Zeki Hayran, Francesco Monticone
IMAGE LICENSED BY INGRAM PUBLISHING
Time-varying systems open intriguing opportunities to explore novel approaches in the design of efficient electromagnetic devices. While such explorations date back to more than half a century ago, recent years have experienced a renewed and increased interest into the design of dynamic electromagnetic systems. This resurgence has been partly fueled by the desire to surpass the performance of conventional devices, and to enable systems that can challenge various well-established performance bounds, such as the Bode–Fano limit, the Chu limit, and others. Here, we give an overview of this emerging research area and provide a concise and systematic summary of the most relevant applications for which the relevant performance bounds can be overcome through time-varying elements. In addition to enabling devices with superior performance metrics, such research endeavors may open entirely new opportunities and offer insight towards future electromagnetic and optical technologies.
The study of electromagnetic wave interactions with time-varying media dates back to more than half a century ago, when several pioneering works [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] revealed the rich physics arising in such systems. Although in the following decades several other works have further explored the implications of wave propagation in dynamic media [11], [12], [13], [14], it was not until recently that the field gained considerable momentum [15], partly owing to the improved fabrication and characterization techniques in various frequency regimes [16], [17], [18]. Consequently, it was soon discovered that time-dependent systems can enable many interesting and novel phenomena, such as interband photonic transitions [19], [20], nonreciprocal systems [21], [22], efficient frequency conversion [23], [24], [25], [26], temporal aiming [27], Fresnel drag [28], negative extinction [29], entangled photon generation [30], [31], parametric oscillations [32], synthetic dimensions [33], topological phase transitions [34], and nonreciprocal gain [35], among others [36]. Several recent publications also further investigated the fundamental aspects of wave propagation in time-varying media, such as [37], [38], [39], [40], [41], [42], thus providing a better understanding of the physical implications of these systems.
Apart from enabling novel effects, time-varying materials and components may also be used to improve a given device to achieve better performance in terms of operational bandwidth, size, efficiency of the relevant process, and other metrics [36]. In some cases, such improvements may even exceed what is normally allowed by well-established performance bounds, as physical bounds and fundamental limits in electromagnetics are typically derived under the assumption of linearity and time invariance (LTI systems). Indeed, as mentioned above, time variance in the system properties may enable various functionalities not present in a conventional setting, such as parametric gain [35] or spectral changes [19], [23], [24], [25], [26], [43], which are not accounted for when deriving bounds for LTI systems, and can therefore be exploited to bypass such physical limits. Consequently, time-varying materials and components provide exceptional opportunities to realize superior devices not restricted by various conventional limits.
In this section we provide a concise overview of the most relevant applications of time-varying designs to overcome various physical bounds of conventional LTI electromagnetic systems. In this regard, this section is divided into various subsections, each of which focuses on a specific application or research topic, with the goal to systematically guide the reader through this emerging research area. We will not discuss some major research directions that also seek to overcome various constraints of conventional systems and broaden their functionality, such as the possibility to break reciprocity [21], [22] or create synthetic dimensions [33] based on time modulation, since these topics are already covered in detail in the literature. Instead, we will focus on certain topics of particular interest to the IEEE Antennas and Propagation community, including how time-varying systems impact the fundamental limitations of antennas [44], broadband impedance-matching networks [45], and radar absorbers [46].
Efficient power transfer from a source to a load with minimal reflections is arguably one of the most important tasks in electromagnetic engineering. Thus, when designing impedance-matching networks to minimize reflections, it becomes essential to identify the limitations and the optimal configurations of the system. In this regard, a matching-bandwidth limit was introduced for linear, time-invariant, passive systems by Bode in 1945 [47], which was later generalized by Fano for arbitrary load impedances [45]. The resulting Bode–Fano limit provides a theoretical upper bound on the bandwidth over which a certain reflection reduction can be attained, independent of the complexity of the employed matching network (only assumed to be LTI, passive, and reactive), and it is therefore a very powerful tool to assess the maximum possible bandwidth of various electromagnetic systems, such as antennas [48] or optical invisibility cloaks [49]. To overcome this limitation, one strategy that has been explored in the literature is to break the underlying assumption of passivity. Indeed, several proposals utilizing active “non-Foster” elements have shown that the bandwidth of small antennas [50] and cloaking devices [51], [52], [53], [54] can be extended beyond what would be possible with a passive system. However, active elements typically increase the noise level of the system [55] and can lead to instabilities (unbounded oscillations in time) [54], [56], [57], thereby disrupting the desired functionality.
An alternative option to go beyond the Bode–Fano limit is to break the assumption of time invariance, i.e., allow temporal variation in the system properties. In this context, [58] employed time-varying matching networks to achieve a better bandwidth performance compared to their time-invariant counterparts by exploiting information about the incident signal. More recently, [59] provided another interesting method, which involves switching the system parameters while a broadband signal (a pulse) is within the matching network and before it reaches the load, as shown in Figure 1(a) and (b). The temporal switching would generally lead to reflected waves due to the temporal discontinuity (temporal interface) [60]; however, it was shown that the matching efficiency (defined as the ratio between the power dissipated on the load and the maximal available power from the source plus the total power provided by the modulation) can still be maximized over a broad bandwidth by optimizing several parameters, including the characteristic impedance of the matching transmission line and the phase velocity of the pulse within the matching line before and after the switching event [59]. Since temporal switching increases the number of free parameters that can be tuned compared to the time-invariant case, one may achieve performance metrics that are otherwise out of reach in a conventional time-invariant setting, especially for large load-source impedance contrast values [see Figure 1(c)] [59]. This proposal was further extended from “hard” (abrupt) to “soft” temporal switching in [61], which could be more feasible to implement in practice. An adverse byproduct of the temporal switching operation is the spectral compression or broadening of the pulse, which necessitates the use of additional analog or digital devices to restore the original pulse spectrum, and thus may complicate the overall matching system. Moreover, this method is feasible only for short time signals (as quasi-monochromatic signals would require impractically long matching lines), and requires knowledge that the pulse is present within the time-switched matching line (which implies the presence of some fast detection mechanism able to trigger the switching event). Even with these limitations, this approach shows a realistic path toward broadband matching networks beyond the Bode–Fano constraints.
Figure 1. (a) A temporally switched transmission line can be employed to match a source impedance RG to a load impedance RL surpassing the Bode–Fano limit. (b) Such a switched matching line can be realized with parallel varactor diodes and banks of series inductors, whose properties are changed abruptly in time at t = ts. (c) The maximum transmission efficiency for various RL/RG values shows that the temporally switched case (solid blue) provides better performance compared to the nonswitched time-invariant case (dashed red line), especially for large source-load impedance contrast values. (d) and (e) A transmission line terminated with a time-varying inductance L(t). (d) Using a suitable time modulation, the inductor can accumulate energy without any reflections. (e) By time-reversing the temporal modulation in (d), the energy trapped by the inductor can be released. (f) Time-modulation profile of the inductance, comprising both the trapping (before t = 2.5 s) and releasing regimes (after t = 2.5 s). (g) Temporal variation of the incident signal V+ and the reflected signal V– at the load position (obtained theoretically and through simulations) for a load inductance temporally modulated as in (f). CNTL: control. Panels (a)–(c) are reproduced with permission from [59], American Physical Society. Panels (d)–(g) are reproduced with permission from [62], American Physical Society.
For the quasi-monochromatic regime, on the other hand, a transmission-line system where the load impedance itself is varied in time has been used to impedance-match a time-harmonic source to a purely reactive load and to accumulate and release energy on demand [see Figure 1(d)–(g)] [62]. Specifically, it was shown that, while a time-invariant reactive load would fully reflect an incident wave, a time-modulated reactance can emulate a resistance and can, therefore, act as a “virtual” absorber [62]. An intuitive explanation of this effect is that temporally modulating the reactance with a suitable tangent function is equivalent to a short-circuited transmission line with the shorted end moving away from the source with the same velocity as the phase velocity of the incident signal [62]. Therefore, the incident wave does not reflect back since it never appears to “reach” the reflecting termination. Consequently, energy from a time-harmonic source can be trapped with no reflections and released on demand [see Figure 1(f) and (g)].
Along similar lines, another interesting application of temporal modulation in the context of impedance matching is related to the problem of capturing and trapping an incident electromagnetic wave into a lossless resonant cavity, with no reflections. This can be achieved by temporally modulating the radiation leakage from the cavity, such that it perfectly cancels the direct reflection from the cavity at all time instants [63]. In this regard, we have also shown that by temporally modulating the shell of a compact core-shell scatterer (acting as an open cavity supporting a nonradiating mode), an incident broadband pulse can be captured within the cavity with no reflection, effectively forcing a broadband signal into a narrowband resonator with ideal efficiency (this may also be interpreted as spectrum compression through temporal modulation with the additional constraint of zero reflections) [64]. Further details can be found in the supplementary material, available at https://doi.org/10.1109/MAP.2023.3236275.
Antenna miniaturization is another major research area within the applied electromagnetics community with many potential benefits for various applications, such as communication systems, sensor networks, implanted medical devices, and navigation systems. While it is generally possible to reduce the size of an antenna down to subwavelength dimensions for a given design frequency, this typically comes at the price of reduced bandwidth, directivity, and radiation efficiency. In this regard, the fundamental tradeoffs between size and bandwidth for electrically small antennas were investigated more than half a century ago in several seminal papers by Wheeler [65], Chu [66], and later by Harrington [67], resulting in, as it is known today, the Chu–Harrington limit [68]. The limit as derived by Chu is based on an equivalent circuit model for the radial wave impedance of the lowest-order spherical wave generated by a generic LTI lossless antenna enclosed within a spherical volume. This equivalent circuit becomes more reactive as the size of the spherical volume is reduced, leading to a higher Q factor (consistent with the fact that, for example, a short wire dipole has a strongly reactive input impedance). Then, as recognized by Chu, while the inverse of the Q factor may be interpreted as the fractional bandwidth of the antenna at a single resonance, a more accurate definition is in terms of the tolerable reflection coefficient over the operating band using a suitable matching network, and therefore the antenna bandwidth is limited by the Bode–Fano limit (see “Impedance Matching” section, above) [69] applied to Chu’s equivalent circuit. In other words, these results define a tradeoff between antenna bandwidth and realized gain (gain taking into account the efficiency associated with reflection losses), and quantify how the product between bandwidth and realized gain is decreased by miniaturization. These analyses were later refined by other researchers (e.g., [44], [70], [71], [72]), further underscoring the fact that for a linear time-invariant passive electrically small antenna one cannot simultaneously minimize size and maximize bandwidth and efficiency.
Following Chu’s work [66], antenna engineers realized early the need for active (i.e., nonpassive) designs, nonlinear materials, or time-varying components to increase the bandwidth of wireless systems based on electrically small antennas. In this context, for example, nonlinear ferrite materials were exploited to realize time-varying inductors, which were used to overcome the limitations of small antennas in very low frequency transmitters by shifting the carrier frequency [73]. Subsequent works (see, for instance, [74], [75], [76], [77], [78]) further demonstrated the unique opportunities enabled by time-varying “switched” elements to design antennas with improved bandwidth characteristics.
More recently, with the renewed surge of interest in temporal modulation in electromagnetic systems, research into electrically small time-varying antennas has been reinvigorated. For example, [79] confirmed that the antenna bandwidth can be increased significantly through impedance modulation. Another more recent work employed “Floquet” impedance matching [80] [Figure 2(a)–(c)] to overcome the Chu–Harrington limit by imparting parametric gain into the system [32] through a periodic modulation of the matching network components at twice the frequency of the antenna resonant frequency. A realistic simulation of the proposed design was also demonstrated based on a small loop antenna [Figure 2(b)] that includes periodically modulated variable capacitors (varactors). Through numerical calculations [Figure 2(c)], it was shown that the bandwidth-efficiency product can be significantly enhanced owing to the temporal modulation compared to the time-invariant case, and can even surpass the predictions of the Bode–Fano limit applied to the Chu–Harrington equivalent circuit, which is a rather remarkable result given the fact that parametric phenomena are typically narrowband.
Figure 2. (a) A “Floquet” matching network with periodically time-varying elements can be employed to increase the efficiency-bandwidth product of an electrically small antenna by imparting parametric gain to the system. (b) An electrically small loop antenna (at the design frequency of 300 MHz) matched with periodically driven variable capacitors (varactors). (c) Transducer gain profiles (efficiency taking into account mismatch losses) obtained through numerical simulations show that the bandwidth of the time-modulated antenna (green dotted line) can exceed the maximum bandwidth predicted by the Chu–Harrington limit (red solid line). The time-invariant antenna (blue dashed line), on the other hand, is bounded by this limit. The incident frequency is normalized to the antenna design frequency. ΔC denotes the capacitance modulation amplitude, while C0 (≈3 pF) is the static capacitance. (d) Schematic for a different example of (receiving) antenna with an enhanced efficiency-bandwidth product. By deliberately inducing a mismatch between the antenna and the load, hence increasing reflection losses, the antenna bandwidth can be widened. The decrease in efficiency associated with mismatch losses can then be compensated through a parametric amplifier. (e) The inherently low-loss nature of parametric amplification can help bring the noise figure down while still maintaining a high bandwidth enhancement. Panels (a)–(c) are reproduced with permission from [80], American Physical Society. Panels (d)–(e) are reproduced with permission from [86], IEEE.
A possible drawback of schemes of this type is that the parametric gain may potentially lead to instabilities. However, unlike antenna systems based on active non-Foster elements [81], [82], [83], the dispersion of the parametric gain can be more easily controlled through the modulation properties to bring the system into the stable regime. Moreover, it was also pointed out that deliberately inducing an impedance mismatch by modifying the source impedance can act as negative feedback to help ensure stability and increase the operational bandwidth at the same time [80]. Indeed, a stable parametric enhancement effect was later experimentally demonstrated in [84] in a fabricated time-varying loop antenna operating at radio frequencies. In this context, other recent studies also investigated the use of time-varying parametric amplifiers with the aim of broadening the bandwidth of electrically small antennas [85], [86] [Figure 2(d) and (e)]. Similar to [80] and [84], this was achieved by first trading realized gain with bandwidth, in line with the Bode–Fano limit, by detuning the system from the ideal impedance-matched condition, hence increasing reflections, and subsequently using a parametric up-converter amplifier based on time-varying reactive elements to compensate for these mismatch losses. Different from [80] and [84], however, in [85] and [86] the authors placed a particular focus on assessing the noise properties of the system and studied a specific circuit configuration in which a receiving antenna is connected to an amplifier [Figure 2(d)] with real input impedance several times greater than the real part of the input impedance of the antenna, as this allows achieving an optimal tradeoff between increased bandwidth and added noise [Figure 2(e)].
Electromagnetic wave absorption plays a crucial role in various electromagnetic and optical systems, ranging from solar energy harvesting to scattering minimization for radar applications or anechoic chamber measurements. A key limitation of conventional electromagnetic absorbers is the typical tradeoff between the absorption bandwidth, i.e., the bandwidth over which a certain level of absorption can be maintained, and the thickness of the absorbing material or structure. Within this context, using the analytical properties of the reflection coefficient, Rozanov [46] showed that this tradeoff is in fact fundamental, and there exists an ultimate thickness to bandwidth ratio ${\Delta}{\lambda} / {d}$ for passive LTI absorbers backed by a perfect conductor. Specifically, for the case of a nonmagnetic absorber, the upper limit for this ratio is ${\Delta}{\lambda} / {d}\,{<}\,{16} / {\left\vert{\ln}{\rho}_{0}\right\vert}$, where ${\rho}_{0}$ is the maximum tolerable reflection within the operating bandwidth. Clearly, the absorption bandwidth can therefore be widened by increasing the thickness of the absorber or the tolerable reflection. Prominent examples of broadband absorbers include cascaded [88], [89] or adiabatically tapered [90], [91], [92] lossy structures, which typically have thicknesses on the order of several wavelengths. However, for many applications, such as radar absorption or solar energy harvesting, creating the thinnest possible absorber with the widest possible bandwidth is highly desirable, which motivates the significant interest in overcoming the “Rozanov bound.” In this direction, it was shown in [87] that temporally switching the relative permittivity of the absorbing material can be used to extend the bandwidth of thin absorbers beyond what is possible with time-invariant structures [Figure 3(a)], similar to the use of temporally switched transmission lines for broadband impedance matching, as discussed in the “Impedance Matching” section, above. An important drawback of these approaches is that, while reflections are reduced within the bandwidth of the incident wave, significant reflections may be induced outside this frequency range as the temporal switching can lead to strong frequency conversion. While this may not pose a problem for certain narrowband applications, it would become an issue in the context of, for example, broadband radar systems and stealth technology, as the target may become even more easily detectable due to the increased reflections outside the original signal bandwidth, which could then be picked up by sufficiently broadband or multiband detectors.
Figure 3. (a) An electromagnetic absorber with a time-switched permittivity can be employed to extend the absorption bandwidth beyond the time-invariant case (shown by the green dashed line). Different reflection spectra are shown, corresponding to incident Gaussian pulses with different widths. (b) Alternatively, one may temporally modulate the complex permittivity (both its real and imaginary parts simultaneously) such that the modulation spectrum becomes one-sided (or “spectrally casual”), which implies that frequency down-conversion and back-reflections due to the temporal modulation are prohibited over a broad bandwidth. (c) Such a complex modulation may potentially increase the bandwidth of electromagnetic absorbers, as in the time-switched case, but without inducing any reflection within or outside the operational bandwidth (as seen in the figure, the field is fully transmitted, frequency up-shifted, and absorbed, with no reflections, at any frequency). Panel (a) is reproduced with permission from [87], Optical Society of America. Panels (b) and (c) are reproduced with permission from [43], Optical Society of America.
To overcome this problem, a possible strategy is to modulate both the real and imaginary part of the permittivity in such a way that the modulation spectrum becomes “unilateral” (or “spectrally causal”), as we theoretically demonstrated in a recent publication [43]. Such an approach ensures that back-reflections due to the temporal modulations are prohibited over a very wide frequency range and, therefore, the incident wave is perfectly absorbed with no reflections over wide bandwidths [Figure 3(b) and (c)]. Along similar lines, a recent work theoretically demonstrated, through a Green’s function analysis and numerical optimizations, that a suitable switching of both the conductivity and permittivity leads to a reduction of reflection over the whole frequency spectrum [93]. As an alternative strategy, a recent experimental work [94] also demonstrated broadband absorption by creating an energy-trap through switched electronic components triggered by the pulse entering the absorbing region. While the strategies described so far work for short broadband pulses and are based on nonperiodic temporal modulations (e.g., “switching”), perfect absorption for quasi-monochromatic signals has also been realized by employing periodic temporal modulations [95]. Specifically, it was shown that perfect absorption can be achieved even with infinitesimal losses due to coherent multichannel illumination with Floquet time-periodic driving schemes.
Our list of applications that can benefit from time-varying systems is certainly not exhaustive. In the supplementary material (available at https://doi.org/10.1109/MAP.2023.3236275), we provide several other application examples that might be of interest to readers also from different backgrounds, such as optics and photonics.
Although the study of time-varying electromagnetic systems dates back more than half a century, recent years have seen a surge of renewed interest in this topic, largely motivated by the need to realize novel electromagnetic systems that are superior to conventional LTI ones for various applications, from high-speed full-duplex communication systems, to broadband stealth technology, to high-efficiency energy harvesting and energy transfer. However, while recent progress in this field has been promising, we believe several challenges should be taken into consideration and carefully addressed before such systems can be adopted into practical use.
Since a time-varying system requires external power to operate, such active systems might become susceptible to instabilities (unbounded oscillations) that can disrupt the desired functionality. Thus, especially for systems involving parametric amplification, a careful analysis is essential to assess and ensure stability [80]. Moreover, the additional energy pumped into the system [96] should be carefully taken into account when determining the figure of merits (such as efficiency) of the design [97] and also to assess whether the external power required by the temporal modulation is beyond practical reach and/or may damage the system [98].
Frequency dispersion is another fundamental property of physical materials, constrained by the implications of the causality principle expressed by the Kramers-Kronig relations. Hence, any realistic system involves, to some degree, frequency variations in the material constitutive parameters. However, especially for numerical studies involving time-varying structures, such dispersion effects are often ignored and the time variation is assumed to be independent of frequency [20]. As a result, any realistic implementation of such proposals will necessarily experience performance deterioration due to frequency dispersion of the material properties. We note that several recent works [98], [99], [100], [101] have now started studying material dispersion effects more systematically in the context of time-varying systems. However, we believe further investigations in this area are important to improve our understanding of time-varying materials and may lead to improved designs and more accurate performance predictions.
For schemes requiring synchronization between the incident wave and the temporal variation, another significant challenge is that fast detectors and switching mechanisms are essential to trigger the modulation in the presence of the incident wave. Such additional components might complicate the design and reduce its practical applicability. Hence, more research is needed to integrate such components into time-varying systems in a practically useful manner.
Dissipation, unless desired to realize absorbers, is another challenge that needs to be addressed for various practical applications that involve time-varying systems. For instance, epsilon-near-zero materials have recently become the subject of increasing interest to realize deep and fast modulations at optical frequencies, as they can be efficiently controlled through external optical pumps [18]. However, such materials still suffer from high losses around their epsilon-near-zero wavelengths. Hence, the search for better epsilon-near-zero materials with lower losses is an important area of research (note that, theoretically, nothing prevents the existence of an epsilon-near-zero material that is dissipationless, albeit frequency-dispersive, within a certain frequency window). Alternatively, one could exploit (rather than eliminate or compensate) the losses in such materials to realize time-dependent loss profiles that accompany the time-varying refractive-index profiles. In this context, we recently provided a general strategy to exploit the synergy between these two time-modulation profiles for various applications [43]. Investigations along these directions may also enhance our understanding of wave dynamics in complex (“non-Hermitian”) time-varying media.
Despite the challenges outlined here, we believe that the use of time-varying materials and components to overcome the limitations of conventional LTI electromagnetic/photonic systems is an exciting and promising research area that might lead to novel and superior devices and, as more research is conducted, a deeper understanding of the actual fundamental limits in electromagnetics. Spatially varying engineered structures, such as frequency-selective surfaces, photonic crystals, and metamaterials have revolutionized the field of electromagnetics in recent decades. Along the same vein, it would be safe to say that the new insights and the new tools provided by the field of time-varying electromagnetic systems will enable further advances and possible revolutions in the next decades.
The authors acknowledge support from the Air Force Office of Scientific Research with Grant FA9550-19-1-0043 through Dr. Arje Nachman. This article has supplementary downloadable material available at https://doi.org/10.1109/MAP.2023.3236275, provided by the authors.
Zeki Hayran (zh337@cornell.edu) is a Ph.D. candidate at the School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA. His research interests include time-varying metamaterials, non-Hermitian optics, and topological photonics. He is a Student Member of IEEE.
Francesco Monticone (francesco.monticone@cornell.edu) is an assistant professor of Electrical and Computer Engineering at Cornell University, Ithaca, New York 14853, USA. His research interests are applied electromagnetics, metamaterials/metasurfaces, and nanophotonics. He is a Member of IEEE and an associate editor of IEEE Transactions on Antennas and Propagation.
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