Nanophotonics 2019; aop Research article Victor Pacheco-Peña and Nader Engheta* Effective medium concept in temporal metamaterials https://doi.org/10.1515/nanoph-2019-0305 Received August 6, 2019; revised October 27, 2019; accepted 1 Introduction November 2, 2019 The arbitrary control of wave-matter interaction by spa- Abstract: Metamaterials are mostly designed in the time- tially designing the electromagnetic properties of media harmonic scenario where wave propagation can be spa- has been of great interest within the research commu- tially manipulated. Tailoring the electromagnetic response nity for decades [1]. Within this context, metamaterials of media in time has also gained the attention of the sci- [and metasurfaces as their two-dimensional (2D) version] entific community in order to achieve further control on have been studied and demonstrated at different spectral wave-matter interaction both in space and time. In the ranges such as microwave, terahertz, and optics [2–11] present work, a temporally effective medium concept in achieving extreme parameters of permittivity (ε) and per- metamaterial is theoretically investigated as a mechanism meability (μ) such as near-zero or negative [12–19]. Meta- to create a medium with a desired effective permittivity. materials have opened new paths to tailor and engineer Similar to spatially subwavelength multilayered metama- wave propagation at will giving rise to new and improved terials, the proposed “temporal multilayered”, or “multi- applications such as sensors [20–22], antennas [23–27], stepped” metamaterial, is designed by alternating in time beam shaping, [28–30] and mathematical operators [31], the permittivity of the medium between two values. In so to name a few. doing, the temporally periodic medium can be modeled Since their conception, metamaterials and metasur- as an effective metamaterial in time with an effective per- faces have been mostly studied within the time-harmonic mittivity initiated by a step function. The analogy between scenario (frequency domain). In this realm, wave-matter the temporal multistepped and the spatial multilayered interaction is controlled by properly engineering the metamaterials is presented demonstrating the duality electromagnetic properties of media in space (i.e. along between both domains. The proposed temporal metama- the x, y, z coordinates). However, to further control wave terial is analytically and numerically evaluated showing propagation at will, there is another dimension, time (t), an excellent agreement with the designed parameters. that can be manipulated in addition to these three dimen- Moreover, it is shown how the effective permittivity can sions. Space-time metamaterials have been studied since be arbitrarily tailored by changing the duty cycle of the several decades ago [32, 33] and they have recently been periodic temporal metamaterial. This performance is also used for exciting applications [34, 35] such as inverse connected to the spatial multilayer scenario in terms of prism [36], frequency conversion [37], nonreciprocity the filling fraction of the different materials used to create [38–40], temporal band-gap, and time reversal [41–48]. the multilayered structures. The idea of “time crystals” has also been introduced and developed [49–52]. Keywords: temporal metamaterials; effective medium; Inspired by the broad opportunities offered by multilayered media; metamaterials. spatial and temporal modulated metamaterials, in this work we investigate (and demonstrate both analytically and numerically) a mechanism to achieve a temporally *Corresponding author: Nader Engheta, Department of Electrical effective permittivity using temporal metamaterials. As and Systems Engineering, University of Pennsylvania, Philadelphia, well known in the spatial case (time-harmonic scenario), PA 19104, USA, e-mail: [email protected]. metamaterials with effective permittivity can be designed https://orcid.org/0000-0003-3219-9520 Victor Pacheco-Peña: School of Engineering, Newcastle University, by alternating subwavelength layers of different materi- Merz Court, Newcastle Upon Tyne NE1 7RU, UK. als in a periodic fashion [14, 53–56]. Here, the tempo- https://orcid.org/0000-0003-2373-7796 ral version of such spatial multilayered metamaterial Open Access. © 2019 Nader Engheta et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. Brought to you by | Newcastle University Authenticated Download Date | 12/13/19 12:37 PM 2 V. Pacheco-Peña and N. Engheta: Effective medium concept in temporal metamaterials is considered by modulating the permittivity of the 2 Theory and results medium in time. Similar to the spatial case, the “mul- tilayered” or multistepped temporal effective medium 2.1 Connection between spatial multilayers is designed by changing the permittivity of the whole and temporal multisteps medium from ε1 to ε2 and returning to ε1 with a periodic- ity much smaller than the period of the incident wave. To begin with, let us consider the well-known spatial By doing so, the multistepped temporal metamaterial is multilayered structure shown in Figure 1A. It consists of able to emulate an effective permittivity as if the permit- tivity was changed in time following a step function. The periodically arranged layers of two different materials (with permittivity ε and ε , both assumed to be real posi- similarities and differences between the spatial and tem- 1 2 poral multilayered effective metamaterials are presented tive quantities) along the propagation z-axis. The differ- here. The effect of the periodicity and duty cycle of the ent layers are considered to be infinitely extent along the time-dependent permittivity for the temporal version of x and y directions with thicknesses much smaller than the multilayered metamaterials is also evaluated, demon- incident wavelength in each medium. The resulting struc- strating that the resulting effective permittivity can be ture is illuminated with a plane wave under normal inci- tailored when changing the duty cycle of the temporal dence. As described in the introduction, it is well known change. All the designs studied here were numerically in the conventional effective medium theory in electro- evaluated using the time-domain solver of the commer- magnetics [56] that such spatial multilayered structures cial software COMSOL Multiphysics®. can be used to create metamaterials with an anisotropic Figure 1: Schematic representation of the spatial and temporal multilayered metamaterials and their corresponding effective media. Top row: Schematic representation of a well known spatial multilayered structure made with alternating subwavelength layers of two media with different permittivity ε1 and ε2 (A), along with the effective medium produced by such spatial multilayered medium (B). Bottom row: Schematic representation of the temporal analogue, i.e. the proposed multistepped temporal metamaterial made by alternating in time subperiod steps of two different permittivities ε1 and ε2 (C), along with the temporally effective medium produced by such temporal multistepped medium (D). Brought to you by | Newcastle University Authenticated Download Date | 12/13/19 12:37 PM V. Pacheco-Peña and N. Engheta: Effective medium concept in temporal metamaterials 3 effective permittivity εeff (εeffxx, εeffyy, εeffzz, for polarization of able to produce a forward and a backward wave with the the E field along the x, y and z axis, respectively) as sym- latter traveling with the same angle, but opposite direc- bolically shown in Figure 1B. In this scenario, the εeff in tion of the incident wave [32]. Interestingly, these exciting each spatial coordinate depends on the values of ε1 and features have been shown to be equivalent of the “reflec- ε2 as well as the thickness of each layer. Based on this, tion” and “transmission”, but in the time domain [60, 61]. the εeff for the spatial multilayered metamaterial can be Such temporal metamaterial has been used for different expressed as follows [56–58]: applications such as time reversal and frequency conver- sion [62]. ε∆==εε 1()−+∆ ε (1a) effexx ffyy 12zz22 By comparing Figure 1A,B and Figure 1C,D, a rela- tion between the spatial and temporal multilayered/ εε multistepped scenarios can be observed. Both cases have ε = 12 (1b) effzz εε∆ +−(1 ∆ ) similar features such as the subwavelength thickness and 12zz22 subperiod temporal change of permittivity, respectively. where Δz1 = 1–Δz2 and Δz2 are the filling fractions for each The main difference relies on the fact that for the temporal subwavelength layer (ε1 and ε2, respectively) defined as case, frequency conversion and unaltered wavelength are the ratio between the absolute thicknesses of each layer expected (conservation of k, i.e. the frequency changes with respect to the total thickness of one spatial period while the wavelength stays unchanged) while for the Δz1,2 = dz1,2/(dz1 + dz2). This spatial multilayered structure spatial scenario the frequency remains the same while the has been used to create hyperbolic and epsilon near- wavelength is changed. However, as it will be discussed in zero metamaterials [14, 53] and has been applied to the following, both spatial and temporal multilayers/mul- different scenarios such as focusing devices [58, 59], tisteps share the analogous feature: the ability to create a demonstrating a successful mechanism to achieve tai- metamaterial with an effective value of εeff. lored electromagnetic parameters. This