Time-modulated meta-atoms

Interaction of electromagnetic radiation with time-variant objects is a fundamental problem whose study involves foundational principles of classical electrodynamics. Such study is a necessary preliminary step for delineating the novel research field of linear time-varying metamaterials and metasurfaces. A closer look to the literature, however, reveals that this crucial step has not been addressed and important simplifying assumptions have been made. Before proceeding to studies of linear time-varying metamaterials and metasurfaces with their effective parameters, we need to rigorously describe the electric and magnetic responses of a temporally modulated meta-atom. Here, we introduce a theoretical model which describes a time-variant meta-atom and its interaction with incident electromagnetic waves in time domain. The developed general approach is specialized for a dipole emitter/scatterer loaded with a time-varying reactive element. We confirm the validity of the theoretical model with full-wave simulations. Our study is of major significance also in the area of nanophotonics and nano-optics because the optical properties of all-dielectric and plasmonic nanoparticles can be varied in time in order to achieve intriguing scattering phenomena.

Figure 1

Schematic view of a metamaterial formed by time-varying meta-atoms. Here, the meta-atom is assumed to exhibit only electric response determined by its electric dipole moment $\mathbf{p}\left(t\right)$. The excitation and temporal variations of the dipole moment can be controlled and engineered using time modulation of the meta-atom.

Figure 2

Engineering of electric current in a series $RC$ circuit. (a) Schematic view of the $RC$ circuit. [(b) and (c)] Time-varying capacitance and the corresponding electric current flowing through the electric circuit described by (a). In this example, $R_0=1\mathrm{\Omega }, V_0=1\mathrm{V}$, and $\omega =1\mathrm{rad}/\mathrm{s}$. The resistance and the capacitance of the circuit are connected to a time-harmonic voltage source $v\left(t\right)=V_{0}cos\left(\omega t\right)$.

Figure 3

Instantaneous power balance for transmitting regime regarding two different dipoles. (a) Electric dipole is fed in the center by a time-harmonic source representing transmitting regime. (b) Resonant dipole: the length of the cylindrical wire, made of perfect conductor, is $l=93.9\mathrm{mm}$ and the radius of the wire is $r=0.3\mathrm{mm}$. (c) Nonresonant dipole: the length of the wire is $l=15\mathrm{mm}$ in this case, which is much smaller than the wavelength that is 200 mm. The radius of the wire remains the same $r=0.3\mathrm{mm}$. In both (b) and (c), the red solid curve shows the instantaneous radiated power, while the blue dot-dashed curve corresponds to a sum of the reactive and supplied powers. Additionally, the dashed orange curve represents the conventional interpretation of the instantaneous radiated power, i.e., $P\left(t\right)=({\mu }_0/6\pi c)\ddot{\mathbf{p}}\left(t\right)·\ddot{\mathbf{p}}\left(t\right)$, which is seen to be incompatible with the instantaneous conservation of energy. In (b), the total reactive power is zero due to the resonance, and the blue dot-dashed curve simply describes the power that is supplied to the resonant dipole.

Figure 4

Instantaneous power balance for a resonant dipole in receiving regime. The cylindrical wire is illuminated by a normal incident electromagnetic plane wave. The length and the radius of the cylindrical wire is $l=93.9\mathrm{mm}$ and $r=0.3\mathrm{mm}$, respectively. (a) The wire is short circuited (zero-ohm load). (b) the wire is loaded by a resistance $R_{\mathrm{load}}=50\mathrm{\Omega }$. (c) The wire is loaded by an inductance $L_{\mathrm{load}}=10\mathrm{nH}$. In both (b) and (c), the loads are lumped elements connected in a small gap at the center of the wire. Red solid curve always expresses the instantaneous radiated power and the blue dot-dashed curve shows the summation of the reactive power, the dissipated power by the load resistance, and the power that is supplied to the dipole by the incident field.

Figure 5

Applications for meta-atoms modulated using time-varying reactive elements. Electrically short dipole ($l=15\mathrm{mm}$ and $r=0.3\mathrm{mm}$) is illuminated by a normally incident plane wave. (a) Schematic representation of a dipole loaded with a time varying inductance that cancels scattering. (b) Current through the inductance and function for the inductance that realize nonscattering regime shown in (a). Note that the current that does not radiate is a linearly growing function. (c) Power balance in the case of linearly growing current. Reactive power grows proportionally to $t^3$ (orange dotted line), amplitude of supplied power grows linearly (blue dot-dashed line) and the power produced by the time-varying inductance (red solid line) is such that it compensates both supplied and reactive powers. The inset figure shows region near zero, when reactive power is negligible. (d) Schematic representation of a dipole loaded with a time varying capacitance that changes the frequency of the scattered wave. (e) Current through the capacitance and function for the capacitance that realize frequency shifting regime in (c). The frequency of the scattered wave is shifted from 1.5 to 0.15 GHz.

Figure 6

Schematic view of an $RC$ circuit with time-modulated capacitance.