1. INTRODUCTION

Time-varying media have recently attracted significant attention in the broad physics and engineering communities, given their opportunities in the context of magnet-free non-reciprocity [1 –5], multifunctional metasurfaces [6], parity–time (PT)-symmetric structures [7,8], amplification [9], impedance matching [10 –12], and symmetry breaking for emission and absorption [13]. Since the pioneering work of Morgenthaler [14], research on dynamic media has focused mainly on time-periodic systems [15 –20], for which analysis is facilitated by the Floquet theorem. Imparting arbitrary time modulation schemes, beyond periodic, to wave–matter interactions can provide new possibilities and significantly broaden the field of dynamic media. Along this line, exotic wave phenomena have been recently shown in non-periodic time-varying structures, such as unlimited accumulation of energy [21,22], emulation of PT symmetry in Hermitian systems [23], and arbitrary transport tuning via reconfigurable effective static potentials [24]. These studies have so far heavily relied on inverse engineering of temporal modulation schemes, enabling only targeted functionalities and/or requiring simplifying assumptions such as weak coupling. At the other extreme compared to periodically varying media, instantaneous temporal switching has drawn increasing attention as a platform for wave engineering [19,25,26]. Abruptly changing in time the properties of an unbounded medium introduces a temporal boundary for wave propagation, dual of a spatial interface, generating forward and backward waves for which, instead of frequency, momentum is conserved and for which analytical solutions exist [26]. When considering more complicated structures, finite in one or more spatial dimensions and open to radiation, however, temporal switching interplays with other scattering phenomena, and the analysis rapidly becomes very challenging in the absence of general constraints such as flux conservation.

To establish a thorough analysis of temporal switching in open systems, input–output techniques appear to be an ideal tool, since they can treat the field redistribution in a scatterer in time domain after the abrupt change of its properties as initial conditions to study the field evolution and associated scattering processes. This approach, initially established in quantum optics [27,28], has been applied to quantum networks [29 –31], photon and charge transport [32 –35], and quantum scattering [36]. As a variant for classical wave physics, coupled-mode theory (CMT) has been used extensively to model coupled resonator systems, well suited in regimes near resonance [37 –39]. Except for a few specific scenarios [40,41], its derivation follows from general principles, leaving many free phenomenological parameters undetermined. Nevertheless, there are two ab initio routes towards a rigorous input–output formalism. The first one is based on Feshbach’s projector technique [42,43], based on the derivation of suitable system-bath Hamiltonians [36,44 –47], whereas the second one involves an expansion in terms of quasi-normal modes (QNMs) [48 –53]. The former, based on a Hermitian Hamiltonian, cannot easily incorporate material loss and dispersion, while the latter typically focuses on scattering in frequency domain and for convenience deals with background and scattered fields, and thus its connection with CMT is vague since, for example, the background fields cannot be identified as inputs in CMT [51].

The goal of this work is twofold: first, we extend the QNM approach to time-switched open systems [51,52], developing a generalized temporal input–output theory that bridges the gap between QNMs and conventional CMT. This approach is ideally suited for time-switched open systems, and yields new physical insights into the exotic wave phenomena arising in them. We then apply this theory to the analysis of temporal switching in thin-layer absorbers to overcome the trade-off between bandwidth and thickness that applies to any time-invariant passive absorber [54].

 

Fig. 1. (a) Gaussian pulse impinges on the Dallenbach absorber. (b) Reflection magnitude $|\rho|$ versus incident (angular) frequency $\omega$ for the optimal thin absorber with $\varepsilon _r^{\rm{opt}} \approx 69.85$, ${\hat \sigma ^{\rm{opt}}} \approx 1.998$, and $d/{\lambda _0} = 0.03$. Red-dashed line represents the frequency spectrum $| {a(\omega)/a({{\omega _0}})} |$ of a pulse with FWHM $\delta \omega = 0.2{\omega _0}$. (c) Trajectories of the first two (normalized) QNM pairs ${\hat y_n}, n = \pm 1, \pm 2$ as ${\varepsilon _r}$ varies from 140 to 2 while $\hat \sigma = {\hat \sigma ^{\rm{opt}}}$. Two special scenarios when ${\varepsilon _r} = \varepsilon _r^{\rm{opt}},\varepsilon _r^{\rm{EP}}({\approx 2.1})$ are marked, respectively, by green plus and red cross symbols. ($d$) Time-domain reflection ${b^{\rm{out}}}({{z_0},t})/{a_{\rm{peak}}}$ (black solid line) for the impinging pulse located initially at ${z_i} = 200{z_0}$. The results calculated at time ${t_s} = 40[{1/{\omega _0}}]$ based on Eq. (7) (green circles) and Eq. (8) with $C = 1$ (red dashed line, see Supplement 1) are also shown.

Download Full Size | PPT Slide | PDF

2. RESULTS AND DISCUSSION

A. Thin Absorbers and Bandwidth Limitations

Absorbing incident radiation over a large bandwidth is an important functionality for a variety of technologies, from radar to anechoic measurements. Broadband absorption comes at the cost of thicker and heavier absorbing layers: consider, for instance, a Dallenbach screen, consisting of a homogeneous lossy nonmagnetic layer of thickness $d$ backed by a perfect electric conductor (PEC), as in Fig. 1(a). For normal incidence, the complex reflection coefficient $\rho (\omega) = \frac{{\tanh ({jn\omega d/{c_0}}) - n}}{{\tanh ({jn\omega d/{c_0}}) + n}}$ [55], where the refractive index $n = \sqrt {{\varepsilon _r}} \;\sqrt {1 - j\sigma /({{\varepsilon _0}{\varepsilon _r}\omega})}$ is written in terms of the dielectric conductivity $\sigma$ and the real-valued (dispersion-less) relative permittivity ${\varepsilon _r}$, and ${c_0} = 1/\sqrt {{\mu _0}{\epsilon _0}}$ is the speed of light in vacuum, with ${{\mu}_0}\;({{\epsilon _0}})$ being the permeability (permittivity). A thin screen with ${\varepsilon _r} \gg 1$ optimally absorbs for $\varepsilon _r^{\rm{opt}} \approx {({\frac{{{\lambda _0}}}{{4d}}})^2}$ and ${\hat \sigma ^{\rm{opt}}} \approx 2\sqrt {\varepsilon _r^{\rm{opt}}} {{\rm \tanh}^{- 1}}\left({1/\sqrt {\varepsilon _r^{\rm{opt}}}}\right)$, where $\hat \sigma = {\sigma}{\eta _0}d$, with ${\eta _0} = \sqrt {{\mu _0}/{\epsilon _0}}$ being the free-space impedance and ${\lambda _0} = 2{\pi}{c_0}/{\omega _0}$ the free-space wavelength at the operating frequency ${\omega _0}$, yielding $\rho ({{\omega _0}}) = 0$. Correspondingly, the maximum bandwidth to least thickness ratio ${\Delta \lambda}/{d} \approx 32{\rho _0}/\pi$ for this geometry [56], where ${\rho _0}$ is the largest acceptable value of $| {\rho (\omega)} |$ within the operating bandwidth ${\Delta \lambda}$ centered around ${\lambda _0}$. This optimal condition falls within the more general Rozanov bound ${\Delta \lambda}/{d} \lt 16/| {{{{\rm ln}\rho}_0}} |$, valid for any nonmagnetic, passive, time-invariant absorber [54]. To reach this limit, further dispersion engineering of the material is generally required. Ultimately, the bandwidth limitation of thin absorbers can be traced back to Kramers–Kronig relations, dictating the temporal dynamics of causal media. Temporal variations and switching change these dynamics, generalizing Kramers–Kronig relations [57], and thus relaxing the bandwidth constraints.

B. QNM Formulation

Given its open resonance, the response of the Dallenbach screen in Fig. 1(a) can be well captured by a QNM formulation. Assuming normal incidence along the positive $z$ direction, we can formulate the problem as

We expand the scattered field $|{{\Psi}_s}({z, t})\rangle = \mathop \sum \nolimits_n {A_n}(t)|{{\tilde \Psi}_n}(z)\rangle$ [with the expansion coefficients ${A_n}(t)$] in terms of QNMs $|{{\tilde \Psi}_n}({z, t})\rangle = |{{\tilde \Psi}_n}(z)\rangle {{e}^{{j}{{\tilde \omega}_n}{t}}}$, which individually satisfy ${\tilde \omega _n}| {{{{\tilde \Psi}}_n}(z)\rangle = \hat H} |{{\tilde \Psi}_n}(z)\rangle$ and the outgoing and PEC boundary conditions at ${z} = {z_0}$ and $z = 0$, yielding $\hat H|{{\Psi}_s}({z,t})\rangle = \mathop \sum \nolimits_n {A_n}(t){\tilde \omega _n}|{{\tilde \Psi}_n}(z)\rangle$ [52,58,59]. $|{{\Psi}_s}({z,t})\rangle$ is obtained by writing the field $|{\Psi}({z,t})\rangle$ inside the layer as the superposition $|{\Psi}({z,t})\rangle = |{{\Psi}_b}({z, t})\rangle + |{{\Psi}_s}({z,t})\rangle$, where the background field $|{{\Psi}_b}({z,t})\rangle = a({t - \frac{z}{{{c_0}}}})\left({\begin{array}{*{20}{c}}1\\1\end{array}}\right) + a({t + \frac{z}{{{c_0}}}})\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}}\right),\;{z_0} \le z \le 0$. Consequently, using Eq. (1), the scattered field $|{{\Psi}_s}({z,t})\rangle$ satisfies

C. Input–Output Formalism

Based on Eqs. (5) and (6), we can introduce an input–output formalism in terms of the modal coefficients ${\psi _n}(t) \equiv \int_{{z_0}}^0 \langle {\tilde \Psi}_n^*(z)|M|{\Psi}({z, t}) \rangle {\rm d}z$ of $|{\Psi}({z, t})\rangle$ with respect to the QNMs. Specifically, using the expression ${A_n}(t) = {\psi _n}(t) - \int_{{z_0}}^0 \langle {\tilde \Psi}_n^*(z)|M|{{\Psi}_b}({z,t}) \rangle {\rm d}z$ in Eqs. (5) and (6), we get

Our input–output formulation Eq. (7) collapses to conventional CMT in the limit in which the response is dominated by a single QNM pair with complex-frequency ${\tilde \omega _{\pm 1}}$, such that the internal fields $|{\Psi}({z, t})\rangle \approx \mathop \sum \limits_{n = \pm 1} {\psi _n}(t)|{{\tilde \Psi}_n}(z)\rangle$. This approximation is valid for our thin absorber when ${\varepsilon _r} \gg 1$ and the operating frequency ${\omega _0}\sim{\rm Re}({{{\tilde \omega}_{- 1}}}) \gt 0.$ In this regime, we find (see Supplement 1)

We apply our formalism to study the scattering from a time-switched Dallenbach screen. Consider [Fig. 1(a)] a wave packet $a({t - \frac{z}{{{c_0}}}}) = f({t - \frac{{z - {z_i}}}{{{c_0}}}})\cos [{{\omega _0}({t - \frac{{z - {z_i}}}{{{c_0}}}})}]$, with normalized envelope $f(t) = {a_{\rm{peak}}}{e^{- 2{t^2}\ln 2/\delta {t^2}}}$, where ${z_i} \ll {z_0}$ determines the initial position of the Gaussian pulse with peak value ${a_{\rm{peak}}} = {({\frac{{2\sqrt {\ln 2}}}{{\sqrt \pi \delta t}}})^{1/2}}$. In the frequency domain, the incident wave packet $a(\omega) = \frac{1}{2}{e^{j\omega {z_i}/{c_0}}}[{\tilde f({\omega - {\omega _0}}) + \tilde f({\omega + {\omega _0}})}],$ where $\tilde f(\omega) = 2{({\frac{{\sqrt {\pi \ln 2}}}{{\delta \omega}}})^{1/2}}{e^{- 2{\omega ^2}\ln 2/\delta {\omega ^2}}}$, with $\delta \omega = 4\ln 2/\delta t$ being the FWHM. We readily calculate the reflection ${b^{\rm{out}}}({{z_0},t})$ and time-dependent internal fields $|{\Psi}({z,t})\rangle,\;{z_0} \le z \le 0$:

Consider the Dallenbach screen with $d/{\lambda _0} = 0.03$ and optimal material parameters $\varepsilon _r^{\rm{opt}} = 69.85$ and ${\hat \sigma ^{\rm{opt}}} = 1.998$. In Fig. 1(b), we show the reflection spectrum (green solid line), together with the frequency spectrum $| {a(\omega)/a({{\omega _0}})} |$ (red-dashed line) for an impinging pulse with FWHM $\delta \omega = 0.2{\omega _0}$ and initial position ${z_i} = 200{z_0}$. In Fig. 1(c), we show the evolution of the first few (normalized) QNM frequencies ${\hat y_n} = {{\tilde \omega}_{n}}{{z}_0}\sqrt {{{\varepsilon}_{ r}}} /{{c}_0},\;n = \pm 1,\; \pm 2$ as we vary ${\varepsilon _r}$ from 140 to 2, with $\hat \sigma = {\hat \sigma ^{\rm{opt}}}$. The green plus and red cross symbols correspond, respectively, to ${\varepsilon _r} = \varepsilon _r^{\rm{opt}}$ for the optimal absorber and ${\varepsilon _r} = \varepsilon _r^{\rm{EP}} \approx 2.1$ for the permittivity value corresponding to an exceptional point (EP) where the $n = \pm 1$ QNMs coalesce (Supplement 1). Around the EP, the complex frequency ${{\hat y}_{\pm 1}}({{\varepsilon _r}})$ of the $n = \pm 1$ QNMs follows a square root behavior, a characteristic signature of second-order EP singularities, as ${{\hat y}_{\pm 1}}({{\varepsilon _r}}) \approx - {j}1.84 \pm 1.53\sqrt {{\varepsilon _r} - \varepsilon _r^{\rm{EP}}}$, whose value changes as $\hat \sigma$ deviates from ${\hat \sigma ^{\rm{opt}}}$ (Supplement 1) (see also the impact of material dispersion on EPs in Refs.  [63,64]). Using Eq. (7) and the calculated QNMs, we evaluate the temporal evolution ${b^{\rm{out}}}({{z_0},t})/{a_{\rm{peak}}}$ of the slab reflection [green circles in Fig. 1(d)], which fits perfectly with the exact result provided by Eq. (9) [black solid line]. In the same subfigure, we also show the result from conventional CMT Eq. (8), which works well in the ${\varepsilon _r} \gg 1$ regime except for an initial short period.

D. Time-switched Absorber

Our ab initio formalism is ideally suited to study the effect of abruptly switching the properties of the Dallenbach screen. Specifically, we consider the case in which the relative permittivity abruptly changes at time ${t_s}$ from ${\varepsilon _1}$ to ${\varepsilon _2}$. The time-domain response of the screen after ${t_s}$ cannot be deduced from Fourier analysis, but it can be readily obtained from our input–output formalism Eq. (7), once knowing the internal field $|{\Psi}({z,\;t_s^ +})\rangle, {z_0} \le z \le 0$ immediately after the switch. To this end, we employ the continuity conditions for the electric displacement and magnetic induction across the temporal boundary at ${{t}_{s}}$ [65,66], which equivalently reads $|{\Psi}({z, t_s^ +})\rangle = \left({\begin{array}{*{20}{c}}{{\varepsilon _1}/{\varepsilon _2}}&0\\0&1\end{array}}\right)|{\Psi}({z, t_s^ -})\rangle$. In turn, the internal field $|{\Psi}({z, t_s^ -})\rangle$ immediately before the switch can be calculated from Eq. (9). Generally, the abrupt change of material properties does not preserve energy, and the injected energy $\Delta {E} \approx {| {{\psi _{- 1}}({t_s^ +})} |^2} - {| {{\psi _{- 1}}({t_s^ -})} |^2}$ using Eq. (8).

In Fig. 2, we study the effect of this abrupt switching on the reflection coefficient of the optimal absorber. In Fig. 2(a), we abruptly change ${\varepsilon _r}$ at an arbitrary switching time ${t_s} \in [{45, 55}]$ (in units of $1/{\omega _0}$) from ${\varepsilon _1} = \varepsilon _r^{\rm{opt}}$ to ${\varepsilon _2} \in [{10, 120}]$. For each pair $({{t_s},{\varepsilon _2}})$, we calculate the relative total reflected energy $10\;{{\rm log}_{10}}| {{E^{\rm{out}}}/{E^{\rm{in}}}} |,$ where ${E^{\rm out/in}} = \int_0^\infty {P^{\rm out/in}}({{z_0},t}){\rm d}t$, with ${P^{\rm{out}}} = \frac{1}{{{\eta _0}}}{({{b^{\rm{out}}}})^2}$ and ${P^{\rm{in}}} = \frac{1}{{{\eta _0}}}{({{a^{\rm{in}}}})^2}$ being the total outgoing/incoming energy. The total reflected energy (upper orange surface) is larger than the optimal static absorber, i.e., the case ${\varepsilon _2} = {\varepsilon _1} = \varepsilon _r^{\rm{opt}}$. When ${\varepsilon _2} \ne {\varepsilon _1} = \varepsilon _r^{\rm{opt}}$, the overall reflection is expected to increase as the switching time ${t_s}$ decreases, since the pulse interacts less with the medium with permittivity $\varepsilon _r^{\rm{opt}}$ before switching. However, the switching operation has the benefit of spreading part of the reflected energy over a wide range of frequencies outside the bandwidth of the incident pulse. Indeed, the rapid temporal switching induces frequency conversion for the fields inside the resonator, as exploited, e.g.,  in time-variant metasurfaces [67,68] and nuclear resonance spectroscopy [69,70]. Therefore, the relative reflected energy, as well as the visibility of the target, is reduced (lower green mesh surface) when we replace the total reflected energy ${E^{\rm{out}}}$ with the filtered energy $E_{\textit{bw}}^{\rm{out}}$ within the pulse frequency window $({{\omega _0} - \Delta \omega ,{\omega _0} + \Delta \omega})$ with $\Delta \omega = \delta \omega = 0.2{\omega _0}$, which covers 98% of the total incoming energy. The switching mechanism effectively minimizes detectability in reflection by spreading the non-absorbed energy over a broad frequency range, as explored in Refs.  [71 –75] for lossless periodically modulated screens. Our approach combines reduced reflection due to absorption with spreading arising through a single switching event, making the functionality efficient and effective. We envision these devices being operated passively as a regular absorber using, e.g., metasurfaces integrating switches at lower frequencies or phase change materials [76,77], with their transition triggered by the arrival of an incoming pulse to improve their bandwidth performance beyond the limits of passive absorbers.

 

Fig. 2. Performance of switched absorbers. (a) The upper orange surface presents the total reflected energy $10\; {{\rm log}_{10}}| {{E^{\rm{out}}}/{E^{\rm{in}}}} |$ versus switching time ${t_s}{\omega _0} \in [{45,55}]$ and relative permittivity ${\varepsilon _2} \in [{10, 120}]$ after switching. The lower green mesh surface shows the reflected energy $10\;{{\rm log}_{10}}| {E_{\textit{bw}}^{\rm{out}}/{E^{\rm{in}}}} |$ within the bandwidth $({{\omega _0} - \delta \omega ,{\omega _0} + \delta \omega})$ of the incoming pulse, with $\delta \omega = 0.2{\omega _0}$. (b) Reflected energy $10\;{{\rm log}_{10}}| {E_{\textit{bw}}^{\rm{out}}/{E^{\rm{in}}}} |$ in (a) as a function of ${\varepsilon _2} \in [{2,\;120}]$ when ${t_s} = 49\;[{1/{\omega _0}}]$. The result from Eq. (8) is also shown (blue dashed line). Left inset: zoom around the EP ${\varepsilon _2} = \varepsilon _r^{\textit{EP}}$. Right inset: normalized switching energy $\Delta E/{E^{\rm{in}}}$ versus ${\varepsilon _2}$, which changes sign at ${\varepsilon _2} = \varepsilon _r^{\rm{opt}}$ [see inset for the portion around $({\varepsilon _r^{\rm{opt}},0})$ (red plus symbol)]. Before switching, the screen operates in the optimal static mode (see Fig. 1).

Download Full Size | PPT Slide | PDF

For given switching time ${t_s}$, the filtered reflection energy $E_{\textit{bw}}^{\rm{out}}$ monotonically decreases with increasing switching strength $| {{\varepsilon _2} - {\varepsilon _1}} |$ when the initial permittivity ${\varepsilon _1} = \varepsilon _r^{\rm{opt}}$, until the system reaches the EP in Fig. 1(c) {see Fig. 2(b) plotted for ${t_s} = 49\;[{1/{\omega _0}}]$}. Around the EP, our input–output formalism converges increasingly slowly, as also seen from the deteriorated comparison with conventional CMT Eq. (8) (blue dashed line). In the left inset, we zoom around ${\varepsilon _2} \approx \varepsilon _r^{\textit{EP}}$, highlighting the rapid change in reflection, and thus the sensitivity around the EP, with a deterioration in overall response. In the right inset, we plot the normalized injected energy $\Delta E/{E^{\rm{in}}}$ by switching as a function of ${\varepsilon _2}$. Depending on whether ${{\varepsilon}_2} \lt {\varepsilon _1} = \varepsilon _r^{\rm{opt}}$, we inject or extract increasing energy as the switching amplitude $| {{\varepsilon _1} - {\varepsilon _2}} |$ grows.

For a given switching amplitude, we find an optimal switching time $t_s^{\rm{opt}}$ that minimizes $E_{\textit{bw}}^{\rm{out}}$. In Fig. 3, we fix ${{\varepsilon}_2} = 8$, and calculate the effective reflection spectrum $| \rho | = | {{{\tilde b}^{\rm{out}}}(\omega)/a(\omega)} |$ for various impinging pulses with FWHM $\delta \omega /{\omega _0} = 0.05,0.1,0.15,0.2$. For each scenario, we employ $t_s^{\rm{opt}}$, which occurs after the peak of each impinging pulse passes through the front surface of the absorber, and can be easily implemented by triggering the switch when the input energy decays. The reflected energy is spread over broader bandwidths, well beyond the optimal reflection achievable in a passive scenario in Fig. 1(b). The bandwidth can be further increased by adjusting the desired minimum attainable reflection level, or by adding a second switching for longer pulses. More generally, we envision the possibility of optimizing the performance by shaping the temporal response of our refractive index variation, especially adjusting it to the incoming pulse shape, if it is known [68,78]. In this figure, we have included also a full-wave simulation with COMSOL considering a finite switching time (purple circles), which validates our theoretical results.

 

Fig. 3. Effective reflection spectrum $| {\rho} |$ of the time-switched absorber for various impinging Gaussian pulses of FWHM $\delta \omega /{\omega _0} = 0.05, 0.1, 0.15, 0.2$, with sufficiently larger initial positions ${{z}_{i}}/{{z}_0} = 750, 350, 300, 200$, respectively. Here, the absorber permittivity switches from ${{\varepsilon}_1} = {\varepsilon}_{r}^{{\rm opt}}$ to ${{\varepsilon}_2} = 8$ at the optimal switching time $t_s^{\rm{opt}} \approx 197,90,71,49 [{1/{\omega _0}}]$. The green-dashed line indicates the reflection spectrum of the optimal static absorber in Fig. 1(b), while purple circles report a simulation result from COMSOL Multiphysics with a short but finite switching time. Other parameters are the same as in Figs. 1 and 2.

Download Full Size | PPT Slide | PDF

3. CONCLUSION

In this work, we have analyzed the effect of abrupt time switching on the overall bandwidth of thin absorbers, showing that the bandwidth of reflection can be largely enhanced compared to passive time-invariant slabs. Our analysis is based on a newly introduced input–output formalism that efficiently captures the scattering phenomena of open systems, ideally suited to analyze temporally switched scenarios. This general formulation, derived from a QNM expansion, is exact and enables a clean first-principle derivation of conventional CMT. Using this approach, we found that temporal switching can effectively reduce the reflection of impinging pulses, and may strengthen the effectiveness of non-magnetic thin absorbers. Our approach can be extended to time-switched metamaterials and metasurfaces, and scattering problems involving nanoparticles, for which the QNM expansion can be efficiently employed. The question on the maximum bandwidth for this response is fundamentally linked to the interplay between the material dynamics and the temporal variations of the wave, and on the available material responses in realistic settings. Furthermore, we believe that our analysis can also be extended to a quantum treatment, involving a proper second quantization scheme based on QNMs, leading to the potential construction of Fock states in dissipative systems [53].