image: Fig. 1 The dielectric response models for conventional dielectric materials and for metamaterials. For conventional dielectric materials, there are three types of polarization models: Drude model for collective oscillation of free electrons, Lorentz model for local resonance of electric dipoles, and Debye model for relaxation process of dipole orientation. In contrast, for metamaterials, there are only two models: Drude model for infinitely long wires, and Lorentz model for resonant structures (e.g., short metallic wires and split-ring resonators (SRRs)). Relaxation model is absent in metamaterial research.
Credit: by Xinmin Fu, Yajuan Han et al.
Polarization is one of the most important electromagnetic (EM) properties of dielectric materials. The essence of polarization is electron movement, manifested as displacements or rotation of electrons with respect to the centers of molecules/atoms in dielectrics or collective oscillation of free electrons in metals, which produces aligned dipole moments along the direction of external electric fields. Polarization is very fundamental in revealing the microscopic mechanisms of macroscopic EM properties of dielectrics. For conventional dielectrics, their constituent particles include atoms, molecules, and others. In general, without external electric fields, the contribution of these particles to macroscopic polarization vanishes due to thermal motion averaging. It is only under the influence of applied electric fields that particles align along the field direction and contribute to dipole moments that can give rise to macroscopic polarization intensity.
In the physics of dielectric materials, the EM properties of dielectrics originate from several crucial polarization mechanisms that can be described by Debye, Drude, and Lorentz models. Metamaterials and their two-dimensional counterparts, metasurfaces, can exhibit extraordinary electromagnetic parameters such as negative permittivity. Notably, the polarization mechanisms responsible for these parameters have long been discussed in dielectric physics. Comparative analysis with dielectric polarization theory reveals that metamaterials, as extensions of dielectric materials, lack a critical component in their electromagnetic response framework: While Drude (free electron collective oscillation) and Lorentz (bound electron local resonance) models are routinely used to describe metamaterial responses, the Debye relaxation model - crucially important in dielectric physics - has remained conspicuously absent (Fig 1). The research team emphasized: "The electromagnetic responses of metamaterials fundamentally originate from polarization behaviors of microstructural, yet their design has long relied on Drude and Lorentz models. Introducing Debye relaxation into metamaterials would complete the missing piece in their electromagnetic response framework, bridge the gap between metamaterials and dielectric physics, enrich theoretical foundations for metamaterial design, and provide robust theoretical support for broadband/ultra-broadband metamaterial applications."
The team proposed a relaxation response model based on co-designed electrical-magnetic resonances. Starting with fundamental theory, they analyzed the generation mechanisms of magnetic and electrical resonances in typical reflective metasurfaces. According to metamaterial design principles, these resonances can be equivalently described as magnetic and electric dipole oscillations (both Lorentz-type resonances). Through theoretical derivation, the team made a breakthrough discovery: The reflection phase variation under single resonance actually follows first-order Debye relaxation, overturning the conventional understanding that metamaterial resonances only exhibit Lorentz-type abrupt phase transitions. More crucially, by carefully designing resonance frequencies, intensities, and Q-factors of magnetic and electrical resonances, they achieved second-order and even higher-order relaxation processes with ultra-wideband gradual phase variations (Fig 2).
To validate this model, the team designed a Quad-Elliptical-Arc (QEA) metallic structure as the meta-atom, demonstrating powerful ultra-broadband dispersion control based on second-order Debye relaxation. Notably, circularly polarized waves were used as excitation sources to intrinsically induce rotational electron movement along the elliptical arcs, effectively simulating orientation polarization processes in dielectric physics. Simulation results revealed: At 8.0 GHz, opposing surface currents between QEA structure and ground plane indicated magnetic resonance; at 12.0 GHz, aligned currents suggested electrical resonance. These cooperative resonances formed a second-order Debye relaxation spanning broader bandwidth. Geometric parameter adjustments enabled customized control of dual first-order relaxations for tailored second-order processes. The team further developed both chromatic and achromatic reflective focusing metasurfaces, with simulations and experiments confirming effective operation across the entire X-band (8.0-12.0 GHz, 40% relative bandwidth), validating the broadband dispersion control capability.
As one of the most important polarization models in dielectric physics, Debye relaxation enables broadband electromagnetic parameter regulation. The team successfully established a Debye relaxation model for metamaterials through electrical-magnetic resonance co-design, theoretically introducing Debye relaxation into metamaterials for the first time. This breakthrough not only bridges metamaterials with classical dielectric physics, enriching their theoretical foundations, but also opens new dimensions for electromagnetic dispersion control. The established framework shows potential for extension to other spectral ranges (THz/optical frequencies) and multidisciplinary applications (e.g., acoustic metamaterials), significantly advancing the frontier of artificial electromagnetic material design.
Journal
Light Science & Applications
Article Title
2nd-Order Debye Relaxation in Electromagnetic Metasurfaces for Wideband Dispersion Engineering