Introduction

Electromagnetic (EM) waves of spatial and guided forms, as widely known, are two crucial intermediaries in extensive wireless and integrated applications respectively. Therefore, various manipulation technologies are usually developed for the two types of waves in parallel. For example, transformation optics has spawned surface-wave1,2,3 and space-wave cloaks4. Negative refractions have led to superlens for space waves5,6,7 and subwavelength focusing for guided waves8. Topology9 roots in both bound states in the radiative continuums10 and robust waveguides11,12.

Decades ago, the concept of parity-time (PT) symmetry13, which originates from non-Hermitian quantum mechanics, has been introduced into photonics, thus igniting intense interests in non-Hermitian photonics. It has resulted in fascinating phenomena i.e., unidirectional transmission for space waves14, as well as coherent perfect absorber15,16, loss-induced transparency17, and high-performance sensors for guided waves18,19. Subsequently, anti-parity-time (APT) symmetry has also been proposed and aroused intense interests across multi disciplines from atomics20,21,22,23, photonics24,25,26,27,28,29,30, classical31 and quantum32 circuits, thermology33 to magnetics34, etc. Fundamentally different from PT-symmetry systems, APT-symmetry systems exhibit exotic dynamics, i.e., energy-difference conservation31,35, thus providing fascinating approaches to harnessing EM waves. On guided waves, APT symmetry has so far resulted in fruitful photonic phenomena, e.g. constant refraction36, mode switching37, and enhanced Sagnac effect38,39. However, the study on harnessing space waves with APT-symmetry systems is still lacking.

Here, we propose a radiative plasmonic APT design scheme to harness space waves (in Fig. 1). The crucial part of such design is the leaky wave, which plays a significant role in two aspects. Its in-plane propagating component provides imaginary coupling channels to enable the APT system, while its out-of-plane radiation nature enables the interaction of APT systems and space waves. We then experimentally demonstrate the design with subwavelength designer surface plasmonic resonators (DSPRs). The DSPRs host low-frequency surface modes which are analogous to the localized surface plasmons at optical frequencies40,41,42,43,44,45. Exploring the degrees of freedom (DoFs) of space waves, we observe two exotic phenomena unrealized previously. Switching polarizations of illuminating space waves, we realize polarization-controlled APT phase transition. Tuning incidence angles, we observe multi-stage APT phase transition in higher-order APT systems, constructed by using the scalability of leaky-wave coupling. Our scheme shows promise in demonstrating novel APT physics, and constructing APT-symmetry-empowered radiative devices.

Fig. 1: Schematic of the radiative plasmonic APT design.
figure 1

The system consists of two plasmonic resonators, whose frequencies are oppositely detuned, i.e. up-detuned (ωH) plasmonH and down-detuned (ωL) plasmonL. ωC denotes the averaged frequency ωC = (ωΗ + ωL)/2. The two resonators are indirectly coupled by leaky waves, whose out-of-plane radiation leakage enables the interaction of the APT system and space waves.

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Results

Designer-plasmonic realization of radiative APT systems

Figure 2a shows a designer-plasmonic radiative APT system consisting of two DSPRs. A purple dashed circle indicates an individual DSPR, which is a cylindrical three-layer structure. The top layer is a groove-textured ultrathin copper disk, the middle layer is a dielectric substrate, and the bottom layer is a complete copper plate attached to the substrate. The top surface of the DSPR exhibits a highly-confined dipole mode ψ1(2)nf, whose horizontal decay length is Lx = 2.9 mm from the edge of the resonator (in Fig. 2b). The subscripts 1 and 2 denote the left and right DSPR respectively. However, this DSPR mode does not radiate in the z-direction, since the upper x-polarized dipole and its image generated by the ground plane conceal their radiations in the far-field, due to the deep subwavelength thickness (h = 2 mm) of the substrate. Removing the top layer, we obtain background structures, which host surface waves ψp propagating along x direction (cross-sectional field in the substrate is shown in Fig. 2c). The dispersion of such surface waves is very close to the light line as shown in Fig. 2d. A slight perturbation (e.g. DSPRs) on the background can transform the surface waves into leaky modes, which not only propagates along x direction (i.e. in-plane radiation), but also radiates in z-direction into the free space46 (i.e. out-of-plane radiation). Consequently, the in-plane component of leaky waves can provide the indirect coupling channels, in addition to the direct coupling through the evanescent field of the two DSPRs. While the out-of-plane radiation leakage enables the interaction of the system and space waves.

Fig. 2: Designer-plasmonic realization of the radiative APT system.
figure 2

a The top view and the front view of the designer-plasmonic APT system. It consists of two DSPRs with a distance d, encircled by the background horizontally. The purple dashed line denotes a single resonator composed of three layers. The top layer is a groove-structured copper plate, standing on a dielectric substrate with the copper coating on its bottom. The background is also constructed with the dielectric substrate coated with copper plating. b The Ez component of the simulated DSPR eigenmode (ψnf) at 3.655 GHz on the XY (left) and XZ (right) planes. c The Ez component of the surface wave (ψp) on the XZ plane. d The dispersions of the surface wave (red) in the background, and the light line (blue) in the air. The leaky waves lie in the gray region. e The distance-dependent strength of calculated direct coupling κ (blue) and indirect couplings χe (magenta) and χo (red). The insets depict the mode-parity modulation on leaky waves.

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In the basis [ψ1nf, ψ2nf]T the coupled designer-plasmonic system is described with a generalized coupled-mode equation:

$$\omega \left[\begin{array}{c}{\psi }_{1}^{{{{{{\rm{nf}}}}}}}\\ {\psi }_{2}^{{{{{{\rm{nf}}}}}}}\end{array}\right]=\left[\begin{array}{cc}{\omega }_{1}+{i\gamma }_{0} & {{\omega }_{0}}({\kappa }_{12}+{i\chi }_{12})\\ {\omega }_{0}({\kappa }_{21}+{i\chi }_{21}) & {\omega }_{2}+i{\gamma }_{0}\end{array}\right]\left[\begin{array}{c}{\psi }_{1}^{{{{{{\rm{nf}}}}}}}\\ {\psi }_{2}^{{{{{{\rm{nf}}}}}}}\end{array}\right]$$
(1)

where ψ1(2)nf, ω1 (ω2) and ω0 represent the complex amplitude, the resonant frequency of the DSPR mode and the averaged frequency ω0 = (ω1 + ω2)/2, respectively. The coefficient κ12 (κ21) quantifies the strength of direct coupling between the two modes, while iχ12 (iχ21) represents their indirect coupling strength through leaky-wave channels47,48,49. Besides the in-plane components of leaky modes which indirectly couple the two DSPRs, the out-of-plane radiation of leaky modes takes the energy of the resonant modes into the ambient space, thus behaving as loss channels (quantified with dissipation rates γ0).

Applying energy conservation50 to the hybrid system including both the plasmonic system and environment, we obtain that coupling coefficients satisfy κ12 = κ21 = κ, iχ12 = iχ21 = iχ (see Supplementary Information I). Therefore, the system Hamiltonian is expressed as H = ω0 + iγ0 + ω0Hint, where the interaction part is Hint = [−δ, κ+iχ; κ+iχ, δ]. δ = |ω1 − ω2 | /(2ω0) represents the frequency detuning.

We further elucidate the synthesis of APT symmetry in Hint, which should satisfy the anti-commutation relation, i.e. {Hint, PT} = 0. The parity operator P, expressed by Pauli matrix σx, exchanges the spatial positions of the two modes. The time-reversal operator T is given by the complex conjugation. Consequently, such anti-commutation relation requires vanished real couplings, while remaining only the imaginary couplings.

For this purpose, we elucidate these two couplings with the field overlap integral51. The indirect coupling through leaky waves is elucidated as iχ ∝ ∫(ψ1(2)nf·ψ2(1)p)/(|CNF | · | CFF | )dV, where normalization constants follow |CNF | 2 = < ψmnf | ψmnf > V and δ (0)|CFF | 2 = < ψmp | ψmp > V (m = 1, 2). The leaky mode excited by the corresponding resonant mode ψ1(2)nf is approximated with the in-plane propagating wave ψ1(2)p. Since the fields of resonant modes are mostly confined in the substrate, the integration is dominated by the parts in the substrate. Since the substrate-field distribution is approximately uniform along the z direction, the volume integral is simplified as iχ ∝ h·∫(ψ1(2)nf·ψ2(1)p)/(|CNF | · | CFF | )dxdy. It is worth noting that there are two forms of indirect couplings, i.e. even-mode and odd-mode couplings, since the propagating wave ψ1(2)p is modulated by the DSPR mode ψ1(2)nf with different parities (shown in Fig. 2e). Taking an arbitrary line along x direction, the field distributions of ψnf is approximated as cos(πy/2 R)sin(πx/2 R) (odd) or sin(πy/2 R)cos(πx/2 R) (even). Due to the parity modulation by ψnf, ψp is approximated as cos(πy/2 R)eikx (odd) or sin(πy/2 R)eikx+iπ/2 (even), respectively. k denotes the propagation constant of the in-plane radiation. Therefore, the non-vanished indirect coupling gives pure imaginary values as shown in Fig. 2e (see Supplementary Information II). In Fig. 2e, the indirect coupling strength is independent of the distance d between two resonators in proper regions (see Supplementary Information II), and χe is less than χο. While the direct coupling is also proportional to the field overlap integral κ ∝ ∫(ψ1nf·ψ2nf)/(|CNF | 2)dV (in Fig. 2e), which decreases as the distance d increases due to the evanescent distance of ψnf. To guarantee vanished direct couplings and retained indirect couplings, we set the edge-edge distance of the two DSPRs sufficiently distant as d = 10 mm, which is larger than 2Lx. Therefore, the interaction Hamiltonian is reduced as Hint = [−δ, iχ; iχ, δ], which satisfies the anti-commutation relation, i.e. {Hint, PT} = 0. It is noteworthy that such photonic indirect coupling, resulting in a plasmonic APT system in the spectral domain, is a mechanism of linear optics, rather than nonlinear optics which usually require high-power input21,26,29.

The reduced Hamiltonian Hint undergoes spontaneous APT-symmetry breaking by sweeping the frequency detuning δ. More specifically, the system changes from APT-symmetric (APT-S) to APT-symmetry-broken (APT-B) phase, when δ changes from | δ / χ| < 1 to | δ / χ| > 1. Three striking features indicate the symmetry-breaking process. Firstly, the eigenvalues λ evolve from two pure imaginary values ±i(χ2 − δ2)1/2 to two pure real values ± (δ2 − χ2)1/2 (as shown in Fig. 3a, b). Such evolution corresponds to the transition from two degenerate modes with different decay rates (the eigenvalues ±i(χ2 − δ2)1/2 manifest as high-Q and low-Q mode respectively) to two non-degenerate modes with the same decay rate (the eigenvalues ± (δ2 − χ2)1/2 correspond to the high-frequency and low-frequency mode respectively). Secondly, the two corresponding eigenmodes of the Hamiltonian change from [ψ1nf, ψ2nf]T = [1, −(iδ ∓ (χ2 − δ2)1/2)/χ]T in the APT-S phase to [1, −i(δ ± ((δ2 − χ2)1/2)/χ)]T in the APT-B phase. Specifically, the APT-S modes exhibit the unit amplitude ratio (|ψ2nf/ψ1nf | = 1) and varied phase differences. However, the APT-B modes exhibit non-unit amplitude ratios with a fixed phase difference arg(ψ2nf) – arg(ψ1nf) = −π/2 (see Supplementary Information III). Thirdly, the effective Hamiltonian becomes defective at the critical point | δ / χ| = 1, which is also termed as the exceptional point (EP), and gives a single eigenvalue λ = 0 and eigenstate [1, −i]T. Therefore, the two-dimensional Hilbert space coalesces into one-dimension space, where the missing dimension is also known as the Jordan vector52.

Fig. 3: Near-field characterizations of plasmonic APT systems.
figure 3

a, b The real part (a) and imaginary part (b) of the calculated eigensolutions of the APT system versus the ratio between the frequency detuning δ and indirect coupling χ. The blue and red regions denote the APT-S and APT-B phases respectively. The red stars indicate the detected resonance frequencies in experiments. c, d The evolution of the transmission spectra (black lines) as | δ / χ| varies, obtained with simulations (c) and near-field-measured experiments (d). Each black line is decomposed into a blue and a red line, according to mode decompositions. The inset shows the setup for both simulations and measurements. The ‘S’ and ‘P’ denote the locations of the near-field source and probe respectively. e–h The measured near-field distributions of APT-S (| δ/χ| = 0), EP( | δ/χ| = 1), and APT-B phases (|δ/χ| = 4.46), respectively. The distributions of Ez and |Ez | are shown with rainbow and hot colors, respectively.

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Near-field demonstration of APT phase transition

Without loss of generality, we take the odd mode (in Fig. 2e) to verify the APT phase transition. Five samples with different δ are realized by changing the inner radius rR of the right DSPR as 6, 6.3, 6.7, 7.0, and 7.3 mm respectively, while keeping the rL = 6 mm for the left DSPR. Their corresponding frequency detuning parameters ω0δ are extracted as 0, 0.025, 0.061, 0.085, and 0.1115 GHz respectively (see Supplementary Information IV). Under the slight detuning and keeping the edge-edge distance as d = 10 mm, the indirect coupling ω0χ is approximately unchanged and extracted as 0.025 GHz from simulation (see Supplementary Information IV for details). Therefore, the sample of ratio | δ / χ| = 0 corresponds to the APT-S phase, |δ / χ| = 1 at the EP and | δ / χ| = 2.44, 3.40, 4.46 in the APT-B phase.

Utilizing near-field measurements (see Supplementary Information V), we further verify the APT phase transition. The near-field transmission spectra are measured by using the point excitation-probe setup (shown in the inset of Fig. 3c). As the ratio | δ / χ| decreases, we experimentally observe that two Lorentz peaks gradually merge into one (shown in Fig. 3d), which is consistent with the simulated spectra in Fig. 3c. The peak evolution corresponds to the change from two real eigenvalues to zero and confirms the phase transition from the APT-B to the APT-S phase. Furthermore, the evolution of spectral linewidth also verifies the APT phase transition. The two peaks in the APT-B phase exhibit the same linewidths, which are extracted as 0.036, 0.044, and 0.045 GHz at | δ / χ| = 2.44, 3.40, and 4.46 respectively. While the spectrum of the APT-S phase (|δ / χ| = 0) is decomposed into two peaks with different linewidths (0.029 and 0.061 GHz). The linewidth evolution corresponds to the change of the imaginary parts of eigenvalues and further confirms the APT phase transition. The noteworthy phenomenon is that the spectrum measured at the APT-S phase (|δ / χ| = 0) is dominated by the high-Q mode, while the degenerate low-Q mode is weak. The phenomenon is due to that the high-Q mode is excited with a larger efficiency (ω0χ + γ0)/2 under the asymmetric near-field excitation in the existence of background loss, thus dominating the measured near-field spectrum (see Supplementary Information VI).

Using near-field imaging technologies, we further observe the APT phase transition by capturing field patterns at resonance peaks. Comparing the samples of | δ / χ| = 0 (APT-S) and 4.46 (APT-B), the field patterns exhibit near-unit and non-unit amplitude ratios respectively (in Fig. 3e, g, h), which are consistent with theoretical results. At the EP ( | δ / χ| = 1), the captured field pattern (in Fig. 3f) deviates from the calculated eigenstates [1, −i]T. Such deviation is due to the emergence of the missing eigenstate [1, i]T in the presence of the background loss γ0 (see Supplementary Information VI). These captured field patterns further confirm the APT phase transition. Also, such the detuning-induced APT phase transition is observed under space-wave illuminations (see Supplementary Information VII).

Note that we use different samples here just for the convenience of experimental characterizations. It does not mean this is the only way to achieve the APT phase transition. Since our plasmonic APT system is an open system, the phase transition is also achieved by changing environmental factors, e.g. permittivity (see Supplementary Information VIII and IX). This approach does not require using different samples, and is promising for sensing applications.

Far-field polarization-controlled APT phase transition

We further investigate the polarization responses of the plasmonic APT system, since the leaky wave links the polarization of space waves to the parity of DSPR modes. Specifically, the horizontal (H) and vertical (V) polarizations are linked to the odd- and even-parity modes, respectively. The even and odd modes exhibit different indirect coupling strengths due to their parity modulations on leaky waves (in Fig. 2e). Therefore, the APT system exhibits two sets of APT phase transitions for odd and even modes (in Fig. 4a). In the blue region of Fig. 4a, the APT system is in the APT-S phase for odd modes while in the APT-B phase for even modes. Since the links between the space-wave polarizations and DSPR-mode parities, we switch the polarizations of illuminating space waves (in Fig. 4b) to induce the APT phase transition. We design the sample (rR = 6.27 mm and rL = 6 mm) in the blue region. A vertical near-field probe is utilized to capture the Ez components of field patterns. We probe the fields at positions A and B (in Fig. 4b) under H- and V- polarized incidence, respectively. When we rotate the space-wave polarization from H to V, the number of resonance peaks changes from one to two (shown in Fig. 4c). The spectral results indicate the polarization-controlled phase transition from APT-S to APT-B, consistent with the theoretical results in Fig. 4a.

Fig. 4: The polarization-controlled APT phase transition.
figure 4

a The evolution of eigen frequencies as the detuning rR − rL changes for odd (red) and even (blue) modes. The stars mark the detected resonance frequencies on the sample rR − rL = 0.27 mm. b The far-field excitation setup. The red and blue arrows denote horizontally (H) and vertically (V) polarized space waves, respectively. Points A and B denote the probe positions for odd and even modes respectively. c The experimental transmission spectra under illuminations of H-polarized (red) and V-polarized (blue) plane waves, respectively. The stars mark the resonance-peak positions. d–f The field patterns Ez by near-field imaging under H-polarized plane-wave illuminations at 3.725 GHz (d), and V-polarized plane-wave illuminations at 3.72 GHz (e) and 3.76 GHz (f). The distributions of Ez and |Ez | are shown with rainbow and hot colors, respectively.

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The polarization-controlled APT phase transition is further verified by the captured field patterns at the resonance peaks. Under H-polarized incidence, the pattern shows the low-Q mode [1, eiπ/3] of odd parity at 3.725 GHz, thus confirming the APT-S phase. While under the V-polarized incidence, the captured patterns, corresponding to the even parity, show [2, −i] at 3.72 GHz and [1, −2i] at 3.76 GHz, respectively, thus confirming the APT-B phase.

Two-stage APT phase transition in higher-order systems

Attributed to the scalability of the leaky-wave coupling, the design mechanism of our radiative APT system is further utilized to study higher-order systems. Moreover, the incoming direction of space waves plays a unique role in identifying high-order APT supermodes. Fig. 5(a) shows the higher-order APT system, which consists of four DSPRs with inner radii [r1, r2, r1, r2], while other parameters remain the same as the above structures. In the basis [ψ1nf, ψ2nf, ψ3nf, ψ4nf]T, the Hamiltonian of this system is expressed as H = ω0 + iγ0 + ω0H4.

$${H}_{4}=\left[\begin{array}{cc}\begin{array}{cc}-\delta & i\chi \\ i\chi & \delta \end{array} & \begin{array}{cc}0 & 0\\ i\chi & 0\end{array}\\ \begin{array}{cc}0 & i\chi \\ 0 & 0\end{array} & \begin{array}{cc}-\delta & i\chi \\ i\chi & \delta \end{array}\end{array}\right]$$

The fourth-order effective Hamiltonian H4 obeys the four-dimensional anti-commutation relation {H4, P4T} = 0, where the four-dimensional parity operator is P4 = [0, 0, 0, 1; 0, 0, 1, 0; 0, 1, 0, 0; 1, 0, 0, 0].

During sweeping the frequency detuning δ, the higher-order APT system undergoes two-stage APT-symmetry breakings (in Fig. 5b), marked by the two distinct EPs, i.e. |δ / χ| = (√5 ∓ 1)/2. When | δ / χ| < (√5 − 1)/2, four degenerate modes differ with the imaginary parts of their eigenvalues λ1,2,3,4 = ±i[−δ2+((3±√5)χ2/2)]1/2. We term this region as the APT-S phase, where all four degenerate modes can be classified into two groups according to their fixed amplitude ratios, i.e. |ψ1nf | : |ψ2nf | : |ψ3nf | : |ψ4nf | = 1: (√5 + 1)/2: (√5 + 1)/2: 1 as Group+ and 1: (√5 − 1)/2: (√5 − 1)/2: 1 as Group−. After crossing the first EP | δ / χ| = (√5 − 1)/2, the system enters the 1st-APT-B phase (√5 − 1)/2 < |δ / χ| < (√5 + 1)/2. The modes in Group+ are still degenerate and keep the fixed amplitude ratio, while the modes in Group− become nondegenerate with real eigenvalues λ3,4 = ±[δ2 + ((√5 − 3)χ2/2)]1/2, exhibiting the fixed phase difference [0, π/2, π, 3π/2]. Tuning δ across the second EP | δ / χ| = (√5 + 1)/2, the system enters the 2nd-APT-B phase. The modes in Group+ also become nondegenerate with the fixed phase difference [0, π/2, 0, π/2] (see Supplementary Information X).

Fig. 5: Observation of multi-stage APT phase transitions under space-wave illuminations.
figure 5

a The top view of the higher-order APT system. From left to right, the inner radii of the four DSPRs are r1, r2, r1, and r2, respectively. b The real (black) and imaginary part (white) of the calculated eigenvalues of the system versus the detuning r2 − r1. The blue, purple and pink regions denote the APT-S, 1st-APT-B and 2nd-APT-B phases, respectively. The red dots represent the EPs. Stars mark the experimentally detected resonance frequencies at r2 − r1 = 0, 0.4, and 0.7 mm. The inset is the schematic of normal (red) and oblique (blue) illuminations to excite modes in Group+ and Group− respectively. c–j The measured field patterns of APT-S at | δ/χ| = 0 (c), 1st-APT-B at | δ/χ| = 1.4 (d–f), and 2nd-APT-B phases at | δ/χ| = 2.4 (g–j), respectively. The distributions of Ez and |Ez | are shown with rainbow and hot colors, respectively.

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To verify the two-stage APT phase transition, we design three samples with different frequency detunings δ. We take r2 as 6, 6.4 and 6.7 mm respectively, while keeping r1 = 6 mm. Their corresponding detunings ω0δ are extracted as 0, 0.035 and 0.06 GHz respectively and the indirect coupling ω0χ is the same as above. Therefore, the sample of ratio | δ / χ| = 0 corresponds to the APT-S phase, |δ / χ| = 1.4 in the 1st-APT-B phase and | δ / χ| = 2.4 in the 2nd-APT-B phase.

We demonstrate the two-stage APT symmetry breaking of the higher-order APT system with space-wave illuminations (see Supplementary Information V for the far-field experimental setup). The detected resonance frequencies are indicated with stars in Fig. 5b, which are consistent with the theoretical eigenvalues. The corresponding captured mode profiles are shown in Fig. 5c–j. Regarding the APT-S sample (|δ / χ| = 0) under normal incidence, Fig. 5c shows the low-Q mode [1, (√5 + 1)/2, (√5 + 1)/2, 1] in Group + . For the samples |δ / χ| = 1.4 (1st-APT-B), modes in Group+ remain degenerate, but the degeneracy in Group− is lifted. Under normal incidence, we clearly identify the low-Q mode [1, (√5 + 1)e(−iπ/3)/2, (√5 + 1)/2, e(−iπ/3)] in Group + (Fig. 5d), but cannot identify any mode in Group − . The reason is that normal incidence does not match the gradient phase difference [0, π/2, π, 3π/2] in Group − , so that these modes cannot be efficiently excited. To address this issue, space waves are illuminated with a specific incidence angle θi = arcsin[λi/4(d + 2 R)] on the sample, where λi denotes the wavelength of incident waves. The nondegenerate modes in Group− are clearly observed in Fig. 5 e, f. In the 2nd-APT-B region (|δ / χ| = 2.4), the high-frequency (low-frequency) modes are very close in spectra. To decern them, we also use normal and oblique incidences to efficiently excite them respectively (in Fig. 5g–j). These results confirm the scalability of our design, and selective excitation by harnessing the incident angle of space waves.

Discussion

We proposed a radiative APT plasmonic design and demonstrated it with designer-plasmonic structures. Unlike previous photonic APT systems restricted in guided waves26,27,28,29, our plasmonic APT systems are capable of harnessing spaces waves, due to the radiative property of leaky waves. Our system, exhibiting polarization-controlled APT phase transition, provides a platform to exploit the polarization of space waves with APT symmetry. The incidence angle of space waves also plays a role in decerning the higher-dimensional APT modes. Furthermore, attributed to the scalability, our system also provides a platform to demonstrate the higher-order APT phenomena, e.g. multi-stage APT phase transitions. Our work opens an avenue to construct APT-empowered radiative devices, and is promising in wireless sensing applications. Also, the subwavelength scale of our designer-plasmonic design is helpful in miniaturizing APT photonic devices. Our plasmonic APT design could be further extended to optical frequencies53 (see Supplementary Information XI) and further interface with gain materials or topological structures12,54, to spawn unusual lasers or intriguing non-Hermitian topological photonic phenomena.

Methods

Simulations

COMSOL Multiphysics is utilized to obtain the eigen-field patterns. The simulated near-field transmission spectra are obtained with CST Microwave Studio. The simulation model consists of two planar designer-plasmonic resonators with a distance d = 10 mm, encircled by the background horizontally. The groove-structured copper plates of 0.018 mm thickness stand on a 200 × 80 × 2 mm dielectric substrate, with a copper coating on its bottom. The outer radius of the copper plate is R = 12 mm, and the groove number is N = 60. The metal is modeled as the perfect electrical conductor (PEC) in the microwave regime. The relative permittivity and loss tangent of the dielectric layer are 2.2 and 0.001 respectively. The near-field excitation source is a discrete port, while plane-wave excitation is utilized in far-field simulation.

Experiments

The sample is fabricated with the technology of printed circuit boards. The dielectric layer of the sample is made of F4BM with the relative permittivity 2.2 ± 0.03 and the loss tangent 0.001. Experimental measurements are carried out in the microwave anechoic chamber. The measurements of S parameters are conducted with the vector network analyzer. Space waves are generated from a horn antenna.