2.1.

Principle and Metasurface Design

The conceptual schematic of the HM for reconfigurable polaritons, which allows for the dynamic switching between the elliptical, flat, and hyperbolic topology of polariton dispersions under real-time control from the external voltage source, is shown in Fig. 1(a). Hence, the spatial wavefront of the surface polaritons is then changed among convex, collimating, and concave. Controlling the anisotropic properties of the metasurface is the key to realizing designer dispersion. Here, we design an anisotropic meta-atom, as shown in the inset of Fig. 1(a), which consists of a copper ground layer with a complementary rectangle-shaped ring, a substrate, a metallic hole, and a top layer with complementary H-shaped pattern. The proposed unit cell is an electromagnetic (EM) resonator, which can be equivalently regarded as an inductor-capacitor (LC) circuit where the central metal pattern mainly functions as an inductor of inductance $L$ and the gaps between the metallic patches as capacitors of capacitance $C$. Obviously, the structure possesses different geometries along $x$ and $y$ directions due to the lack of C4 symmetry, leading to the EM anisotropy of the unit cell. Anisotropic polariton topologies arise when a surface satisfies specific constitutive parameter conditions, for example, an inductive surface with a positive imaginary part of both orthogonal conductivity components [$\mathrm{Im}({\sigma }_{xx})>0$ and $\mathrm{Im}({\sigma }_{yy})>0$] can support elliptical topology, while hyperbolic dispersion topology arises when the surface behaves as a capacitive and inductive one along the orthogonal direction, respectively $(\mathrm{sgn}[\mathrm{Im}({\sigma }_{xx})]·\mathrm{sgn}[\mathrm{Im}({\sigma }_{yy})]<0)$, according to the rigorously derived dispersion equation.¹⁷ Analogously, the proposed unit cell with well-designed geometries can support the elliptical or hyperbolic dispersion topologies over certain frequency bands, depending on the anisotropic EM responses (or resonance properties) of the LC circuit. To this aim, an electrically driven varactor diode (namely, $C_v$) is loaded to the meta-atom structure, serving as the key tunable element to provide continuous control of the resonance properties, thus changing the polaritons’ dispersion and enabling the reconfigurable topological transition over the certain frequency band of interest. More details of the equivalent circuit analysis can be found in Sec. S1 in the Supplementary Material.

Fig. 1

Electrically reconfigurable topology of polariton dispersions with tunable metasurface. (a) Conceptual schematic of topologically reconfigurable polaritons, where the proposed reconfigurable metasurface allows for surface wavefront control from convex to collimating and eventually to concave and convex, by applying different external bias-voltage. The black arrow represents the excitation source. Bottom-right panels show the schematic of the unit cell. (b), (c) Dispersions of the polaritons along the (b) $x$ and (c) $y$ propagating directions. The right panels show the dispersion diagrams of the polaritons with different loaded capacitance values $C_v$. (d) Reconfigurable topology of IFCs of the polaritons with loaded capacitances of 0, 0.24, 0.30, and 0.52 pF, respectively. The dashed lines show the IFCs of the first band in the first Brillouin zone at different frequencies. The color maps for the dispersion of polaritons are obtained by simulations in the commercial software Computer Simulation Technology Microwave Studio.

2.2.

Numerical Calculation and Experimental Characterization

We numerically investigate the dispersion characteristics of the designer polaritons in reconfigurable HM by using the eigenvalue module of a commercial software Computer Simulation Technology Microwave Studio. The optimized parameters are $p=8.5\mathrm{mm}$, $L_y=6.7\mathrm{mm}$, $L_x=3.8\mathrm{mm}$, $w=0.3\mathrm{mm}$, $l_a=1.2\mathrm{mm}$, and $l_b=1.8\mathrm{mm}$. The thickness of the substrate (permittivity 2.2) and copper ground layer (conductivity $\sigma =5.96\times {10}^7\mathrm{S}/\mathrm{m}$) is $t_s=1\mathrm{mm}$ and $t_m=0.035\mathrm{mm}$, respectively. To gain an insight into the underlying mechanism, we have established the effective circuit models of the unit cell and analyzed the tunable dispersion characteristics of the polaritons, as detailed in Sec. S1 in the Supplementary Material. Based on the equivalent circuit theory, the spoof surface plasma frequencies of the designer polaritons can be calculated as the resonance frequency of the meta-atom.^28,51–53 We denote the spoof surface plasma frequencies as $f_x$ and $f_y$ when the polaritons propagate along the $x$ and $y$ directions of the meta-atom, respectively. The dispersion topologies depend on the relationship among $f_0$, $f_x$, and $f_y$; here, $f_0$ represents the observational frequency. The polaritons can propagate along both the $x$ and $y$ directions with the elliptical dispersion topology when $f_0<f_x<f_y$, while a hyperbolic one can be realized when $f_x<f_{0 }<f_y$, and the polariton propagating along the $x$ direction will no longer exist. The dispersions of the designer polaritons as functions of the capacitance $C_v$ are plotted in Figs. 1(b) and 1(c). Parameters $k_x$ and $k_y$ represent the wave vectors of the designer polaritons along the $x$ and $y$ propagating directions, respectively. The dispersion diagrams with four typical capacitance values, i.e., 0, 0.24, 0.30, and 0.52 pF are illustrated in detail in the right panels of Figs. 1(b) and 1(c), where the dotted lines show unconventional dispersion effects like those natural surface plasmon polaritons with high confinement in optical frequencies. Increasing (decreasing) loaded capacitance $C_v$ can decrease (increase) the spoof surface plasma frequencies, which are 5.07, 4.65, 4.53, and 4.13 GHz for $f_x$, and 8.00, 6.70, 6.35, and 5.33 GHz for $f_y$, with the loaded capacitance of 0, 0.24, 0.30, and 0.52 pF, respectively.

The strong in-plane anisotropy consistently occurs with various capacitance values, resulting in distinct dispersion diagrams of designer polaritons along orthogonal propagation directions. These properties can also be evidenced by the surface current distributions of the unit cell, illustrating different modes of polariton propagation along the $x$ and $y$ directions (see Sec. S2 in the Supplementary Material). Thus, we can flexibly engineer the dispersion properties of the unit cell and achieve reconfigurable topology of polariton dispersions. Figure 1(d) presents the polariton IFCs of the first band in the first Brillouin zone with different loaded capacitances (see Sec. S3 in Supplementary Material). As anticipated, topological transitions of the designer polaritons occur with the increase of the loaded capacitances, in which we can see obvious variation tendency of the IFCs. For instance, the red lines indicate that when the observational frequency is fixed as 6.1 GHz, the IFC curve can be sequentially changed from elliptical to flat and then hyperbolic with increasing momentum.

The polariton dispersion at 6.1 GHz is clearly presented in Fig. 2(a), where the polariton IFC starts from open elliptical topology (0 pF), to a flat line (0.24 pF), and then it changes to hyperbola (0.3 pF). Eventually, the polariton IFC features an approximate circle when the loaded capacitance $C_v$ is changed to 0.52 pF. In this case, the IFCs of the first band in the first Brillouin zone no longer exist at 6.1 GHz [right panel of Fig. 1(d)] because the observational frequency $f_0$ (6.1 GHz) is larger than the spoof surface plasma frequencies of $f_x$ (4.13 GHz) and $f_y$ (5.33 GHz). Here, the topology of polariton dispersion is extracted from numerically simulated results of dispersion diagrams of the second band in the first Brillouin zone (see Sec. S4 in the Supplementary Material). Based on the uniaxial surface conductivity tensor, we calculated and analyzed the dispersions of the anisotropic surface based on the dispersion equations (see Sec. S5 in the Supplementary Material).

Fig. 2

Simulations and experiments of the topologically active polaritons by the proposed reconfigurable metasurface. (a) Elliptical, flat, hyperbolic, and circular topology of IFC of the polaritons at the observation frequency of 6.1 GHz. (b), (c) Simulated field distributions [real part of the $z$ component of the electric field, Re ($E_z$)] in the $xy$ plane that is 7 mm above the proposed reconfigurable metasurface (upper panels) and the corresponding dispersions in momentum space [FFT of Re ($E_z$)] at 6.1 GHz (bottom panels). The observation area covers $54\times 36$ unit cells with a size of $459\mathrm{mm}\times 306\mathrm{mm}$. (d) Photograph of the fabricated sample and experimental setup. (e), (f) Measured field distributions [Re ($E_z$)] in the $xy$ plane that is 1 mm above the proposed reconfigurable metasurface (upper panels) and the corresponding dispersions in momentum space [FFT of Re ($E_z$)] at 6.1 GHz (bottom panels). The observation area covers $24\times 16$ unit cells with a size of $204\mathrm{mm}\times 136\mathrm{mm}$. The DC voltages are set as 8 V ($C_v=0.25\mathrm{pF}$), 6.5 V ($C_v=0.29\mathrm{pF}$), and 3 V ($C_v=0.52\mathrm{pF}$) for the measured results of the last three columns. The abbreviation a.u. represents arbitrary units.

Next, we carry out numerical simulations to confirm the electrically tunable polariton dispersions, including field distributions in position space and corresponding polariton IFCs in momentum space ($k$-vector space) based on the proposed metasurface; here, $k$ represents the wave vector of the designer polaritons (see Sec. S3 in the Supplementary Material). In the full-wave simulations, the metasurface consists of $54\times 36$ unit cells with a total size of $459\mathrm{mm}\times 306\mathrm{mm}$. To efficiently excite the designer polaritons, a dipole source is put between the two metal layers of the unit cell located at the bottom edge of the metasurface [Fig. 1(a)].

The wavefront of the polaritons [Fig. 2(b)], obtained by full-wave simulations, undergoes intriguing transitions from convex to collimating, to concave, and eventually to convex patterns with the increase of the loaded capacitance $C_v$, just as indicated by the polariton IFCs.⁵⁴ The propagation characteristics of polaritons can be explained through the analysis of group velocity $v_g$, defined as $v_g=\partial \omega /\partial k$, which is perpendicular to the IFCs. Here, $\omega$ represents angular frequency. Thus, as shown in the left panel of Fig. 2(b), the polariton is guided into two main propagating directions when it comes to an open elliptical topology of IFC (also see Video 1 in the Appendix). When the loaded capacitance increases, the polaritons can still propagate in a splitting manner with a decreasing radiation angle. Until the flat topology of IFC appears, the second panel of Fig. 2(b) shows that the polariton propagates in a self-collimating manner without diffraction (also see Video 2 in the Appendix); such special property was originally studied in photonic crystals.⁵⁵ As the loaded capacitance $C_v$ continues to increase, indicating extreme anisotropy of the unit cell, we can see that the polariton propagates in a convergent manner due to the hyperbolic dispersion [third panel of Fig. 2(b)], where the wavefront of the polariton is concave (also see Video 3 in the Appendix). When reaching a capacitance value of 0.52 pF [last panel of Fig. 2(b)] or larger, the polariton propagation vanishes along the $y$ direction (see Video 4 in the Appendix). The corresponding in-plane dispersion of the designer polaritons can be intuitively viewed by the fast Fourier transform (FFT) of Re ($E_z$), as shown in Fig. 2(c). Topological transitions are observed in momentum space where open elliptical shapes transition into flat lines and then into open hyperbolas before closing into circles. The circle-shaped IFCs in $k$-vector space in first three panels of Fig. 2(c) originate from the electric fields radiated by the excitation source, which locate in the center position of $k$-space but with much smaller momentum than that of the designer polariton.

Experiments were conducted to validate the reconfigurable topology of polaritons using the proposed metasurface. The samples were fabricated through a standard printing circuit board technology. Complementary H-shaped patterns are periodically printed on an F4B substrate (permittivity ${\epsilon }_r=2.2$ and tangential loss $\tan \delta =0.001$). An off-the-shelf commercial varactor diode of type MA46H120 was selected as the tunable capacitive element in each unit cell due to its flexible tunability and stability,⁵⁶ with a tunable capacitance range from 0.149 to 1.304 pF, a parasitic resistance of $<2\mathrm{\Omega }$, and a parasitic inductance of 0.05 nH extracted from the experimental measurements (see Sec. S6 in Supplementary Material).⁵⁷ The fabricated reconfigurable metasurface is composed of $24\times 16$ unit cells, with a total size of $204\mathrm{mm}\times 136\mathrm{mm}$. Due to the limitations in achieving minimum realizable capacitance values, an additional sample composed of identical unit cells but without varactor diodes was fabricated to verify the case of $C_v=0\mathrm{pF}$, namely, open elliptical topology of the polariton dispersion. In fact, there are alternative options for varactor diodes based on specific manipulation requirements, for instance, the diode type MGV125-08 (0.055 to 0.6 pF) offers a wide range of controllable capacitance, while the MAVR-011020-1411 (0.03 to 0.24 pF) covers the low capacitance region.

For the measurements, the fabricated samples are placed on a 3D movement platform, as shown in Fig. 2(d). In this setup, the top and bottom layers of the metasurface are connected to negative “−” electrode and positive “+” electrode of a DC voltage source, respectively. All the unit cells of the metasurface are in parallel connections that share the same DC bias voltage. A subminiature version A convertor, connected to the transmitting port of the vector network analyzer (VNA), is used to connect the top and bottom layers of the metasurface to excite the polaritons propagating on the metasurface. A monopole antenna serving as the detector is connected to the receiving port of VNA for probing the electric field distributions during the sample movement. The measured electric field distributions of the metasurface are shown in Fig. 2(e), where the wavefront of the designer polariton changes from convex to collimating and to concave. Simultaneously, the corresponding polariton IFC changes from an open ellipse to a flat line and then to a hyperbola with increasing momentum in $k$-vector space, as shown in the first three panels of Figs. 2(e) and 2(f). In addition, a circle topology of polariton dispersion is also experimentally observed, when a large capacitance $C_v$ is realized by applying a smaller bias voltage onto the metasurface, as shown in the last panel of Fig. 2(f). In general, it might be slightly different between the realistic capacitance and fitting value of the varactor diode due to the measurement error, as well as the fabrication tolerance of the varactor diode. Within a reasonable margin of error, the measured results are consistent with the simulated results, fully validating the reconfigurable topological transition of the metasurface polaritons. In addition, experimental measurements demonstrate that the response time of the topology switching is around $1.6\mu \mathrm{s}$, enabling high-speed responsive reconfigurability of the proposed metasurface (see Sec. S7 in the Supplementary Material).

2.3.

Controllable Field Canalization and Tunable Planar Focusing

As the polariton propagation is highly correlated with the dispersion characteristic that can be flexibly manipulated at a certain frequency band, the proposed reconfigurable metasurface may find plenty of applications for polariton manipulation. As the proof of concept, we here experimentally present and discuss the intriguing phenomena of controllable field canalization and tunable planar focusing enabled by the tunable polariton dispersions. When the polariton dispersion is switched to flat line fashion, the metasurface can yield diffraction-free propagation wave due to the flat topology of IFC [the second column of Fig. 2(b)]. This phenomenon is known as canalized propagation,^39,58–60 with excellent potential in realizing hyperlensing,^10,61 subdiffraction-resolution imaging,²⁶ retroreflection,⁶² and position sensing.⁶³ Here, the proposed reconfigurable metasurface benefits from the utilization of voltage-controllable varactor diodes, enabling us to achieve a broad bandwidth of field canalization. Figure 3(a) clearly depicts the topological transition regions as functions of frequency and loaded capacitance $C_v$, in which blue, light green, and red regions represent the closed elliptical, open elliptical, and hyperbolic polariton dispersions, respectively. The gray region denotes that the polariton IFCs of the first band in the first Brillouin zone no longer exist. For a fixed capacitance or bias voltage, we can only observe the field canalization phenomena in a very limited frequency band, due to the inherent dispersion of the metasurface structure. Differently, by adaptively applying different bias voltage, such dispersion can be counteracted to yield canalization propagation with enhanced bandwidth, as depicted by the green dashed line, the presence of flat polariton IFCs. The green circles in Fig. 3(a) denote the measured canalization points under different bias voltages. Figure 3(b) shows the measured electric field distributions at 5.5 GHz ($V_C=4\mathrm{V}$, $C_v=0.42\mathrm{pF}$), 5.9 GHz ($V_C=6\mathrm{V}$, $C_v=0.31\mathrm{pF}$), 6.2 GHz ($V_C=10\mathrm{V}$, $C_v=0.21\mathrm{pF}$), and 6.3 GHz ($V_C=12\mathrm{V}$, $C_v=0.19\mathrm{pF}$), where we observe obviously the polaritons propagating in a self-collimating manner at different frequencies. More simulated and measured results can be found in the Supplementary Material (see Sec. S8 in the Supplementary Material). Considering the ohmic loss of the parasitic resistance of the varactor diode, detailed information about the propagation loss of the field canalization mode is further provided (see Sec. S9 in the Supplementary Material).

Fig. 3

Controllable field canalization and tunable planar focusing based on topological transition of polariton dispersions. (a) Topological transition regions as a function of frequency and loaded capacitance value $C_v$, which is extracted from the numerical simulations of dispersion diagrams of both the first and the second band in the first Brillouin zone. (b) Measured field distributions [Re ($E_z$)] in the $xy$ plane that is 1 mm above the reconfigurable metasurface at different frequencies. The observation areas consist of $24\times 16$ unit cells with a size of $204\mathrm{mm}\times 136\mathrm{mm}$. (c) Schematic of the planar focusing as the polaritons propagate from the metasurface into the surrounding air medium. The inset shows the hyperbolic topology of polariton dispersion at 6.1 GHz with the $C_v=0.32\mathrm{pF}$. The red arrows indicate the propagation directions of the polaritons and the red dot represents the focal point. (d) Measured intensity distributions (${|E_z|}^2$) in the $xy$ plane that is 1 mm above the reconfigurable HM at different frequencies. The observation areas have a size of $136\mathrm{mm}\times 136\mathrm{mm}$.

As the landmark feature of hyperbolic polaritons, hyperbolic dispersion has garnered significant attention due to its exotic optical properties and potential applications in all-angle negative refraction,^27,64 planar focusing,^28,30 waveguiding,⁶⁵ and spin control.⁶⁶ Here, we demonstrate tunable planar focusing based on the proposed reconfigurable metasurface. Figure 3(c) presents the schematic of the tunable polariton focusing, where light blue and light yellow blocks are the reconfigurable metasurface and air regions, respectively, and the green dashed circle placed at the bottom side of the metasurface represents the excitation source. The metasurface is switched to the state of hyperbolic dispersion by the bias voltage. The propagation directions of the polariton, illustrated by the red arrows, are included within the IFCs to schematically show the dispersion-induced hyperbolic propagation, which will be focused into a hotspot as the polariton propagates into the surrounding air medium. Measurements are conducted in the air region represented by the red dotted frame to confirm the tunable planar focusing and excellent performance can be observed from 6.04 to 6.6 GHz. Figure 3(d) presents the measured intensity distributions ($|E_z{|}^2$) at 6.1 GHz ($V_C=6\mathrm{V}$, $C_v=0.31\mathrm{pF}$), 6.3 GHz ($V_C=7.3\mathrm{V}$, $C_v=0.27\mathrm{pF}$), 6.43 GHz ($V_C=8.8\mathrm{V}$, $C_v=0.23\mathrm{pF}$), and 6.6 GHz ($V_C=11.8\mathrm{V}$, $C_v=0.19\mathrm{pF}$), where the efficient energy focusing of the designed polaritons is verified by the focal points in the air, with a full width at half-maximum of around $0.5\lambda$. More simulated and measured results are provided in the Supplementary Material (see Sec. S10 in the Supplementary Material).

2.4.

Planar Reconfigurable Integrated Polariton Circuit

Furthermore, the proposed metasurface can realize the concept of planar reconfigurable polariton circuits, where the individual control of each meta-atom IFC is achieved through external voltages. This allows for regional control over the dispersion curves of the meta-atoms and enables the combination of different anisotropic topologies to generate intriguing polariton functions. As the design examples, Fig. 4(a) shows the schematic of the planar reconfigurable polariton circuit composed of two different metasurface regions. The blue and orange blocks represent the arrays of unit cells with the loaded capacitance of $C_{v1}$ and $C_{v2}$, respectively, which can be realized by applying different voltages on the corresponding regions. At the interface [dashed line in Fig. 4(a)] of two regions, the on-demand manipulation and guidance of propagation of polaritons can be engineered by tailoring the distributions of topologies of polariton dispersions with flexibility and diversity. The green dashed circle is the excitation source placed at the bottom side of the metasurface. Simulations of several extraordinary propagation phenomena are conducted at 6.1 GHz, as shown in Figs. 4(b)–4(d). The polariton IFCs in two regions are presented in the right panels, where the black solid arrows represent the group velocity $v_g$, corresponding to the propagation directions of designer polaritons (black dotted arrows in left panels). As shown in Fig. 4(b), the negative refraction occurs at the interface when the polariton IFC is set to be hyperbolic dispersion in region 2 (also see Video 5 in the Appendix). Figure 4(c) shows the wave-splitting functionality (also see Video 6 in the Appendix), where the designer polariton propagates along two symmetric directions at the interface by combining the open elliptical and flat topology of polariton dispersions. In contrast, the polariton propagation can be suppressed along the $y$ direction when the varactor diodes of region 2 are tuned as large capacitance value (e.g., $C_v=0.52\mathrm{pF}$ in Fig. 2), as shown in Fig. 4(d). In this case, the collimating wavefront is launched on region 1 and propagates toward region 2, but eventually spreads out as convex wavefront in region 2 (also see Video 7 in the Appendix). It should be noted that the proposed reconfigurable integrated polariton circuit in Fig. 4(a) is a reciprocal system, and the demonstrated transformation of the polariton propagation regimes in Figs. 4(b)–4(d) is still feasible in reverse. In addition to the results presented above, other functionalities for manipulation of polaritons can be realized, such as collimating refraction [see Fig. S16(a) in Sec. S11 in the Supplementary Material and also Video 8 in the Appendix] and the suppression of polariton propagation for convex wavefront [see Fig. S16(b) in Sec. S11 in the Supplementary Material and also Video 9 in the Appendix]. These results show promise for their use in designing reconfigurable spatial multiplexers. Notably, unlike the passive metasurfaces where polariton propagation is fixed once fabricated, our designs offer on-demand reconfigurability through voltage-driven electronic elements, which represents a fundamental difference from the existing approaches. Serving as the illustration examples, here we only consider the proposed reconfigurable polariton circuit composed of two different regions of unit cells for simplification. In fact, there is much potential for other extraordinary propagation phenomena of polaritons and advanced functions by a polariton circuit with more complex combination of topologies of polariton dispersion (see Sec. S12 in the Supplementary Material), and even by a planar programmable integrated polariton circuit (see Sec. S13 in Supplementary Material). Particularly for the planar programmable integrated polariton circuit, a series of cases for polariton propagation (supporting multi-excitation sources, multi-ports, and flexibly designable propagation routes) has been designed and simulated, which significantly expanded the possibilities for polariton manipulation and guidance and may shed new light on the design of subwavelength surface waveguides. The proposed metasurface that supports topologically reconfigurable polaritons makes it a promising candidate for on-chip advanced photonics systems and may shed new light on designing planar multifunctional plasmonic devices and integrated optoelectronic circuits, etc.

Fig. 4

Reconfigurable integrated polariton circuit. (a) Schematic of the planar reconfigurable polariton circuit composed of two regions (region 1 and region 2) represented by blue and orange blocks, respectively. The loaded capacitances of unit cells are different in region 1 and region 2, leading to different topologies of polariton dispersions. (b)–(d) Simulated field distributions [Re ($E_z$)] in the $xy$ plane that is 7 mm above the polariton circuit. The right panels show the polariton IFCs in two different regions. The loaded capacitances in region 1 and region 2 are (b) 0.03 and 0.32 pF for realizing the negative refraction, (c) 0.28 and 0.03 pF for realizing the wave splitting, and (d) 0.28 and 0.52 pF for realizing the suppression of polariton propagation. The green dashed circle represents the excitation source. The dotted lines indicate the main propagating directions of the polaritons. The observation areas consist of $64\times 48$ unit cells with a size of $544\mathrm{mm}\times 408\mathrm{mm}$.