Quasi Normal Modes in Dispersive Photonic Time-Crystals

TL;DR

This paper extends the quasi-normal mode framework to dispersive photonic time crystals by formulating Floquet Quasi-Normal Modes (FQNMs) for periodically driven slabs. Using an operator-based Floquet formalism, it analyzes the trajectories of FQNM quasifrequencies under continuous changes of system parameters, revealing ubiquitous exceptional points driven by RC-symmetry and non-perturbative frequency coupling. It identifies two limiting structures in large slabs: static-limit points associated with zero-index-like behavior and exceptional-limit points arising from frequency coupling, with a Drude-slab example showing an exceptional point that corresponds to the maximum bulk gain $\omega^*\sim \frac{\Omega}{2}+0.0072i\,\Omega$. The results warn against naive perturbative treatments and offer concrete predictions for experiments on time-varying, dispersive cavities and their gain properties.

Abstract

Quasinormal modes characterise the transient response of static optical cavities. Here, we introduce the notion of a Floquet quasinormal mode to describe transient responses in photonic time crystals. Contrasting their static counterparts, exceptional points associated with symmetry transitions are an inherent feature, as modes spontaneously and non-perturbatively lock their phase to the oscillations of the material. We further investigate the limiting behaviour of the Floquet quasinormal modes in large cavities. Distinct non-perturbative behaviour arises in time-modulated systems as increasingly large time-crystal cavities come closer to achieving the maximum gain predicted from a bulk wavenumber bandgap.

Figures (6)

• Figure 1: The QNMs and FQNMs of a static Drude metal slab: The parameters are those given in the main text, with $\eta = 0$, setting the time modulation to zero, and a slab length of $L = 85\ c\Omega^{- 1}$ (aside from the inset of panel c). In all panels we show a colour plot of the indicator function $\ln( | \det( \widetilde{\mathrm{Q}}) |)$, the divergence of this quantity indicating a non--empty kernel, and thus a solution to (\ref{['eq:FQNMCondition']}). Numerically determined roots of $\det( \widetilde{\mathrm{Q}})=0$ are plotted with white dots. Here $\widetilde{\mathrm{Q}}$ approximates the operator $\mathrm{\widehat{Q}}$ derived in the main text, but truncated to a $5 \times 5$ matrix. (a) The QNMs of a static Drude metal. The band of large $| \det( \widetilde{\mathrm{Q}})|$ around $\Re(\omega_0)=0$ corresponds to a region of suppressed transmission. The modes within this region are thus mostly confined to the material, and only weakly couple to electromagnetic waves. (b) The FQNMs of a static Drude metal. These can be immediately identified as shifted copies of the QNMs of panel (a), shifted in frequency by integer multiples of $\Omega$. Denoted with i-iii are $3$ such copies, with their associated eigenvectors plotted component-wise in the inset. Note that the frequency shifts of each eigenvector precisely cancel the shifted quasifrequency $\omega_0$, such that each replica FQNM corresponds to the same time-domain field. (c) The crossing of FQNM quasifrequencies about symmetry axes (labeled with $A_{n}$). Inset: FQNMs crossing a symmetry axis. The FQNM quasifrequencies are plotted as coloured points denoting slab lengths between $L=83\ c\Omega^{- 1}$ and $L=89\ c\Omega^{- 1}$. By definition of the symmetry axis, as a given mode crosses an $\mathcal{RC}$-axis $A_{n}$, it must collide with its $\mathcal{RC}$-symmetric pair. In static media, these modes aren't coupled, and so no additional behaviour arises at these crossings, a fact expected to change for finite modulations.
• Figure 2: $\mathcal{RC}$-symmetry (un)breaking of FQNMs in a photonic time crystal: In all plots, we consider frequency offsets $\omega_{\rm offset}$ relative to an $\mathcal{RC}$-symmetry axis $A_{n}$ (specified as $A_1$ in panel d). In panels a-c we plot the possible qualitative behaviours allowed by symmetry for FQNMs approaching a symmetry axis $A_{n}$. (a) If the modes are not coupled by time-variations in the system, they will pass one another unperturbed. This is most likely for static media (see Figure \ref{['fig:StaticFQNMs']}). Away from the crossing point, the behaviour of later modes can be expected to line up with this example. (b) As is commonly expected from Hermitian systems, a pair of modes may experience avoided crossing. (c) Our system obeys $\mathcal{RC}$-symmetry rather than Hermitian symmetry, which possesses the same capacity for spontaneous symmetry transitions as the well-studied $\mathcal{PT}$-symmetry Bender1998. Thus, a pair of modes may be pulled together by coupling, colliding in an exceptional point where they transition from possessing $\mathcal{RC}$-symmetry only as a pair to each individually preserving it, with each lying directly on the symmetry axis $A_{n}$. (d) An example of (c) in the FQNM trajectories of a Drude metal (parameters given below Eq. (\ref{['eq:ExampleDrudeModel']})) about the $A_1$ symmetry axis. These trajectories are plotted as a function of length between $L = 26\ c\Omega^{- 1}$ and $L = 34\ c\Omega^{- 1}$.
• Figure 3: The QNMs of a Drude slab approaching a limit point as slab length increases: Plotted as points: sampled trajectories of QNMs for slab lengths increasing from $L = 50\ c\Omega^{- 1}$ to $L = 200\ c\Omega^{- 1}$. The white cross corresponds to a zero wavenumber point of the Drude model (discussed further in the main text), apparently attracting the QNMs of the system.
• Figure 4: Exceptional limit point in a large slab: For illustrative purposes, in this example, rather than a Drude model, we consider the $4 \times 4$ truncation of $\widehat{\chi} = 1 + 0.2{\widehat{\Delta}}_{1}(0)$. We plot the FQNM indicator function as a colourmap, with FQNM quasifrequencies highlighted by white dots, and their approximations plotted in smaller green dots. All frequencies are plotted relative to the $\mathcal{RC}$-symmetry axis $A_{1}$ with $\omega_\mathrm{offset}=\omega_0-\frac{\Omega}{2}$. For a very thick slab ($L = 5 \times 10^{7}\ c\Omega^{- 1}$), many modes have clustered around this exceptional limit point, associated with the weak third wavenumber bandgap in the dispersion relation of $\widehat{K}^2(\omega_0)$. Indeed, our approximation remains valid despite the wavenumber bandgap under consideration actually lying entirely within (in terms of its complex frequency content), the much stronger first wavenumber bandgap.
• Figure 5: FQNMs in a Drude slab for a large range of increasing lengths: Sampled FQNM trajectories are plotted as points coloured by the imaginary part of their quasifrequency. For readability, only modes for a $2\times2$ truncation of $\widehat{\chi}$ are plotted. The $\mathcal{RC}$-symmetry axis $A_{1}$ is shown as a translucent yellow plane, with a light yellow outline. (i) An otherwise smooth FQNM trajectory (qualitatively presented with a transparent white line) in a static system is broken in time-varying media by $\mathcal{RC}$-symmetry transitions about the $A_{1}$ axis. (ii) Most FQNM trajectories have limiting behaviour similar to static systems, approaching refractive index zeros (white dashed lines) as slab length increases. (iii) However, for sufficiently thick slabs, pairs of modes become permanently confined to $A_1$ and isolated from the static limit points (ii). These modes are associated closely with the bulk wavenumber bandgap, and responsible for the diverging transmission coefficients of the slabs presented in Hooper2025. Noting their isolation, one might predict that they tend towards an exceptional limit point, which is validated when considering much larger slabs, as in Figure \ref{['fig:RCStagnationPoint']}.