The advent of electromagnetic meta-atoms (EMAs) significantly alters the interaction between electromagnetic waves and subwavelength particles, enabling the emergence of novel phenomena associated with scattering and absorption, leading to promising applications. The physics behind these properties is rooted in the unique configurations of EMAs, which enable selective excitation of multipolar modes with custom-made amplitudes, resulting in phenomena such as superscattering, invisibility, Fano resonance, and Kerker effect. Ferrite materials possess intrinsic magnetic responses that allow the design of magnetic EMAs with nonreciprocal features, arising from time-reversal symmetry breaking in ferrites under a bias magnetic field (BMF). By periodically arranging an array of EMAs, magnetic metamaterials (MMs) can be constructed to manipulate electromagnetic waves nonreciprocally, particularly near the magnetic surface plasmon (MSP) resonance. By designing magnetic EMAs with two types of yttrium iron garnet (YIG) ferrite materials with different saturation magnetizations, two MSP resonances can be achieved. As a result, MMs constructed from these magnetic EMAs exhibit a nonreciprocal perfect absorption effect at two different frequencies for incident Gaussian beams with transverse magnetic (TM) polarization. Specifically, in one direction, the structure acts as a perfect absorber, while in the mirror-symmetric direction, the beam is mainly reflected. This nonreciprocal phenomenon is closely related to the lattice Kerker effect and nonreciprocal Fano resonance. The magnetic EMAs serve as fundamental elements for nonreciprocal optics and microwave photonics.

The scattering properties of magnetic EMAs can be solved using the generalized Mie theory, which relates the scattering field to the incident field via Mie coefficients. The scattering cross sections of the magnetic EMAs are calculated based on these coefficients, and nonreciprocal Fano resonances are visualized by examining the scattering cross sections. By incorporating multiple scattering theory, the scattering field generated by multiple magnetic EMAs is rigorously calculated, enabling a deeper analysis of nonreciprocal scattering behavior. In addition, photonic band diagrams, absorbance, and reflectance are calculated to optimize the configurations of magnetic EMAs and the direction of the incident Gaussian beam, ensuring perfect absorption at a specified direction while achieving substantial reflection in the mirror-symmetric direction. Effective-medium theory is also employed to retrieve the effective constitutive parameters of MMs, identifying the MSP resonance frequencies, which are compared with photonic band diagrams.

By periodically arranging an array of magnetic EMAs in a square lattice, with a lattice constant a=9 mm, we construct MMs that serve as nonreciprocal perfect absorbers. The saturation magnetizations of the two ferrite materials in the magnetic EMAs are M_s1=0.300 T and M_s2=0.175 T, with an inner radius r_c=0.9 mm, an outer radius r_s=2.3 mm, and damping factors set as a₁=a₂=2×10^-2. The BMF is set to H₀=600 Oe. By plotting transmittance and reflectance as functions of frequency f and incident angle θ_inc, we identify two operating frequencies f₁=3.90 GHz and f₂=5.56 GHz, with corresponding incident angles θ_inc=±70° (Fig. 2). Next, keeping other parameters constant, we plot the transmittance and reflectance as functions of a₁ and a₂ and optimize the damping factor to a₁=a₂=1.2×10^-2 (Fig. 3). Further optimization of the inner and outer radii yields r_c=0.8 mm and r_s=2.25 mm, achieving an absorbance greater than 97% and a reflectance exceeding 80% in the mirror-symmetric direction (Fig. 3). Using multiple scattering theory, full-wave simulations reveal nonreciprocal perfect absorption (NPA) at two different frequencies. The lower-frequency NPA corresponds to the resonant mode in the outer layer, while the higher-frequency NPA corresponds to the resonant mode in the core. By plotting the angular scattering amplitude of the unit cell at the central position, the nonreciprocal lattice Kerker effect is observed. Specifically, perfect absorption for the rightward incident beam corresponds to backward scattering, while strong reflection occurs for the leftward incident beam due to forward scattering (Fig. 4). By comparing the photonic band diagrams with effective-medium theory, MSP resonances at two different frequencies are confirmed, with the operating frequencies falling within the vicinity of the MSP resonances. In addition, the operating frequencies can be flexibly tuned upwards or downwards by adjusting the BMF, as evidenced by the frequency shift of the MSP resonances (Fig. 5). To further investigate the scattering properties of magnetic EMAs and their connection to NPA, the scattering cross section and the amplitude of Mie coefficients are calculated as functions of frequency. Two asymmetric Fano dips are identified near the operating frequencies, resulting from the interference between the broadband 0th-order mode and the narrowband -1st-order mode. Moreover, the tunability of both the Fano resonances and operating frequencies via the BMF introduces an extra degree of freedom (Figs. 6 and 7).

Magnetic EMAs composed of two types of ferrite materials with different saturation magnetizations have been designed, serving as building blocks for MMs to achieve dual-band NPA. At a specified incident angle, the Gaussian beam is absorbed with an absorbance exceeding 97%, while in the mirror-symmetric direction, it is strongly reflected with a reflectance over 80%. The NPA effect arises from the time-reversal symmetry breaking nature of MSP resonance and the nonreciprocal lattice Kerker effect. The phenomenon is also closely related to the nonreciprocal Fano resonances of isolated magnetic EMAs, originating from the interaction between the broadband bright mode and the narrowband dark mode associated with angular momentum channels m=0 and m=-1. In addition, both the operating frequencies and Fano resonances can be flexibly controlled by the BMF, enhancing potential applications in nonreciprocal optics and microwave photonics.