Homogenization Theory of Space-Time Metamaterials

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Homogenization Theory of Space-Time Metamaterials

P.A. Huidobro, M.G. Silveirinha, E. Galiffi, and J.B. Pendry

Phys. Rev. Applied 16, 014044 – Published 19 July 2021

Abstract

We present a general framework for the homogenization theory of space-time metamaterials. By mapping to a frame comoving with the space-time modulation, we derive analytical formulas for the effective material parameters for traveling-wave modulations in the low-frequency limit: electric permittivity, magnetic permeability, and magnetoelectric coupling. In doing so, we provide a recipe for the calculation of effective parameters of space–time-modulated media where the parameters follow a traveling-wave form of any shape and we show how synthetic motion can result in giant bianisotropy. This allows us to deepen the understanding of how nonreciprocity can be achieved in the long-wavelength limit and to completely characterize the different nonreciprocal behaviors available in space–time-modulated media. In particular, we show how the modulation speed, which can be subluminal or superluminal, together with the relative phase between electric and magnetic modulations, provide tuning knobs for the nonreciprocal response of these systems. Furthermore, we apply the theory to derive exact formulas for the Fresnel drag experienced by light traveling through traveling-wave modulations of electromagnetic media, providing insight into the differences and similarities between synthetic motion and moving matter. Since we exploit a series of Galilean coordinate transformations, the theory may be generalized to other kinds of waves.

Received 22 September 2020
Revised 20 June 2021
Accepted 24 June 2021

DOI:https://doi.org/10.1103/PhysRevApplied.16.014044

Physics Subject Headings (PhySH)

Electromagnetism Light-matter interaction Metamaterials Optical & microwave phenomena Photonic crystals

PT-symmetry

Classical electromagnetism Effective medium approach

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

P.A. Huidobro ^1,*, M.G. Silveirinha ¹, E. Galiffi², and J.B. Pendry ²

¹Instituto de Telecomunicações, Instituto Superior Tecnico—University of Lisbon, Avenida Rovisco Pais 1, Lisboa 1049-001, Portugal
²The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom

^*p.arroyo-huidobro@lx.it.pt

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Vol. 16, Iss. 1 — July 2021

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Figure 1
Synthetic motion in space--time-modulated metamaterials enables nonreciprocity in the effective-medium limit. (a),(d) Traveling-wave modulations of the electromagnetic parameters as seen from the laboratory frame (a, unprimed coordinates) and from a frame comoving with the modulations (d, primed coordinates). (b) A sketch of a representative band diagram of space--time-modulated media: the bands are displaced by the space-time reciprocal lattice vector ( $g, Ω$ ). Nonsymmetric band gaps open in non-impedance matched systems, such as with space-time modulations of only the permittivity. (c) Homogenization breaks down for modulation speeds approaching the speed of waves in the medium: the luminal regime is characterized by nonreciprocal wave amplification at any frequency. (e),(f) In the long-wavelength limit, the response is nonreciprocal only if both the permittivity and the permeability are modulated in space and time. The modulation speed and the phase between the electric and magnetic modulations are the knobs for tuning the nonreciprocal response through the interplay between effective permittivity, effective permeability, and effective magnetoelectric coupling.
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Figure 2
The group velocity and effective parameters for a traveling bilayered medium as a function of the speed, $v$ . (a) The group velocity, in absolute value, of forward (blue, solid) and backward (orange, dashed) waves, for matched space-time modulations, $α_{e} = α_{m} = 0.1$ . The range of modulations where a homogenization picture is not valid, $c_{1} < v / c_{0} < c_{2}$ , is marked with a gray area. (b) The effective permittivity, $ϵ_{⊥}^{eff}$ , and permeability, $μ_{⊥}^{eff}$ (left axis), and the magnetoelectric coupling $ξ_{eff}$ (right axis). (c) The effective magnetoelectric coupling as a function of the space-time modulation speed, $v$ , and the values of $(α_{e}, α_{m})$ , parametrized by $ϕ$ , as shown in the inset panel. We take $ϵ_{m} = μ_{m} = 1.3$ and $d_{1} = d_{2} = d / 2$ . We show relative values of the wave velocity and effective parameters, i.e., $v_{eff}$ is in units of $c_{0}$ , $ϵ_{eff}$ and $μ_{eff}$ are in units of $ϵ_{0}$ and $μ_{0}$ , respectively, and $ξ_{eff}$ is in units of $c_{0}^{- 1}$ .
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Figure 3
The group velocity and effective parameters for impedance-matched sinusoidal space-time modulations ( $α_{e} = α_{m} = 0.05$ ) as a function of the modulation speed, $v$ . The shaded area represents the unstable regime where a band description loses meaning. (a) The group velocity, $v_{eff}$ , from exact analytical theory (black line), homogenization (black dots), and Floquet-Bloch mode expansion numerics (green circles). (b),(c) The effective permittivity and permeability, $ϵ_{⊥}^{eff} = μ_{⊥}^{eff}$ (b) and magnetoelectric coupling $ξ_{eff}$ (c). Numerical homogenization (blue dots) and the exact formula (blue line) are compared against a three-mode approximation in the Floquet-Bloch expansion (dashed orange line). In all panels, the limiting values of the effective parameters at the threshold of the luminal regime are depicted with gray dashed horizontal lines. We take $ϵ_{m} = μ_{m} = 1$ . We show relative values of the wave velocity and effective parameters, i.e., $v_{eff}$ is in units of $c_{0}$ , $ϵ_{eff}$ and $μ_{eff}$ are in units of $ϵ_{0}$ and $μ_{0}$ , respectively, and $ξ_{eff}$ is in units of $c_{0}^{- 1}$ .
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Figure 4
The effective magnetoelectric coupling for nonmatched systems. (a) $ξ_{eff}$ as a function of the modulation speed, $v / c_{0}$ , and the values of $(α_{e}, α_{m})$ , parametrized by $ϕ$ (see inset). The effective $ξ$ is zero whenever only one of the parameters is modulated (marked with horizontal dashed lines). The gray region centered at $v = c_{0}$ corresponds to the luminal regime. (b) The effective parameters for antiphased modulations ( $+ α, - α$ ), for a small range close to the luminal regime. The permittivity, $ϵ_{eff} / ϵ_{0}$ , and permeability, $μ_{eff} / μ_{0}$ , are shown with a solid line and magnetoelectric coupling as a dashed line. We take $α = 0.05$ , $ϵ_{m} = μ_{m} = 1$ , and $c_{m} = 1$ , as in Fig. 3. We use units of $c_{0}^{- 1}$ for $ξ_{eff}$ .
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Figure 5
The electromagnetic parameters, (a) the permittivity and permeability and (b) the velocity, $v_{D}$ , of the equivalent uniaxial moving medium as a function of the speed modulation. Numerical homogenization (blue dots) and the exact formula (blue line) are compared against a three-mode approximation in the Floquet-Bloch expansion (dashed orange line). We take $ϵ_{m} = μ_{m} = 1.3$ and $α_{e} = α_{m} = 0.05$ . The permittivity and permeability are given in units of $ϵ_{0}$ and $μ_{0}$ , respectively, and the drag velocity in units of $c_{0}$ .
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