Nonlocal Microwave Engineering: Constructing Dispersion Relations and Enabling On-Demand Frequency-Momentum Transformations via Time-Switched Long-Distance Interactions

Abstract

Nonlocal metamaterials (MTMs) have recently attracted significant research attention across different areas of wave physics, owing to their ability to translate long-range interactions among meta-atoms into a wide array of wavevector-dependent responses and functionalities. In this work, we introduce nonlocal transmission line metamaterials (TL MTMs) as a versatile platform to investigate and engineer nonlocality in the microwave frequency regime. We first establish a concise theoretical framework for nonlocal TL MTMs based on circuit and network theory, from which we derive the general dispersion relation for TL MTMs having arbitrarily complex nonlocal coupling configurations. Building upon this foundation, we demonstrate how such structures can be used to synthesize nearly arbitrary, even, dispersion functions within their first Brillouin zone, effectively linking nonlocal circuit parameters to prescribed dispersion profiles. Finally, we introduce time-switched nonlocal TL MTMs, a new class of metamaterials with time-varying nonlocality in which the nonlocal branches are dynamically activated as an electromagnetic pulse propagates through the structure. This platform enables complex, nearly arbitrary frequency-momentum transformations on a propagating pulse, as well as the simultaneous excitation of modes with positive, negative, and zero group velocity within the first Brillouin zone. Our results offer new physical insights into the behavior of nonlocal MTMs, a versatile platform to investigate the interplay of frequency dispersion, spatial dispersion and time modulation, and a general theoretical foundation for the design of more advanced nonlocal and time-varying electromagnetic and photonic systems.

Figures (4)

• Figure 1: Theory of microwave nonlocal transmission-line metamaterials (TL MTMs).(A) Standard, local, LC Ladder network composed of shunt capacitors, $C_{l}$ each connected to their nearest neighbor through an inductor $L_{1}$. (B) Nonlocal TL MTM having local shunt components, $C_l$ connected to their nearest neighbor through an inductor $L_{1}$ and their $m =2,3$ longer-distance neighbors via arbitrary admittances $Y_{2}$, in blue, and $Y_{3}$ in green. (C) "Super-cell" of a nonlocal TL MTM: the local shunt admittance $Y_l$ is connected to $M$ long-distance neighbors through nonlocal admittances, $Y_m$. Using Kirchhoff’s current law, we can simplify the orange boxed part into an effective admittance given by a simple sum of nonlocal admittances $Y_m$ each modulated by the wavenumber. The equivalent unit cell of the nonlocal MTM is shown with the effective admittance on the top-right corner. The inset of the figure shows the variation of the modulated nonlocal admittance as a function of the wavenumber, at a fixed frequency $\omega_{*}$, within the first Brillouin Zone of the MTM. The modulated nonlocal admittance achieves a maximum value of $4Y_m$ (boxed in blue), which then corresponds to a "maxon" in its dispersion relation in Fig. \ref{['fig2']}(A), and a minimum value of 0 (boxed in red), and therefore an open circuit, which then corresponds to a "roton" in the dispersion relation.
• Figure 2: Inductive nonlocal TL MTM.(A) Illustration of an inductive nonlocal TL MTM with two orders of nonlocality $m = 2,3$. Each shunt local capacitor is connected to its $1^{\text{st}}$(in gray), $2^{\text{nd}}$ (in blue) and $3^{\text{rd}}$ (in green) neighbor, with the last two representing nonlocal connections via an inductor having inductance $L_{m}$. (B) Dispersion relation of a nonlocal inductive TL MTM: theoretical (green line) and numerical (contour plot) results. A “super-cell” of the nonlocal TL MTM used to implement the dispersion relation is shown in the top-right inset of the panel, with the local components in blue and the nonlocal ones in green. For comparison, the dispersion relation of the local LC ladder is plotted in blue. The nonlocal dispersion is always above that of the local one due to the squared sinusoidal term introduced in the numerator of the expression, a feature of nonlocal TL MTM of inductive type. At the roton point shown in Fig. \ref{['fig1']}(C) the nonlocal dispersion coincides with the local one since the wave does not “see” the nonlocal inductance. (C) High-pass-filter-like (HPF-like) dispersion relation implemented with an inductive nonlocal TL MTM of order $M = 6$ (theoretical results in green, and contour plot for the numerical results). In black we show the ideal high-pass-filter-like dispersion relation implemented using an ideal Fourier Series. A “super-cell” of the nonlocal TL MTM is shown in the inset of the figure. Due to passivity, and therefore the inability to realize positive Fourier coefficients, this is the best approximation of the desired dispersion relation achievable using only nonlocal inductors. (D) Plot showing the Fourier coefficients (stem plot; right orange $y$-axis) needed to realize the desired dispersion relation pictured in (C) and the nonlocal inductance values (circles) required to implement such coefficients (left blue $y$-axis). As indicated by Eq. \ref{['eq7']}, only the first 6 Fourier coefficients, which are negative, can be implemented using passive nonlocal inductors. This strongly restricts the range of synthetizable functions using nonlocal TL MTM of inductive type.
• Figure 3: Constructing more complex dispersion curves with capacitive and inductive nonlocality.(A) Dispersion relation of a nonlocal capacitive TL MTM: theoretical (green line) and numerical (contour plot) results. . A “super-cell” of the nonlocal capacitive TL MTM used to implement the dispersion relation is shown in the top-right inset of the panel, with the local components in blue and the nonlocal ones in green. For comparison, the dispersion relation of the local LC ladder is shown in blue. The nonlocal dispersion relation always lies below that of the local case because the sinusoidal term appears in the denominator of the expression, in contrast to the inductive case, where it appears in the numerator. (B) High-pass-filter-like dispersion relation implemented with a capacitive nonlocal TL MTM of order $M$ = 11 (theoretical results in green, and contour plot for the numerical results). In black we show the ideal HPF-like dispersion relation implemented using an ideal Fourier Series. A “super-cell” of the nonlocal TL MTM is shown in the inset of the figure: we utilize a parallel LC resonator both in the shunt central component, as well as for local and nonlocal connections. (C) Plot showing the Fourier coefficients (stem plot; right orange y-axis) needed to realize the desired dispersion relation pictured in (B) and the nonlocal inductance values (circles) and capacitance values (asterisks) required to implement such coefficients (left blue y-axis). The first 6 coefficients, which are negative, can be implemented with inductors only. For $m > 6$, instead, both nonlocal inductors and capacitors are needed. As discussed in the main text, the introduction of nonlocal capacitors allows for the implementation of positive Fourier coefficients and, therefore, the realization of a broader range of periodic dispersion functions. (D) Nonlocal TL MTM whose dispersion relation follows the profile of the Florence-Duomo. The first Brillouin Zone of the MTM is shown with theoretical (green dashed line) and numerical (contour plot) results. In the top-left inset (I), we show a picture of the Florence Duomo view from Piazzale Michelangelo and in the top-middle inset (II) we show the silhouette of the Duomo used to design the MTM. The MTM “super-cell” is identical to that shown in the inset of (B), but here with $M = 50$. This highlights how higher orders of nonlocality enable the realization of increasingly complex dispersion features, albeit at the cost of increased structural size.
• Figure 4: Time-switched nonlocal TL MTM.(A) Local LC transmission line excited with a sinc pulse of bandwidth 66-74 MHz (shown in time domain in inset I, and in frequency domain in inset II). The spatio-temporal Fourier transform of the propagating pulse coincides with the local dispersion relation (dashed blue line) given by Eq. \ref{['eq2']}. While the pulse is fully contained within the LC ladder, the nonlocal inductive branch is switched on as shown in inset I (orange line shows the switching waveform). The pulse undergoes an $\omega-$transition from a region of positive to a region of negative group velocity. The spatio-temporal Fourier transform of the pulse after switching agrees falls exactly on the dispersion curve of the nonlocal structure, given by Eq. \ref{['eq5']} for $L_3$ = $L_1$ (dashed green line). The “super-cell” of the time-switched nonlocal TL MTM is shown in the bottom right corner, with the nonlocal branch in green and the local one in blue. In red we show the switch that can dynamically connect and disconnect the nonlocal branch from the TL MTM. (B) Similar to panel (A) but for a sinc pulse of bandwidth 30-60 MHz and a time-switching-induced $\omega-$transition into a flat-band region of the dispersion curve of the nonlocal structure. To implement a large flat band, the nonlocal branch is designed with a parallel LC with: $L_3 = 0.3L_1$ and $C_3 = C_l$. (C) Local LC transmission line excited with a sinc pulse of bandwidth 5-40 MHz. While the pulse is fully contained within the LC ladder, $M = 50$ nonlocal connections are switched on. In this case, the $\omega-$transition shapes the pulse to follow the dispersion curve approximating the profile of the Florence Duomo (I). A close-up view is shown in (II). In both the local and nonlocal case, the agreement between theoretical predictions (blue and green dashed lines) and numerical experiments (contour plots) is excellent. These results highlight the ability of time-switched nonlocal TL MTMs to shape propagating modes on demand. Time-domain field animations for the cases in panels (A) and (B) are provided as Supplementary Materials.