Figure 1

The Möbius-ring resonator pair. (a) An idealized one-dimensional low-pass ladder network consisting of a series of inductances $L$ and capacitances $C$ along the line. Mutual inductances are considered between the loop elements defined by $I_n$ (see text). There are three options for making a periodic structure from the segment shown. They include the ring resonator in (b), the Möbius resonator in (c), and a Möbius resonator of opposite chirality (not shown). The discrete inductive and capacitive elements in (b) and (c) are omitted for clarity.Reuse & Permissions

Figure 2

Eigenvalues and eigenfunctions for one-dimensional low-pass Möbius-ring resonators. (a) Measured spectra (eigenvalues) are shown for the Möbius (red) and ring (black) resonators. The spectral amplitudes are the voltage standing wave ratios (SWR) observed while driving the resonators inductively at a single location. Using this method all resonant frequencies are visible, but the degenerate doublets are unresolved in most cases (see text for details). The insets show theoretical [black, following Eq. (4)] and measured current amplitudes (blue). (b) If the measured eigenvalues of the Möbius resonator are assigned to half-integral mode indices in Eq. (5), they lie along the identical theoretical dispersion curve (in bold gray, $\kappa =0.07$) describing the conventional ring resonator. The dotted curves represent one half (long-dashed curve) and twice (short-dashed curve) the observed coupling constant. Inset (color plots): The 15 predicted eigenfunctions for the eight element Möbius-ring pair are represented by Biot-Savart plots of the magnetic field amplitude at each eigenvalue.Reuse & Permissions

Figure 3

Eigenvalues and eigenfunctions for a $2\times 8$ high-pass Möbius (a) and ring resonator (b). Theoretical eigenvalues are given by gray lines, and experimental eigenvalues by symbols. In addition, for each data point the theoretical eigenfunction is displayed by rendering the current amplitudes for the 16 elements in gray scale. Model parameters for the numerical computation of theoretical eigenvalues for both resonators were obtained by measurements of the long- and short-sided inductive coupling constants and the single element resonance frequency.Reuse & Permissions