Metasurfaces have attracted significant research interest in recent years because they allow versatile control of electromagnetic wave propagation with a flat, low-profile form factor. Their planar structure facilitates scalable fabrication of metasurfaces based on printed circuit boards (PCB) or silicon photonic technologies. Considerable progress has been made in realizing functionalities such as polarization conversion1-4, perfect absorbtion5, 6 and wavefront control7-10. However, most metasurfaces demonstrated to date obey the Lorentz reciprocity. Nonreciprocal metasurfaces (NRMs) have only recently emerged as a new type of metasurfaces only recently11-15. Unlike conventional metasurfaces whose characteristics are symmetric with respect to the wave propagation direction, NRMs are uniquely capable of performing direction-dependent functions. For instance, NRM-integrated radomes can act as nonreciprocal antennas, which can not only curtail antenna echoes16 but also enable arbitrary emission and receiving characteristics of electromagnetic waves. NRMs also facilitate full-duplex wireless communications by doubling the bandwidth of wireless communications17.
Several mechanisms can be used to achieve nonreciprocity18. A transistor-loaded circuit allows unidirectional gain of electromagnetic waves owing to its rectifying effect, thereby breaking Lorentz reciprocity19, 20. These devices can exhibit a broad operational bandwidth and forward transmission gain. However, they suffer from low operation frequencies (limited by transistor speed), low power-handling capability, high biasing power, and poor signal-to-noise ratio17, 21, 22. Nonlinearity in metasurfaces with asymmetric structures is another means of generating nonreciprocity, although they are sensitive to incident power and are limited by dynamic reciprocity to pulsed-wave applications only23, 24. Reciprocity can also be lifted by spatiotemporal modulation, although this approach incurs a large constant power consumption, a trade-off between modulation frequency and operation bandwidth, and undesired high-harmonic generation12, 17, 25. Finally, magnetic materials are naturally nonreciprocal given their asymmetric permittivity or permeability tensors. Even though these materials are widely used in commercial isolators or circulators, they are surprisingly less explored for NRM applications, likely because bulky biasing magnets preclude the local control of individual meta-atoms. To summarize, bias-free, linear, and passive NRMs are lacking for applications in free-space microwave or photonic systems.
Here, we theoretically propose and experimentally demonstrate a new class of NRM consisting of self-biased magnetic meta-atoms. Using all-dielectric Mie resonators made of M-type La-doped BaFe12O19 hexaferrite (La:BaM), the local magnetization direction can be locked by the strong magnetocrystalline anisotropy of this material, which eliminates the requirement of magnetic field biasing. Moreover, each meta-atom can be individually magnetized along arbitrary directions, providing unprecedented design flexibility to attain bidirectional amplitude and phase profiles. Following a nonreciprocal digital coding metasurface (NDCM) design methodology, we designed and fabricated NRMs in a “LEGO-like” manner with a diverse set of functions, such as unidirectional transmission, nonreciprocal beam deflection, nonreciprocal beam focusing, and nonreciprocal holography in the Ku band. This NRM platform represents a new class of free-space nonreciprocal devices ideally poised for applications such as electromagnetic wave isolation, circulation, nonreciprocal antennas and radomes, and full-duplex transmission.
Device structure and operation principles
The meta-atom of the NRM is shown in Fig. 1a. We chose La:BaM as the magnetic material because of its large off-diagonal component of the permeability tensor (see details in Supplementary Fig. S1) and the high remanent magnetization (Mr) resulting from its strong magnetocrystalline anisotropy field (Ha = 1.46 T). The meta-atom consists of a subwavelength La:BaM cuboid pillar on top of a Teflon substrate, with Mr along the surface normal (z or -z) direction, as depicted in Fig. 1a. The magnetic hysteresis, permittivity, and permeability tensor of La:BaM in the remanence state are measured and calculated in Supplementary Note 1. In this configuration, a circular polarized electromagnetic wave “sees” different refractive indices and extinction coefficients when the wavevector is parallel or antiparallel to the magnetization direction, which is attributed to magnetic circular birefringence (MCB) and magnetic circular dichroism (MCD) effects (for low loss frequencies MCB dominates). Therefore, the magnetic Mie resonators exhibit different resonance frequencies for the forward and backward propagating waves, leading to an asymmetric transmission amplitude (Fig. 1b) and phase (Fig. 1c). Fig. 1d shows a schematic diagram of the nonreciprocal phase gradient metasurface design, with the orange color indicating Mr along the +z direction and the purple color indicating Mr along the -z direction. The arrangement of these cells can generate arbitrary direction-dependent phase and amplitude profiles by leveraging MCB/MCD effects.
One-way transmission NRMs
The concept of a one-way transmission NRM is illustrated in Fig. 2a. This device was realized by detuning the Mie resonance wavelength owing to the MCB effect. The metasurface consists of periodically arranged cuboid pillars of La:BaM, which are magnetized under 2.4 T magnetic field and left at remanence before attaching to a 20 cm× 20 cm× 0.2 cm Teflon substrate. The device was designed to operate in the Ku band (~15 GHz, λ = 2 cm). The width and height of each cuboid were 3 mm and 6 mm, respectively. The period of the meta-atoms was 1 cm. Fig. 2b shows the fabricated sample. We first simulated the transmission spectra of the metasurface under right circular polarized (RCP) incidence for forward and backward transmissions, as shown in Fig. 2c. The forward transmission spectrum shows two resonance peaks at 13.2 GHz and 14.6 GHz, respectively. The two peaks shifted to higher frequencies at 13.6 GHz and 15.4 GHz respectively for backward transmission. Multipole decomposition indicates that the two resonances correspond to a magnetic dipole resonance (MD) mode at 13.2 GHz and a hybrid electric-dipole magnetic-quadrupole (ED-MQ) mode at 14.6 GHz, respectively (see details in Supplementary Fig. S2). Nonreciprocal one-way transmission with an isolation ratio of 10.9 dB and insertion loss of 0.6 dB at 15.4 GHz, and an isolation ratio of 10.4 dB and insertion loss of 5 dB at 13.6 GHz were theoretically predicted. Fig. 2d shows the experimental transmission spectra of the metasurface (see the measurement setup details in Supplementary Fig. S3). An isolation ratio of 7.0 dB and insertion loss of 8.4 dB at 13.7 GHz, and an isolation ratio of 5.6 dB and insertion loss of 1.1 dB at 15.7 GHz were experimentally observed, consistent with the simulation results. The minor difference between the experiment and simulation was likely attributable to the imperfect circular polarization of the source antenna with an axis ratio of 2.5 dB. We further simulated (Fig. 2e) and measured (Fig. 2f) the transmission contrasts at incident angles ranging from 0° to 64°. The results showed that the resonance frequencies (especially those of the ED-MQ mode) were almost independent of the incident angle. The maximum angle was only limited by the finite size of the NRM and our experimental set-up. The results would also be identical for 0° to -64° incidence for symmetry considerations. This wide-angle tolerance suggests that such NRM is compatible with integration on curved surfaces, for instance, on antenna radomes.
Phase-gradient NRMs, whose phase profiles differ for forward and backward incidence, allow full on-demand control of bidirectional electromagnetic wave propagation. The design of such metasurfaces is based on the concept of NDCM: each element has both forward and backward propagation phases, which can be written as (ϕf(ω), ϕb(ω)), where ϕf(ω) and ϕb(ω) are the forward and backward transmission phases, respectively, at frequency ω. To construct a 2-bit phase gradient NRM for arbitrary forward and backward propagation phase profiles, 16 elements were required26 (Supplementary Fig. S4a). However, because the magnetic meta-atoms can be placed with their magnetization up or down, the phase of the forward and backward incidence can be switched by flipping the direction of Mr. This reduces the library elements to only 10 for arbitrary and nonreciprocal phase profiles, as detailed in Supplementary Notes 4 and 5. The optimal meta-atom design must minimize the phase error while maximizing the transmission efficiency for both forward and backward waves. To balance the trade-off between phase error and transmission efficiency, we used a single figure-of-merit to evaluate the meta-atom designs27 (see details in Supplementary Note 4). The 10 elements were selected in a parameter space consisting of 9,192 parameter combinations (Supplementary Note 4). The transmission spectra of each of the ten fabricated elements were characterized. The transmittance and phase as a function of frequency were characterized by measuring the S21 and S12 parameters, which agreed well with the simulation results presented in Supplementary Note 5. The optimal 10 elements yielded an average transmittance of 42.4% and 38.5%, and an average phase error of ±11.9° and ±12.0° for forward and backward incidence, respectively.
Nonreciprocal beam deflector
After finding the 10 nonreciprocal elements, it is possible to construct on-demand bi-directional phase profiles for the NRM in a “LEGO-like” manner. We demonstrated this by first assembling a nonreciprocal beam deflector using meta-atoms (Fig. 3b). Fig. 3a shows the concept of the device. At 14.6 GHz, the device allows forward normal transmittance for the RCP with plane wave incidence, whereas it diffracts the incident wave away from the surface normal for the backward wave. The entire device area measured 17 cm × 10 cm. Four elements with (forward, backward) phase delays of (0, 0), (0, π/2), (0, π), and (0, 3π/2) constituted one period of 10 mm. Therefore, the metasurface exhibited an almost flat phase distribution for forward transmission and a sawtooth phase profile for backward transmission, as shown in Figs. 3c and 3d. The experimentally assessed phase map (stars) agreed well with the target phase profile (dashed lines). Figs. 3e and 3f show the simulated electric field profile at 14.6 GHz, confirming the proposed nonreciprocal diffraction functionality. In our simulation, the diffraction angle reached 29°, with a diffraction efficiency of 17.7% for backward incidence. For forward incidence, the transmission efficiency reached 33.5%. The relatively low transmission efficiency was attributed to the large impedance mismatch at the air/metasurface interface as well as the energy loss into different diffraction orders (<4%). These imperfections induce ~24% and ~34% reflected energy for forward and backward incidence, respectively (see Supplementary Fig. S7). Experimentally, we demonstrate a 31.3% transmission efficiency for the 0th order diffraction for forward incidence, and 13.8% transmission efficiency for the 1st order diffraction at 30° ± 0.5° for backward incidence. As shown in Figs. 3g and 3h, the radiation patterns exhibited excellent agreement between the experiment and simulation (see experimental details in Supplementary Fig. S8). The slightly lower measured transmission efficiency compared with the simulation may be attributed to the lower transmittance of the experimental meta-atoms and the spherical wavefront of the incident wave (see details in Supplementary Information Fig. S9).
The nonreciprocal metalens exhibit different focal lengths for forward and backward transmission (Fig. 4a). Fig. 4b shows a photograph of the fabricated nonreciprocal metalens sample. The sample area was 17 cm × 17 cm. The meta-atoms were arrayed with a 1 cm period. The hyperbolic phase profiles of the metalens are given by:28
where r is the radial coordinate, λ is the designed wavelength, f1/2 is the designed focal length for forward and backward incidence, and φf/b(r) is the phase profile at position r for forward and backward incidence. In our prototype, focal lengths of f1 = 6 cm and f2 = 13 cm were implemented for forward and backward transmission, respectively. The ten elements with proper forward and backward transmission phases were mapped onto the targeted phase profiles of the metalens, as shown in Figs. 4c and 4d. Fig. 4e displays the electric field intensity profiles in the y-z plane for forward RCP incidence at 15 GHz. The simulated focal length was 5.4 cm with a focusing efficiency of 46.9%. The discrepancy between the simulated (6 cm) and measured (5.4 cm) focal lengths stemmed from coupling between adjacent meta-atoms (see details in Supplementary Fig. S10). For backward incidence, the focal length and focusing efficiency were 11 cm and 42.3%, respectively. The focusing efficiency was defined as the ratio of the power on the focal plane within a radius of 3×FWHM of the focal spot to the total incident power. The electric field intensity profiles (Figs. 4g and 4h) were measured using a near-field scanning setup via antenna scanning in the x-y plane at different z-positions (see details in Supplementary Fig. S11). The shadowed areas in both figures could not be probed because of the finite size of the scanning antenna. The intensity profiles showed clear focal spots with focal lengths of 55 ± 1.5 mm and 105 ± 1.5 mm for forward and backward transmission, respectively. The ripple patterns were caused by interference between the incident and reflected waves29. The measured focusing efficiencies were 30.5% and 30% for forward and backward incidence, respectively. The lower focusing efficiency in the experiment compared to the simulation was attributed to the spherical instead of the plane wavefront of the incident wave (~7% and ~3% focusing efficiency attenuation for forward and backward incidence, respectively; see details in Supplementary Note 8), as well as the lower transmission of experimentally fabricated meta-atoms compared to simulation (~9% lower transmittance on average, see details in Supplementary Table 1). Fig. 4i shows the line scans of the normalized intensity distributions at the focal plane. The insets of Fig. 4i show the simulated and measured focal-spot images. The dashed curve shows the focal-spot profile of an ideal aberration-free lens with the same aperture size and focal length. The Strehl ratio of the experiment reached 0.81, indicating diffraction-limited focusing of our metalens. The focal spot profiles of backward incidence are presented in Fig. 4j. In this case, the Strehl ratio reached 0.91. The evolution of the focal spot profiles in the x-y plane along the z-axis was also measured, as shown in Supplementary Fig. S12.
Holography is a promising technology for three-dimensional displays30, beam shaping31 and artificial intelligence32. Here, we utilize the NRM to achieve nonreciprocal holography, which displays different holograms for forward and backward propagations. To design phase-only computer-generated holograms, we selected two image planes 10 cm from the surface of each side of the sample. Hand-written Greek letters “ɛ” and “µ” were taken as the holographic patterns, as illustrated in Fig. 5a. The Gerchberg–Saxton algorithm was used to retrieve the phase profiles of the hologram (Supplementary Fig. S13). Figs. 5c and 5d depict the discrete phase profiles of the “ɛ” and “µ” patterns for forward and backward incidences at 15 GHz, respectively. Figs. 5e and 5f show holographic images simulated with the phase profiles in Fig. 5c and 5b, showing clear “ɛ” and “µ” letters in the respective image planes. Using the same measurement setup used for metalens characterization, intensity profiles of “ɛ” and “µ” letters (Figs. 5g and 5h) were mapped, confirming good agreement with the design. The overall efficiencies33, 34 (the fraction of the incident energy that contributes to the transmitted holographic image) for the two incident directions were 27.0% and 26.8%, respectively. Compared to the simulated efficiencies of 44.0% and 42.4%, the difference was caused by the spherical wave front (~5.3% and ~4% deviation from the plane wave), lower transmittance of the meta-atoms in the experiment (~9% lower than the simulation in average), and phase errors of the meta-atoms (~3% deviation from the simulation). In addition, because there were only 17 × 17 meta-atoms in this demo, the energy outside the holographic images could not be completely suppressed, limiting the maximum efficiency to 75% (see details in Supplementary Note 12). These considerations can be improved in future studies to achieve nonreciprocal holography with an improved resolution and efficiency.