Near-Field Spectroscopy of Individual Asymmetric Split-Ring Terahertz Resonators

Metamaterial resonators have become an efficient and versatile platform in the terahertz frequency range, finding applications in integrated optical devices, such as active modulators and detectors, and in fundamental research, e.g., ultrastrong light–matter investigations. Despite their growing use, characterization of modes supported by these subwavelength elements has proven to be challenging and it still relies on indirect observation of the collective far-field transmission/reflection properties of resonator arrays. Here, we present a broadband time-domain spectroscopic investigation of individual metamaterial resonators via a THz aperture scanning near-field microscope (a-SNOM). The time-domain a-SNOM allows the mapping and quantitative analysis of strongly confined modes supported by the resonators. In particular, a cross-polarized configuration presented here allows an investigation of weakly radiative modes. These results hold great potential to advance future metamaterial-based optoelectronic platforms for fundamental research in THz photonics.

This supplementary Information has 6 pages, 7 figures and 2 tables.

Resonator parameters
The main parameters for the resonators used in this work are reported in Table S1 and Table   S2. Figure S1 shows an optical image of the 9 resonators sample used for the near-field measurements.
Table S1: Size of the 3*3 mm 2 far-field arrays corresponding to the measurement in Fig. 1  The parameters defining the resonators are showed in Figure 1 in the main text. The central gap , is the distance between the two halves, is the radius of the outer ring, is the width of the ring, is the sweep angle of the center of the asymmetric gap from y -axis centre bar, is the open angle of the gap and is the periodicity of unit cells. Figure S1: Optical image of the 9 different resonators prepared for the near-field measurement.

Finite element Simulations
Further normalized transmission (S21) simulations performed with Comsol Multiphysics for resonators 5, 7 and 9 are reported in Figure S2, together with the respective calculated Q factors for the mode A.

Far-Field measurements
Far-field transmission measurements for a 3x3 mm 2 array having as unit cell the same design of resonator 5 and a periodicity of 60 μm, for incident 0 incoming polarization are presented in Figure S3.

Q factor measurements
There is a clear difference in both the spectral amplitude and Q-factor of the three resonators.
The time-domain information from the individual resonances allows us to directly extract the resonance Q-factor using the single harmonic oscillation (SHO) model 1 . The classic SHO time domain waveform y(t) is: Where α is amplitude, γ is decay rate, ω is the natural frequency of the oscillation. The Q-factor of the oscillation can be evaluated by: Where ξ is the damping ratio, it can be expressed as: The experimental result and its exponential fitting of the resonator 5 are shown in Fig. 2(

Near-field Spatial mapping
Near-field spatial mapping for the 90 configuration are reported in Figure S6. The probe was scanned across the resonator area for resonators 5, 7 and 9, respectively reported in Fig. S6   a), b) and c). The normalized E-field for mode A is showed in the panel of Figure S6 d), together with the current density J. Finally, Figure S6 e) reports the time delay position (red dashed line) which was kept fixed during these measurements for resonator 5 (similar procedure was used for the other 2 resonators). It is possible to clearly observe the difference in the "d" parameter between resonators 7 and 9, as reported in Table 2. Figure S6: a)-c). Spatial mapping of resonator 5, 7 and 9. Mode A E-field norm distribution for resonator 5 and d) the relative fixed time delay used for the acquisition.

Sample-probe approach measurements
The sample-probe distance was estimated to be ~ 5 m, as discussed in the main text and in 2,3 . An exemplar set of measurements acquired at different relative aperture/probe Figure S7: a) Time waveforms acquired on resonator 5 at different relative position z with respect to the minimal distance, allowing raster scanning. b) Maximum current signal recorded for different relative distances distances z on resonator 5 is showed in Figure S7. It is possible to notice in Fig. S7 that the amplitude of the signal increases as the sample is brought closer to the probe. For all positions with z =2-10 m, the waveform remains the same, and only the amplitude increases. Therefore, spectroscopic measurements can be performed at different distances within that range without affecting the results In this range, the resonant characteristics are not affected, but the maximum amplitude of the waveform reduces with increasing resonator-probe separation. The final spatial resolution is anyway limited by the probe dimension, which has size of 10x10 m 2 .
We also note that there is no position feedback in our system and the position is controlled with ~1 m precision.