Metamaterials are mostly designed in the time-harmonic scenario where wave propagation can be spatially manipulated. Tailoring the electromagnetic response of media in time has also gained the attention of the scientific community in order to achieve further control on wave-matter interaction both in space and time. In the present work, a temporally effective medium concept in metamaterial is theoretically investigated as a mechanism to create a medium with a desired effective permittivity. Similar to spatially subwavelength multilayered metamaterials, the proposed “temporal multilayered”, or “multistepped” metamaterial, is designed by alternating in time the permittivity of the medium between two values. In so doing, the temporally periodic medium can be modeled as an effective metamaterial in time with an effective permittivity initiated by a step function. The analogy between the temporal multistepped and the spatial multilayered metamaterials is presented demonstrating the duality between both domains. The proposed temporal metamaterial is analytically and numerically evaluated showing an excellent agreement with the designed parameters. Moreover, it is shown how the effective permittivity can be arbitrarily tailored by changing the duty cycle of the periodic temporal metamaterial. This performance is also connected to the spatial multilayer scenario in terms of the filling fraction of the different materials used to create the multilayered structures.
The arbitrary control of wave-matter interaction by spatially designing the electromagnetic properties of media has been of great interest within the research community for decades . Within this context, metamaterials [and metasurfaces as their two-dimensional (2D) version] have been studied and demonstrated at different spectral ranges such as microwave, terahertz, and optics , , , , , , , , ,  achieving extreme parameters of permittivity (ε) and permeability (μ) such as near-zero or negative , , , , , , , . Metamaterials have opened new paths to tailor and engineer wave propagation at will giving rise to new and improved applications such as sensors , , , antennas , , , , , beam shaping, , ,  and mathematical operators , to name a few.
Since their conception, metamaterials and metasurfaces have been mostly studied within the time-harmonic scenario (frequency domain). In this realm, wave-matter interaction is controlled by properly engineering the electromagnetic properties of media in space (i.e. along the x, y, z coordinates). However, to further control wave propagation at will, there is another dimension, time (t), that can be manipulated in addition to these three dimensions. Space-time metamaterials have been studied since several decades ago ,  and they have recently been used for exciting applications ,  such as inverse prism , frequency conversion , nonreciprocity , , , temporal band-gap, and time reversal , , , , , , , . The idea of “time crystals” has also been introduced and developed , , , .
Inspired by the broad opportunities offered by spatial and temporal modulated metamaterials, in this work we investigate (and demonstrate both analytically and numerically) a mechanism to achieve a temporally effective permittivity using temporal metamaterials. As well known in the spatial case (time-harmonic scenario), metamaterials with effective permittivity can be designed by alternating subwavelength layers of different materials in a periodic fashion , , , , . Here, the temporal version of such spatial multilayered metamaterial is considered by modulating the permittivity of the medium in time. Similar to the spatial case, the “multilayered” or multistepped temporal effective medium is designed by changing the permittivity of the whole medium from ε1 to ε2 and returning to ε1 with a periodicity much smaller than the period of the incident wave. By doing so, the multistepped temporal metamaterial is able to emulate an effective permittivity as if the permittivity was changed in time following a step function. The similarities and differences between the spatial and temporal multilayered effective metamaterials are presented here. The effect of the periodicity and duty cycle of the time-dependent permittivity for the temporal version of multilayered metamaterials is also evaluated, demonstrating that the resulting effective permittivity can be tailored when changing the duty cycle of the temporal change. All the designs studied here were numerically evaluated using the time-domain solver of the commercial software COMSOL Multiphysics®.
2 Theory and results
2.1 Connection between spatial multilayers and temporal multisteps
To begin with, let us consider the well-known spatial multilayered structure shown in Figure 1A. It consists of periodically arranged layers of two different materials (with permittivity ε1 and ε2, both assumed to be real positive quantities) along the propagation z-axis. The different layers are considered to be infinitely extent along the x and y directions with thicknesses much smaller than the incident wavelength in each medium. The resulting structure is illuminated with a plane wave under normal incidence. As described in the introduction, it is well known in the conventional effective medium theory in electromagnetics  that such spatial multilayered structures can be used to create metamaterials with an anisotropic effective permittivity εeff (εeffxx, εeffyy, εeffzz, for polarization of the E field along the x, y and z axis, respectively) as symbolically shown in Figure 1B. In this scenario, the εeff in each spatial coordinate depends on the values of ε1 and ε2 as well as the thickness of each layer. Based on this, the εeff for the spatial multilayered metamaterial can be expressed as follows , , :
where Δz1=1–Δz2 and Δz2 are the filling fractions for each subwavelength layer (ε1 and ε2, respectively) defined as the ratio between the absolute thicknesses of each layer with respect to the total thickness of one spatial period Δz1,2=dz1,2/(dz1+dz2). This spatial multilayered structure has been used to create hyperbolic and epsilon near-zero metamaterials ,  and has been applied to different scenarios such as focusing devices , , demonstrating a successful mechanism to achieve tailored electromagnetic parameters. This technique has been developed within the frequency domain by using different materials in the spatial coordinates. However, an interesting question may be asked: would it be possible to design metamaterials using such techniques in the time domain? More specifically, would it be possible to achieve an arbitrary tailored effective permittivity by simply modulating the permittivity of the entire (spatially unbounded) medium in time, much faster than the frequency of the signal?
To answer this question, let us discuss the relation and analogy between the spatial multilayered metamaterial and the proposed temporal scenario shown in Figure 1A and C, respectively. For the temporal case shown in Figure 1C, we have considered a monochromatic wave traveling in a spatially unbounded medium with the relative permeability μ=1 and a time-dependent relative permittivity ε(t). This ε(t) is temporally changed between ε1 and ε2 (both positive real quantities) with a period much smaller than the period of the incident wave. The aim here is to relate this temporal multistepped structure to a temporally effective permittivity initiated with a step function in time [as shown in Figure 1D]. A single step change of permittivity has been a research topic since the last century , . It has been shown that with such temporal change of permittivity the wavenumber k is preserved while the frequency is modified from ω1 to (with μ1=μ2). Moreover, the temporal boundary created at the time of the rapid change of ε is able to produce a forward and a backward wave with the latter traveling with the same angle, but opposite direction of the incident wave . Interestingly, these exciting features have been shown to be equivalent of the “reflection” and “transmission”, but in the time domain , . Such temporal metamaterial has been used for different applications such as time reversal and frequency conversion .
By comparing Figure 1A,B and Figure 1C,D, a relation between the spatial and temporal multilayered/multistepped scenarios can be observed. Both cases have similar features such as the subwavelength thickness and subperiod temporal change of permittivity, respectively. The main difference relies on the fact that for the temporal case, frequency conversion and unaltered wavelength are expected (conservation of k, i.e. the frequency changes while the wavelength stays unchanged) while for the spatial scenario the frequency remains the same while the wavelength is changed. However, as it will be discussed in the following, both spatial and temporal multilayers/multisteps share the analogous feature: the ability to create a metamaterial with an effective value of εeff.
To evaluate the performance of the proposed temporal multistepped metamaterial, let us consider a time-dependent permittivity with a step function, as shown in Figure 2A,B. Here, the permittivity of the whole medium is rapidly changed (with a rise/fall time much smaller than the period of the incident wave) from ε1=1 to ε2=1.25 and is kept at this value. With this setup, the numerical results of the electric field for the forward wave at a single spatial location within the spatially unbounded medium (right-most position of a 20λ1 simulation box with λ1=0.2 mm [f1=1.5 GHz] as the wavelength of the incident monochromatic wave, see method Section for more details of the numerical model) are shown, as the green curve in Figure 2C, where it can be seen how the period of the signal after the permittivity change, i.e. (1.11T), is different from the one before the change of permittivity, i.e. (T), as expected due to the frequency change produced at the temporal boundary. Moreover, note that the amplitude of the electric field after the permittivity change is reduced from 1 to ~0.85. These results are in agreement with the expected change of the E field at the temporal boundary in the time domain  where an amplitude of the electric field of 0.85 is theoretically predicted when the permittivity is modified from 1 to 1.25. Note that in these results we are only plotting the field distribution for the forward wave, but a backward wave is also excited once inducing the temporal boundary. However, since the change of permittivity is small (ε1=1 to ε2=1.25 for a step function of permittivity) the “temporal equivalent” of Fresnel coefficient for this backward wave , ,  is almost negligible with a value of −0.047. From now on we will show the results of the generated forward wave and a discussion regarding the backward waves will be presented in Section 2.3.
Now, what if we use a temporally periodic permittivity, i.e. rapidly alternating it between 1 and 1.25? To evaluate this scenario, we can consider the time-dependent permittivities shown in Figure 2A,B as blue lines with a periodicity of (1/10)T and (1/20)T, respectively. A schematic representation of the periodic change is shown in the first row of Figure 2 to guide the eye. With this configuration, the numerical results of the electric field distribution at the same location as that of the single step function are shown in Figure 2C,D. As observed, the electric field distribution for the temporally periodic multistepped metamaterial present some “ripples” which can be explained as follows: when the permittivity is first changed from ε1=1 to ε2=1.25, the amplitude of the temporal E field is modified to a value that can be calculated using the “temporal equivalent” of the Fresnel coefficient for the forward wave , , . i.e. the amplitude is reduced because ε2>ε1. Now, when the permittivity is returned to its initial value (from ε2=1.25 to ε1=1) the amplitude of the temporal E field will be increased following Hence, the “ripples” are produced due to the fact that the temporal E field will experience multiple changes in its amplitude, because of the multiple temporal boundaries induced in the periodic scenario. Moreover, such E field possesses higher frequency harmonics (as it will be further discussed in the following Sections), but our interest is mostly on its fundamental frequency. By comparing these results with those obtained with the single step function, we can see how the period (and hence the frequency) of the wave for the temporally periodic change of permittivity is different from that of the single step change of ε with a clear mismatch between the results. Moreover, note that the period of the wave for the case with a step function of permittivity is longer (1.11T) compared to the temporal multistepped cases (~1.055T). This implies that the new frequency of the wave with the temporally periodic ε(t) is actually higher than the one produced when using a single step change of ε in time from 1 to 1.25.
Based on these results, in order to match the converted frequency of the temporal multisteps with that of the single change of permittivity in time, it is necessary to modify the value ε2 for the case of the single step change to a value εeff smaller than that used for ε2 in the temporal multistepped case. By doing this, the converted frequency of the single step change of permittivity will be increased (see previous discussion about the frequency conversion at the temporal boundary) and the period of the converted wave will match that of the temporal multistepped metamaterial. But, what should be the value of εeff for such single step change? The answer to this question can be obtained by exploiting the similarities between the spatial multilayered metamaterial Eq. (1) and the temporal multistepped scenario: it can be suggested that the temporally effective permittivity created by the latter case would be equivalent to a step function with a single temporal change of permittivity from ε1 to εeff such that εeff for the step function will be an “effective” value between the ε1 and ε2 used for the temporal multistepped case. Hence, the temporally effective permittivity can be expressed as follows (the analytical derivation can be found in the methods section below):
where Δt1=1–Δt2 and Δt2 are the filling fractions (in time) for each temporal step (times where the permittivity is ε1 and ε2, respectively).
By comparing Eq. (1b) and Eq. (2) it can be seen how the temporal and spatial multistepped/multi-layererd metamaterials are connected with a similar expression for the effective permittivity in both domains. This duality can also be seen in terms of the boundary conditions for the spatial and temporal scenarios: in the former case, the εeffzz defined by Eq. (1b), the normal component of vector D is preserved at each spatial interface between the two materials with ε1 and ε2. Similarly, in the temporal scenario, D should be preserved at each temporal boundary produced when the permittivity is rapidly changed from ε1 to ε2 , , , .
To evaluate and test Eq. (2), the numerical results of the electric field distribution at the same location as Figure 2 for both temporal multistepped metamaterials with period (1/10)T and (1/20)T with Δt1=Δt2=0.5 are shown in Figure 3B,D along with the results using a single step function for εeff following Eq. (2), with ε1=1 and εeff=1.111. Note that the results for the temporal multisteps are those also shown in Figure 2C,D, but are re-plotted here for a better comparison. From these results, it can be clearly seen how similarity is achieved between the periodicity of the wave for the case of a single step change of permittivity and that obtained using the temporally periodic case.
For the sake of completeness, the results of the electric field at the same location as before considering a temporal multistepped metamaterial with a period of 0.4T are shown in Figure 3F and the time-dependent permittivity for this case is shown in Figure 3E. By comparing these results with those using smaller periods (such as 0.1T and 0.05T), the good matching with the single step change is not achieved. This is expected because the effective permittivity described in Eq. (2) is valid for periods much smaller than the period of the incident wave (an analogous feature holds for the spatial scenario where Eq. (1) is valid for layers with subwavelength thicknesses ). These results demonstrate that temporal multistepped metamaterials can produce a temporally effective permittivity (similar to the case of a spatial multilayer metamaterial) which can be modeled as a single step change of permittivity in time.
2.2 Tailoring the temporal effective permittivity
To further study the performance of the temporal multistepped metamaterial, we put forward the following question: would it be possible to tailor the effective permittivity to any arbitrary value between the two positive ε1 and ε2? As is known, this can be achieved with the spatial multilayered case by simply changing the filling fraction of the two materials used for each of the layers , see Eq. (1a,b). Hence, since the proposed temporal multistepped metamaterial is the dual of the spatial case described by Eq. (1b), one can also tailor the effective permittivity in the time domain by changing the temporal filling fraction for each temporal layer (Δt1 and Δt2) in Eq. (2).
To verify this, the numerical results of the electric field using different values of duty cycle (DC) ranging from 0.2 to 0.8 with a period of (1/10)T for the temporally periodic permittivity of the temporal multisteps are shown in the third row of Figure 4. Here, a DC of 0.2 and 0.8 mean Δt2=0.2×0.1T and Δt2=0.8×0.1T, respectively. The temporally periodic permittivities for each case are shown in the first and second rows of Figure 4. As observed, the period (and hence the effective permittivity) of each case is changed depending on the value of DC used. To better observe this performance, the effective permittivity for each case can be calculated using Eq. (2) and the resulting single step functions for each DC are plotted as red curves in the second row of Figure 4. It is shown for a DC of 0.2, 0.4, 0.6 and 0.8 the effective permittivity is equivalent to a single step function that is rapidly changed from 1 to εeff of 1.041, 1.087, 1.136 and 1.190, respectively. With these parameters, the numerical results for the electric field distribution using each effective step function of permittivity are shown as green curves in the third row of Figure 4. As observed, an excellent agreement is obtained between the results, demonstrating how the temporal multistepped metamaterial can be used to create an arbitrary temporally effective permittivity between ε1 and ε2 by simply changing the DC.
2.3 Results for higher values of ε2
In the previous sections, the temporal multistepped scenario was demonstrated by periodically changing the permittivity between the two values of ε1=1 to ε2=1.25. To better appreciate the role of these values, here we give another example with higher ε2=2 in order to evaluate a bigger change of permittivity. The total duration of each periodic multistep is the same as in the previous studies and two different DCs are considered (0.2 and 0.8) in order to further study the effect of the filling fraction in the temporal εeff.
With this configuration the numerical results of the electric field distribution considering a time-dependent permittivity rapidly changing between ε1=1 to ε2=2 are shown as blue lines in Figure 5B,F considering a DC of 0.2 and 0.8, respectively. These results are then compared with those obtained considering a time-dependent permittivity using a single step with effective permittivity calculated using Eq. (2) (εeff=1.111 for a duty cycle of 0.2 and εeff=1.667 for a duty cycle of 0.8). For completeness the time-dependent permittivity functions in both cases are shown in the Figure 5A,E for both duty cycles.
As observed, a good agreement is achieved between the results of the temporal multistepped metamaterials and the single steps, demonstrating the capability of using such temporal multisteps as an effective medium in the time domain. Moreover, note that the amplitude of the electric field for the case with a DC of 0.8 (~0.72 in Figure 5F) is smaller than the one obtained with a DC of 0.2 (~0.95 Figure 5B). As explained before, these are expected results because of the temporal coefficient for the electric field after the temporal change which theoretically predicts an amplitude of the electric field of ~0.7 and ~0.92 for a DC of 0.8 and 0.2, respectively. Moreover, we also note higher “ripples” in the E field, again expected due to the fact that the permittivity undergoes higher changes between 1 and 2, as explained in the previous sections. For completeness, the results of the normalized spectral contents calculated from the temporal E field from Figure 5B,F are shown in Figure 5C,G for DC values of 0.2 and 0.8, respectively. As it is shown, the ripples observed in the time domain (Figure 5B,F) are translated into higher-order harmonics, as discussed before. However, note that their amplitude is much smaller than that of the fundamental frequency demonstrating that the temporal effective medium is still valid (for the fundamental frequency) in temporal multistepped metamaterials for cases when the change of permittivitty is increased. For completeness, a zoom-in of the spectral contents around the fundamental harmonic extracted from Figure 5C,G is shown in Figure 5D,H, respectively. As observed, a good agreement is again noticeable between the results of the temporal multisepped metamaterial and the effective temporal medium modeled as a single step. From these results, the frequency is changed from f1 to f2≈0.94f1 and to f2≈0.77f1 when using a DC of 0.2 and 0.8, respectively, in agreement with theoretical calculations  which predict a frequency change of f2≈0.948f1 and f2≈0.774f1 for each DC, respectively.
In all the results discussed in the previous section (Figures 2–5) we have shown the results of the distribution of the temporal E field of the forward wave (FW) at a single location. However, since we have demonstrated how the multistepped temporal metamaterial is able to emulate an effective permittivity as if the permittivity was changed in time using a single step function, it is important to show that a backward wave (BW) is also produced for the case when such a multistepped temporal metamaterial is implemented.
To do this, let us consider a multistepped temporal metamaterial with a time-dependent permittivity that is initially ε1=2 and then it is periodically changing between ε1=2 and ε2=1. The results of the spatial distribution of the electric field at different times are shown in Figure 6 considering a DC of 0.2 (Figure 6A,B) and 0.8 (Figure 6C,D). The field distribution using the multistepped temporal metamaterials are shown in Figure 6A,C along with their effective medium using a single step of permittivity, Figure 6B,D. From these results a good agreement is observed for the field distribution of the multistepped temporal metamaterials resembling the one generated by their effective version using a single step. Moreover, note that the BW wave is created in all the cases shown in Figure 6 with their spatial distributions showing the same wavelength as the FW waves, i.e. after the temporal change of permittivity, k remains the same while the frequency is changed. Finally, note that the BW is more evident for the case with DC=0.8 which is due to the fact that the effective permittivity (εeff=1.111) is smaller than the one produced with a DC of 0.2 (εeff=1.667). Hence, it is expected to have a larger amplitude for the BW wave when the permittivity is effectively changed in a single step from ε1 to εeff=1.111 since , .
Such multistepped temporal metamaterials could be realized at microwave frequencies by using circuit elements loaded in transmission lines  in order to change the permittivity between two different values. This can become more challenging when working at higher frequencies (such as the IR and visible spectrum) due to the reduced period of the signal. However, within the past few years great efforts have been devoted to rapidly tuning the electromagnetic properties of materials at optical frequencies. In this realm, the frequency conversion using temporal boundaries has been recently demonstrated experimentally at such spectral range , . Hence, we are optimistic that our concept of multistepped temporal metamaterials as a way to produce an effective temporal permittivity could be realized not just at microwaves, but at higher frequencies in the near future.
3.1 Analytical solution for the effective permittivity in temporal multisetpped metamaterials
As explained in the Introduction, the interaction of electromagnetic waves with time-dependent media has gained much attention in the scientific community. In this realm, time-dependent periodic metamaterials have been recently studied demonstrating their capability to achieve the temporal version of a photonic band-gap structure by using different functions of permittivity and/or permeability , , . Here, we focused our attention in the analytical formulation of the effective permittivity for the temporal scenario analogous with the spatial multilayered metamaterial using the transfer matrix method. The structure under study is the one shown in Figure 1C,D.
where j represents each temporal multistep with permittivity ε1 and ε2, respectively. In this realm, because (a) the change of permittivity is applied to the whole medium where the wave is traveling (unbounded medium) and (b) the wavenumber k is preserved at each temporal boundary  the characteristic matrix for the temporal multisteps in Figure 1C can be then calculated as follows:
where τ1 and τ2 are the absolute temporal duration of each multistep. The total period of the multistep is then τtotal=τ1+τ2<<T. As the result, the filling factors (Δt1 and Δt2) in Eq. (2) are Δt1=τ1/τtotal and Δt2=τ2/τtotal. As observed, Eqs. (4a,b) only depend on the frequencies at each temporal step ω1,2 and their absolute temporal duration τ1,2, meaning that each induced temporal step does not affect the spatial distribution of the wave since its wavelength is not modified (conservation of k).
Now, we can calculate an equivalent matrix M(τtotal) for the whole period τtotal by simply multiplying Eqs. (4a,b). Based on this, M(τtotal) can be expressed in the following manner:
Now we can define a characteristic matrix for an equivalent medium made of an effective permittivity (εeff) as follows:
with ωeff as the effective angular frequency of the equivalent medium. Based on this, the equivalent medium described by the equation above will be the same as the one defined by Eq. (5) by defining the equalities between each element of both matrices: a11total=a11eff, a12total=a12eff, a21total=a21eff and a22total=a22eff. After applying these equivalences between Eqs. (5–6), the effective permittivity can then be extracted using the following equality between both characteristic matrices:
where=τ1 and τ2<<T, and Based on this, Eq. (7b) is reduced to:
Retrieving the expression shown in Eq. (2).
3.2 Simulation setup
The numerical analysis shown in Figures 2–6 was carried out using the time-domain solver of the commercial software COMSOL Multiphysics®. A simulation box of 20λ1×4λ1 was implemented and the FW waves were recorded at the right-most location of this box (x=20λ1, y=0). To avoid undesirable reflections, top and bottom perfect electric conductor (PEC) boundaries were used with the scattering boundary condition at the right boundary in all the simulations. The incident field was applied from the left boundary of the simulation box via a scattering boundary condition. A triangular mesh was implemented with a minimum and maximum size of 0.13×10−3λ1 to ensure accurate results. The rapid changes of ε in all these studies were modeled by implementing rectangular analytical functions with smooth transitions using two continuous derivatives to ensure convergence in the calculations.
In this work, temporal multistepped metamaterials have been investigated in order to achieve an effective permittivity in the time domain. It has been shown how temporal multisteps can produce a temporally effective permittivity modeled and initiated with a single step function of change of permittivity in time. The analogy between the spatial and temporal multilayered/multistepped metamaterials have been presented demonstrating the relation between both temporal and spatial domains. Within this context, it has been shown how the effective permittivity of the temporal multistepped metamaterial can be arbitrarily engineered by simply changing the duty cycle of the temporally periodic permittivity. This performance has been related and analogous to the spatial multilayered scenario where the effective permittivity can be manipulated by changing the filling fraction of the two subwavelength materials used as building blocks of the multilayered structure. The proposed technique has been analytically derived and numerically evaluated demonstrating an excellent agreement with the designed parameters. The results presented here may be applied to the design of effective permittivities in the time domain and may open new paths in the study of the exciting phenomena that may be achieved by using temporal metamaterials.
The authors would like to acknowledge the partial support from the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through grant N00014-16-1-2029, Funder Id: http://dx.doi.org/10.13039/100000006. V.P.-P. acknowledges support from the Newcastle University (Newcastle University Research Fellow).
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