] F. Gao, H. Xue, Z. Yang. Topologically protected refraction of robust kink states in valley photonic crystals. Nat. Phys., 14, 140-144(2018).

    [18] R. Jia, S. Kumar, T. C. Tan. Valley-conserved topological integrated antenna for 100-Gbps THz 6G wireless. Sci. Adv., 9, eadi8500(2023).

    [19] J.-C. Lu, X.-D. Chen, W.-M. Deng. One-way propagation of bulk states and robust edge states in photonic crystals with broken inversion and time-reversal symmetries. J. Opt., 20, 075103(2018).

    [20] Y. Liu, G. P. Wang, J. B. Pendry. All-angle reflectionless negative refraction with ideal photonic Weyl metamaterials. Light Sci. Appl., 11, 276(2022).

    [21] H. He, C. Qiu, L. Ye. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature, 560, 61-64(2018).

    [22] Y. Yang, H. X. Sun, J. P. Xia. Topological triply degenerate point with double Fermi arcs. Nat. Phys., 15, 645-649(2019).

    [23] Z. Zhu, X. Huang, J. Lu. Negative refraction and partition in acoustic valley materials of a square lattice. Phys. Rev. Appl., 12, 024007(2019).

    [24] Z. Zhang, Y. Tian, Y. Wang. Directional acoustic antennas based on valley-Hall topological insulators. Adv. Mater., 30, 1803229(2018).

    [25] E. Colomés, M. Franz. Antichiral edge states in a modified haldane nanoribbon. Phys. Rev. Lett., 120, 086603(2018).

    [26] J. Chen, W. Liang, Z. Y. Li. Antichiral one-way edge states in a gyromagnetic photonic crystal. Phys. Rev. B, 101, 214102(2020).

    [27] P. Zhou, G. G. Liu, Y. Yang. Observation of photonic antichiral edge states. Phys. Rev. Lett., 125, 263603(2020).

    [28] T. Wu, Y. Li, X. Feng. Topological photonic lattice for uniform beam splitting, robust routing, and sensitive far-field steering. Nano Lett., 23, 3866-3871(2023).

    [29] C. A. Rosiek, G. Arregui, A. Vladimirova. Observation of strong backscattering in valley-Hall photonic topological interface modes. Nat. Photonics, 17, 386-392(2023).

    [30] A. Kumar, Y. J. Tan, N. Navaratna. Slow light topological photonics with counter-propagating waves and its active control on a chip. Nat. Commun., 15, 926(2024).

    [31] D. Yu, S. Arora, L. Kuipers. Impact of transforming interface geometry on edge states in valley photonic crystals. Phys. Rev. Lett., 132, 116901(2024).

    [32] Y. Lumer, N. Engheta. Topological insulator antenna arrays. ACS Photonics, 7, 2244-2251(2020).

    [33] S. Xu, Y. Wang, R. Agarwal. Absence of topological protection of the interface states in photonic crystals. Phys. Rev. Lett., 131, 053802(2023).

    [34] D. M. Pozar. Microwave Engineering(2011).

    [35] J. Chen, Z. Y. Li. Prediction and observation of robust one-way bulk states in a gyromagnetic photonic crystal. Phys. Rev. Lett., 128, 257401(2022).

    [36] Z. Zhang, P. Delplace, R. Fleury. Anomalous topological waves in strongly amorphous scattering networks. Sci. Adv., 9, eadg3186(2023).

    [37] T. Qu, M. Wang, X. Cheng. Topological photonic alloy. Phys. Rev. Lett., 132, 223802(2024).

    [38] X. Cui, R. Y. Zhang, Z. Q. Zhang. Photonic topological Anderson insulators. Phys. Rev. Lett., 129, 043902(2022).

    Tools

    Get Citation

    Copy Citation Text

    Yidong Zheng, Jianfeng Chen, Zitao Ji, Zhi-Yuan Li, "Continuously tunable topological negative refraction via a tailorable Bloch wavevector in momentum space," Photonics Res. 13, 2159 (2025)

    Download Citation

    EndNote(RIS)BibTexPlain Text
    Set citation alerts for article
    Save article for my favorites
    Paper Information

    Category: Nanophotonics and Photonic Crystals

    Received: Feb. 24, 2025

    Accepted: May. 4, 2025

    Published Online: Jul. 25, 2025

    The Author Email: Zhi-Yuan Li (phzyli@scut.edu.cn)

    DOI:10.1364/PRJ.560388

    CSTR:32188.14.PRJ.560388

    Topics

    laser devices and laser physics
    Lasers and Laser Optics
    Laser physics
    laser manufacturing
    Instrumentation, Measurement and Metrology