Holding and amplifying electromagnetic waves with temporal non-foster metastructures

Abstract

We introduce a mechanism that can both hold and amplify electromagnetic waves by rapidly changing the permittivity of the medium during the wave travel from a positive to a dispersionless (i.e. non-Foster) negative value and then back again. The underlying physics behind this phenomenon is theoretically explored by considering plane wave and Gaussian pulse propagation in an unbounded medium. Interestingly, we show that a rapid positive-to-negative temporal change of ε(t) causes the propagation of the wave to stop (observed by a frozen phase in time) while the amplitude of the frozen field exponentially grows. Stepping the permittivity back to the original (or a new) positive value will cause the wave to thaw and resume propagation with the original (or the new) frequency, respectively. We numerically study the case of dipole radiation in such time-varying non-Foster structures. As a possible implementation, we propose a parallel plate waveguide platform loaded with time-dependent media emulating parallel lumped non-Foster negative capacitors. Such non-Foster time-varying structures may open new venues in controlling and manipulating wave-matter interaction.

Introduction

Slowing down wave propagation has been of great interest to the scientific community. Multiple approaches have been demonstrated to reduce the speed of light and spatially trap it, such as ultracold atomic gases¹, optical resonators², photonic crystals³, and wave interference in disordered media⁴ inspired by the Anderson localization in electron transport⁵. Also of interest is light amplification which has been achieved using several effects such as gain and nonlinear media. On a different front, the idea of four-dimensional (4D) metamaterials in which the material not only varies in three-dimensional space (3D) but also in time (1D) has seen growing interest in recent years^{6,7,8,9,10,11,12,13,14,15}. The time-dependent electromagnetic platforms have a long history, dating back to the middle part of the 20^th century^16,17. In past studies, the relative permittivity (ε) of the medium was rapidly changed in time from one positive to another positive value (both greater than unity). This effectively acted as the temporal analog of the spatial interface between two semi-infinite materials with different ε, resulting in a forward (FW) (i.e., time-refraction) and a backward (BW) (i.e., time-reflection) waves^18,19,20. This analogy between spatial and temporal interfaces has been recently exploited to explore various interesting phenomena^13,21. For instance, the notion of photonic time crystals (PTC) has been developed wherein the dielectric permittivity is modulated in time between two positive values, functioning as the temporal counterpart of spatial photonic crystals²². In this context, it has been recently shown that this can produce light amplification^23,24 with Lyubarov et al. proposing the exciting idea of nonresonant PTC lasers²⁵. Further examples include the implementation of different periodic functions of the EM parameters in time^26,27 and also in space-time media^28,29, where it has been shown how such parametric modulation could enable parametrically amplified waves.

Inspired by the opportunities offered by spatiotemporal metamaterials for wave-matter interaction, here we introduce a mechanism that can hold and amplify a propagating electromagnetic wave. Fundamentally different from PTCs, the effect is based on a single temporal transition between a dielectric material to a dispersionless negative permittivity material based on non-Foster elements. The preliminary stages of our early findings were orally presented in the conference³⁰. First, the underlying physics of this approach is explored. We consider an initial monochromatic plane wave traveling within an unbounded, spatially homogeneous, isotropic medium having a time-dependent relative permittivity ε(t). A temporal boundary is then induced by rapidly changing ε(t) in a single step from a positive (to consider a passive dispersionless medium) to a dispersionless negative value. It is analytically shown that the phase of the initial monochromatic electromagnetic wave freezes in space. While the material is held in this state and the total wave is spatially frozen, the amplitudes of the FW and the BW waves excited at the temporal boundary exponentially grows and shrinks in time, respectively, enabling a total wave within the medium (FW + BW wave) to also grow exponentially. In this scenario, the originally monochromatic nature of the wave is transformed into a signal that is no longer periodic in time (while it is still periodic in space) but which has an exponential variation in time, and thus it has no single frequency. (Note that these two aspects are fundamentally different from the techniques proposed in³¹ where surface waves were frozen by an effective temporal change of the surface impedance, leading the frequency of the wave to a zero value after applying a temporal interface). Now, to achieve non-dispersive negative permittivity (as required to keep the permittivity negative in time), one must consider non-Foster elements and therefore provide the material with an external energy source. After sufficient time, the permittivity is rapidly rendered back to a positive value greater than unity (again assuming a passive dispersionless medium). At this point, the waves are thawed and resume propagation in both directions. If the new positive permittivity is different from the original positive value, frequency conversion will also occur. Therefore, light amplification and frequency conversion can be achieved with a properly designed temporal function of the permittivity with a negative value over a limited duration of time, forced by an external energy source (as required for non-Foster structures). In addition, we numerically demonstrate a case of a two-dimensional (2D) dipole radiation in such a non-Foster time-varying medium. As an example of the proposed mechanism, a parallel-plate waveguide loaded with switchable parallel thin time-dependent dielectrics emulating capacitors with non-Foster negative capacitances is considered and numerically evaluated demonstrating the ability to achieve the above features.

Results

Holding and amplifying waves: rapid change of permittivity from a positive to a negative value

To begin with, let us discuss the differences and similarities between single-interface spatial and temporal boundaries and between traditional (i.e., passive) and non-Foster media (as a type of active media with external source) by examining four cases (see Fig. 1). In all cases, and in the rest of this manuscript, we assume the relative permeability of all materials involved to be that of free space (i.e. µ₁ = µ₂ = 1), and all permittivity values are real-valued. Our analyzes always begin with the traditional incident field, a propagating monochromatic continuous wave (CW) in medium 1, defined as ${E}_{1}=\hat{y}{e}^{i\left[{{\omega }}_{1}t-{k}_{1}x\right]}$, with$\,{k}_{1}={{\omega }}_{1}/{v}_{1}$ as the wave number in medium 1, ${{\omega }}_{1}$ is the angular frequency, ${v}_{1}=c/\sqrt{{\varepsilon }_{1}}$ is the phase velocity, c is the speed of light in vacuum, and ${\varepsilon }_{1}\ge \,1$ is the relative permittivity. Figure 1A shows the conventional spatial interface between two semi-infinite media defined such that $\varepsilon \left(x\right)={\varepsilon }_{1}\left[x < {x}_{1}\right]+{\varepsilon }_{2}\left[x > {x}_{1}\right]$ where ε₂ ≥ ε₁, defined using Iverson notation. It is well known that upon encountering the spatial interface at $x={x}_{1}$, a reflected and transmitted wave is produced. The transmitted wave number is “modified” to ${k}_{2}={{\omega }}_{1}/{v}_{2}$, with ${v}_{2}=c/\sqrt{{\varepsilon }_{2}}$, but the frequencies remain the same (i.e. ${{\omega }}_{2}={{\omega }}_{1}$).

Fig. 1: Schematic representation of analogy between the spatial and temporal boundaries.

A spatial boundary between two semi-infinite media with two different positive relative permittivities showing the spatial transmitted, T_s, and reflected, R_s, waves. B spatial boundary between two semi-infinite media with two different oppositely-signed relative permittivities (one positive ε₁ ≥ 1 and one negative real value ε₂ < 0). C temporal boundary in a spatially unbounded medium, in which the relative permittivity is temporally changed from a positive value to another positive value. D similar to (C) but with relative permittivity is being changed from a positive value to a negative real value (possible only with non-Foster concept) (one positive ε₁ ≥ 1 and one negative ε₂ < 0). Note that the sketches of FW and BW waves in (C and D) represent the field at a single time in the whole space. The background colors in (A, B) represent the spatial regions x < x₁ (yellow) and x > x₁ (green). Similarly, the shaded regions in C, D represent the permittivity at times t < t₁ (blue) and t > t₁ (orange).

Let us now consider the temporal analog^6,16,17,18 of such a spatial boundary wherein the medium is spatially unbounded and defined with a time-dependent permittivity $\varepsilon \left({{\rm{t}}}\right)={\varepsilon }_{1}\left[t < {t}_{1}\right]+{\varepsilon }_{2}\left[t > {t}_{1}\right]$. This scenario is schematically shown in Fig. 1C. When the permittivity of the medium rapidly changes at $t={t}_{1}$ with a rise time much smaller than the period of the signal T, a temporal boundary is introduced^{13,16,17,18,19,20,32,33}. Analogous to the spatial boundary, this produces a set of two waves: one traveling forward (FW) (i.e., time-refraction) and one backwards (BW) (i.e., time-reflection). However, in the case of the temporal boundary, the wave number remains unchanged (${k}_{1}={k}_{2}=k$) but the frequency of both FW and BW waves is changed from ω₁ to ω₂ (where ω₂ = (v₂/v₁)ω₁)^16,17_. These results are well known³⁴.

Returning to a spatial boundary, consider now the case in which the permittivity of medium 2 is characterized by a negative real value, i.e. ε₂ < 0 (Fig. 1B). In this case, it is well known that the wave number of the transmitted wave in medium 2 becomes imaginary, generating a spatially evanescently decaying field such that there is no propagation. The frequencies of the transmitted, reflected and incident waves remain the same. However, what happens when the permittivity of the medium on the other side of the temporal boundary, ε₂, is negative? See Fig. 1D for a schematic representation of this case. It is expected that, similar to the positive–to–positive temporal interface, a set of FW and BW waves will be generated wherein the wave numbers will be preserved. However, what will happen to the new frequency of these two new waves and how will their propagation be? In this scenario, the frequency is also changed but now from a real ω₁ to an imaginary value (given that ${{\omega }}_{2}=\left({v}_{2}/{v}_{1}\right){{\omega }}_{1}=\left(\sqrt{{\varepsilon }_{1}}/\sqrt{{\varepsilon }_{2}}\right){{\omega }}_{1}$) leading to an exponential variation in time for the FW and BW waves, with one growing and one decaying in time. In this realm, the total electric (E_{2_t1} = E_FW1 + E _BW1) and magnetic (H_{2_t1} = H_FW1 - H _BW1) fields after inducing the positive–to–negative temporal boundary can be calculated considering the conservation of vectors D and B at the temporal interface as the known boundary conditions¹⁶ (see Supplementary Materials section 1 and 2 for the detailed formulation):

$$\,\begin{array}{c}{E}_{2{{\rm{\_}}}t1}=\hat{y}\left[-\frac{1}{2}\frac{{{{\varepsilon }}}_{1}}{\left|{{{\varepsilon }}}_{2}\right|}\cos \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)-\frac{1}{2}\frac{\sqrt{{{{\varepsilon }}}_{1}}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}\sin \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)\right]{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left(t-{t}_{1}\right)\right]}\\+\hat{y}\left[-\frac{1}{2}\frac{{{{\varepsilon }}}_{1}}{\left|{{{\varepsilon }}}_{2}\right|}\cos \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)+\frac{1}{2}\frac{\sqrt{{{{\varepsilon }}}_{1}}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}\sin \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)\right]{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left(t-{t}_{1}\right)\right]}\end{array}$$

(1a)

$$\begin{array}{c}{H}_{2{{\rm{\_}}}t1}=\hat{z}\left[\frac{1}{2}\frac{\sqrt{{{{{\varepsilon }}}_{1}{{\varepsilon }}}_{0}}}{\sqrt{{{{\rm{\mu }}}}_{1}{{{\rm{\mu }}}}_{0}}}\cos \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)-\frac{1}{2}\frac{\sqrt{{{{\varepsilon }}}_{0}}}{\sqrt{{{{\rm{\mu }}}}_{1}{{{\rm{\mu }}}}_{0}}}\frac{{{{\varepsilon }}}_{1}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}\sin \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)\right]{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left(t-{t}_{1}\right)\right]}\\+\hat{z}\left[\frac{1}{2}\frac{\sqrt{{{{{\varepsilon }}}_{1}{{\varepsilon }}}_{0}}}{\sqrt{{{{\rm{\mu }}}}_{1}{{{\rm{\mu }}}}_{0}}}\cos \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)+\frac{1}{2}\frac{\sqrt{{{{\varepsilon }}}_{0}}}{\sqrt{{{{\rm{\mu }}}}_{1}{{{\rm{\mu }}}}_{0}}}\frac{{{{\varepsilon }}}_{1}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}\sin \left({k}_{1}x-{{{\omega }}}_{1}{t}_{1}\right)\right]{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left(t-{t}_{1}\right)\right]}\end{array}$$

(1b)

where µ₀ and ε₀ are the permeability and permittivity of free-space, respectively. The first and second terms of each equation correspond to the FW and BW waves, respectively, although strictly speaking, they are no longer waves, as they are not propagating. Nevertheless, we continue to use the terminology FW and BW for simplicity. The choice of the sign in front of the $i\sqrt{|{\varepsilon }_{2}|}$ is justified as we have independently derived the relevant expressions entirely in the time domain using the Laplace transform, as shown in the Supplementary Materials section 8. One immediately notices, from Eq. 1, that the electromagnetic wave is no longer sinusoidally periodic in time (while it is still periodic in space). Rather, the two fields (FW and BW waves) exponentially grow/decay in time and are frozen in space. In other words, the electromagnetic wave changes its nature from being monochromatic to one that varies purely exponentially with time with the total electric field E_{2_t1} not actually moving in the unbounded medium for t > t₁. However, as we show later, this field distribution does have a non-zero net time-average Poynting vector, which guarantees the conservation of momentum. This performance is schematically shown in Fig. 1D.

Since the electromagnetic field described by the total electric field in Eq. 1 for t > t₁ is no longer monochromatic (and instead it is changing with time purely exponentially) while we desire to keep the value of permittivity negative in this time period, we conclude that this medium is required to be non-dispersive with a constant permittivity forced to be kept negative in this time period. This is unphysical for passive media according to the Foster Reactance theorem³⁵. (In the Supplementary Materials section 9, we show an example concerning the impossibility of such negative permittivity for a passive medium with the Drude dispersion. Other forms of dispersion for passive media may exhibit similar feature.) So far, we have not made any assumption regarding the passive or active nature of the time-dependent metamaterial. However, at this point, it is important to ask: is it indeed possible to have such exponential growth of E_{2_t1} in a real passive physical system? Interestingly a similar exponential growth has been analytically investigated in a medium with non-dispersive negative permittivity in the time-harmonic scenario³⁶. So, what is special about non-dispersive negative permittivity materials? As is known, the basic energy dispersion constraint together with causality requirements forces a passive material to have dispersive parameters satisfying (for the time convention ${e}^{i\omega t}$) $\left\{\partial \left[{\omega \; \varepsilon }\left({\omega }\right)\right]/\partial {\omega }\right\} > {\varepsilon }_{0}$ and $\left\{\partial \left[{\omega }{\mu }\left({\omega }\right)\right]/\partial {\omega }\right\} > {\mu }_{0}$³⁷. What these two expressions tell us is that the overall reactive electromagnetic energy stored in any passive lossless/low-loss material is always greater than the energy stored in vacuum. So, for passive media, negative permittivity must be dispersive and cannot be constant over a range of frequencies. Hence, it is straightforward to conclude that, to achieve a non-dispersive negative ε, an active medium with an external source of energy is needed. Interestingly, similar expressions apply to any reactive element in the field of circuit theory. In this realm, the Foster reactance theorem requires that $\left\{\partial \left[{{\rm{X}}}\left({\omega }\right)\right]/\partial {\omega }\right\} > 0$ and $\left\{\partial \left[{{\rm{Y}}}\left({\omega }\right)\right]/\partial {\omega }\right\} > 0$ (with X and Y as the reactance and susceptance, respectively)³⁵. This connection between the Foster reactance theorem and the constraints for energy dispersion in passive materials has been exploited to design non-dispersive metamaterials^38,39,40 such as broadband epsilon-near-zero (ENZ) response⁴¹ by using non-Foster (i.e., active) elements^42,43. Based on this information, our proposed frozen electromagnetic waves mechanism mentioned above may become possible using time-dependent platforms with non-Foster structures with an external source of energy.

Results for epsilon-positive-to-negative temporal boundary: step function of ε(t)

Let us now evaluate the response of the proposed electromagnetic wave holding phenomenon described by Eq. 1 with some quantitative values. Let us assume that the relative permittivity of the entire medium where the plane wave is traveling is rapidly changed from ε₁ = 1 to ε₂ = –20 at t₁ = 37.1 T, where T is the period of the original monochromatic wave, and is kept at this value for t > t₁ (The issue of transitioning through ε₂ = 0, the effect of losses and their influence in a potential experiment are discussed in detail in Section 10 and 11 of the Supplementary Materials). (A study considering a wideband Gaussian pulse is shown in the supplementary materials section 12 for completeness. In such case, it is shown that the amplification rate, linearly dispersive in k, leads to a distortionless amplified replica of the input pulse with blue-shifted central frequency. Otherwise, in the general scenario of arbitrary spectrum of the initial wave, the spectral shift is replaced by a spectral distortion wherein higher frequencies become preponderant, as dictated by the increasing amplification rate with increasing k). Now the time-dependent ε(t) for the first study is shown in Fig. 2. With this configuration, the analytical results of the total electric and magnetic fields as a function of space and time are shown in Fig. 2D, G, respectively. From these results, we can see how the electromagnetic wave propagates along the positive x-axis for t < t₁ while it is effectively frozen and exponentially growing once the temporal boundary is introduced (for t > t₁). This performance can be also corroborated by looking at the instantaneous Poynting vector (Fig. 2J) as a function of space and time. As observed, it is positive for t < t₁ meaning that the wave is propagating, whereas it is spatially frozen and exponentially growing in time for t > t₁. (As we show later, when t > t₁ the Poynting vector is not strictly zero but instead has a non-zero net value.)

Fig. 2: Analytical results for epsilon-positive -to- negative temporal boundary: step function of ε(t).

A, D, G, J Time-dependent permittivity, electric field, magnetic field, and instantaneous Poynting vector distributions as a function of space (x) and time (t) considering that relative ε of the entire medium is changed from a positive ε₁ = 1 to a negative real value ε₂ = −20 at t₁ = 37.1 T and is kept at this value (based on non-Foster concept) for t > t_1. (Note different ranges of scale bars are used in the two different temporal ranges to show details of the distributions.) B Space average of the instantaneous Poynting vector (blue) over one-wavelength distance (${\int }_{0}^{{\lambda }_{0}}{S}_{{total}}\left(x,t\right){dx}$) along with the momentum (red) as a function of time. C Phase of the E (blue) and H fields (green) along with their difference (yellow) for times t > t₁ calculated using the theory shown in the Supplementary Materials section 2. E, F Electric field, H, I magnetic field and (k, l) instantaneous Poynting vector distributions as a function of space (x) extracted from D, G, J at a time t = 37 T (second column) and t = 39 T (third column) corresponding to times before and after the change of permittivity from a positive to a negative value, respectively.

To further investigate these theoretical results, the analytically-evaluated values of the electric and magnetic field distributions along with the instantaneous Poynting vector at t = 37 T (before t₁) and t = 39 T (after t₁) are shown in the second (Fig. 2E, H, K) and third columns (Fig. 2F, I, L) of Fig. 2, respectively. These two times correspond to an instant before and after the temporal boundary is introduced at t₁ = 37.1 T, respectively. As observed, for t = 37 T (Fig. 2E, H, K) the electric and magnetic field distributions correspond to those of a single propagating wave along the positive x-axis where the electric and magnetic fields are in-phase meaning that the instantaneous Poynting vector is positive, as described before. However, at t = 39 T (Fig. 2F, I, L), since the ε has been changed to a negative value and has been kept at this value (using the notion of non-Foster), the electric and magnetic fields are not spatially in synch, meaning that where the E is maximum, the H is almost zero, and vice versa. In fact, as shown in our detailed calculation in the Supplementary Materials section 4, as we change the permittivity from a positive to a negative value at t = t₁ the phase difference between the E and H is 180 degrees at t₁₊ (see Fig. 2C), but as time goes on this phase difference approaches 90 degrees (but it never gets to 90 degrees exactly), while the magnitude of E and H grows exponentially. (Although for t > t₁ we no longer have a monochromatic signal but instead we have exponentially growing and decaying fields), nevertheless we still use the notion of phase difference between the E and H to show the spatial separation between their maxima and their zero values, i.e., by approaching 90 degree phase difference here we mean at locations where E is maximum, H is approaching toward its zero (but it will never get there exactly) and vice versa). The combination of these two asymptotic behaviors provides a non-zero net Poynting vector for t > t₁, which as we will show later, guarantees the conservation of the momentum. For the sake of completeness, the analytical results of the electric and magnetic field distributions for the FW and BW at different times before and after t = t₁ are shown in the Supplementary Materials section 5 Fig. S2. From those results it is shown how an exponential growth and decay is experienced by the FW and BW waves, respectively, while they are spatially almost frozen. It is worth mentioning that in the work of Lyubarov et al. ²⁵, they achieve amplification of light by periodically modulating the permittivity of the medium in time while the light still propagates, whereas here in our approach we change the permittivity only once in a single step in time, and the wave is also essentially frozen in space.

Results for epsilon-positive-to-negative-to-positive temporal slab: square function of ε(t)

In the previous section, we discussed the case where ε was rapidly changed from a positive to a negative value and is kept at this value using the concept of the non-Foster phenomenon. However, what would happen if ε is then changed back to a positive value after being negative? This section addresses this question by considering a time dependent ε(t) defined by two single steps (similar to a square function in time, i.e., two consecutive temporal boundaries) i.e., a temporal slab³². We consider again a monochromatic plane wave traveling in an unbounded medium with the time-dependent ε as shown in Fig. 3 (the general analytical solution for this scenario can be found in the Supplementary Materials section 1 and 3).

Fig. 3: Analytical results for epsilon-positive-to-negative-to-positive temporal slab: square function of ε(t).

A Electric field distribution as a function of space (x) and time (t) when relative ε is changed from ε₁ = 1 to a positive real value ε₂ = 5 at t₁ = 37 T and it is then changed to ε₃ = 10 at t₂ = 40 T. B Electric field distribution as a function of time (t) at two spatial locations x = 2λ₀ (black) and x = 2.25λ₀ (purple) extracted from panel (A). C Space average of the instantaneous Poynting vector over one-wavelength distance (${\int }_{0}^{{\lambda }_{0}}{S}_{{total}}\left(x,t\right){dx}$, blue) along with the momentum (red) as a function of time calculated from panel (A). D Electric field distribution as a function of space (x) and time (t) when relative ε is changed from ε₁ = 1 to a real negative value ε₂ = −15 (based on the notion of non-Foster) at t₁ = 37 T and then to ε₃ = 10 at t₂ = 40 T. (Note in panel D three different ranges of scale bars are used in the three different temporal ranges to show details of the distributions.) E Electric field distribution at x = 2λ₀ (black) and x = 2.25λ₀ (purple) extracted from panel (D). F same as (C) but for the case discussed in (D).

We first examine the case where ε₁ = 1 and ε₃ > ε₂ > ε₁ meaning that all ε are positive (see Fig. 3A). Here, ε is changed from ε₁ = 1 to ε₂ = 5 at t₁ = 37 T and then to ε₃ = 10 at t₂ = 40 T. With this configuration, the ε of the whole medium is kept as ε₂ for a duration of t₂–t₁ = 3 T. The plots of the analytical results of the electric field distribution as a function of space and time for this setup are shown in Fig. 3A. As observed, a set of FW₁ and BW₁ waves are produced at the first temporal boundary (t = t₁) traveling with a new frequency ${{\omega }}_{2}=\left(\sqrt{{\varepsilon }_{1}}/\sqrt{{\varepsilon }_{2}}\right){{\omega }}_{1}$, as mentioned before. Then, these two waves are again split when the second temporal change of ε is introduced at t₂. These four waves (two traveling FW and two traveling BW) propagate now with a single frequency ${{\omega }}_{3}=\left(\sqrt{{\varepsilon }_{2}}/\sqrt{{\varepsilon }_{3}}\right){{\omega }}_{2}=\left(\sqrt{{\varepsilon }_{1}}/\sqrt{{\varepsilon }_{3}}\right){{\omega }}_{1}$, as expected. The results of the electric field distribution at different times t < t₁, t₁ < t < t₂, and t > t₂ are shown in Supplementary Materials section 6 (Figs. S3B–D), respectively. From these results, it is clear how the frequency of the wave is modified at each temporal boundary but the wave number k, and hence the wavelength, is kept the same throughout the process. For the sake of completeness, and to better appreciate the change of frequency after inducing each temporal boundary at t₁ and t₂, the electric field distribution was calculated at two different locations (x/λ₀ = 2 and x/λ₀ = 2.25) and the results are shown in Fig. 3B where it is seen how the period of the wave is different within the time regions t < t₁, t₁ < t < t₂ and t > t₂.

Let us now evaluate the case where ε is brought down to a negative value as shown in Fig. 3D. For this case, ε is rapidly changed from ε₁ = 1 to ε₂ = –15 at t₁ = 37 T and it is then changed to a positive value ε₃ = 10 (ε₃ ≠ ε₁) at t₂ = 40 T. Here, the ε of the entire medium is kept negative (using the of non-Foster scenario) for a duration t₂ – t₁ = 3 T. The results of the electric field distribution as a function of space and time for this time-dependent ε are shown in Fig. 3D. We see that the electromagnetic wave propagates along the positive x for t < t₁. Then, the wave is essentially frozen for the time interval t₁ < t < t₂ and its amplitude exponentially grows (with the FW and BW waves exponentially growing and decaying, as shown in the Supplementary Materials Fig. S3 and Fig. S4) (while it still have a non-zero net Poynting vector, see Fig. 3F), in agreement with the results discussed in the previous section for a step function of ε.

Now, at t > t₂, the permittivity of the medium is changed back to a positive value ${\varepsilon }_{3}$. Remarkably, this causes the wave inside the medium to thaw, allowing it to propagate again. However, we note that for t > t₂, in addition to the frequency change from ${{\omega }}_{1}$ to ${{\omega }}_{3}=\left(\sqrt{{\varepsilon }_{1}}/\sqrt{{\varepsilon }_{3}}\right){{\omega }}_{1}$, two interesting things happen: i) the wave behaves almost like a “standing wave” since the FW and BW waves have approximately (but not exactly) the same amplitude and ii) the amplitude of the electromagnetic wave has been increased up to ~30 times compared to its value for t < t₁. It is important to highlight the fact that for t > t₂, we do not have a perfect standing wave, because strictly speaking the amplitudes of the FW and the BW waves, albeit very high, are not exactly the same. In fact, we show analytically that there is a net non-zero Poynting vector value in the direction of +x, which guarantees the conservation of momentum. To better observe these results, the electric field distribution at times t < t₁, t₁ < t < t₂, and t > t₂ are also shown in Supplementary Materials section 6 (Figs. S3F–H, respectively). For the sake of completeness, the evaluated electric field at the two single locations x/λ₀ = 2 and x/λ₀ = 2.25 are also shown in Fig. 3E. Additionally, the instantaneous Poynting vector for the total, FW and BW waves as a function of space and time are shown in the Supplementary Materials Fig. S5. By comparing the results for the negative square function (Fig. 3D–F) with those of the positive square function (Fig. 3A–C) we can see how the phenomena of holding and amplifying the wave and the frequency conversion can be achieved with the proposed temporal slab involving non-dispersive negative permittivity (based on the non-Foster notion). See Supplementary Movie 1, 2 for the cases shown in Fig. 3A–C and Fig. 3D–F, respectively.

Effect of time interval t₂ – t₁ on frozen waves

Since the frozen waves in the time interval t₂ – t₁ experience exponential growth and decay, one may ask: is there any connection between this time interval and the amplitude of the FW and BW waves in t > t₂? To answer these questions, we first consider the positive square function of ε as shown in Fig. 4A. For the sake of simplicity, here ε is changed from a positive ε₁ ≥ 1 to another positive value ε₂ > ε₁ at t = t₁, it is kept to this value for a time t₂ – t₁ and then is returned to ε₃ = ε₁ ≥ 1 at t = t₂. As discussed in the previous section and as schematically shown in Fig. 4A, A set of waves (FW_1, BW₁) are produced at the first temporal boundary (t = t₁) with coefficients T₁ and R₁, respectively, which can be easily calculated by applying the temporal continuity of D and B at t = t₁, i.e., D_t1-δ = D_t1+δ and B_t1-δ = B_t1+δ in the limit when δ → 0 (see Supplementary Materials Section 1 for full details). Then these two waves are split into a second set (FW_2, BW₂) at t = t₂ with coefficients T₂ and R₂ calculated following the same process. According to ref. ¹⁸, these coefficients T₁, R₁, T₂ and R₂ are then defined as $\left({T}_{1},{R}_{1}\right)=0.5\left[\left({\varepsilon }_{1}/{\varepsilon }_{2}\right)\pm \left(\sqrt{{\varepsilon }_{1}}/\sqrt{{\varepsilon }_{2}}\right)\right]$ and $\left({T}_{2},{R}_{2}\right)=0.5\left[\left({\varepsilon }_{2}/{\varepsilon }_{3}\right)\pm \left(\sqrt{{\varepsilon }_{2}}/\sqrt{{\varepsilon }_{3}}\right)\right]$. The propagating nature of the waves during this whole process has been recently exploited to achieve time reversal for water waves³² and also antireflection temporal coatings for electromagnetic waves⁴⁴ by properly engineering the time interval t₂–t_1. Now, for the negative square function (Fig. 4B) we have seen before in Fig. 3D, E how spatially frozen FW₁ and BW₁ waves are produced at t = t₁ when ε is changed to a negative value. These frozen waves grow/decay exponentially until they are split into a FW₂ and a BW₂ at t = t₂ when the permittivity is changed to a positive value again. Considering the scenarios from Fig. 4A, B, the general solution for the total electric field (E_{3_t2}) for a square function of permittivity can be defined as follows (see Supplementary Materials Section 3 for the full formulation along with the complete solution for the magnetic field):

$${E}_{3{\_}t2}= \left\langle \frac{1}{4}\left(\frac{{{{\varepsilon }}}_{1}}{{{{\varepsilon }}}_{3}}+\frac{\sqrt{{{{\varepsilon }}}_{1}}}{\sqrt{{{{\varepsilon }}}_{3}}}\right)\cos \left[{k}_{1}x-\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{{\varepsilon }_{3}}}{{{\omega }}}_{1}\left({t-t}_{2}\right)-{{{\omega }}}_{1}{t}_{1}\right] \right. \\ \left. \left\{{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}+{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}\right\}\right.\\ \left.+\frac{1}{4}\left(-\frac{{{{\varepsilon }}}_{1}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}\sqrt{{{{\varepsilon }}}_{3}}}+\frac{\sqrt{{{{\varepsilon }}}_{1}}\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}{{{{\varepsilon }}}_{3}}\right){\mathrm{sin}}\left[{k}_{1}x-\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{{\varepsilon }_{3}}}{{{\omega }}}_{1}\left({t-t}_{2}\right)-{{{\omega }}}_{1}{t}_{1}\right] \right. \\ \left. \left\{{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}-{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}\right\}\right\rangle \\ \left\langle \frac{1}{4}\left(\frac{{{{\varepsilon }}}_{1}}{{{{\varepsilon }}}_{3}}-\frac{\sqrt{{{{\varepsilon }}}_{1}}}{\sqrt{{{{\varepsilon }}}_{3}}}\right)\cos \left[{k}_{1}x+\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{{\varepsilon }_{3}}}{{{\omega }}}_{1}\left({t-t}_{2}\right)-{{{\omega }}}_{1}{t}_{1}\right] \right. \\ \left. \left\{{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}+{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}\right\}\right.\\ \left.+\frac{1}{4}\left(\frac{{{{\varepsilon }}}_{1}}{\sqrt{\left|{{{\varepsilon }}}_{2}\right|}\sqrt{{{{\varepsilon }}}_{3}}}+\frac{\sqrt{{{{\varepsilon }}}_{1}}\sqrt{\left|{{{\varepsilon }}}_{2}\right|}}{{{{\varepsilon }}}_{3}}\right){\mathrm{sin}}\left[{k}_{1}x+\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{{\varepsilon }_{3}}}{{{\omega }}}_{1}\left({t-t}_{2}\right)-{{{\omega }}}_{1}{t}_{1}\right] \right. \\ \left. \left\{{e}^{\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}-{e}^{-\left[\frac{\sqrt{{\varepsilon }_{1}}}{\sqrt{\left|{\varepsilon }_{2}\right|}}{{{\omega }}}_{1}\left({t}_{2}-{t}_{1}\right)\right]}\right\}\right\rangle$$

(2)

where the first and second terms represent the final FW and BW waves (each of them formed by two sets of waves) for t > t₂. From Eq. 2, we can see how the exponential terms inside braces define the exponential decay/grow of the frozen waves within the time interval t₂ – t₁. Interestingly, it can be seen how the terms $-\left[\left(\sqrt{{\varepsilon }_{1}}/\sqrt{\left|{\varepsilon }_{2}\right|}\right){{\omega }}_{1}\left({t}_{2}-{t}_{1}\right)\right]$ and $\left[\left(\sqrt{{\varepsilon }_{1}}/\sqrt{\left|{\varepsilon }_{2}\right|}\right){{\omega }}_{1}\left({t}_{2}-{t}_{1}\right)\right]$ will make the exponentials decay and grow in time, respectively, with the decaying term getting smaller for large enough values of t₂ – t₁, as discussed in Eq. 1 and Figs. 1–3. Hence, the new set of waves created at t = t₂ from this decaying term will be very small (but not equal to zero). The other set of waves created at t = t₂ from the growing term will be large. The combination of these sets of waves leads to the FW and BW waves for t > t₂ whose amplitudes are approximately (but not exactly) the same, as can be seen from the results in Fig. 3 and Fig. S3 of the Supplementary Materials.

Fig. 4: Effect of the time interval t₂-t₁.

A, B schematic representation of a square function of permittivity considering (ε₁ = ε₃ = 1, ε₂ > 1) and (ε₁ = ε₃ = 1, ε₂ < 0), respectively. C, D amplitude of the FW₁ and BW₁ waves created at the first temporal boundary as a function of the time interval t₂ - t₁ where ε = ε₂ < 0 calculated just before the second temporal boundary (t₂^-), respectively. Two values of ε₂ are considered here: ε₂ = –4 (dark blue and dark red) and ε₂ = –8 (light blue and light red). E amplitude of the final FW and BW waves as a function of the time interval t₂ - t₁ where ε = ε₂ < 0 calculated after the second temporal boundary is introduced (t > t₂). F, G, numerical results of the electric field of FW (blue) and BW (red) waves (calculated at x/λ₀ = 7.5 and x/λ₀ = −7.5, respectively) for times t > t₂ considering that ε₁ = 1, ε₂ = –8 and ε₃ = 1 using the time intervals t₂ - t₁ = 0.2 T and t₂ - t₁ = 0.4 T where ε = ε₂ < 0, respectively. The shaded regions in A–E represent times t < t₁ (light yellow), t₁ < t < t₂ (light orange) and t > t₂ (light green).

To quantitatively study this performance, let us consider a square function of ε where it is changed from ε₁ = 1 to a negative real value (ε₂) at t = t₁ and then returned to ε₃ = ε₁ = 1 at t = t₂. The results of the amplitude of the frozen FW₁ and BW₁ waves created after the first temporal boundary at a time just before t₂ (t = t₂^-) are shown in Fig. 4C, D, respectively, considering different intervals where ε is kept negative (t₂ – t₁). In these plots, two values of ε₂ are considered (ε₂ = –4 and ε₂ = –8) for comparison. The first aspect we observe from these results is that the exponential growth/decay of the frozen FW₁ and BW₁ is faster for ε₂ = –4 as compared with the case of ε₂ = –8, as expected from Eq. 1. Moreover, we note that when ε₂ = –4, the amplitude of the BW₁ decays to ~0.21 when t₂ – t₁ = 0.2 T while it is strongly reduced to a value of ~0.01 when increasing t₂ – t₁ to 0.8 T.

As explained before, this performance has a direct consequence on the amplitude of the total FW and BW waves created after the second temporal boundary at t = t₂ (see Eq. 2). Following the examples shown in Fig. 4C, D, the amplitude of the total FW and BW waves for t > t₂ as a function of the time interval t₂ – t₁ are shown in Fig. 4E considering the same two values of negative ε₂ (ε₂ = –4 and ε₂ = –8). From these results, it is clear how the FW and BW have almost (but not exactly) the same amplitude when using time intervals t₂ – t₁ larger than 0.7 T whereas this would not be the case when reducing this interval. For the sake of completeness, the simulation results using COMSOL Multiphysics® of the electric field distribution of the FW and BW waves are calculated and shown in Fig. 4F, G, considering a square function of ε with ε₁ = ε₃ = 1 and ε₂ = –8 using t₂ – t₁ = 0.2 T and t₂ – t₁ = 0.4 T, respectively. As observed, the amplitudes of the waves are different/similar (but not identical) when using t₂ – t₁ = 0.2 T and t₂ – t₁ = 0.4 T, respectively, in agreement with the discussion from Fig. 4E. For completeness, additional numerical simulations and analytical calculations showing different cases of square functions of permittivity where wave amplification and frequency conversion is achieved with our approach are shown in the Supplementary Materials Fig. S6. Moreover, the analytical results of the instantaneous Poynting vector for cases where ε₁ = ε₃ = 1 and ε₂ = –4 considering different values of t₂ – t₁ (namely 0.2 T, 0.4 T, and 0.6 T) are presented in the Supplementary Materials section 7 Fig. S7, showing how for t > t₂ and when t₂ – t₁ = 0.6 T the FW and BW wave have similar (but not equal) amplitudes, producing an almost (but not exact) perfect standing wave, as explained above, while it is not the case when a reduced t₂ – t₁ = 0.2 T is implemented.

Electromagnetic momentum consideration

In our temporal, spatially unbounded, homogeneous non-Foster metastructures, the electromagnetic momentum must be conserved. This is indeed the case here. Considering ${{\bf{D}}}{{{\times }}}{{\bf{B}}}$ as the electromagnetic momentum per unit volume⁴⁵, from our analysis (in the Supplementary Materials) it can be seen from Figs. 2 and 3 that this term stays unchanged throughout all three time regions t < t₁, t₁ < t < t₂, and t > t₂. In temporal regions t < t₁ and t > t₂, this momentum density, which is directly related to the Poynting vector, is clearly noticed because in these two regions we have propagating waves (i.e., in t < t₁ we have a single propagating wave, and in t > t₂, we have FW and BW waves with similar (but slightly different) amplitudes, thus we have a net non-zero Poynting vector, see Fig. 2B and Fig. 3F.) However, the temporal region t₁ < t < t₂, in which we have frozen waves with growing and decaying fields in time, needs further discussion here. In this time interval, we have these two frozen waves each of which carries zero “space-averaged” Poynting vector (by “space-averaged” we mean the average of the instantaneous Poynting vector over a one-wavelength distance). However, when both frozen waves, one growing and one decaying in time, are considered, this combination exhibits a non-zero net space-averaged Poynting vector that guarantees the conservation of the momentum. The presence of this net non-zero Poynting vector in the combined frozen waves can be interpreted as the temporal analog of the spatial case of exponentially decaying and growing evanescent waves, where each evanescent wave carries no time-average Poynting vector, whereas the presence of both indeed reveals a net time-average Poynting vector tunneling through these two evanescent waves.

Temporal metastructure and non-Foster elements

As described before, owing to the dispersion constraints for passive media, an active temporal metastructure with an external source of energy is needed to change its permittivity from a positive to a negative non-dispersive value and kept at that value for a range of time. In this section, we propose and numerically evaluate the performance of a 2D parallel-plate waveguide loaded with switchable parallel negative non-Foster elements (emulated by thin time-dependent layers where the ε is changed from a positive, larger than one, to a negative value) to show how the electromagnetic wave in this waveguide can be spatially held and amplified after introducing such active non-Foster elements. In this numerical study, a waveguide filled with air (ε₁ = 1) is used working at the frequency of f = 1.5 GHz (λ₁ = 200 mm). The two metallic perfect electric conducting (PEC) plates are separated by a distance h = 0.5λ₁, the total length along the x-axis is 6λ₁ and it is considered that the width (along the z-axis, out-of-plane) is W = 1 m to approximate it as a 2D waveguide. To emulate the negative permittivity, an array of 75 negative non-Foster capacitors are inserted within the waveguide and separated from each other by an air layer (ε₁ = 1) of thickness 0.08λ₁. These negative capacitors are emulated using thin dielectrics (thickness 0.01λ₁) having a time-dependent ε that is changed from a positive (ε₁ = 1) to a negative value ε₂ = -4. With this setup, the emulated negative non-Foster elements can be switched on, changing the permittivity of the medium from a positive to a negative value⁴¹. The negative capacitors are switched on at t₁ = 9 T and then kept on for t > t₁. The signal from the source is switched off at the same time as the first temporal boundary (t = t₁) to ensure the evaluation of the metastructure using the signal already present in the waveguide.

With this configuration, the numerical results of the electric field distribution at different time regions are shown in Fig. 5C–E (for completeness zoom-in figures of the electric field distributions are shown as insets on the right-hand side of Fig. 5C–E). As observed, the wave is traveling along the positive x direction for t < t₁ (Fig. 5C). Once the temporal boundary is induced at t₁ = 9 T, the wave inside the region with non-Foster capacitors is essentially frozen in space, and subsequently its amplitude grows in time (see Fig. 5D–E). Note how the electric field inside the waveguide changes its sign when inducing the temporal boundary, in agreement with the discussion above. The energy for such growth is supplied by the external source of energy required for non-Foster elements. These results demonstrate how electromagnetic waves can be held and amplified due to the change in the capacitor values. For completeness, section 11 of the supplementary materials performs a detailed analytical and numerical study of a potential experiment setup, based on a transmission line loaded with lumped nonideal op-amp unit cells operating as negative impedance converters. Additionally, section 13 shows how our amplifying instability can be further tailored, making the practical realization of our concept robust against capacitance imperfections, due to fabrication tolerance.

Fig. 5: Proposed idea for a time-dependent parallel-plate waveguides using non-Foster elements.

A schematic representation of the two-dimensional (2D) parallel-plate waveguide having a height of h = 0.5λ₁ and a total length along the x axis of 6λ₁. A total of 75 non-Foster negative capacitors (C_N) are placed within the waveguide separated by a distance Δx = 0.08λ₁. The capacitors are emulated using thin dielectrics of thickness W_CN = 0.01λ₁ having a time dependent ε that is changed from ε₁ = 1 to ε₂ = −4 at t = t₁ = 9 T. The equivalent circuit of the parallel-plate waveguide is shown in the bottom of panel (A). B schematic representation of the change of ε as a function of time. C, D, E numerical results of the electric field distribution at times before and at the temporal boundary (t < t₁^- and t = t₁^-), just after the temporal boundary when the capacitors are just activated (t = t₁⁺), and longer times after the temporal boundary (t > t₁⁺), respectively. The insets on the right-hand side of panels C–E represent zoom-in figures of the electric field distribution at different times.

Dipole radiation in temporal non-Foster metastructures

As a final study, here we demonstrate the performance of a two-dimensional (2D) line dipole immersed within the proposed time-varying non-Foster platform. The schematic representation is shown in Fig. 6A. (note that here a 2D line dipole is a source that is infinitely extended along the z axis (i.e., it is z independent) while its dipole vector is in the x-y plane). For times t < t₁ the electric 2D dipole is radiating within the unbounded medium of ε₁ = 1. At t = t₁, the dipole is switched off and the ε is rapidly changed to ε₂ < 0, meaning that the electromagnetic signal already present in the medium will be held in space and its magnitude will increase exponentially. The numerical results (using COMSOL Multiphysics ®) of this scenario considering that ε₁ = 1 for t < t₁ and it is changed to ε₂ = -4 at t = t₁ = 9.4 T are shown in Fig. 6B (where the E field, H field, and power distributions as a function of time are shown, all calculated at x/λ₀ = y/λ₀ = 0.75. To enable the convergence of the results, the source was switched off first, and then 0.7 T later ε was rapidly changed (see method section for further details). As observed, for times t < t₁ the wave emitted by the 2D dipole propagates within the unbounded medium while it is frozen and exponentially increases for t > t₁, in agreement with the discussion presented in the previous sections.

Fig. 6: Immersing a two-dimensional (2D) dipole within the temporal non-Foster platform.

A Schematic representation showing the dipole radiating for times t < t₁ where ε₁ ≥ 1 and then the signal is spatially frozen and exponentially increasing its magnitude for times t > t₁ where ε₂ < 0. B Numerical results of the in-plane components of the E field (E_x, dark blue, and E_y, blue), out-of-plane H field (green) and power distribution (purple) calculated at x/λ₀ = y/λ₀ = 0.75. The time-dependent ε function is shown on the top panel from B. The background colors in (A) represent the permittivity of the medium at times t < t₁ (blue) and t > t₁ (orange).

Discussion

In conclusion, we have explored scenarios where temporal boundaries are induced by using temporal metastructures with a rapid change of permittivity from a positive to a negative value, requiring non-Foster structures with external energy. The underlying physics of such temporal structures have been studied here showing how the electromagnetic wave present within such a medium can be forced to effectively “stop”, achieving a frozen status, and then grow in time, as demonstrated, analytically and numerically, here. Moreover, if the permittivity of the temporal metastructure is returned to a different positive value, propagation is again allowed while also achieving frequency conversion. It has been shown how an active medium with external energy is required to achieve such non-dispersive negative permittivity in a temporal structure, which has a direct relation with the well-known Foster reactance theorem in circuit theory. An example of our proposed theory for such “filling stations for waves” has been studied numerically using a parallel plate waveguide loaded with parallel thin time-dependent media emulating non-Foster negative capacitors demonstrating agreement between the analytical and numerical calculations. This concept may have exciting potential applications in slowing and stopping light combined with amplification, which may lead to new possibilities for coherent sources.

Methods

All the numerical simulations were carried out using the time-domain solver of the commercial software COMSOL Multiphysics®. For the configuration shown in Figs. 4–5 along with those results shown in the Supplementary Materials Fig. S6, PEC boundaries were used for the top and bottom boundaries while scattering boundary conditions were implemented at the right boundary to avoid undesirable reflections. The incident field was applied from the left boundary of the simulation box via a scattering boundary condition. For the results shown in Fig. 4 and Fig. S6 a refined Mapped mesh was implemented with a maximum and minimum element size of 0.14 λ₀ and 0.15 × 10⁻³ λ₀, respectively, with curvature factor of 0.2 and a resolution of narrow regions of 1. The rapid changes of ε from these figures were modeled by implementing an analytical function that combined two built-in single step functions (i.e., emulating square functions ε) having a smooth transition of 7.5 × 10⁻⁶ T and one derivative to ensure convergence of results. For the results discussed in Fig. 5, a Mapped mesh was also applied with a maximum and minimum element size of 0.16λ₀ and 0.15 × 10⁻³λ₀, and the same curvature factor and resolution of narrow region values as for the results in Fig. 4. An extra automatic mesh refinement was also implemented. The rapid change of ε was modeled using an analytical single step function having a smooth transition of 1.2 × 10⁻⁵T and one continuous derivative to ensure convergence of results. For the results shown in Fig. 6, an electric dipole was placed at x/λ₀ = y/λ₀ = 0 and embedded within a circular geometry (modeling the unbounded medium) with a radius of 3λ₀. Scattering boundary conditions around the circle were implemented to avoid undesirable reflections. The dipole was modeled using +1 and −1 dipole moment along the x- and y-axes, respectively, to model a 45 degrees rotated dipole. The electric current dipole moment magnitude was modeled using a sinusoidal signal being switched off at t/T = 8.7 using a smooth transition of ~1.49 T and one continuous derivative to aid in the convergence of the simulation. The ε of the whole medium was rapidly from ε₁ = 1 to ε₂ = −8 at t₁ = 9.4 T using an analytical step function with a smoothing of 2.9 × 10⁻⁴T and one continuous derivative.