Temporal Switching to Extend the Bandwidth of Thin Absorbers

Wave absorption in time-invariant, passive thin films is fundamentally limited by a trade-off between bandwidth and overall thickness. In this work, we investigate the use of temporal switching to reduce signal reflections from a thin grounded slab over broader bandwidths. We extend quasi-normal mode theory to time switching, developing an ab initio formalism that can model a broad class of time-switched structures. Our formalism provides optimal switching strategies to maximize the bandwidth over which minimal reflection is achieved, showing promising prospects for time-switched nanophotonic and metamaterial systems to overcome the limits of time-invariant, passive structures.

Time-varying media have been recently raising 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], PT-symmetric structures [7]- [8], amplification [9] and symmetry breaking for emission and absorption [10]. Since the pioneering work of Morgenthaler [11], research on dynamic media has mainly focused on time-periodic systems [12]- [17], for which the 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 [18]- [19] and arbitrary transport tuning via reconfigurable effective static potentials [20]. These studies have so far heavily relied on inverse engineering of the temporal modulation schemes, enabling only targeted functionalities and/or requiring simplifying assumptions like weak couplings. At the other extreme compared to periodically varying media, instantaneous temporal switching has drawn increasing attention as a platform for wave engineering [16], [21]- [22]. 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 [22]. 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 constrains 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 [24]- [25], has been applied to quantum networks [26]- [28], photon and charge transport [29]- [32] and quantum scattering [33]. As a variant for classical wave physics, coupled-mode theory (CMT) has been extensively used to model coupled resonator systems, well suited in regimes near resonance [34]- [36]. Except for few specific scenarios [37]- [38], 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 [39]- [40], based on the derivation of suitable system-bath Hamiltonians [33], [41]- [44], whereas the second one involves an expansion in terms of quasinormal modes (QNMs) [45]- [50]. The former, involving Hermitian system 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 [48].
The goal of this work is twofold: first we extend the QNM approach to time-switched open systems [48]- [49], 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, in order to overcome the trade-off between bandwidth and thickness that applies to any time-invariant passive absorber [23].
| * = * sin * , − tan * : cos * • , : ≤ ≤ 0 − * sin * :  (6)]. We emphasize that the infinite summation in τ typically does not converge, but the observable quantities, such as AxC : , , do [33]. The ab initio input-output formulation Eq. (7) manifests causality, and can be applied beyond the weak-coupling limit, in contrast with the conventional CMT. To the best of our knowledge, Eq. (7) is the first example of rigorous inputoutput formalism based on QNMs, ideally suited to tackle time-switched open resonators as discussed later.
Our input-output formulation Eq. (7)   in the lossless scenario and compare it with the general principle formulation. As expected, Eq. (8) converges to the conventional CMT [36] in the weak-coupling regime. Yet, when strong coupling and/or material dissipation is present, for which CMT cannot apply, our ab initio input-output formalism Eq. (7) provides a generalized tool to analytically study the scattering problem.
We apply our formalism to study the scattering from a time-switched Dallenbach screen. Consider, Fig. 1(a) where is the complex reflection coefficient derived above, and the complex wave number . We can employ Eq. (7) to evaluate the reflection AxC : , starting from an arbitrary time j , after knowing the initial condition * j = Ψ * * | |Ψ , j : Z / . Note that * 0 = 0, since initially the incoming wave packet is far away from the screen, and * j , j > 0 can be calculated exactly from the knowledge of the internal field |Ψ , j in Eq. (9) [58].
Consider the Dallenbach screen with / : = 0.03 and optimal material parameters 4 ABC = 69.85 and ABC = 1.998. In Fig. 1(b), we show the reflection spectrum (green-solid line), together with the frequency spectrum / : (red-dashed line) for an impinging pulse with FWHM = 0.2 : and initial position d = 200 : . In Fig. 1(c), we show the evolution of the first few (normalized) QNM frequencies * = ω ' z : ε¸/c : , = ±1, ±2 as we vary 4 from 140 to 2, with = ABC . The green plus and red cross symbols correspond respectively to 4 = 4 ABC for the optimal absorber and 4 = 4 ¹º ≈ 2.1 for the permittivity value corresponding to an exceptional point (EP) where the = ±1 QNMs coalesce [57]. Around the EP, the complex frequency y ±I 4 of the = ±1 QNMs follow a square root behavior, a characteristic signature of secondorder EP singularities, as y ±I 4 ≈ −j1.84 ± 1.53 4 − 4 ¹º [57]. Using Eq. (7) and the calculated QNMs, we evaluate the temporal evolution AxC : , / B©ªŽ of the slab reflection [green circles in Fig. 1(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 j from I to G . The time-domain response of the screen after j cannot be deduced from Fourier analysis, but it can be readily obtained from our input-output formalism Eq. (7), once knowing the internal field |Ψ , j 1 , : ≤ ≤ 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 OE [59] - [60], which equivalently reads |Ψ , j 1 = I / G 0 0 1 |Ψ , j 0 . In turn, the internal field |Ψ , j 0 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 ∆E ≈ 0I j 1 G − 0I j 0 G 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) The switching mechanism effectively minimizes detectability in reflection by spreading the nonabsorbed energy over a broad frequency range, as explored in Refs. [61]- [65] 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 phase change materials, with its phase transition being triggered by the arrival of an incoming pulse to improve their bandwidth performance beyond the limits of passive absorbers. For given switching amplitude, we find an optimal switching time j ABC that minimizes pÃ AxC . In Fig. 3, we fix ε G = 8, and calculate the effective reflection spectrum = AxC / for various impinging pulses with FWHM / : = 0.05, 0.1, 0.15, 0.2 . For each scenario, we employ j ABC , which occurs right 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. [red dashed line and see Ref. [57]] are also shown. indicates the reflection spectrum of the optimal static absorber in Fig. 1(b). Other parameters are the same as Fig. 1 and 2.

I. First-principle derivation of conventional coupled-mode theory (CMT)
In this section, we present a first-principle derivation of the conventional CMT using the inputoutput formulation Eq. (7) of the main text. We first note that the QNMs appear in pairs [66] and their complex wave characteristics obey 0* = − * * , 0* = − * * and 0* G = * * G with the label = ±1, ±2, ±3, ⋯. Therefore, to ensure the real-valued nature of the outgoing radiation Next, we consider lossless screens and analytically examine the derived conventional CMT.
We start our discussion by rewriting Eq. (8) [see main text] in the conventional form [68] where the resonant (angular) frequency of the single mode Ω = Re 0I , its decay rate Γ = Im 0I , the coupling between the mode and the port K = D = − 0I sin 0I : . Therefore, we can easily calculate the (approximated) direct path C in Eq.
(S10), which turns out to be C = 1 + j Ù − I ‰ v + 1/ 4 in the weak-coupling limit when 4 → ∞. We point out that in the process of decoupling the positive and negative frequencies for the derivation of the conventional CMT in Eq. (S10), we exclude the contribution of QNM n = 1 for the direct path C. This contribution is I τ 0)¥+ / : / : τ : 0GZ / /. / , : = 0I and supposed to be of higher order since the operation frequency : is far away from the complex frequency I of the QNM = 1. Indeed, in the scenario here, it approaches to 2 − 16/ G / 4 when 4 → ∞.
In the weak-coupling regime where 4 → ∞ and when σ = 0 for the lossless screen, we can also expand the other system parameters in Eq. (S10 We can easily see that in the leading order the system parameters as derived satisfy all the requirements imposed by the timereversal symmetry and energy conservation principles, which are 1 = 2Γ, K = D and C * = − [68]. Finally, we remark that the performance of the conventional CMT as Eq. (S10) will not be improved when we incorporate higher-order terms in system parameters, since in its derivation the crucial (rotating-wave) approximation has been already made by decoupling the positive and negative frequencies. Typically, the approximated equation under the constraints of general principles will behave relatively better.

II. Formation of a QNM EP
We analyze the formation of the QNM EP y 4 ¹º = − , p > 0 as seen in Fig. 1(c)  (S12) The emergence of the EP requires that I ≠ 0, and then combining Eqs. (S12) and (S13) allows us to evaluate the EP y 4 ¹º = − with In turn, with a given value for the normalized conductivity, substituting the expression for p in Eq. (S14) into Eq. (S12) allows us to determine the parameter 4 ¹º . For example, in the case of the main text when = ABC = 1.998, we get 4 ¹º ≈ 2.1 and correspondingly p ≈ 1.84 using Eq.
(S14). To obtain the expansion coefficient I of y ±I 4 around the EP with respect to ∆ , we need to resort to the next order O ∆ , which after considering Eqs. (S12) and (S13) can be simplified to be Interestingly, the expansion coefficient G is absence in Eq. (S15) due to the relationships in Eqs.
(S12) and (S13), which enables an analytical result for the expansion coefficient I :