## Abstract

We show that magnetic response at terahertz frequencies can be achieved in a planar structure composed of nonmagnetic conductive resonant elements. The effect is realized over a large bandwidth and can be tuned throughout the terahertz frequency regime by scaling the dimensions of the structure. We suggest that artificial magnetic structures, or hybrid structures that combine natural and artificial magnetic materials, can play a key role in terahertz devices.

The range of electromagnetic material response found in nature represents only a small subset of that which is theoretically possible. This limited range can be extended by the use of artificially structured materials, or metamaterials, that exhibit electromagnetic properties not available in naturally occurring materials. For example, artificial electric response has been introduced in metallic wire grids or cell meshes, with the spacing on the order of wavelength (*1*); a diversity of these meshes are now used in THz optical systems (*2*). More recently, metamaterials with subwavelength scattering elements have shown negative refraction at microwave frequencies (*3*), for which both the electric permittivity and the magnetic permeability are simultaneously negative. The negative-index metamaterial relied on an earlier theoretical prediction that an array of nonmagnetic conductive elements could exhibit a strong, resonant response to the magnetic component of an electromagnetic field (*4*). In the present work, we show that an inherently nonmagnetic metamaterial can exhibit magnetic response at THz frequencies, thus increasing the possible range in which magnetic and negative-index materials can be realized by roughly two orders of magnitude.

Conventional materials that exhibit magnetic response are far less common in nature than materials that exhibit electric response, and they are particularly rare at THz and optical frequencies. The reason for this imbalance is fundamental in origin: Magnetic polarization in materials follows indirectly either from the flow of orbital currents or from unpaired electron spins. In magnetic systems, resonant phenomena, analogous to the phonons or collective modes that lead to an enhanced electric response at infrared or higher frequencies, tend to occur at far lower frequencies, resulting in relatively little magnetic material response at THz and higher frequencies.

Magnetic response of materials at THz and optical frequencies is particularly important for the implementation of devices such as compact cavities, adaptive lenses, tunable mirrors, isolators, and converters. A few natural magnetic materials that respond above microwave frequencies have been reported. For example, certain ferromagnetic and antiferromagnetic systems exhibit a magnetic response over a frequency range of several hundred gigahertz (*5*–*7*) and even higher (*8*, *9*). However, the magnetic effects in these materials are typically weak and often exhibit narrow bands (*10*), which limits the scope of possible THz devices. The realization of magnetism at THz and higher frequencies will substantially affect THz optics and their applications (*11*).

From a classical perspective, we can view a magnetic moment as being generated by microscopic currents that flow in a circular path. Such solenoidal currents can be induced, for example, by a time-varying magnetic field. Although this magnetic response is typically weak, the introduction of a resonance into the effective circuit about which the current flows can markedly enhance the response. Resonant solenoidal circuits have been proposed as the basis for artificially structured magnetic materials (*12*), although they are primarily envisaged for lower radio-frequency applications. With recent advances in metamaterials, it has become increasingly feasible to design and construct systems at microwave frequencies with desired magnetic and/or electric properties (*3*, *13*, *14*). In particular, metamaterials promise to extend magnetic phenomena because they can be designed to work at high frequencies with broad bandwidth and tunability and can attain large positive or negative values of the magnetic permeability.

A magnetic metamaterial can be formed from an array of nonmagnetic, conducting, split-ring resonators (SRRs) (Fig. 1). An SRR consists of two concentric annuli of conducting material, each with a gap situated oppositely. The gaps enable the structure to be resonant at wavelengths much larger than its physical dimensions, and the combination of many SRRs into a periodic array allows the material to behave as a medium with an effective magnetic permeability μ_{eff}(ω), where ω is frequency. The origin of the effective permeability enhancement stems from a resonance in the SRR, associated with the inductance corresponding to the rings and the capacitance corresponding to the gaps within and between the rings.

The effective permeability can be expressed in the form (*4*, *15*) (1) where *F* is a geometrical factor, ω_{0} is the resonance frequency, Γ is the resistive loss in the resonating SRR, and μ′_{eff} and μ″_{eff} are the real and imaginary magnetic permeability functions. In the quasi-static limit, the qualitative picture of this magnetic response is straightforward: The external magnetic field with a varying flux normal to the metallic loop will induce a current flow, which in turn results in a local magnetic dipole moment. Well below the resonance frequency, ω_{0}, the strength of the magnetic dipole increases with frequency, and this dipole response stays in phase with the excitation field, i.e., it has a paramagnetic response. As the frequency of the incident field approaches ω_{0}, the currents generated in the loops can no longer keep up with the external field and begin to lag. As the frequency increases above ω_{0}, the induced dipole moment lags further until it is completely out of phase with the excitation field, which results in a magnetic permeability smaller than unity (i.e., a diamagnetic response), including values less than zero. In contrast to conventional ferromagnetism, the magnetic activity associated with these conductive elements is completely devoid of any permanent magnetic moment.

In order to obtain magnetic resonant behavior in the THz range, the appropriate dimensions of the SRRs can be first approximated by analytical methods (*4*) and then confirmed by numerical simulation. We designed and constructed three different SRR samples on a 400-μm-thick quartz substrate by a self-aligned microfabrication technique called photo-proliferated process (*16*). The SRRs are made from copper and are 3 μm thick. Their periodicity is subwavelength (λ/7 in our samples, where λ is the wavelength of the excited field at resonance frequency), which allows the composite to behave as an effective medium to external THz radiation (λ = 300 μm at 1 THz).

Most reported works on microwave metamaterials have focused on characterizing bulk one- and two-dimensional structures, in which waveguide configurations are frequently used. At the submillimeter wavelengths associated with THz frequencies, optical components such as lenses and mirrors are commonly used, making a free-space characterization more convenient to pursue (Fig. 1). We performed the measurements here using spectroscopic ellipsometry at oblique incidence. A Fourier transform infrared spectrometer adapted for S-polarized (Fig. 1) and Ppolarized light from 0.6 THz to 1.8 THz was used for the measurements, with a silicon bolometer as the detector. We placed the sample within an evacuated compartment, then focused light from a mercury arc lamp source on the substrate at an angle of 30° from the surface normal.

In the frequency-dependent ellipsometry measurements (Fig. 2), the parameter plotted, tan^{–2}(Ψ), represents the inverse absolute square of the ellipsometric parameter ρ(ω) = tan(Ψ)·exp(*i*Δ), where Ψ is the amplitude ratio and Δ is the phase difference. This ellipsometric parameter displays the reflectance ratio of two polarizations. The SRRs are expected to respond magnetically when the magnetic field penetrates the rings (S-polarization) (Fig. 1) and to exhibit no magnetic response when the magnetic field is parallel to the plane of the SRR (P-polarization). Thus, the reflectance ratio (Fig. 2) is the natural function to use, because this parameter provides the ratio of the magnetic to electric response from the SRRs (*17*).

The reflectance ratio for sample D1 (Fig. 2, red curve) exhibits a resonant peak, centered at ∼1.25 THz in the spectrum. The resonance in the reflectance was broad, nearly 30% of the bandwidth of its center frequency. If the magnetic response was due to the constituent SRRs, then this resonance should scale with dimensions in accordance to Maxwell's equations. In order to elucidate our findings, two more SRRs with different dimensions were characterized (Fig. 2). These SRRs all exhibit a similar magnetic mode, and their resonant frequencies occur between 0.8 and 1.0 THz. We found an expected monotonic redshifting of resonant frequencies as the dimensions of SRRs were scaled up. In addition, the bandwidth of the magnetic response could be tuned by adjusting the parameters of the SRR element.

As further verification that the peaks in Fig. 2 were due to the magnetic response of the SRRs, we performed a numerical simulation using High-Frequency Structure Simulator (HFSS), a commercial electromagnetic mode solver. S-parameter transmission and reflection were calculated as a function of frequency for a periodic infinite array of copper SRRs with the dimensions of the three designed samples (Fig. 2). The calculation was performed to determine the frequency of the resonant magnetic response of the SRRs, for comparison to the ellipsometry measurements (*18*). The results of the simulation and experimental curves were in good agreement. In Fig. 3, we display the simulated real and imaginary portions of the effective magnetic permeability that corresponds to samples D1, D2, and D3. For sample D1, the onset of the simulated imaginary permeability peak occurs at ∼1.15 THz, which corresponds well with the onset of the experimental peak in tan^{–2}(Ψ). The noticeable difference in the actual peak locations is to be expected, because tan^{–2}(Ψ) consists of ratios of absolute values; i.e., the resonance width observed is dependent on the strength of the oscillator. Thus, it is important when considering tan^{–2}(Ψ) and μ_{eff} (ω) to compare the onset of the resonances.

The scalability of these magnetic metamaterials throughout the THz range and potentially into optical frequencies promises many applications, such as biological (*19*) and security imaging, biomolecular fingerprinting, remote sensing, and guidance in zero-visibility weather conditions. Additionally, the effect is nearly an order of magnitude larger than that obtained from natural magnetic materials (*20*). Structures with a negative magnetic response, when combined with plasmonic wires that exhibit negative electrical permittivity (*21*–*24*), should produce a negative refractive index material at these very high frequencies, enabling the realization of needed devices in the THz regime.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/303/5663/1494/DC1

Materials and Methods

Figs. S1 and S2