Abstract

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.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [15], multifunctional metasurfaces [6], parity–time (PT)-symmetric structures [7,8], amplification [9], impedance matching [1012], 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 [1520], 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 [2931], photon and charge transport [3235], 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 [3739]. 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 [36,4447], whereas the second one involves an expansion in terms of quasi-normal modes (QNMs) [4853]. 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

$$\begin{split}{-}{j}\frac{\partial}{{\partial {t}}} | {{\Psi}({z,t} )\rangle = \hat H} |{\Psi}({z,t} )\rangle,\quad \hat H = \left({\begin{array}{*{20}{c}}{j\sigma /\left({{\varepsilon _0}{\varepsilon _r}} \right)}&{\hat p/{\varepsilon _r}}\\{\hat p}&0\end{array}} \right),\end{split}$$
where the differential operator $\hat p = j{c_0}\frac{\partial}{{\partial z}}$, and, without loss of generality, the state vector $|{\Psi}({z, t})\rangle = \left({\begin{array}{*{20}{c}}{{E_x}}\\{{\eta _0}{H_y}}\end{array}}\right)$ consists of the electric and magnetic field components ${E_x}$ and ${H_y}$ within the layer ${-} {d} = {z_0} \lt z \lt 0$. The incident radiation is generally expressed as $|{{\Psi}^{\rm{in}}}({z \le {z_0}, t})\rangle \equiv a({t - \frac{z}{{{c_0}}}})\left({\begin{array}{*{20}{c}}1\\1\end{array}}\right)$, and $|{{\Psi}^{\rm{in}}}({{z_0},t})\rangle$ determines the inhomogeneous incoming-wave boundary condition for Eq. (1) at all instants $t$.

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

$${-}{j}\frac{\partial}{{\partial {t}}}| {{{\Psi}_s}({z,t} )\rangle = \hat H} |{{\Psi}_s}({z,t} )\rangle + |{s^ +}({z,t} )\rangle,$$
where the inhomogeneous term $|{s^ +}({z,t}) = \left({\begin{array}{*{20}{c}}{\frac{{j\sigma}}{{{\varepsilon _0}{\varepsilon _r}}}}&{({\frac{1}{{{\varepsilon _r}}} - 1})\hat p}\\0&0\end{array}}\right)|{{\Psi}_b}\rangle$. The reflected wave $|{{\Psi}^{\rm{out}}}({z \le {z_0},t})\rangle = b({t + \frac{z}{{{c_0}}}})\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}}\right)$ can then be obtained from the continuity of $|{\Psi}({{z_0},t})\rangle$ at ${z_0}$, leading to
$$|{{\Psi}^{\rm{out}}}\left({{z_0},t} \right)\rangle = a\left({t + \frac{{{z_0}}}{{{c_0}}}} \right)\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}} \right) + |{{\Psi}_s}\left({{z_0}, t} \right)\rangle.$$
Our QNMs constitute a complete basis [60,61] and satisfy the orthonormality condition $\int_{{z_0}}^0 \langle {\tilde \Psi}_m^*(z)|M|{{\tilde \Psi}_n}(z) \rangle {\rm d}z = {\delta _{\textit{mn}}},\quad {M} = {\varepsilon _0}\left({\begin{array}{*{20}{c}}{- {\varepsilon _r}}&0\\0&1\end{array}}\right)$, where $\langle {\tilde \Psi}_m^*(z)| = |{{\tilde \Psi}_m}{(z)\rangle^T}$, with the superscript ${T}$ indicating the transpose operation. Explicitly, the QNMs of our geometry are
$$\begin{split}&|{\tilde \Psi _n}\left(z \right)\rangle =\\& \left\{{\begin{array}{*{20}{c}}{{c_n}{{\left[{{{\sin}\left({{{\tilde k}_n}z} \right),}\quad {- {\tan}\left({{{\tilde k}_n}{z_0}} \right){\cos}\left({{{\tilde k}_n}z} \right)}} \right]}^T},\quad {z_0} \le z \le 0}\\{- {c_n}{\sin}\left({{{\tilde k}_n}{z_0}} \right){e^{j\frac{{{{\tilde \omega}_n}}}{{{c_0}}}\left({z - {z_0}} \right)}}{{\left[{{- 1,}1} \right]}^T},\quad z \le {z_0}}\end{array},} \right.\end{split}$$
where the complex wavenumbers ${\tilde k_n} = \frac{{\sqrt {{\varepsilon _r}}}}{{{c_0}}}{\tilde \omega _n}\sqrt {1 - j\sigma /({{\varepsilon _0}{\varepsilon _r}{{\tilde \omega}_n}})}$ are written in terms of complex (angular) frequencies ${\tilde \omega _n}$ that satisfy the transcendental equation ${\tan}({{{\tilde k}_n}{z_0}}) = - {j}{c_0}{\tilde k_n}/{\tilde \omega _n}$, and the normalization factor ${c_n} = \sqrt {\frac{2}{{d{\varepsilon _0}}}} {\{{{{\tan}^2}({{{\tilde k}_n}d}) - {\varepsilon _r} + \frac{{\sin ({2{{\tilde k}_n}d})}}{{2{{\tilde k}_n}d}}[{{{\tan}^2}({{{\tilde k}_n}d}) + {\varepsilon _r}}]} \}^{- 1/2}}$. These properties, combined with Eq. (2), provide a relation for the expansion coefficients ${A_n}(t) = \int_{{z_0}}^0 \langle {\tilde \Psi}_n^*(z)|M|{{\Psi}_s}({z,\;t}) \rangle {\rm d}z$ of the scattered field:
$${-}{j}\frac{{\rm d}}{{{\rm dt}}}{A_n}(t ) = {\tilde \omega _n}{A_n}(t ) + \int_{{z_0}}^0 \langle {\tilde \Psi}_n^*\left(z \right)|M|{s^ +}({z,t} ) \rangle {\rm d}z.$$
We can then derive the outgoing field $|{{\Psi}^{\rm{out}}}({{z_0},t})\rangle = b({t + \frac{{{z_0}}}{{{c_0}}}})\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}}\right)$ using Eqs. (3) and (4):
$$b\left({t + \frac{{{z_0}}}{{{c_0}}}} \right) = a\left({t + \frac{{{z_0}}}{{{c_0}}}} \right) - \mathop \sum \limits_n {A_n}(t ){c_n}{\rm \sin}\left({{{\tilde k}_n}{z_0}} \right).$$

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

$$\begin{split}{-}{ j}\frac{{\rm d}}{{{\rm dt}}}{\psi _n}(t )& = {\tilde \omega _n}{\psi _n}(t ) + 2j{c_0}{\varepsilon _0}{c_n}\sin \left({{{\tilde k}_n}{z_0}} \right){a^{\textit{in}}}\left({{z_0},t} \right), \\ {b^{\rm{out}}}\left({{z_0},t} \right) &= \int_{{{\left({t + \frac{{2{z_0}}}{{{c_0}}}} \right)}^ -}}^t K\left({t - {t^\prime}} \right){a^{\rm{in}}}\left({{z_0},{t^\prime}} \right){\rm d}{t^\prime} \\&\quad - \mathop \sum \limits_n {c_n}\sin \left({{{\tilde k}_n}{z_0}} \right){\psi _n}(t ),\\[-1.3pc]\end{split}$$
where ${a^{\rm{in}}}({{z_0},t}) \equiv a({t - \frac{{{z_0}}}{{{c_0}}}}) = \frac{1}{{2\pi}}\int_{- \infty}^\infty a(\omega){e^{j\omega ({t - {z_0}/{c_0}})}}d\omega$ and ${b^{\rm{out}}}({{z_0},t}) \equiv b({t + \frac{{{z_0}}}{{{c_0}}}}) = \frac{1}{{2\pi}}\int_{- \infty}^\infty b(\omega){e^{j\omega ({t + {z_0}/{c_0}})}}{\rm d}\omega$ specify the incoming and outgoing fields at ${z_0}$ and time $t$, i.e., $|{{\Psi}^{\rm{in}}}({{z_0},t})\rangle = {a^{\rm{in}}}({{z_0},t})\left({\begin{array}{*{20}{c}}1\\1\end{array}}\right)$ and $|{{\Psi}^{\rm{out}}}({{z_0},t})\rangle = {b^{\rm{out}}}({{z_0},t})\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}}\right)$, the integral kernel $K({\tau}) = \delta ({{\tau} + 2\frac{{{z_0}}}{{{c_0}}}}) - \frac{{{c_0}}}{{{z_0}}}\mathop \sum \limits_n {\hat K_n}({\tau})$, with ${\hat K_n}(\tau) \equiv c_n^2{\varepsilon _0}{z_0}\sin ({{{\tilde k}_n}{z_0}})\cos [{{{\tilde k}_n}({{c_0}\tau + {z_0}})}]\{\tan ({{{\tilde k}_n}{z_0}}) + {\varepsilon _r}\tan $ $[{{{\tilde k}_n}({{c_0}\tau + {z_0}})}] \}$, and the integration range $[{{{({t + \frac{{2{z_0}}}{{{c_0}}}})}^ -},t}]$ covers the lower limit $t + \frac{{2{z_0}}}{{{c_0}}}$, ensuring that the Dirac delta $\delta ({{\tau} + 2\frac{{{z_0}}}{{{c_0}}}})$ within $K({\tau})$ produces the expected term $a({t + \frac{{{z_0}}}{{{c_0}}}})$ [Eq. (6)]. We emphasize that the infinite summation in $K({\tau})$ typically does not converge, but the observable quantities, such as ${b^{\rm{out}}}({{z_0},t})$, do [36]. The ab initio input–output formulation Eq. (7) manifests causality, and can be applied beyond the weak-coupling limit, in contrast to conventional CMT. To the best of our knowledge, Eq. (7) is the first example of rigorous input–output formalism based on QNMs, ideally suited to tackle time-switched open resonators, as discussed later.

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)

$$\begin{split}{-}{\rm j}\frac{{{\rm d}{\psi _{- 1}}(t )}}{{{\rm dt}}} &= {\tilde \omega _{- 1}}{\psi _{- 1}}(t ) + j{c_{- 1}}\sin \left({{{\tilde k}_{- 1}}{z_0}} \right)\sqrt {\frac{2}{{{\eta _0}}}} a_ + ^{\rm{in}}\left({{z_0},t} \right), \\ b_ + ^{\rm{out}}\left({{z_0},t} \right) &= {C}a_ + ^{\rm{in}}\left({{z_0},t} \right) - {c_{- 1}}\sin \left({{{\tilde k}_{- 1}}{z_0}} \right)\sqrt {\frac{2}{{{\eta _0}}}} {\psi _{- 1}}(t ),\end{split}$$
where the complex wave amplitudes $a_ + ^{\rm{in}}({{z_0},t}) = \sqrt {2/{\eta _0}} \frac{1}{{2\pi}}\int_0^\infty a(\omega){e^{j\omega ({t - {z_0}/{c_0}})}}{\rm d}\omega$ and $b_ + ^{\rm{out}}({{z_0},t}) = \sqrt {2/{\eta _0}} \frac{1}{{2\pi}}\int_0^\infty b(\omega){e^{j\omega ({t + {z_0}/{c_0}})}}{\rm d}\omega$ are normalized to the time-averaged power flux under the slowly varying envelope approximation. The direct coupling coefficient C connecting the complex excitation with the outgoing fields is ${C} \approx {{e}^{2{j}{\omega _0}{z_0}/{c_0}}} + \int_{- 2{z_0}/{c_0}}^0 {\hat K_{- 1}}({\tau}){e^{- j\tau {\omega _0}}}{c_0}/{z_0}{\rm d}{\tau}$, with ${C} \to 1$ for our thin absorber in the limit ${\varepsilon _r} \gg 1$. In Supplement 1, we examine Eq. (8) analytically in the lossless scenario and compare it with the general principle formulation. As expected, Eq. (8) converges to conventional CMT [39] 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)] 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 $|{\Psi}({z,t})\rangle,\;{z_0} \le z \le 0$:

$$\begin{split}{b^{\rm{out}}}\left({{z_0},t} \right) &= \frac{1}{{2\pi}}\int_{- \infty}^\infty \left[{- a\left(\omega \right){e^{- \frac{{j\omega {z_0}}}{{{c_0}}}}}} \right]\rho \left(\omega \right){e^{j\omega t}}{\rm d}\omega , \\ |{\Psi}\left({z,\;t} \right)\rangle &= \frac{1}{{2\pi}}\int_{- \infty}^\infty \frac{{2a\left(\omega \right){e^{j\omega \left({t - {z_0}/{c_0}} \right)}}}}{{\sin \left({\tilde k{z_0}} \right) + jn\cos \left({\tilde k{z_0}} \right)}}\\&\quad\times\left({\begin{array}{*{20}{c}}{\sin \left({\tilde kz} \right)}\\{jn\cos \left({\tilde kz} \right)}\end{array}} \right){\rm d}\omega ,\end{split}$$
where $\rho (\omega)$ is the complex reflection coefficient derived above, and the complex wave number $\tilde k \equiv \frac{{\omega n}}{{{c_0}}}$. We can employ Eq. (7) to evaluate the reflection ${b^{\rm{out}}}({{z_0},t})$ starting from an arbitrary time ${t_s}$, after knowing the initial condition ${\psi _n}({{t_s}}) = \int_{{z_0}}^0 \langle{\tilde \Psi}_n^*(z)|M|{\Psi}({z,{t_s}})\rangle{\rm d}z$. Note that ${\psi _n}(0) = 0$, since initially the incoming wave packet is far away from the screen, and ${\psi _n}({{t_s}}),{t_s} \gt 0$ can be calculated exactly from the knowledge of the internal field $|{\Psi}({z,{t_s}})\rangle$ in Eq. (9) [62].

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.  [7175] 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].

Funding

Air Force Office of Scientific Research; Simons Foundation; U.S. Department of Defense.

Acknowledgment

We thank Ahmed Mekawy for his help on the COMSOL simulation in Fig. 3.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

REFERENCES AND NOTES

1. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009). [CrossRef]  

2. D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014). [CrossRef]  

3. 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 96, 165144 (2017). [CrossRef]  

4. Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017). [CrossRef]  

5. N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014). [CrossRef]  

6. X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020). [CrossRef]  

7. R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012). [CrossRef]  

8. T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018). [CrossRef]  

9. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018). [CrossRef]  

10. A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018). [CrossRef]  

11. H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019). [CrossRef]  

12. Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020). [CrossRef]  

13. Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016). [CrossRef]  

14. F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958). [CrossRef]  

15. D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966). [CrossRef]  

16. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]  

17. J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016). [CrossRef]  

18. N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018). [CrossRef]  

19. T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018). [CrossRef]  

20. P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019). [CrossRef]  

21. M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019). [CrossRef]  

22. D. L. Sounas, “Virtual perfect absorption through modulation of the radiative decay rate,” Phys. Rev. B 101, 104303 (2020). [CrossRef]  

23. H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020). [CrossRef]  

24. H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018). [CrossRef]  

25. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020). [CrossRef]  

26. C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials,” arXiv:1905.00560 (2019).

27. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984). [CrossRef]  

28. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef]  

29. B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984). [CrossRef]  

30. J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013). [CrossRef]  

31. A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015). [CrossRef]  

32. S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015). [CrossRef]  

33. T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015). [CrossRef]  

34. J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020). [CrossRef]  

35. J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020). [CrossRef]  

36. D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020). [CrossRef]  

37. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

38. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]  

39. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004). [CrossRef]  

40. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24, 2378–2388 (2007). [CrossRef]  

41. H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020). [CrossRef]  

42. H. Feshbach, “Unified theory of nuclear reactions,” Ann. Phys. (N.Y.) 5, 357–390 (1958). [CrossRef]  

43. H. Feshbach, “A unified theory of nuclear reactions. II,” Ann. Phys. (N.Y.) 19, 287–313 (1962). [CrossRef]  

44. C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003). [CrossRef]  

45. C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004). [CrossRef]  

46. D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003). [CrossRef]  

47. W. Domcke, “Projection-operator approach to potential scattering,” Phys. Rev. A 28, 2777–2791 (1983). [CrossRef]  

48. S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000). [CrossRef]  

49. B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014). [CrossRef]  

50. E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016). [CrossRef]  

51. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018). [CrossRef]  

52. W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018). [CrossRef]  

53. S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019). [CrossRef]  

54. K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000). [CrossRef]  

55. D. M. Pozar, Microwave Engineering (Wiley, 1998).

56. G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

57. D. M. Solis and N. Engheta, “A generalization of the Kramers-Kronig relations for linear time-varying media,” arXiv:2008.04304 (2020).

58. R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Prentice-Hall, 1987).

59. As a comparison, for a valid QNM expansion $|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t})|{{\tilde \Psi}_{n}}({z})\rangle$ of the state vector $|{\Psi}({{z},{t}})\rangle$, a seemingly similar equality ${\hat H}|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t}){{\tilde \omega}_{n}}|{{\tilde \Psi}_{n}}({z})\rangle$ does not hold, which hinders a construction of an input–output formalism using QNMs [see Eq. (7)] directly from Eq. (1).

60. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994). [CrossRef]  

61. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998). [CrossRef]  

62. ${\psi _n}({{t}_s}), {{t}_s} \gt {0}$ can also be obtained by solving Eq. (7) with the initial condition ${\psi _n}({0})= {0}$.

63. S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015). [CrossRef]  

64. N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016). [CrossRef]  

65. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic fields from a temporal boundary,” Opt. Lett. 39,574–577 (2014). [CrossRef]  

66. A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016). [CrossRef]  

67. K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018). [CrossRef]  

68. M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019). [CrossRef]  

69. K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017). [CrossRef]  

70. F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014). [CrossRef]  

71. A. Tennant, “Reflection properties of a phase modulating planar screen,” Electron. Lett. 33, 1768–1769 (1997). [CrossRef]  

72. B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002). [CrossRef]  

73. B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004). [CrossRef]  

74. B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005). [CrossRef]  

75. B. Chambers and A. Tennant, “Active Dallenbach radar absorber,” in IEEE International Symposium on Antennas and Propagation, Albuquerque, New Mexico, 2006.

76. M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017). [CrossRef]  

77. K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008). [CrossRef]  

78. M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
    [Crossref]
  2. D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
    [Crossref]
  3. 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 96, 165144 (2017).
    [Crossref]
  4. Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
    [Crossref]
  5. N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
    [Crossref]
  6. X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
    [Crossref]
  7. R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
    [Crossref]
  8. T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
    [Crossref]
  9. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
    [Crossref]
  10. A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
    [Crossref]
  11. H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
    [Crossref]
  12. Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020).
    [Crossref]
  13. Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
    [Crossref]
  14. F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
    [Crossref]
  15. D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
    [Crossref]
  16. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
    [Crossref]
  17. J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
    [Crossref]
  18. N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
    [Crossref]
  19. T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
    [Crossref]
  20. P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019).
    [Crossref]
  21. M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
    [Crossref]
  22. D. L. Sounas, “Virtual perfect absorption through modulation of the radiative decay rate,” Phys. Rev. B 101, 104303 (2020).
    [Crossref]
  23. H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
    [Crossref]
  24. H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
    [Crossref]
  25. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020).
    [Crossref]
  26. C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials,” arXiv:1905.00560 (2019).
  27. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
    [Crossref]
  28. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [Crossref]
  29. B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
    [Crossref]
  30. J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
    [Crossref]
  31. A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
    [Crossref]
  32. S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015).
    [Crossref]
  33. T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
    [Crossref]
  34. J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020).
    [Crossref]
  35. J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020).
    [Crossref]
  36. D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020).
    [Crossref]
  37. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  38. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [Crossref]
  39. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
    [Crossref]
  40. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24, 2378–2388 (2007).
    [Crossref]
  41. H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
    [Crossref]
  42. H. Feshbach, “Unified theory of nuclear reactions,” Ann. Phys. (N.Y.) 5, 357–390 (1958).
    [Crossref]
  43. H. Feshbach, “A unified theory of nuclear reactions. II,” Ann. Phys. (N.Y.) 19, 287–313 (1962).
    [Crossref]
  44. C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
    [Crossref]
  45. C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004).
    [Crossref]
  46. D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
    [Crossref]
  47. W. Domcke, “Projection-operator approach to potential scattering,” Phys. Rev. A 28, 2777–2791 (1983).
    [Crossref]
  48. S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
    [Crossref]
  49. B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
    [Crossref]
  50. E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
    [Crossref]
  51. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
    [Crossref]
  52. W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
    [Crossref]
  53. S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
    [Crossref]
  54. K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000).
    [Crossref]
  55. D. M. Pozar, Microwave Engineering (Wiley, 1998).
  56. G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).
  57. D. M. Solis and N. Engheta, “A generalization of the Kramers-Kronig relations for linear time-varying media,” arXiv:2008.04304 (2020).
  58. R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Prentice-Hall, 1987).
  59. As a comparison, for a valid QNM expansion $|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t})|{{\tilde \Psi}_{n}}({z})\rangle$|Ψ(z,t)⟩=∑n⁡ψn(t)|Ψ~n(z)⟩ of the state vector $|{\Psi}({{z},{t}})\rangle$|Ψ(z,t)⟩, a seemingly similar equality ${\hat H}|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t}){{\tilde \omega}_{n}}|{{\tilde \Psi}_{n}}({z})\rangle$H^|Ψ(z,t)⟩=∑n⁡ψn(t)ω~n|Ψ~n(z)⟩ does not hold, which hinders a construction of an input–output formalism using QNMs [see Eq. (7)] directly from Eq. (1).
  60. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
    [Crossref]
  61. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
    [Crossref]
  62. ${\psi _n}({{t}_s}), {{t}_s} \gt {0}$ψn(ts),ts>0 can also be obtained by solving Eq. (7) with the initial condition ${\psi _n}({0})= {0}$ψn(0)=0.
  63. S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
    [Crossref]
  64. N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
    [Crossref]
  65. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic fields from a temporal boundary,” Opt. Lett. 39,574–577 (2014).
    [Crossref]
  66. A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
    [Crossref]
  67. K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
    [Crossref]
  68. M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
    [Crossref]
  69. K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
    [Crossref]
  70. F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
    [Crossref]
  71. A. Tennant, “Reflection properties of a phase modulating planar screen,” Electron. Lett. 33, 1768–1769 (1997).
    [Crossref]
  72. B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002).
    [Crossref]
  73. B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004).
    [Crossref]
  74. B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005).
    [Crossref]
  75. B. Chambers and A. Tennant, “Active Dallenbach radar absorber,” in IEEE International Symposium on Antennas and Propagation, Albuquerque, New Mexico, 2006.
  76. M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
    [Crossref]
  77. K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
    [Crossref]
  78. M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
    [Crossref]

2020 (9)

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020).
[Crossref]

D. L. Sounas, “Virtual perfect absorption through modulation of the radiative decay rate,” Phys. Rev. B 101, 104303 (2020).
[Crossref]

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020).
[Crossref]

D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020).
[Crossref]

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

2019 (6)

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
[Crossref]

P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

2018 (9)

N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
[Crossref]

H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
[Crossref]

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
[Crossref]

2017 (4)

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
[Crossref]

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 96, 165144 (2017).
[Crossref]

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
[Crossref]

2016 (5)

J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
[Crossref]

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

2015 (4)

S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015).
[Crossref]

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

2014 (5)

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic fields from a temporal boundary,” Opt. Lett. 39,574–577 (2014).
[Crossref]

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

2013 (1)

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

2012 (1)

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

2009 (2)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

2008 (1)

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

2007 (1)

2005 (1)

B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005).
[Crossref]

2004 (3)

B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004).
[Crossref]

C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004).
[Crossref]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

2003 (3)

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
[Crossref]

C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
[Crossref]

2002 (1)

B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002).
[Crossref]

2000 (2)

S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
[Crossref]

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000).
[Crossref]

1998 (1)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

1997 (1)

A. Tennant, “Reflection properties of a phase modulating planar screen,” Electron. Lett. 33, 1768–1769 (1997).
[Crossref]

1994 (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
[Crossref]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref]

1984 (2)

B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

1983 (1)

W. Domcke, “Projection-operator approach to potential scattering,” Phys. Rev. A 28, 2777–2791 (1983).
[Crossref]

1966 (1)

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

1962 (1)

H. Feshbach, “A unified theory of nuclear reactions. II,” Ann. Phys. (N.Y.) 19, 287–313 (1962).
[Crossref]

1958 (2)

H. Feshbach, “Unified theory of nuclear reactions,” Ann. Phys. (N.Y.) 5, 357–390 (1958).
[Crossref]

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

Agrawal, G. P.

Alú, A.

T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

Alù, A.

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
[Crossref]

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Antonov, V.

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

Asadchy, V. S.

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Barrick, D. E.

G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

Becerra-Fuentes, O. M.

J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

Benson, T. M.

Bhaskaran, H.

M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
[Crossref]

Busch, K.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Caloz, C.

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 96, 165144 (2017).
[Crossref]

C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials,” arXiv:1905.00560 (2019).

Caneva, T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Cervantes-González, J. C.

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Chamanara, N.

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 96, 165144 (2017).
[Crossref]

Chambers, B.

B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005).
[Crossref]

B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004).
[Crossref]

B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002).
[Crossref]

B. Chambers and A. Tennant, “Active Dallenbach radar absorber,” in IEEE International Symposium on Antennas and Propagation, Albuquerque, New Mexico, 2006.

Chan, C. T.

N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
[Crossref]

Chang, D. E.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Chowdhury, E.

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Christodoulides, D. N.

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

Cirac, J. I.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Commandré, M.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Creagh, S. C.

Deck-Léger, Z.-L.

C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials,” arXiv:1905.00560 (2019).

Denker, J. S.

B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[Crossref]

Dezfouli, M. K.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Diaz-Rubio, A.

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

Domcke, W.

W. Domcke, “Projection-operator approach to potential scattering,” Phys. Rev. A 28, 2777–2791 (1983).
[Crossref]

Douglas, J. S.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Dutra, S. M.

S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

Engheta, N.

V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020).
[Crossref]

D. M. Solis and N. Engheta, “A generalization of the Kramers-Kronig relations for linear time-varying media,” arXiv:2008.04304 (2020).

Estep, N. A.

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Evers, J.

D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020).
[Crossref]

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Faggiani, R.

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
[Crossref]

Fan, S.

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
[Crossref]

S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

Fan, Z.

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Feshbach, H.

H. Feshbach, “A unified theory of nuclear reactions. II,” Ann. Phys. (N.Y.) 19, 287–313 (1962).
[Crossref]

H. Feshbach, “Unified theory of nuclear reactions,” Ann. Phys. (N.Y.) 5, 357–390 (1958).
[Crossref]

Fleury, R.

T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

Franke, S.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Fritzsche, S.

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Goerttler, S.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Götte, J. B.

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Gradoni, G.

Grigoryan, K. K.

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Haber, J.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Haberman, R.

R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Prentice-Hall, 1987).

Hackenbroich, G.

C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004).
[Crossref]

C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
[Crossref]

Hadad, Y.

Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020).
[Crossref]

A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
[Crossref]

Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
[Crossref]

Halevi, P.

J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Han, S.

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Hayrapetyan, A. G.

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Heeg, K. P.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Holberg, D. E.

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Hong, M.

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

Hughes, S.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Hugonin, J.-P.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Jacobs, K.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

Jeon, W.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Joannopoulos, J. D.

Kaldun, A.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Kang, B.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Keitel, C. H.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Knorr, A.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Kocharovskaya, O.

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

Kottos, T.

H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
[Crossref]

Koutserimpas, T. T.

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

Krasnok, A.

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

Kremers, S.

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Krichbaum, C. K.

G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

Kristensen, P. T.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Kunz, K. S.

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Lalanne, P.

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Langbein, W.

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

Lee, K.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Lencer, D.

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Lentrodt, D.

D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020).
[Crossref]

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Leung, P. T.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
[Crossref]

Li, H.

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
[Crossref]

Liu, J.

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020).
[Crossref]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
[Crossref]

Liu, Y.-X.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

Maassen van den Brink, A.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Maier, S. A.

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

Makris, K. G.

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

Manzoni, M. T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Martínez-Romero, J. S.

J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

Maywar, D. N.

Mekawy, A.

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

Min, B.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Mirmoosa, M. S.

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Morgenthaler, F. R.

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

Moussa, H.

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

Muljarov, E. A.

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

Nguyen, N. B.

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

Nicolet, A.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Nienhuis, G.

S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
[Crossref]

Nori, F.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

Ott, C.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Oulton, R. F.

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

Pacheco-Peña, V.

Pantazopoulos, P. A.

P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019).
[Crossref]

Park, J.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Petrosyan, R. G.

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Pfeifer, T.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Phang, S.

Poon, J. K. S.

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Ptitcyn, G. A.

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Radeonychev, Y. V.

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

Reiser, P.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Reiserer, A.

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

Rempe, G.

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

Richter, M.

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

Robertson, J.

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Röhlsberger, R.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Rotermund, F.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Rozanov, K. N.

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000).
[Crossref]

Ruck, G.

G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

Rüffer, R.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Sauvan, C.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Savin, D. V.

D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
[Crossref]

Segal, D.

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020).
[Crossref]

Sewell, P. D.

Shafirin, P.

M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
[Crossref]

Shakhmuratov, R. N.

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

Shapiro, B.

H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
[Crossref]

Shcherbakov, M. R.

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
[Crossref]

Shi, T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Shi, Y.

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
[Crossref]

Shlivinski, A.

Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020).
[Crossref]

A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
[Crossref]

Shportko, K.

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Shvets, G.

M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
[Crossref]

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Sokolov, V. V.

D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
[Crossref]

Solis, D. M.

D. M. Solis and N. Engheta, “A generalization of the Kramers-Kronig relations for linear time-varying media,” arXiv:2008.04304 (2020).

Sommers, H.-J.

D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
[Crossref]

Son, J.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Soric, J.

Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
[Crossref]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Sounas, D.

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

Sounas, D. L.

D. L. Sounas, “Virtual perfect absorption through modulation of the radiative decay rate,” Phys. Rev. B 101, 104303 (2020).
[Crossref]

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Stefanou, N.

P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019).
[Crossref]

Strohm, C.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Stuart, W. D.

G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

Subramanian, R.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Suen, W. M.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

Talisa, N.

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Taravati, S.

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 96, 165144 (2017).
[Crossref]

Taubner, T.

M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
[Crossref]

Tennant, A.

B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005).
[Crossref]

B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004).
[Crossref]

B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002).
[Crossref]

A. Tennant, “Reflection properties of a phase modulating planar screen,” Electron. Lett. 33, 1768–1769 (1997).
[Crossref]

B. Chambers and A. Tennant, “Active Dallenbach radar absorber,” in IEEE International Symposium on Antennas and Propagation, Albuquerque, New Mexico, 2006.

Tong, S. S.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Tretyakov, S. A.

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Vagizov, F.

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

Vial, B.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Viviescas, C.

C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004).
[Crossref]

C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
[Crossref]

Vukovic, A.

Vynck, K.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Wang, N.

N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
[Crossref]

Wang, X.

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

Werner, K.

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Wille, H.-C.

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Woda, M.

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Wu, R.-B.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

Wuttig, M.

M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
[Crossref]

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Xiao, Y.

Xu, S.

S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015).
[Crossref]

Yan, W.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
[Crossref]

Yariv, A.

Young, K.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
[Crossref]

Yu, Z.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Yurke, B.

B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[Crossref]

Zhang, J.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

Zhang, Z.-Q.

N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
[Crossref]

Zolla, F.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Zurita-Sánchez, J. R.

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

ACS Photon. (2)

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photon. 4, 1639–1645 (2017).
[Crossref]

Ann. Phys. (N.Y.) (2)

H. Feshbach, “Unified theory of nuclear reactions,” Ann. Phys. (N.Y.) 5, 357–390 (1958).
[Crossref]

H. Feshbach, “A unified theory of nuclear reactions. II,” Ann. Phys. (N.Y.) 19, 287–313 (1962).
[Crossref]

Electron. Lett. (1)

A. Tennant, “Reflection properties of a phase modulating planar screen,” Electron. Lett. 33, 1768–1769 (1997).
[Crossref]

IEE Proc. Radar Son. Nav. (1)

B. Chambers and A. Tennant, “General analysis of the phase-switched screen. Part 1: The single layer case,” IEE Proc. Radar Son. Nav. 149, 243–247 (2002).
[Crossref]

IEEE Antennas Propag. Mag. (1)

B. Chambers and A. Tennant, “The phase-switched screen,” IEEE Antennas Propag. Mag. 46(6), 23–37 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

IEEE Trans. Antennas Propag. (5)

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000).
[Crossref]

B. Chambers and A. Tennant, “A smart radar absorber based on the phase-switched screen,” IEEE Trans. Antennas Propag. 53, 394–403 (2005).
[Crossref]

Y. Hadad and A. Shlivinski, “Soft temporal switching of TL parameters: wave-field, energy balance, applications,” IEEE Trans. Antennas Propag. 68, 1643–1654 (2020).
[Crossref]

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

IRE Trans. Microwave Theory Tech. (1)

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

J. Opt. B (1)

C. Viviescas and G. Hackenbroich, “Quantum theory of multimode fields: applications to optical resonators,” J. Opt. B 6, 211–223 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. G. Hayrapetyan, J. B. Götte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Laser Photon. Rev. (1)

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Nat. Commun. (1)

M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10, 1345 (2019).
[Crossref]

Nat. Mater. (1)

K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, “Resonant bonding in crystalline phase-change materials,” Nat. Mater. 7, 653–658 (2008).
[Crossref]

Nat. Photonics (3)

M. Wuttig, H. Bhaskaran, and T. Taubner, “Phase-change materials for non-volatile photonic applications,” Nat. Photonics 11, 465–476 (2017).
[Crossref]

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Nat. Phys. (1)

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Nature (1)

F. Vagizov, V. Antonov, Y. V. Radeonychev, R. N. Shakhmuratov, and O. Kocharovskaya, “Coherent control of the waveforms of recoilless γ-ray photons,” Nature 508, 80–83 (2014).
[Crossref]

New J. Phys. (2)

N. B. Nguyen, S. A. Maier, M. Hong, and R. F. Oulton, “Recovering parity-time symmetry in highly dispersive coupled optical waveguides,” New J. Phys. 18, 125012 (2016).
[Crossref]

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input-output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (16)

H. Li, T. Kottos, and B. Shapiro, “Driving-induced metamorphosis of transport in arrays of coupled resonators,” Phys. Rev. A 97, 023846 (2018).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref]

B. Yurke and J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[Crossref]

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, “Non-Markovian quantum input-output networks,” Phys. Rev. A 87, 032117 (2013).
[Crossref]

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

T. T. Koutserimpas, A. Alú, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982–3989 (1994).
[Crossref]

S. Xu and S. Fan, “Input-output formalism for few-photon transport: a systematic treatment beyond two photons,” Phys. Rev. A 91, 043845 (2015).
[Crossref]

C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
[Crossref]

W. Domcke, “Projection-operator approach to potential scattering,” Phys. Rev. A 28, 2777–2791 (1983).
[Crossref]

S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

M. R. Shcherbakov, P. Shafirin, and G. Shvets, “Overcoming the efficiency-bandwidth tradeoff for optical harmonics generation using nonlinear time-variant resonators,” Phys. Rev. A 100, 063847 (2019).
[Crossref]

Phys. Rev. Appl. (3)

X. Wang, A. Diaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and design of multifunctional space-time metasurfaces,” Phys. Rev. Appl. 13, 044040 (2020).
[Crossref]

H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14, 031002 (2020).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Phys. Rev. B (8)

D. L. Sounas, “Virtual perfect absorption through modulation of the radiative decay rate,” Phys. Rev. B 101, 104303 (2020).
[Crossref]

P. A. Pantazopoulos and N. Stefanou, “Layered optomagnonic structures: time Floquet scattering-matrix approach,” Phys. Rev. B 99, 144415 (2019).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. I. General formulation,” Phys. Rev. B 101, 155406 (2020).
[Crossref]

J. Liu and D. Segal, “Generalized input-output method to quantum transport junctions. II. Applications,” Phys. Rev. B 101, 155407 (2020).
[Crossref]

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 96, 165144 (2017).
[Crossref]

N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media with a complex time-periodic permittivity,” Phys. Rev. B 98, 085142 (2018).
[Crossref]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (2018).
[Crossref]

Phys. Rev. E (1)

D. V. Savin, V. V. Sokolov, and H.-J. Sommers, “Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering,” Phys. Rev. E 67, 026215 (2003).
[Crossref]

Phys. Rev. Lett. (5)

H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124, 193901 (2020).
[Crossref]

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with Floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

Phys. Rev. X (1)

D. Lentrodt and J. Evers, “Ab initio few-mode theory for quantum potential scattering problems,” Phys. Rev. X 10, 011008 (2020).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

Y. Hadad, J. Soric, and A. Alù, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. USA 113, 3471–3475 (2016).
[Crossref]

Rev. Mod. Phys. (2)

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Science (1)

K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, “Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances,” Science 357, 375–378 (2017).
[Crossref]

Other (9)

B. Chambers and A. Tennant, “Active Dallenbach radar absorber,” in IEEE International Symposium on Antennas and Propagation, Albuquerque, New Mexico, 2006.

${\psi _n}({{t}_s}), {{t}_s} \gt {0}$ψn(ts),ts>0 can also be obtained by solving Eq. (7) with the initial condition ${\psi _n}({0})= {0}$ψn(0)=0.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

G. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

D. M. Solis and N. Engheta, “A generalization of the Kramers-Kronig relations for linear time-varying media,” arXiv:2008.04304 (2020).

R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Prentice-Hall, 1987).

As a comparison, for a valid QNM expansion $|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t})|{{\tilde \Psi}_{n}}({z})\rangle$|Ψ(z,t)⟩=∑n⁡ψn(t)|Ψ~n(z)⟩ of the state vector $|{\Psi}({{z},{t}})\rangle$|Ψ(z,t)⟩, a seemingly similar equality ${\hat H}|{\Psi}({{z},{t}})\rangle = \mathop \sum \nolimits_{n} {{\psi}_{n}}({t}){{\tilde \omega}_{n}}|{{\tilde \Psi}_{n}}({z})\rangle$H^|Ψ(z,t)⟩=∑n⁡ψn(t)ω~n|Ψ~n(z)⟩ does not hold, which hinders a construction of an input–output formalism using QNMs [see Eq. (7)] directly from Eq. (1).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials,” arXiv:1905.00560 (2019).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplementary Materials

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
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.
Fig. 2.
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).
Fig. 3.
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.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

$$\begin{split}{-}{j}\frac{\partial}{{\partial {t}}} | {{\Psi}({z,t} )\rangle = \hat H} |{\Psi}({z,t} )\rangle,\quad \hat H = \left({\begin{array}{*{20}{c}}{j\sigma /\left({{\varepsilon _0}{\varepsilon _r}} \right)}&{\hat p/{\varepsilon _r}}\\{\hat p}&0\end{array}} \right),\end{split}$$
$${-}{j}\frac{\partial}{{\partial {t}}}| {{{\Psi}_s}({z,t} )\rangle = \hat H} |{{\Psi}_s}({z,t} )\rangle + |{s^ +}({z,t} )\rangle,$$
$$|{{\Psi}^{\rm{out}}}\left({{z_0},t} \right)\rangle = a\left({t + \frac{{{z_0}}}{{{c_0}}}} \right)\left({\begin{array}{*{20}{c}}{- 1}\\1\end{array}} \right) + |{{\Psi}_s}\left({{z_0}, t} \right)\rangle.$$
$$\begin{split}&|{\tilde \Psi _n}\left(z \right)\rangle =\\& \left\{{\begin{array}{*{20}{c}}{{c_n}{{\left[{{{\sin}\left({{{\tilde k}_n}z} \right),}\quad {- {\tan}\left({{{\tilde k}_n}{z_0}} \right){\cos}\left({{{\tilde k}_n}z} \right)}} \right]}^T},\quad {z_0} \le z \le 0}\\{- {c_n}{\sin}\left({{{\tilde k}_n}{z_0}} \right){e^{j\frac{{{{\tilde \omega}_n}}}{{{c_0}}}\left({z - {z_0}} \right)}}{{\left[{{- 1,}1} \right]}^T},\quad z \le {z_0}}\end{array},} \right.\end{split}$$
$${-}{j}\frac{{\rm d}}{{{\rm dt}}}{A_n}(t ) = {\tilde \omega _n}{A_n}(t ) + \int_{{z_0}}^0 \langle {\tilde \Psi}_n^*\left(z \right)|M|{s^ +}({z,t} ) \rangle {\rm d}z.$$
$$b\left({t + \frac{{{z_0}}}{{{c_0}}}} \right) = a\left({t + \frac{{{z_0}}}{{{c_0}}}} \right) - \mathop \sum \limits_n {A_n}(t ){c_n}{\rm \sin}\left({{{\tilde k}_n}{z_0}} \right).$$
$$\begin{split}{-}{ j}\frac{{\rm d}}{{{\rm dt}}}{\psi _n}(t )& = {\tilde \omega _n}{\psi _n}(t ) + 2j{c_0}{\varepsilon _0}{c_n}\sin \left({{{\tilde k}_n}{z_0}} \right){a^{\textit{in}}}\left({{z_0},t} \right), \\ {b^{\rm{out}}}\left({{z_0},t} \right) &= \int_{{{\left({t + \frac{{2{z_0}}}{{{c_0}}}} \right)}^ -}}^t K\left({t - {t^\prime}} \right){a^{\rm{in}}}\left({{z_0},{t^\prime}} \right){\rm d}{t^\prime} \\&\quad - \mathop \sum \limits_n {c_n}\sin \left({{{\tilde k}_n}{z_0}} \right){\psi _n}(t ),\\[-1.3pc]\end{split}$$
$$\begin{split}{-}{\rm j}\frac{{{\rm d}{\psi _{- 1}}(t )}}{{{\rm dt}}} &= {\tilde \omega _{- 1}}{\psi _{- 1}}(t ) + j{c_{- 1}}\sin \left({{{\tilde k}_{- 1}}{z_0}} \right)\sqrt {\frac{2}{{{\eta _0}}}} a_ + ^{\rm{in}}\left({{z_0},t} \right), \\ b_ + ^{\rm{out}}\left({{z_0},t} \right) &= {C}a_ + ^{\rm{in}}\left({{z_0},t} \right) - {c_{- 1}}\sin \left({{{\tilde k}_{- 1}}{z_0}} \right)\sqrt {\frac{2}{{{\eta _0}}}} {\psi _{- 1}}(t ),\end{split}$$
$$\begin{split}{b^{\rm{out}}}\left({{z_0},t} \right) &= \frac{1}{{2\pi}}\int_{- \infty}^\infty \left[{- a\left(\omega \right){e^{- \frac{{j\omega {z_0}}}{{{c_0}}}}}} \right]\rho \left(\omega \right){e^{j\omega t}}{\rm d}\omega , \\ |{\Psi}\left({z,\;t} \right)\rangle &= \frac{1}{{2\pi}}\int_{- \infty}^\infty \frac{{2a\left(\omega \right){e^{j\omega \left({t - {z_0}/{c_0}} \right)}}}}{{\sin \left({\tilde k{z_0}} \right) + jn\cos \left({\tilde k{z_0}} \right)}}\\&\quad\times\left({\begin{array}{*{20}{c}}{\sin \left({\tilde kz} \right)}\\{jn\cos \left({\tilde kz} \right)}\end{array}} \right){\rm d}\omega ,\end{split}$$