Time-varying metamaterials based on graphene-wrapped microwires: Modeling and potential applications

The successful realization of metamaterials and metasurfaces requires the judicious choice of constituent elements. In this paper, we demonstrate the implementation of time-varying metamaterials in the terahertz frequency regime by utilizing graphene-wrapped microwires as building blocks and modulation of graphene conductivity through exterior electrical gating. These elements enable enhancement of light-graphene interaction by utilizing optical resonances associated with Mie scattering, yielding a large tunability and modulation depth. We develop a semianalytical framework based on transition-matrix formulation for modeling and analysis of periodic and aperiodic arrays of such time-varying building blocks. The proposed method is validated against full-wave numerical results obtained using the finite-difference time-domain method. It provides an ideal tool for mathematical synthesis and analysis of space-time gradient metamaterials, eliminating the need for computationally expensive numerical models. Moreover, it allows for a wider exploration of exotic space-time scattering phenomena in time-modulated metamaterials. We apply the method to explore the role of modulation parameters in the generation of frequency harmonics and their emerging wavefronts. Several potential applications of such platforms are demonstrated, including frequency conversion, holographic generation of frequency harmonics, and spatiotemporal manipulation of light. The presented results provide key physical insights to design time-modulated functional metadevices using various building blocks and open up new directions in the emerging paradigm of time-modulated metamaterials.

Figure 1

The schematic of an isolated microwire wrapped by time-varying graphene.

Figure 2

The schematic of an aperiodic arrangement of microwires wrapped by time-varying graphene layers.

Figure 3

The schematic of a periodic array of microwires wrapped by time-varying graphene layers.

Figure 4

The block diagram of the steps and equations involved in the calculation of transmitted and reflected fields for a periodic array of wires wrapped by time-modulated graphene layers.

Figure 5

The comparison of near-field results obtained with the T-matrix and FDTD method corresponding to a periodic array of dielectric cylinders coated by sinusoidally modulated graphene layers incident normally by a TE-polarized plane wave.

Figure 6

Frequency conversion in a time-modulated graphene-wrapped microwire. (a) The schematic depiction of a graphene-wrapped ${\mathrm{Si}/\mathrm{SiO}}_2$ microwire modulated sinusoidally in time. (b) The TE scattering efficiency spectrum of time-invariant microwire with a Fermi energy level of $E_f=0.3$ eV. The peaks correspond to TE resonant modes. Parts (c) and (d) represent the scattering efficiencies of the fundamental and first-order generated frequency harmonics as functions of Fermi energy modulation depth and wavelength. (e) The magnetic-field profiles corresponding to the resonant modes of the fundamental and first-order frequency harmonics.

Figure 7

(a) The amplitude and phase shifts of transmission coefficients corresponding to fundamental and first-order frequency harmonics as functions of wavelength and modulation depth of the Fermi energy for a modulation phase delay of $\alpha =0$. The phase shift is depicted as pseudocolor and is measured at each wavelength with respect to the phase at $\beta =0$. Part (b) demonstrates the same quantities as functions of wavelength and modulation phase delay of the Fermi energy for a modulation depth of $\beta =0.75$. The phase shift at each wavelength is measured with respect to the phase at $\alpha =0$.

Figure 8

Holographic generation of frequency harmonics using time-modulated metasurfaces. (a) The schematic of a time-modulated metasurface consisted of graphene-wrapped microwires biased via an active RF circuit incorporating amplifiers and phase shifters to provide required modulation depth and phase delay. Parts (b) and (c) demonstrate the required amplitude and phase modulation at the metasurface plane, respectively, which are mapped to the modulation depth and phase delay of the elements. (d) The magnetic field amplitude corresponding to the first-up modulated frequency harmonic generated in the forward direction (transmission). (e) Comparison of the ideal and realized patterns at the target plane located $5\lambda$ away from the metasurface plane.

Figure 9

(a) The effective permittivity of the deeply subwavelength graphene-wrapped microwire as a function of the Fermi energy level of graphene. Parts (b) and (c) compare the near-field distributions of electric field corresponding to graphene-wrapped microwires and homogenized counterparts for Fermi energy levels of $E_f=0.2$ and 0.5 eV, respectively.

Figure 10

Spatiotemporal manipulation of light using a time-modulated metamaterial. (a) The schematic depiction of a space-time gradient metamaterial consisted of graphene-wrapped microwires biased from the top using an RF biasing grid incorporating amplifiers to provide required modulation depth and offset for each element. Parts (b) and (c) demonstrate the required average value and modulation depth of the effective permittivity, respectively, as functions of the radial position within the lens, which are mapped to the average value and modulation depth of the Fermi energy of graphene-wrapped microwires. (d) The steady-state electric-field amplitude distribution at different time steps within a scanning cycle.