Single-shot near-field reconstruction of metamaterial dispersion

TL;DR

This work tackles the problem of experimentally mapping full 3D metamaterial dispersion by introducing a single-shot near-field technique in a multi-mode resonator. A fixed source excites multiple TM modes, whose $H_z$ near-field distributions are scanned and Fourier-analyzed to yield in-plane wavevectors; Fabry-Pérot resonances along the stack height supply discrete $k_z$ values, enabling construction of $H_z(k_x,k_y,k_z,f)$ and extraction of isofrequency surfaces across frequencies. Applied to a double non-connected wire metamaterial, the method reconstructs hyperbolic isofrequency surfaces in the low-frequency regime and shows strong agreement with analytical and numerical models, providing a rapid, conceptually insightful tool for spatially dispersive metamaterials. The approach has broad potential for fast dispersion characterization across microwave to optical domains, particularly for high-index, bounded samples where Fabry–Pérot-type sampling is feasible.

Abstract

We present a single-shot near-field technique to reconstruct the isofrequency surfaces of metamaterials in the microwave regime. In our approach, we excite resonant modes using a fixed source in a resonator composed of the material under test and map the in-plane field distribution with a movable probe. Applying a fast Fourier transform (FFT) to the measured field reveals the sample's in-plane dispersion. By extending this analysis over multiple frequencies and comparing the results with Fabry-Pérot resonances, we retrieve the full three-dimensional dispersion relation. When we apply the method to a double non-connected wire metamaterial, it accurately captures the low-frequency hyperbolic isofrequency surface, providing both a precise experimental tool and conceptual insight into spatially dispersive metamaterials.

Figures (7)

• Figure 1: (a) Field-scanning setup with a fixed source and a movable probe that scans the field in the $x$–$y$ plane above the sample. (b) Each resonance corresponds to a peak in the transmission spectrum. Applying a FFT to the scanned field yields the $k_x$ dependence of the field, assuming a fixed $k_y$ (i.e., slab waveguide with $k_y = 0$). (c) At low frequencies ($f = f_0$), only the fundamental mode along $z$ is supported, characterized by $k_z = n_z \pi / D_z$ with $n_z = 1$. (d) As the frequency increases ($f = f_1$), a higher-order mode ($n_z = 2$) appears. (e) Tracking the guided modes across several frequencies, e.g., the first two guided modes, provides discrete points in constant frequency cuts of the dispersion relation $\omega(k_x, k_z)$.
• Figure 2: (a) Unit cell of the double non-connected wire metamaterial. (b) Isofrequency surface $\omega=0.2 (2\pi c/a)$ (below the plasma frequency) plotted according to the effective permittivity tensor derived in simovski2004low.
• Figure 3: Double non-connected wire media multi-mode resonator: a brick consisting of $N_x \times N_y \times N_z$ unit cells, with $N_x = 30$, $N_y = 30$, and $N_z = 3$. The metamaterial period is $a = 5.7~\mathrm{mm}$, and the wire radius is $r = 0.6~\mathrm{mm}$. The wire holder was 3D printed from ABS plastic (dielectric constant $\varepsilon' = 2.4$, loss tangent $\tan \delta \approx 0.01$, 100% fill).
• Figure 4: Measurements of the $H_z$ field near the sample. (a) Transmission coefficient between the source and an arbitrarily positioned receiver; the numbers indicate the number of half-wavelengths across the wire array. (b) Frequency range covering all responses with the same $k_y = 2\pi / (N_y a)$ order. (c) Examples of near-field distributions in real space for selected frequencies, along with the corresponding Fourier spectra.
• Figure 5: Dispersion branches of different waveguide modes. (a) Experimentally obtained Fourier spectrum showing frequency versus $k_x$ for $k_y a / \pi = 2/30$. (b) Numerically calculated dispersion relation for the same $k_y$, obtained by solving the eigenmode problem. (c–e) Numerically calculated $H$-field distributions of the corresponding waveguide modes, with arrows indicating the magnetic field in the $x$–$z$ plane.