Grigorii Ptitcyn, Mohammad Sajjad Mirmoosa, Amirhosein Sotoodehfar, Sergei A. Tretyakov
IMAGE LICENSED BY INGRAM PUBLISHING
During the last decade, possibilities to realize new phenomena and create new applications by varying system properties in time have gained increasing attention in many research fields. Although the interest in using time-modulation techniques for engineering electromagnetic (EM) response has become revitalized only in recent years, the field originates from the middle of the previous century, and a multitude of works have been published ever since. In this tutorial article, we provide a historical picture and review the basic concepts in this field. In particular, we introduce the general theory of linear time-varying (LTV) systems and discuss the means to properly account for frequency dispersion of nonstationary systems. Also, we elucidate models of time-varying electrical circuits and materials and discuss some useful effects that can be achieved by time modulation of circuit or material parameters.
Dynamic changes of properties of a system made by an external force remove limitations on system response imposed by invariance under time translations. This is sometimes an unwanted feature that is inevitably present and whose effects should be mitigated by some means. For instance, cars in motion and people walking in the streets dynamically alter propagation channels of telecommunication systems, or, control equipment on a plane constantly experiences dynamic changes of the environment. Researchers in fields such as signal processing, communications, radio wave propagation, information theory, and control engineering are familiar with the practical importance of this problem [1]. In EMs and microwave and antenna engineering, however, nonstationarity can be importantly used as an additional degree of freedom for controlling signals and waves in the desired way. In the past two decades, applied physicists and EM engineers have attempted to manipulate waves by designing artificial materials and surfaces: metamaterials and metasurfaces. In particular, they studied how the geometry of inclusions (forming an artificial material or surface) and the distances between them modify and engineer wave-matter interactions. As a follow-up step, why not assume that the properties of such inclusions change in time. Taking this fourth dimension into account helps us realize new and exotic effects in interactions of waves with materials or surfaces and enhance the functionalities of devices. For instance, in the microwave range, one can use mechanically moving parts of electrical devices or employ electronic components such as varactors and transistors. At short wavelengths, it can be optical pumping at an extremely high speed, which can even be considered a step-like change. An illustration of various means that realize time modulation is given in Figure 1.
Figure 1. The possible ways to realize time modulation of circuits and media.
Note that despite the broad variety of problems and various frequency ranges, the fundamental formulation of EM problems for time-varying systems is common. In this tutorial article, we overview the foundations and basic methods used for the analysis and engineering of microwave and optical time-varying systems. We begin by presenting a historical overview of the development of time-modulation techniques from approximately 1950 until 2022. We start from the work published by Lotfi A. Zadeh in 1950 and gradually move toward developments in antenna engineering, electric circuits, and EM media, where researchers investigated time-varying systems for enhancement of bandwidth, amplification, realizing nonreciprocity, frequency mixing, and so forth. Next, we present and discuss the general basic concepts associated with linear and causal time-varying systems. We scrutinize systems that possess memory and are temporally dispersive. In the last part, we discuss the general theory of electric circuits, consisting of time-varying elements or dielectrics with time-varying permittivity. In this introductory tutorial, we focus on only circuits, but actually, this knowledge can be effectively used in many branches of antenna and microwave engineering.
Before continuing, it is important to clarify the difference between time modulation and tuning (reconfiguration) of devices. Tunable systems are capable of switching between several operational regimes, and in each of the regimes, parameters of the system remain constant. For example, reconfigurable antennas can be shifted to different resonance frequencies using a switchable capacitance bank. Another example are reconfigurable intelligent surfaces that can be dynamically adjusted, optimizing propagation channels. For tunable systems, intermediate (transient) states are usually not taken into account, and only steady-state operational regimes are of interest. In contrast to that, time-modulated structures capitalize exactly on the transient regime, which is realized when system parameters change in time. Such systems are within the focus of this tutorial.
Despite the recent burst of research interest in the field of time modulation, its historical origin goes back to the middle of the previous century. Interestingly, the problem of dynamic variations of a system’s parameters was not allocated to a single research field. Probably one of the first papers on this topic appeared in communication theory. Zadeh (1921–2017), the father of fuzzy logic and one of the pioneers of artificial intelligence, described “frequency analysis of variable networks” in 1950 [2], [3]. Zadeh extended the frequency analysis techniques that were conventionally used and proposed a “system function” defined in a joint time-frequency domain to characterize LTV systems. Zadeh comprehensively tackled the problem and studied such aspects as stability [4], pulse response [3], and power spectra of variable networks [5]. These ideas were further developed by Kailath by discussing possible simplifications [6] and experimental issues [7]. General theoretical investigations of time-varying systems were further continued by Gersho [8], [9], where he studied such a fundamental property as “time-frequency” duality, which was also later investigated by Bello [10]. In addition to that, Bello considered randomly time-variant channels and introduced the assumption of wide-sense, stationary uncorrelated scattering, which gathered huge attention [11].
In the newly born field of radio engineering, scientists were focused on overcoming practical issues such as small bandwidth and large antenna size, and time modulation of antennas was considered one of the possible resolutions to these problems. Several antenna geometries were proposed and tested for efficient information transfer, however, only in the 1950s was a general understanding of fundamental limitations on small antennas formulated by Wheeler [12] and Chu [13]. In these works, the authors proved that for linear, passive, and time-invariant antennas, their size, bandwidth, and efficiency cannot be optimized simultaneously. Naturally, to overcome Chu’s limitation, one needs to utilize either active, nonlinear, or time-varying systems. One of the first attempts to overcome Chu’s limitation was performed by Jacob and Brauch [14]. They used so-called “antenna keying,” where nonlinear properties of ferrite materials are utilized to modulate the resonance frequency of a small antenna using a time-varying inductor. It indeed helped transmit signals in the kilohertz frequency range beyond the fundamental limitation, however, the presence of a ferrite core in the structure hindered applicability of this system at higher frequencies because the saturation effect of ferrite cores sets the maximal modulation speed of such inductors. This problem was later resolved using a switchable capacitor bank that was used as a time-varying capacitor [15]. Time-varying components were used also in [16] and [17]. Worth mentioning is that the antenna-keying approach was later reinvented and renamed direct antenna modulation [18], [19], [20], keeping all of its essential features.
In the years to come, antenna engineers preferred to move (and still are moving) toward higher frequencies where higher bandwidth and smaller antenna sizes could be obtained, often sacrificing antenna efficiency. Obviously, this development cannot continue indefinitely. In addition to that, low-frequency applications as well as the need for electrically small antennas with a wide bandwidth still create a strong demand for nonlinear, active, and/or time-varying antennas.
Studies of electrical circuits with time-varying parameters probably began with the works of Cullen [21], Tien [22], and Tien and Suhl [23]. Cullen and Suhl considered time-modulated electrical circuits with distributed parameters and noticed that they were capable of parametric amplification when the phase velocity of the modulation wave is the same as the wave-phase velocity (called luminal metamaterials in [24]). Tien found frequency mixing and envisaged many other phenomena that can occur in parametric circuits with spatially distributed parameters. Studies of amplification in these structures continued [25], together with the discovery of a new functionality: nonreciprocity. In [26], Kamal proposed and experimentally tested a parametric device made of just two time-varying circuit elements that was capable of nonreciprocal wave transmission. Nonreciprocity is essential for many applications, and time modulation brings a new way to realize it [27].
Possibly the earliest work on media with time-varying parameters dates back to 1958, when Morgenthaler [28] considered propagation of EM waves through a dielectric material sample with time-varying permittivity or permeability and found a solution for the case when the impedance of the medium remains constant. He noticed that an abrupt jump of permittivity creates forward- and backward-propagating waves, analogous to a spatial discontinuity. Traveling-wave modulation was studied from the general perspectives of Simon [29] and Oliner et al. [30] in 1961. They considered the problem of EM wave propagation in media with a progressive sinusoidal disturbance. Later, Felsen and Whitman [31] considered slow and abrupt modulations of unbounded media and focused their attention on the excitation problem, and on the source-dependent phenomena. Interestingly, they even incorporated special cases of dispersion in their analysis and found a focusing effect in an abruptly changing medium. In the same year, Fante [32] studied a time-varying half-space and considered slow and abrupt modulations as well as included dispersion in his description. A very peculiar property of time-varying materials was found in [33] by Holberg and Kunz. They studied fields inside a time-varying slab and discovered exponential amplification of waves with certain wavenumbers.
In the Soviet Union, the field of time-varying media was heavily studied in the plasma community. The community studied the problem from various different perspectives, such as wave propagation in moving media [34], [35], [36], transformation of waves on moving boundaries [37] and inside media with slowly varying parameters [37], [38], [39], reflection and refraction of waves on moving inhomogeneous plasma structures [40], [41], [42], energy relations in time-varying media [43], and more. In one of his pioneering papers [44], Stepanov considered the dielectric constant of unsteady plasma. For convenience of the analysis, Stepanov performed a Fourier transform of plasma susceptibility and introduced a new material parameter of plasma that simultaneously depends on time and frequency, thus incorporating frequency dispersion in his model of nonstationary plasma.
All of these studies (and others) provide a solid platform for current research. Many of these works are being developed further or reworked for technologies available today. Let us now briefly review the current status of research on time-modulated structures.
Various phenomena and effects have been investigated theoretically and experimentally over the last 40 years. For instance, the role of frequency dispersion in time-modulated systems is of utmost importance because materials whose properties can be altered dynamically are usually highly dispersive. This problem was actively studied by Mirmoosa et al. [45], Engheta [46], Solis and Engheta [47], and Sloan et al. [48]. In [45], Mirmoosa et al. further developed the theoretical approach proposed by Zadeh [2]. They coined the term temporal complex polarizability to describe light-particle interactions when the properties of a subwavelength particle change in time. Their definition is exactly what Zadeh wrote in equation (2) in his paper [2]. Pacheco-Pen´a and Engheta [49], [50] exploit complementary roles of space and time in Maxwell’s equations to find temporal analogues to spatial phenomena. Other very distinct methods are investigated in Derode et al. [51], Fink [52], Przadka et al. [53], and Popoff et al. [54], where time-reversal symmetry of EM waves are uniquely capitalized. Many research techniques try to use time-modulation techniques to overcome certain fundamental limitations imposed for passive structures [12], [13], [55]. For instance, breaking limitations on antenna performance was studied, for example, by Hadad et al. [56], Li et al. [57], and Mostafa et al. [58]. The breaking of Lorentz reciprocity using time-varying structures was studied actively in many research groups, including Galiffi et al. [24], Taravati et al. [59], Chamanara et al. [60], Caloz et al. [61], Sounas and Alù [62], Correas-Serrano et al. [63], Dinc et al. [64], Fleury et al. [65], Song et al. [66], Lira et al. [67], Yu and Fan [68], Shi et al. [69], Wang et al. [70], Asadchy et al. [71], Shaltout et al. [72], Shaltout et al. [73], Koutserimpas and Fleury [74], Zhang et al. [75], Taravati and Kishk [76], Ramaccia et al. [77], Ramaccia et al. [78], Li et al. [79], and many more.
In addition to nonreciprocity, numerous other applications and models were studied, such as frequency conversion [80], [81], [82], [83], [84], [85], [86], [87], [88], amplification [74], [89], [90], Doppler shift [91], [92], [93], extreme accumulation of energy [94], instantaneous control of radiation and scattering [95], [96], Fresnel drag [97], negative refraction [98], effective medium models [99], [100], control of diffusion [101], pulse compression [102], camouflage [103], [104], temporal birefringence [105], [106], temporal photonic crystals [107], [108], [109], space-time crystals [110], [111], [112], power combiners [113], Wood anomalies [114], and more. This is not a complete list of applications and groups working on time modulation. The number of works that have appeared in the recent decades is massive, and the general interest toward this research topic has only grown (see, for example, the review papers in [115], [116]).
Any classical physical system can be modeled by a differential equation or set of differential equations. That is, temporal variations of a certain dynamic variable or variables $\text{O}(\text{t})$ (for example, an electric field at some point of space or current in a conductor) under the action of some external excitation $\text{I}(\text{t})$ (for example, an external electromotive force of a voltage source, or the field of an incident wave) can be found as the solution of equation \[{\mathcal{L}}_{t}{O}{(}{t}{)} = {I}{(}{t}{)} \tag{1} \] where ${\mathcal{L}}_{t}$ is a differential operator that acts on function ${O}{(}{t}{)}{.}$ The solution to this equation for an excitation in form of the Dirac delta function is called Green’s function ${G}{(}{\gamma},{t}{):}$ \[{\mathcal{L}}_{t}{G}{(}{\gamma},{t}{)} = {\delta}{(}{\gamma}{)}{.} \tag{2} \]
Here, t represents the observation time, and ${\gamma}$ is the delay between t and time moment ${\tau}$ when the excitation arrives, i.e., ${\gamma} = {t}{-}{\tau}{.}$ Note that the following derivations can be performed, equivalently, also in terms of the time variable ${\tau}$ instead of the delay parameter ${\gamma}{.}$ Sometimes, G is called the impulse response of a system as it is a reaction of the system to the input in the form of a delta-function pulse [117]. Sometimes, it is also called the fundamental solution of an operator as any input can be viewed as a summation of weighted delta functions.
Most importantly, for linear systems, the response to excitations that arbitrarily vary in time can be found in terms of the Green function. Here we discuss LTV systems. Let us prove this property of Green’s function by multiplying (2) by an arbitrary function of time ${I}{(}{t}{-}{\gamma}{)}$ and integrating over ${\gamma}{:}$ \[\mathop{\int}\nolimits_{{-}\infty}\nolimits^{\infty}{{\mathcal{L}}_{t}}{G}{(}{\gamma},{t}{)}{I}{(}{t}{-}{\gamma}{)}{d}{\gamma} = \mathop{\int}\nolimits_{{-}\infty}\nolimits^{\infty}{\delta}{(}{\gamma}{)}{I}{(}{t}{-}{\gamma}{)}{d}{\gamma} = {I}{(}{t}{)}{.} \tag{3} \]
Because the operator ${\mathcal{L}}_{t}$ acts at the variable t and it is a linear operator (we consider linear systems), we can take the operator outside of the integral: \[{\mathcal{L}}_{t}\mathop{\int}\nolimits_{{-}\infty}\nolimits^{\infty}{G}{(}{\gamma},{t}{)}{I}{(}{t}{-}{\gamma}{)}{d}{\gamma} = {I}{(}{t}{)}{.} \tag{4} \]
By comparing (1) and (4), we see that the output of the system is given by \[{O}{(}{t}{)} = \mathop{\int}\nolimits_{0}\nolimits^{\infty}{G}{(}{\gamma},{t}{)}{I}{(}{t}{-}{\gamma}{)}{d}{\gamma}{.} \tag{5} \]
Mathematically, the integration in (5) would be performed over the retardation time ${\gamma}$ from ${-}\infty$ to $\infty,$ but we restrict the integration to only nonnegative values. Negative retardation time means that excitations that will happen at future moments of time affect the output at present. This is against an extremely fundamental empirical law of nature: causality. The causality principle says that the state of a system is defined by its evolution in the past, and it cannot be affected by events at future moments of time.
Let us see how this equation simplifies for linear time-invariant systems when the operator ${\mathcal{L}}_{t}$ that describes the system does not change in time. It means that shifting the input by some time interval also shifts the output by the same interval because the system is immutable. Thus, the output depends only on the time interval ${\gamma}$ between the moment when the input is applied and the moment when we observe the resulting output, which allows us to write \[{O}{(}{t}{)} = \mathop{\int}\nolimits_{0}\nolimits^{\infty}{G}{(}{\gamma}{)}{I}{(}{t}{-}{\gamma}{)}{d}{\gamma}{.} \tag{6} \]
Despite the very similar forms of (5) and (6), the first is far more complicated to use because the response function is explicitly dependent on two time variables. Because for stationary systems the integral in (6) is a convolution, we can make a Fourier transform and write in the frequency domain a simple linear relationship between the input and output variables: ${O}{(}\Omega{)} = {G}{(}\Omega{)}{I}{(}\Omega{)}{.}$ This familiar frequency-domain method is not possible to use for time-varying systems because the integral in (5) is not a conventional convolution.
To be able to use (5), one can make assumptions about the system and simplify the theory accordingly. One of these assumptions is instantaneous response, which simplifies the theory immensely. Under this assumption, Green’s function is expressed as \[{G}^{\text{ir}}{(}{\gamma},{t}{)} = {\delta}{(}{\gamma}{)}{G}{(}{t}{)} \tag{7} \] where superscript “ir” stands for “instantaneous response.” One can immediately see how easy and compact the relationship between the input and output becomes \[{O}{(}{t}{)} = {G}{(}{t}{)}{I}{(}{t}{)}{.} \tag{8} \]
Written in this form, the equation is very convenient and easy to use. However, this simplicity comes at a high cost. For example, regarding EM media, instantaneous response means that atoms get polarized instantaneously without inertia, which, strictly speaking, is never the case. In systems where the transition time is negligible compared to the period of the incident field oscillations, it is possible to use the instantaneous response approximation. In other words, (8) remains valid if the input interacts with the system that is modulated slowly enough such that it can be practically treated as static at all moments of time. In this approximation, frequency dispersion of materials is completely neglected.
Next we explain how these basic concepts are used to analyze electric circuits with lumped time-varying elements. We concentrate on time-varying capacitors and describe their salient characteristics. The interested reader can repeat the same procedure for other elements such as inductors and resistors.
Conventionally, capacitance is defined as the electric charge divided by the voltage. However, in general, this definition does not provide a complete picture, even in the time-invariant scenario, as it is valid only in the assumption that the capacitor exhibits an instantaneous response. In general, the electric charge $\text{Q}(\text{t})$ and the voltage $\text{v}(\text{t})$ over a capacitor should be related as in (5): \[{Q}{(}{t}{)} = \mathop{\int}\nolimits_{0}\nolimits^{\infty}{C}{(}{\gamma},{t}{)}{v}{(}{t}{-}{\gamma}{)}{d}{\gamma}{.} \tag{9} \]
Based on this general linear and causal relationship, we aim to find a formula that connects the electric current and voltage fully in the frequency domain. Using the Fourier transform, we write the instantaneous voltage based on its frequency-domain counterpart. Accordingly, (9) becomes ${Q}{(}{t}{)} = {1}{/}{{2}{\pi}}\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{{C}_{T}}{(}{\omega},{t}{)}{V}{(}{\omega}{)}\exp{(}{j}{\omega}{t}{)}{d}{\omega},$ where we define the temporal complex capacitance as \[{C}_{T}{(}{\omega},{t}{)} = \mathop{\int}\nolimits_{0}\nolimits^{ + \infty}{C}{(}{\gamma},{t}{)}\exp{(}{-}{j}{\omega}{\gamma}{)}{d}{\gamma} \tag{10} \] which is the Fourier transform of the capacitance kernel ${(}{C}{(}{\gamma},{t}{)} = {0}$ for ${\gamma}\,{<}\,{0}$ due to causality). The electric current flowing through the capacitor is the time derivative of the electric charge. Thus, \[{i}{(}{t}{)} = \frac{1}{{2}{\pi}}\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{{Y}_{T}}{(}{\omega},{t}{)}{V}{(}{\omega}{)}\exp{(}{j}{\omega}{t}{)}{d}{\omega} \tag{11} \] where \[{Y}_{\text{T}}{(}{\omega},{t}{)} = \frac{{d}{C}_{T}{(}{\omega},{t}{)}}{{\text{d}}{t}} + {j}{\omega}{C}_{\text{T}}{(}{\omega},{t}{)} \tag{12} \] can be called temporal complex admittance modeling of the nonstationary capacitance. Interestingly, this parameter is complex valued. In the aforementioned relationship, the first term is associated with the time derivative of the temporal complex capacitance. In the stationary scenario, this time derivative vanishes and the admittance is determined only by the conventional second term.
Suppose that a voltage source ${v}_{s}(\text{t})$ with a series resistance R is connected to a time-varying capacitor and a stationary inductor. From the circuit theory, we know that ${v}_{s}{(}{t}{)} = {Ri}{(}{t}{)} + {L}{d}{i}{/}{d}{t} + {v}{(}{t}{),}$ where $\text{v}(\text{t})$ is the voltage across the capacitor, and L represents the inductance. The current and voltage in a time-varying capacitor are related to each other as in (11). Taking the Fourier transform of (11), we have \[{I}{(}\Omega{)} = \frac{1}{{2}{\pi}}\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{Y}{(}{\omega},\Omega{-}{\omega}{)}{V}{(}{\omega}{)}{d}{\omega} \tag{13} \] in which \[{Y}{(}{\omega},\Omega{)} = \mathop{\int}\nolimits_{{-}\infty}\nolimits^{\infty}{{Y}_{T}}{(}{\omega},{t}{)}{e}^{{-}{j}\Omega{t}}{\text{d}}{t} = {j}{(}\Omega + {\omega}{)}{C}{(}{\omega},\Omega{)}{.} \tag{14} \]
Here, ${Y}{(}{\omega},\Omega{)}$ and ${C}{(}{\omega},\Omega{)}$ are the Fourier transforms of the temporal complex admittance and capacitance, respectively. How the admittance and the capacitance are associated with each other fully in the frequency domain is something intriguing. In fact, it is very similar to the time-invariant scenario. However, in a nonstationary case, we have two angular frequencies that are added. Equation (13) can be written as \[{I}{(}\Omega{)} = \frac{{j}\Omega}{{2}{\pi}}\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{C}{(}{\omega},\Omega{-}{\omega}{)}{V}{(}{\omega}{)}{d}{\omega}{.} \tag{15} \]
Using this equation, we can finally find that \[{V}_{s}{(}\Omega{)} = {j}\Omega\left({\frac{{R} + {j}\Omega{L}}{{2}{\pi}}}\right)\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{C}{(}{\omega},\Omega{-}{\omega}{)}{V}{(}{\omega}{)}{d}{\omega} + {V}{(}\Omega{)}{.} \tag{16} \]
This is an integral equation that is solvable, at least numerically, if the capacitance kernel is known.
If we assume the instantaneous response of a time-varying capacitor, the capacitance kernel becomes a multiplication of Dirac delta function ${\delta}{(}{\gamma}{)}$ and a function of time variable t. Thus, ${C}{(}{\omega},\Omega{)}$ is invariant with respect to the first argument, and therefore, (16) becomes \[{V}_{s}{(}\Omega{)} = {j}\Omega\left({\frac{{R} + {j}\Omega{L}}{{2}{\pi}}}\right)\mathop{\int}\nolimits_{{-}\infty}\nolimits^{ + \infty}{C}{(}\Omega{-}{\omega}{)}{V}{(}{\omega}{)}{d}{\omega} + {V}{(}\Omega{)}{.} \tag{17} \]
The integrals in (15)–(17) reduce to convolutions, and the inverse Fourier transform of (15) gives us the usual, instantaneous relationship ${i}{(}{t}{)} = {d}{/}{d}{t}{[}{C}{(}{t}{)}{v}{(}{t}{)]}{.}$ Many works on time-modulated structures start from assuming that circuit or material parameters (such as capacitance, inductance, permittivity, and permeability) are functions of time, as in this relationship between the current and voltage, and solve the corresponding equations in the time domain. It is important to remember that this approach completely neglects frequency dispersion of all materials from which the device is made. As an example, for a capacitor, this is a valid assumption if the capacitor is made of perfectly conducting plates with a vacuum between them, and the capacitance is modulated by mechanical movements of the plates.
As another limiting case, let us suppose that the capacitor is static (time invariant). In this scenario, the capacitance kernel depends on only the time-delay variable ${\gamma},$ and, as a result, ${C}{(}{\omega},\Omega{)} = {2}{\pi}{C}{(}{\omega}{)}{\delta}{(}\Omega{)}{.}$ Hence, (16) is simplified to ${V}_{s}{(}\Omega{)} = {(}{1} + {j}\Omega{RC}{(}\Omega{)}{-}{\Omega}^{2}{LC}{(}\Omega{))}{V}{(}\Omega{),}$ which is the usual frequency-domain expression for an RLC circuit.
Finally, we note that the same theory is applicable to other linear and causal relations of EM theory. For example, the general relationship between displacement vector (D) and electric field (E) in time-modulated dielectrics or metamaterials has the form \[{\bf{D}}{(}{\bf{r}},{t}{)} = \mathop{\int}\nolimits_{0}\nolimits^{\infty}{\epsilon}{(}{\gamma},{t}{)}{\bf{E}}{(}{\bf{r}},{t}{-}{\gamma}{)}{d}{\gamma} \tag{18} \] which is exactly the same as the general relationship defining time-modulated capacitance [see (9)].
Many useful features in time-modulated systems are realized with time-periodic modulations. Importantly, periodicity of modulation allows us to use the Floquet theorem, which was introduced by Gaston Floquet (1847–1920) in 1883 [118]. The Floquet theorem is applicable to any linear structure with periodically varied properties, including spatial and temporal variations. In the case of spatially varied parameters, this theorem is known as Bloch’s theorem [119], [120]. The theorem is quite general, and it can be applied to vector and scalar quantities for unbounded media, slabs, and boundaries. Here, we consider an unbounded homogeneous dielectric medium whose permittivity ${\epsilon}{(}{t}{)}$ periodically varies in time with the period T. For simplicity, we assume an instantaneous response of the medium. As the medium is homogeneous in space and there is no preferred direction, let us assume that there is an EM wave propagating in the z direction. Fields are governed by the wave equation \[\frac{{\partial}^{2}}{\partial{z}^{2}}{\bf{u}}{(}{z},{t}{)}{-}\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial{t}^{2}}{[}{\epsilon}{(}{t}{)}{\bf{u}}{(}{z},{t}{)]} = {0} \tag{19} \] where c is the speed of light in a vacuum, and ${\bf{u}}{(}{z},{t}{)}$ represents a transverse electric field in the time domain. Without loss of generality, we can assume a certain polarization of ${\bf{u}}{(}{z},{t}{)}$ that lies in the xy-plane and further consider only the field amplitude. As a mathematical solution of (19), ${\bf{u}}{(}{z},{t}{)}$ can be a complex-valued function. As the medium is modulated only in time, we can separate the variables and consider spatial dependency as that of a plane wave, ${e}^{{-}{jkz}},$ where k denotes the wavenumber. Thus, we look for solutions in form \[{u}{(}{z},{t}{)} = {u}_{t}{(}{t}{)}{e}^{{-}{jkz}} \tag{20} \] which reduces (19) to \[\frac{1}{{c}^{2}}\frac{{\text{d}}^{2}}{\text{d}{t}^{2}}{[}{\epsilon}{(}{t}{)}{u}_{t}{(}{t}{)]} + {k}^{2}{u}_{t}{(}{t}{)} = {0}{.} \tag{21} \]
This equation is convenient to rewrite as \[\frac{{\text{d}}^{2}}{\text{d}{t}^{2}}{y}{(}{t}{)} + {K}{(}{t}{)}{y}{(}{t}{)} = {0} \tag{22} \] where ${y}{(}{t}{)} = {u}_{t}{(}{t}{)}{\epsilon}{(}{t}{)}$ and ${K}{(}{t}{)} = {k}^{2}{c}^{2}{/}{\epsilon}{(}{t}{)}{.}$ Interestingly, in this form, (22) as a second-order differential equation with a time-dependent coefficient can be clearly recognized as Hill’s equation [121]. The Floquet theorem, whose detailed proof is given in the supplementary materials available at https://www.doi.org/10.1109/MAP.2023.3261601, states that the two linearly independent solutions of this second-order equation, ${y}_{1}(\text{t})$ and ${y}_{2}(\text{t}),$ can be found in form \[{y}_{1}{(}{t}{)} = {e}^{{j}{\alpha}{t}}{p}_{1}{(}{t}{),}{y}_{2}{(}{t}{)} = {e}^{{-}{j}{\alpha}{t}}{p}_{2}{(}{t}{)} \tag{23} \] where ${\alpha}$ is a constant, and ${p}_{1,2}$ are periodic functions with the period T.
As ${p}_{1,2}(\text{t})$ are periodic, it is possible to present them as an infinite Fourier series of frequency harmonics: \[{p}_{1,2}{(}{t}{)} = \mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{p}_{n}{e}^{{-}{jn}\frac{{2}{\pi}}{T}{t}}} \tag{24} \] where ${p}_{n}$ denote the amplitudes of harmonics. For convenience, let us represent ${2}{\pi}{/}{T}$ as the modulation angular frequency ${\omega}_{M}{.}$ By using the expression in (24), the solution ${y}_{1}(\text{t})$ in (23), for example, can be rewritten as \[{y}_{1}{(}{t}{)} = \mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{p}_{n}{e}^{{j}{(}{\alpha}{-}{n}{\omega}_{M}{)}{t}}{.}} \tag{25} \]
Worth noting is that ${y}_{1}(\text{t})$ is a complex-valued function, and the real-valued solution can be obtained by taking the real part of it. Equation (25) shows that the solution of the wave (19) can be written as an infinite series of frequency harmonics that are separated in frequency by ${\omega}_{M}{.}$ Let us further consider a simple example that shows how useful the Floquet theorem is.
The most noticeable case that people have been studying is associated with approximately instantaneous and periodic time-varying capacitance or permittivity as this allows us to create devices such as parametric amplifiers. In the following, we use the general theory presented in the previous section and draw important conclusions for this particular case. Because the response is assumed to be instantaneous, the capacitance kernel is equal to the Dirac delta function, which is multiplied by a function that depends on only the time variable t. In other words, we have ${C}{(}{\gamma},{t}{)} = {\delta}{(}{\gamma}{)}{h}{(}{t}{),}$ in which $\text{h}(\text{t})$ is periodic, meaning that it can be expanded into a Fourier series as \[{C}_{T}{(}{t}{)} = {h}{(}{t}{)} = \mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{a}_{n}\exp{(}{jn}{\Omega}_{p}{t}{)}{.}} \tag{26} \]
It is clear that the temporal complex capacitance is real valued, and it does not depend on the angular frequency ${\omega}$ because the system is dispersionless. In (26), ${a}_{n}$ are the Fourier coefficients, and ${\Omega}_{p}$ represents the angular frequency associated with the periodicity of ${h}{(}{t}{)}{.}$ Now we need to make a Fourier transform of the temporal complex capacitance. This simply results in a summation of the Dirac delta functions as \[{C}{(}\Omega{)} = {2}{\pi}\mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{a}_{n}{\delta}{(}\Omega{-}{n}{\Omega}_{p}{)}{.}} \tag{27} \]
Next, we substitute this relation into (15) to deduce a general expression that relates the electric current to the voltage. After some small manipulations, we arrive at \[{I}{(}\Omega{)} = \mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{j}\Omega{a}_{n}{V}{(}\Omega{-}{n}{\Omega}_{p}{)}{.}} \tag{28} \]
Let us suppose that the capacitor is connected to a time-harmonic voltage source, i.e., ${v}{(}{t}{)} = {A}\cos{(}{\omega}_{0}{t} + {\phi}{)}$ or ${V}{(}\Omega{)} = {\pi}{A}{[}\exp{(}{j}{\phi}{)}{\delta}{(}\Omega{-}{\omega}_{0}{)} + \exp{(}{-}{j}{\phi}{)}{\delta}{(}\Omega + {\omega}_{0}{)]}{.}$ Using this expression, we see that \begin{align*}{I}{(}\Omega{)} & = {j}{\pi}{A}\mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{\Omega{a}_{n}{[}\exp{(}{j}{\phi}{)}{\delta}{(}\Omega{-}{n}{\Omega}_{p}{-}{\omega}_{0}{)}} \\ & \quad + \exp{(}{-}{j}{\phi}{)}{\delta}{(}\Omega{-}{n}{\Omega}_{p} + {\omega}_{0}{)]}{.} \tag{29} \end{align*}
Making the inverse Fourier transform, we find \begin{align*}{i}{(}{t}{)} & = \frac{A}{2}\mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{[}{j}{(}{\omega}_{0} + {n}{\Omega}_{p}{)}{a}_{n}\exp{(}{j}{(}{\omega}_{0}{t} + {n}{\Omega}_{p}{t} + {\phi}{))}} \\ & \quad {-}{j}{(}{\omega}_{0}{-}{n}{\Omega}_{p}{)}{a}_{n}\exp{(}{-}{j}{(}{\omega}_{0}{t}{-}{n}{\Omega}_{p}{t} + {\phi}{))]}{.} \tag{30} \end{align*}
At a quick glance, we observe that there are two exponential functions (multiplied by some coefficients) that are not complex conjugate to each other due to the opposite signs of the terms $\pm{n}{\Omega}_{p}{.}$ However, we notice that the integer n changes from ${-}\infty$ to $ + \infty{.}$ By having this small but important point in mind, and by paying attention to the fact that ${a}_{n}^{\ast} = {a}_{{-}{n}}$ (because the time-varying capacitance is a real-valued function), we conclude that \[{i}{(}{t}{)} = {A}\cdot{\text{Re}}\left[{\mathop{\sum}\limits_{{n} = {-}\infty}^{ + \infty}{{j}{(}{\omega}_{0} + {n}{\Omega}_{p}{)}\exp{(}{j}{\phi}{)}{a}_{n}\exp{(}{j}{(}{\omega}_{0} + {n}{\Omega}_{p}{)}{t}{)}}}\right] \tag{31} \] where ${\text{Re}}{[}\ldots{]}$ denotes the real part. In (31), if we define ${I}_{n}$ as the complex coefficient that is multiplied by the exponential function, the equation finally reduces to \[{i}{(}{t}{)} = {\text{Re}}\left[{\mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{I}_{n}\exp(\text{j}{\omega}_{n}\text{t})}}\right] \tag{32} \] in which ${\omega}_{n} = {\omega}_{0} + {n}{\Omega}_{p}{.}$ This gives us the message that in such instantaneous periodic systems there are oscillations at infinitely many frequencies that become far from the fundamental frequency ${\omega}_{0}$ with the increasing index n. Using this result, if this time-varying capacitance is part of a more complex electric circuit, we can write the voltage over the capacitance as \[{v}{(}{t}{)} = {\text{Re}}\left[{\mathop{\sum}\limits_{{n} = {-}\infty}\limits^{ + \infty}{{V}_{n}\exp(\text{j}{\omega}_{n}\text{t})}}\right]{.} \tag{33} \]
Equations (32) and (33) are used to introduce an admittance matrix modeling time-modulated circuit components. Let us derive this matrix, which helps solve complicated problems. We should substitute the expressions in (32) and (33) into (28) and attempt to find a way to relate ${I}_{n}$ to ${V}_{n}{.}$ By doing this, we realize that for every integer l $\left({{l} = {0},\pm{1},\pm{2},\ldots}\right)$ we have ${I}_{l} = {\Sigma}_{m}{\Sigma}_{n}{j}{[}{\omega}_{0} + {(}{n} + {m}{)}{\Omega}_{p}{]}{a}_{m}{V}_{n}{.}$ Here, the key point is that the summations are with respect to all possible values of m and n such that ${m} + {n} = {l}{.}$ For example, if ${l} = {2},$ the possible values are ${(}{m},{n}{)} = \ldots{(}{-}{1}, + {3}{),}{(}{0}, + {2}{),}{(} + {1}, + {1}{)}\ldots{.}$ As a result, if we restrict ourselves to N values for m, there are N different values for n. It is simple to play with the aforementioned relationship and present it as the following: \begin{align*}\begin{array}{l}{\left({\begin{array}{c}{{I}_{{-}{N}}}\\{{I}_{{1}{-}{N}}}\\{\vdots}\\{{I}_{N}}\end{array}}\right) = \left({\begin{array}{cccc}{j{\omega}_{{-}{N}}{a}_{0}}&{j{\omega}_{{-}{N}}{a}_{{-}{1}}}&{\cdots}&{j{\omega}_{{-}{N}}{a}_{{-}{2}{N}}}\\{j{\omega}_{{1}{-}{N}}{a}_{1}}&{j{\omega}_{{1}{-}{N}}{a}_{0}}&{\cdots}&{\vdots}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{j{\omega}_{N}{a}_{2N}}&{j{\omega}_{N}{a}_{{2}{N}{-}{1}}}&{\cdots}&{j{\omega}_{N}{a}_{0}}\end{array}}\right)\left({\begin{array}{c}{{V}_{{-}{N}}}\\{{V}_{{1}{-}{N}}}\\{\vdots}\\{{V}_{N}}\end{array}}\right){.}}\end{array} \tag{34} \end{align*}
The matrix multiplied by the voltage vector on the right side is the admittance matrix. Contemplating this matrix, we see that it is a square matrix $\left({{2}{N} + {1}\times{2}{N} + {1}}\right)$ whose diagonal is associated with the dc component of the modulated signal ${a}_{0}.$
In a similar way, matrix relations can be written for other periodically time-modulated circuit components [122] or impedance parameters of metasurfaces [123]. We see that circuits and metasurfaces with time-modulated elements can be studied using familiar circuit-theory methods, except that scalar voltages and currents become vector amplitudes of frequency harmonics, and scalar relations between voltages and currents are replaced by matrix relations. In the supplementary materials available at https://www.doi.org/10.1109/MAP.2023.3261601, we also discuss the power and energy relationships in time-varying capacitors.
In this short introductory tutorial, we focused on the most fundamental notions of the theory of LTV circuits and systems. Under the assumption of linearity, the general causal relationships between currents and voltages or fields and polarizations take the form of integral relationships where the kernel depends on two time variables: observation and delay time. In the frequency domain, we deal with two frequency variables: one to account for frequency dispersion and the other for the signal spectrum. Importantly, currents oscillating at a certain frequency become coupled to voltages at all frequencies, and the circuit-theory relationships take a matrix form. Although some phenomena, like frequency conversion, are similar to nonlinear phenomena, it is important to stress that all of the circuit and field equations remain linear, which greatly simplifies their solutions but does not allow realization of nonlinear effects. On the other hand, external time modulations break the time-reversal symmetry of field equations, allowing realization of nonreciprocal devices. In addition, time-modulated systems are not conservative because power can be exchanged with the external device that changes some of the system components in time. This property allows realization of various parametric amplifiers. Solutions of the linear matrix or integral equations for time-varying structures reveal possibilities for multiple applications of time-modulated circuit components and materials, starting from classical mixers or parametric amplifiers, to magnetless nonreciprocal devices or antennas working beyond fundamental limits set for linear passive devices.
This work was supported by the Academy of Finland under grant 330260. This article has supplementary downloadable material available at https://www.doi.org/10.1109/MAP.2023.3261601, provided by the authors.
Grigorii Ptitcyn (ptitcyn@seas.upenn.edu) is with Aalto University, 02150 Espoo, Finland. His main research interests include electromagnetics of complex media, metasurfaces, time-modulated structures, and reconfigurable surfaces.
Mohammad Sajjad Mirmoosa (mirmoosa@protonmail.com) is with Aalto University, 02150 Espoo, Finland. His main research interests include theories of electromagnetism and wave-matter interaction.
Amirhosein Sotoodehfar (amirhossein.sotoodeh@ucalgary.ca) is with the University of Calgary, Calgary, AB T2N 1N42500, Canada. His main research interests are quantum materials and quantum chemistry.
Sergei A. Tretyakov (sergei.tretyakov@aalto.fi) is with Aalto University, 02150 Espoo, Finland. His main scientific interests are electromagnetic field theory, complex media electromagnetics, and microwave engineering.
[1] G. Matz and F. Hlawatsch, “Fundamentals of time-varying communication channels,” in Wireless Communications Over Rapidly Time-Varying Channels. New York, NY, USA: Academic, 2011, pp. 1–63.
[2] L. A. Zadeh, “Frequency analysis of variable networks,” Proc. IRE, vol. 38, no. 3, pp. 291–299, Mar. 1950, doi: 10.1109/JRPROC.1950.231083.
[3] L. A. Zadeh, “The determination of the impulsive response of variable networks,” J. Appl. Phys., vol. 21, no. 7, pp. 642–645, 1950, doi: 10.1063/1.1699724.
[4] L. A. Zadeh, “On stability of linear varying-parameter systems,” J. Appl. Phys., vol. 22, no. 4, pp. 402–405, 1951, doi: 10.1063/1.1699972.
[5] L. A. Zadeh, “Correlation functions and power spectra in variable networks,” Proc. IRE, vol. 38, no. 11, pp. 1342–1345, Nov. 1950, doi: 10.1109/JRPROC.1950.234427.
[6] T. Kailath, “Sampling models for linear time-variant filters,” Res. Lab. Electron., Massachusetts Inst. Technol., Cambridge, MA, USA, Tech. Rep. 352, May 1959.
[7] T. Kailath, “Measurements on time-variant communication channels,” IRE Trans. Inf. Theory, vol. 8, no. 5, pp. 229–236, Sep. 1962, doi: 10.1109/TIT.1962.1057748.
[8] A. Gersho, “Characterization of time-varying linear systems,” School Elect. Eng., Cornell Univ., Ithaca, NY, USA, Tech. Rep. AD0420573, Feb. 1963.
[9] A. Gersho, “Properties of time-varying linear systems,” School Elect. Eng., Cornell Univ., Ithaca, NY, USA, Tech. Rep. EE552, 1962.
[10] P. Bello, “Time-frequency duality,” IEEE Trans. Inf. Theory, vol. 10, no. 1, pp. 18–33, Jan. 1964, doi: 10.1109/TIT.1964.1053640.
[11] P. Bello, “Characterization of randomly time-variant linear channels,” IEEE Trans. Commun. Syst., vol. 11, no. 4, pp. 360–393, Dec. 1963, doi: 10.1109/TCOM.1963.1088793.
[12] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, no. 12, pp. 1479–1484, Dec. 1947, doi: 10.1109/JRPROC.1947.226199.
[13] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, May 1948, doi: 10.1063/1.1715038.
[14] M. Jacob and H. Brauch, “Keying VLF transmitters at high speed,” Electronics, vol. 27, no. 12, pp. 148–151, 1954.
[15] H. Wolff, “High-speed frequency-shift keying of LF and VLF radio circuits,” IRE Trans. Commun. Syst., vol. 5, no. 3, pp. 29–42, Dec. 1957, doi: 10.1109/TCOM.1957.1097513.
[16] J. Galejs, “Switching of reactive elements in high-Q antennas,” IEEE Trans. Commun. Syst., vol. 11, no. 2, pp. 254–255, Jun. 1963, doi: 10.1109/TCOM.1963.1088740.
[17] H. Hartley, “Analysis of some techniques used in modern LF-VLF radiation systems,” IEEE Trans. Commun. Technol., vol. 16, no. 5, pp. 690–700, Oct. 1968, doi: 10.1109/TCOM.1968.1089913.
[18] V. F. Fusco and Q. Chen, “Direct-signal modulation using a silicon microstrip patch antenna,” IEEE Trans. Antennas Propag., vol. 47, no. 6, pp. 1025–1028, Jun. 1999, doi: 10.1109/8.777127.
[19] W. Yao and Y. Wang, “Direct antenna modulation - A promise for ultra-wideband (UWB) transmitting,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., 2004, vol. 2, pp. 1273–1276, doi: 10.1109/MWSYM.2004.1339221.
[20] A. Babakhani, D. B. Rutledge, and A. Hajimiri, “Near-field direct antenna modulation,” IEEE Microw. Mag., vol. 10, no. 1, pp. 36–46, Feb. 2009, doi: 10.1109/MMM.2008.930674.
[21] A. Cullen, “A travelling-wave parametric amplifier,” Nature, vol. 181, no. 4605, pp. 332–332, Feb. 1958, doi: 10.1038/181332a0.
[22] P. Tien, “Parametric amplification and frequency mixing in propagating circuits,” J. Appl. Phys., vol. 29, no. 9, pp. 1347–1357, Sep. 1958, doi: 10.1063/1.1723440.
[23] P. Tien and H. Suhl, “A traveling-wave ferromagnetic amplifier,” Proc. IRE, vol. 46, no. 4, pp. 700–706, Apr. 1958, doi: 10.1109/JRPROC.1958.286770.
[24] E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband nonreciprocal amplification in luminal metamaterials,” Phys. Rev. Lett., vol. 123, no. 20, Nov. 2019, Art. no. 206101, doi: 10.1103/PhysRevLett.123.206101.
[25] M. Currie and R. Gould, “Coupled-cavity traveling-wave parametric amplifiers: Part I-analysis,” Proc. IRE, vol. 48, no. 12, pp. 1960–1973, Dec. 1960, doi: 10.1109/JRPROC.1960.287564.
[26] A. Kamal, “A parametric device as a nonreciprocal element,” Proc. IRE, vol. 48, no. 8, pp. 1424–1430, Aug. 1960, doi: 10.1109/JRPROC.1960.287569.
[27] B. Anderson and R. Newcomb, “On reciprocity and time-variable networks,” Proc. IEEE, vol. 53, no. 10, pp. 1674–1674, Jul. 1965, doi: 10.1109/PROC.1965.4321.
[28] F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Techn., vol. 6, no. 2, pp. 167–172, Apr. 1958, doi: 10.1109/TMTT.1958.1124533.
[29] J.-C. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Techn., vol. 8, no. 1, pp. 18–29, Jan. 1960, doi: 10.1109/TMTT.1960.1124657.
[30] A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Techn., vol. 9, no. 4, pp. 337–343, 1961, doi: 10.1109/TMTT.1961.1125340.
[31] L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag., vol. 18, no. 2, pp. 242–253, Mar. 1970, doi: 10.1109/TAP.1970.1139657.
[32] R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag., vol. 19, no. 3, pp. 417–424, May 1971, doi: 10.1109/TAP.1971.1139931.
[33] D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag., vol. 14, no. 2, pp. 183–194, Mar. 1966, doi: 10.1109/TAP.1966.1138637.
[34] S. Averkov and N. Stepanov, “Wave propagation in systems with a traveling parameter,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, vol. 2, no. 2, pp. 203–212, Mar. 1959.
[35] V. Gavrilenko, G. Lupanov, and N. Stepanov, “Dynamo-optical effects in a plasma,” Radiophys. Quantum Electron., vol. 15, no. 2, pp. 138–143, Feb. 1972, doi: 10.1007/BF02209107.
[36] V. Gavrilenko, G. Lupanov, and N. Stepanov, “Electromagnetic waves in nonuniformly moving media (electromagnetic wave reflection from region with variable drift velocity and polarization of waves propagating in nonuniformly moving medium, applying to isotropic plasma),” in Proc. Electromagn. Wave Theory, 1971, pp. 518–525.
[37] L. Ostrovskii and N. Stepanov, “Nonresonance parametric phenomena in distributed systems,” Radiophys. Quantum Electron., vol. 14, no. 4, pp. 387–419, Apr. 1971, doi: 10.1007/BF01030725.
[38] N. Stepanov, “Adiabatic transformation of a wave spectrum in a nonstationary medium with dispersion,” Radiophys. Quantum Electron., vol. 12, no. 2, pp. 227–234, Feb. 1969, doi: 10.1007/BF01031285.
[39] V. Pikulin and N. Stepanov, “The kinetic theory of electromagnetic waves in an electron plasma with slowly varying parameters,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, vol. 16, pp. 1138–1145, Aug. 1973, doi: 10.1007/BF01031619.
[40] N. Stepanov, “On wave reflection from arbitrary moving inhomogeneity,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, vol. 5, pp. 908–916, May 1962.
[41] Y. M. Sorokin and N. Stepanov, “The reflection and refraction of electromagnetic waves in a moving inhomogeneous plasma,” Radiophys. Quantum Electron., vol. 14, no. 1, pp. 17–23, Jan. 1971, doi: 10.1007/BF01032902.
[42] N. Stepanov and Y. M. Sorokin, “Kinetic theory of electromagnetic wave reflection from a moving inhomogeneous plasma sheath,” Soviet Phys. Tech. Phys., vol. 17, p. 456, Sep. 1972.
[43] Y. M. Sorokin, “On a certain energy relationship for waves in systems having traveling parameters,” Radiophys. Quantum Electron., vol. 15, no. 1, pp. 36–39, Jan. 1972, doi: 10.1007/BF02209239.
[44] N. Stepanov, “Dielectric constant of unsteady plasma,” Radiophys. Quantum Electron., vol. 19, no. 7, pp. 683–689, Jul. 1976, doi: 10.1007/BF01034233.
[45] M. Mirmoosa, T. Koutserimpas, G. Ptitcyn, S. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” New J. Phys., vol. 24, no. 6, Jun. 2022, Art. no. 063004, doi: 10.1088/1367-2630/ac6b4c.
[46] N. Engheta, “Metamaterials with high degrees of freedom: Space, time, and more,” Nanophotonics, vol. 10, no. 1, pp. 639–642, 2021, doi: 10.1515/nanoph-2020-0414.
[47] D. M. Solís and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: A generalization of the Kramers-Kronig relations,” Phys. Rev. B, vol. 103, no. 14, Apr. 2021, Art. no. 144303, doi: 10.1103/PhysRevB.103.144303.
[48] J. Sloan, N. Rivera, J. D. Joannopoulos, and M. Soljacˇic´, “Casimir light in dispersive nanophotonics,” Phys. Rev. Lett., vol. 127, no. 5, Jul. 2021, Art. no. 053603, doi: 10.1103/PhysRevLett.127.053603.
[49] V. Pacheco-Peña and N. Engheta, “Temporal equivalent of the Brewster angle,” Phys. Rev. B, vol. 104, no. 21, Dec. 2021, Art. no. 214308, doi: 10.1103/PhysRevB.104.214308.
[50] V. Pacheco-Peña and N. Engheta, “Temporal aiming,” Light, Sci. Appl., vol. 9, no. 1, pp. 1–12, Jul. 2020, doi: 10.1038/s41377-020-00360-1.
[51] A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett., vol. 90, no. 1, Jan. 2003, Art. no. 014301, doi: 10.1103/PhysRevLett.90.014301.
[52] M. Fink, “From Loschmidt daemons to time-reversed waves,” Philosophical Trans. Roy. Soc. A, Math., Phys. Eng. Sci., vol. 374, no. 2069, Jun. 2016, Art. no. 20150156, doi: 10.1098/rsta.2015.0156.
[53] A. Przadka, S. Feat, P. Petitjeans, V. Pagneux, A. Maurel, and M. Fink, “Time reversal of water waves,” Phys. Rev. Lett., vol. 109, no. 6, Aug. 2012, Art. no. 064501, doi: 10.1103/PhysRevLett.109.064501.
[54] S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A.-C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett., vol. 107, no. 26, Dec. 2011, Art. no. 263901, doi: 10.1103/PhysRevLett.107.263901.
[55] M. Manteghi, “Fundamental limits, bandwidth, and information rate of electrically small antennas: Increasing the throughput of an antenna without violating the thermodynamic Q-factor,” IEEE Antennas Propag. Mag., vol. 61, no. 3, pp. 14–26, Jun. 2019, doi: 10.1109/MAP.2019.2907892.
[56] Y. Hadad, J. C. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Nat. Acad. Sci., vol. 113, no. 13, pp. 3471–3475, Mar. 2016, doi: 10.1073/pnas.1517363113.
[57] H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett., vol. 123, no. 16, Oct. 2019, Art. no. 164102, doi: 10.1103/PhysRevLett.123.164102.
[58] M. H. Mostafa, G. Ptitcyn, and S. Tretyakov, “Dipole antennas with time-varying body and shape,” in Proc. 14th Int. Congr. Artif. Mater. Novel Wave Phenomena (Metamater.), 2020, pp. 63–65, doi: 10.1109/Metamaterials49557.2020.9285128.
[59] S. Taravati, N. Chamanara, and C. Caloz, “Nonreciprocal electromagnetic scattering from a periodically space-time modulated slab and application to a quasisonic isolator,” Phys. Rev. B, vol. 96, no. 16, Oct. 2017, Art. no. 165144, doi: 10.1103/PhysRevB.96.165144.
[60] N. Chamanara, S. Taravati, Z.-L. Deck-Léger, and C. Caloz, “Optical isolation based on space-time engineered asymmetric photonic band gaps,” Phys. Rev. B, vol. 96, no. 15, Oct. 2017, Art. no. 155409, doi: 10.1103/PhysRevB.96.155409.
[61] C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl., vol. 10, no. 4, Oct. 2018, Art. no. 047001, doi: 10.1103/PhysRevApplied.10.047001.
[62] D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon., vol. 1, no. 3, pp. 198–204, Feb. 2014, doi: 10.1021/ph400058y.
[63] D. Correas-Serrano, J. Gomez-Diaz, D. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 1529–1532, 2016, doi: 10.1109/LAWP.2015.2510818.
[64] T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alù, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nature Commun., vol. 8, no. 1, pp. 1–9, Oct. 2017, doi: 10.1038/s41467-017-00798-9.
[65] R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt., vol. 20, no. 3, Feb. 2018, Art. no. 034007, doi: 10.1088/2040-8986/aaaa3e.
[66] A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and nonreciprocal amplification with dynamic gain-loss modulation,” Phys. Rev. A, vol. 99, no. 1, Jan. 2019, Art. no. 013824, doi: 10.1103/PhysRevA.99.013824.
[67] H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett., vol. 109, no. 3, Jul. 2012, Art. no. 033901, doi: 10.1103/PhysRevLett.109.033901.
[68] Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon., vol. 3, no. 2, pp. 91–94, Jan. 2009, doi: 10.1038/nphoton.2008.273.
[69] Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon., vol. 4, no. 7, pp. 1639–1645, Jun. 2017, doi: 10.1021/acsphotonics.7b00420.
[70] X. Wang et al., “Nonreciprocity in bianisotropic systems with uniform time modulation,” Phys. Rev. Lett., vol. 125, no. 26, Dec. 2020, Art. no. 266102, doi: 10.1103/PhysRevLett.125.266102.
[71] V. S. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” Proc. IEEE, vol. 108, no. 10, pp. 1684–1727, Oct. 2020, doi: 10.1109/JPROC.2020.3012381.
[72] A. M. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Exp., vol. 5, no. 11, pp. 2459–2467, Jul. 2015, doi: 10.1364/OME.5.002459.
[73] A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science, vol. 364, no. 6441, May 2019, Art. no. eaat3100, doi: 10.1126/science.aat3100.
[74] T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett., vol. 120, no. 8, Feb. 2018, Art. no. 087401, doi: 10.1103/PhysRevLett.120.087401.
[75] L. Zhang et al., “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater., vol. 31, no. 41, Aug. 2019, Art. no. 1904069, doi: 10.1002/adma.201904069.
[76] S. Taravati and A. A. Kishk, “Dynamic modulation yields one-way beam splitting,” Phys. Rev. B, vol. 99, no. 7, Feb. 2019, Art. no. 075101, doi: 10.1103/PhysRevB.99.075101.
[77] D. Ramaccia, D. L. Sounas, A. V. Marini, A. Toscano, and F. Bilotti, “Electromagnetic isolation induced by time-varying metasurfaces: Nonreciprocal Bragg grating,” IEEE Antennas Wireless Propag. Lett., vol. 19, no. 11, pp. 1886–1890, Nov. 2020, doi: 10.1109/LAWP.2020.2996275.
[78] D. Ramaccia, D. L. Sounas, A. Alù, F. Bilotti, and A. Toscano, “Nonreciprocity in antenna radiation induced by space-time varying metamaterial cloaks,” IEEE Antennas Wireless Propag. Lett., vol. 17, no. 11, pp. 1968–1972, Sep. 2018, doi: 10.1109/LAWP.2018.2870688.
[79] A. Li, Y. Li, J. Long, E. Forati, Z. Du, and D. Sievenpiper, “Time-modulated nonreciprocal metasurface absorber for surface waves,” Opt. Lett., vol. 45, no. 5, pp. 1212–1215, Mar. 2020, doi: 10.1364/OL.382865.
[80] M. Liu, D. A. Powell, Y. Zarate, and I. V. Shadrivov, “Huygens’ metadevices for parametric waves,” Phys. Rev. X, vol. 8, no. 3, Sep. 2018, Art. no. 031077, doi: 10.1103/PhysRevX.8.031077.
[81] K. Lee et al., “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nature Photon., vol. 12, no. 12, pp. 765–773, Dec. 2018, doi: 10.1038/s41566-018-0259-4.
[82] M. M. Salary, S. Jafar-Zanjani, and H. Mosallaei, “Electrically tunable harmonics in time-modulated metasurfaces for wavefront engineering,” New J. Phys., vol. 20, no. 12, Dec. 2018, Art. no. 123023, doi: 10.1088/1367-2630/aaf47a.
[83] K. Lee et al., “Electrical control of terahertz frequency conversion from time-varying surfaces,” Opt. Exp., vol. 27, no. 9, pp. 12,762–12,773, Apr. 2019, doi: 10.1364/OE.27.012762.
[84] X. Zou, Q. Wu, and Y. E. Wang, “Monolithically integrated parametric mixers with time-varying transmission lines (TVTLs),” IEEE Trans. Microw. Theory Techn., vol. 68, no. 10, pp. 4479–4490, Oct. 2020, doi: 10.1109/TMTT.2020.3011116.
[85] Y. Zhou et al., “Broadband frequency translation through time refraction in an epsilon-near-zero material,” Nature Commun., vol. 11, no. 1, pp. 1–7, May 2020, doi: 10.1038/s41467-020-15682-2.
[86] K. Pang et al., “Adiabatic frequency conversion using a time-varying epsilon-near-zero metasurface,” Nano Lett., vol. 21, no. 14, pp. 5907–5913, Jul. 2021, doi: 10.1021/acs.nanolett.1c00550.
[87] J. Yang et al., “Simultaneous conversion of polarization and frequency via time-division-multiplexing metasurfaces,” Adv. Opt. Mater., vol. 9, no. 22, Sep. 2021, Art. no. 2101043, doi: 10.1002/adom.202101043.
[88] B. Apffel and E. Fort, “Frequency conversion cascade by crossing multiple space and time interfaces,” Phys. Rev. Lett., vol. 128, no. 6, Feb. 2022, Art. no. 064501, doi: 10.1103/PhysRevLett.128.064501.
[89] N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B, vol. 98, no. 8, Aug. 2018, Art. no. 085142, doi: 10.1103/PhysRevB.98.085142.
[90] S. Lee et al., “Parametric oscillation of electromagnetic waves in momentum band gaps of a spatiotemporal crystal,” Photon. Res., vol. 9, no. 2, pp. 142–150, 2021, doi: 10.1364/PRJ.406215.
[91] Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett., vol. 36, no. 4, pp. 505–507, Feb. 2011, doi: 10.1364/OL.36.000505.
[92] D. Ramaccia, D. L. Sounas, A. Alù, A. Toscano, and F. Bilotti, “Doppler cloak restores invisibility to objects in relativistic motion,” Phys. Rev. B, vol. 95, no. 7, Feb. 2017, Art. no. 075113, doi: 10.1103/PhysRevB.95.075113.
[93] Z. Wu and A. Grbic, “Serrodyne frequency translation using time-modulated metasurfaces,” IEEE Trans. Antennas Propag., vol. 68, no. 3, pp. 1599–1606, Mar. 2020, doi: 10.1109/TAP.2019.2943712.
[94] M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl., vol. 11, no. 1, Jan. 2019, Art. no. 014024, doi: 10.1103/PhysRevApplied.11.014024.
[95] G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res., vol. 1, no. 2, Sep. 2019, Art. no. 023014, doi: 10.1103/PhysRevResearch.1.023014.
[96] M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A, vol. 102, no. 1, Jul. 2020, Art. no. 013503, doi: 10.1103/PhysRevA.102.013503.
[97] P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, and J. Pendry, “Fresnel drag in space–time-modulated metamaterials,” Proc. Nat. Acad. Sci., vol. 116, no. 50, pp. 24,943–24,948, Nov. 2019, doi: 10.1073/pnas.1915027116.
[98] V. Bruno et al., “Negative refraction in time-varying strongly coupled plasmonic-antenna–epsilon-near-zero systems,” Phys. Rev. Lett., vol. 124, no. 4, Jan. 2020, Art. no. 043902, doi: 10.1103/PhysRevLett.124.043902.
[99] V. Pacheco-Peña and N. Engheta, “Effective medium concept in temporal metamaterials,” Nanophotonics, vol. 9, no. 2, pp. 379–391, 2020, doi: 10.1515/nanoph-2019-0305.
[100] P. A. Huidobro, M. G. Silveirinha, E. Galiffi, and J. Pendry, “Homogenization theory of space-time metamaterials,” Phys. Rev. Appl., vol. 16, no. 1, Jul. 2021, Art. no. 014044, doi: 10.1103/PhysRevApplied.16.014044.
[101] M. Camacho, B. Edwards, and N. Engheta, “Achieving asymmetry and trapping in diffusion with spatiotemporal metamaterials,” Nature Commun., vol. 11, no. 1, pp. 1–7, Jul. 2020, doi: 10.1038/s41467-020-17550-5.
[102] H. Barati Sedeh, M. M. Salary, and H. Mosallaei, “Optical pulse compression assisted by high-Q time-modulated transmissive metasurfaces,” Laser Photon. Rev., vol. 16, no. 5, Mar. 2022, Art. no. 2100449, doi: 10.1002/lpor.202100449.
[103] M. Liu, A. B. Kozyrev, and I. V. Shadrivov, “Time-varying metasurfaces for broadband spectral camouflage,” Phys. Rev. Appl., vol. 12, no. 5, Nov. 2019, Art. no. 054052, doi: 10.1103/PhysRevApplied.12.054052.
[104] X. Wang and C. Caloz, “Spread-spectrum selective camouflaging based on time-modulated metasurface,” IEEE Trans. Antennas Propag., vol. 69, no. 1, pp. 286–295, 2020, doi: 10.1109/TAP.2020.3008621.
[105] A. Akbarzadeh, N. Chamanara, and C. Caloz, “Inverse prism based on temporal discontinuity and spatial dispersion,” Opt. Lett., vol. 43, no. 14, pp. 3297–3300, 2018, doi: 10.1364/OL.43.003297.
[106] V. Pacheco-Peña and N. Engheta, “Spatiotemporal isotropic-to-anisotropic meta-atoms,” New J. Phys., vol. 23, no. 9, Sep. 2021, Art. no. 095006, doi: 10.1088/1367-2630/ac21df.
[107] E. Lustig, Y. Sharabi, and M. Segev, “Topological aspects of photonic time crystals,” Optica, vol. 5, no. 11, pp. 1390–1395, Nov. 2018, doi: 10.1364/OPTICA.5.001390.
[108] Y. Sharabi, E. Lustig, and M. Segev, “Disordered photonic time crystals,” Phys. Rev. Lett., vol. 126, no. 16, Apr. 2021, Art. no. 163902, doi: 10.1103/PhysRevLett.126.163902.
[109] A. M. Shaltout, J. Fang, A. V. Kildishev, and V. M. Shalaev, “Photonic time-crystals and momentum band-gaps,” in Proc. Conf. Lasers Electro-Opt., 2016, pp. 1–2.
[110] N. Chamanara, Z.-L. Deck-Léger, C. Caloz, and D. Kalluri, “Unusual electromagnetic modes in space-time-modulated dispersion-engineered media,” Phys. Rev. A, vol. 97, no. 6, Jun. 2018, Art. no. 063829, doi: 10.1103/PhysRevA.97.063829.
[111] Z.-L. Deck-Léger, N. Chamanara, M. Skorobogatiy, M. G. Silveirinha, and C. Caloz, “Uniform-velocity spacetime crystals,” Adv. Photon., vol. 1, no. 5, Oct. 2019, Art. no. 056002, doi: 10.1117/1.AP.1.5.056002.
[112] J. Park and B. Min, “Spatiotemporal plane wave expansion method for arbitrary space–time periodic photonic media,” Opt. Lett., vol. 46, no. 3, pp. 484–487, Feb. 2021, doi: 10.1364/OL.411622.
[113] X. Wang, V. S. Asadchy, S. Fan, and S. Tretyakov, “Space-time metasurfaces for perfect power combining of waves,” ACS Photon., vol. 8, no. 10, pp. 3034–3041, May 2021, doi: 10.1021/acsphotonics.1c00981.
[114] E. Galiffi, Y.-T. Wang, Z. Lim, J. B. Pendry, A. Alù, and P. A. Huidobro, “Wood anomalies and surface-wave excitation with a time grating,” Phys. Rev. Lett., vol. 125, no. 12, Sep. 2020, Art. no. 127403, doi: 10.1103/PhysRevLett.125.127403.
[115] D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nature Photon., vol. 11, no. 12, pp. 774–783, Nov. 2017, doi: 10.1038/s41566-017-0051-x.
[116] E. Galiffi et al., “Photonics of time-varying media,” Adv. Photon., vol. 4, no. 1, Jan. 2022, Art. no. 014002, doi: 10.1117/1.AP.4.1.014002.
[117] A. V. Oppenheim, A. S. Willsky, and S. Nawab, Signals and Systems. Englewood Cliffs, NJ, USA: Prentice Hall, 1996.
[118] G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Annales Scientifiques de L’École Normale Supérieure, vol. 12, no. 2, pp. 47–88, 1883, doi: 10.24033/asens.220.
[119] F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Zeitschrift für Physik, vol. 52, nos. 7–8, pp. 555–600, Jul. 1929, doi: 10.1007/BF01339455.
[120] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering: From Fundamentals to Applications. Hoboken, NJ, USA: Wiley, 2017.
[121] W. Magnus and S. Winkler, Hill’s Equation. New York, NY, USA: Dover, 1979.
[122] P. Jayathurathnage, F. Liu, M. S. Mirmoosa, X. Wang, R. Fleury, and S. A. Tretyakov, “Time-varying components for enhancing wireless transfer of power and information,” Phys. Rev. Appl., vol. 16, no. 1, Jul. 2021, Art. no. 014017, doi: 10.1103/PhysRevApplied.16.014017.
[123] X. Wang, A. Díaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl., vol. 13, no. 4, Apr. 2020, Art. no. 044040, doi: 10.1103/PhysRevApplied.13.044040.
Digital Object Identifier 10.1109/MAP.2023.3261601