Exceptional Points, Lasing, and Coherent Perfect Absorption in Floquet Scattering Systems

TL;DR

This work develops a universal Floquet scattering framework for finite photonic time crystals with time-periodic, dispersionless permittivity, revealing that the Floquet scattering matrix $S_ ext{F}$ is pseudounitary and that its eigenvalues undergo symmetry-breaking transitions at exceptional points as modulation strength grows. Time-symmetric driving leads to a breaking of time-reversal symmetry, with an anti-linear operator $X$ mapping unimodular states to themselves and broken states to their partners; at parametric resonance, CPA and lasing occur simultaneously when a zero and a diverging eigenvalue appear. The authors demonstrate these phenomena across diverse geometries—slabs, spheres, and metasurfaces—showing that qBICs in metasurfaces can drastically reduce the required modulation strength to reach CPA and lasing. A unified treatment via TBM and resonant-state expansion yields analytic predictions for EP thresholds and CPA-lasing conditions, with supplementary analysis confirming robustness across multispectral channels. The results provide a rigorous, geometry-agnostic route to dynamically engineer scattering properties in time-modulated photonic devices, with potential impact on active control of light in next-generation optical systems.

Abstract

Periodically time-varying media, known as photonic time crystals (PTCs), provide a promising platform for observing unconventional wave phenomena. We analyze the scattering of electromagnetic waves from spatially finite PTCs using the multispectral Floquet scattering matrix, which naturally incorporates the frequency-mixing processes intrinsic to such systems. For dispersionless, real, and time-periodic permittivities, this matrix is pseudounitary. Here we demonstrate that this property leads to multiple symmetry-breaking transitions: for increasing driving strength, scattering matrix eigenvalues lying on the unit circle (unbroken symmetry regime) meet at exceptional points (EPs), where they break up into inverse complex conjugate pairs (broken symmetry regime). We identify the symmetry operator associated with these transitions and show that, in time-symmetric systems, it corresponds to the time-reversal operator. Remarkably, at the parametric resonance condition, one eigenvalue vanishes while its partner diverges, signifying simultaneous coherent perfect absorption (CPA) and lasing. Since our approach relies solely on the Floquet scattering matrix, it is not restricted to a specific geometry but instead applies to any periodically time-varying scattering system. To illustrate this universality, we apply our method to a variety of periodically time-modulated structures, including slabs, spheres, and metasurfaces. In particular, we show that using quasi-bound states in the continuum resonances sustained by a metasurface, the CPA and lasing conditions can be attained for a minimal modulation strength of the permittivity. Our results pave the way for engineering time-modulated photonic systems with tailored scattering properties, opening new avenues for dynamic control of light in next-generation optical devices.

Figures (8)

• Figure 1: Behavior of the eigenvalues $\lambda$ of the Floquet scattering matrix $S_\mathrm{F}$ for a time-varying slab (see text for parameter values). a, We consider light scattering off a slab with a time-periodic permittivity with period $T=2\pi/\Omega$. The frequency of the incident light field can change during scattering such that the output light consists of several frequency components $\omega+n\Omega$. b, We assume a time-harmonic permittivity of the slab. c-e, We track four eigenvalues of $S_\mathrm{F}$ as a function of the modulation strength $M_\mathrm{s}$ for different choices of the quasifrequency $\omega$. In all cases, for small $M_\mathrm{s}$ all eigenvalues are unimodular and thus lie on the unit circle (blue). At a critical modulation strength, EPs are formed, where two respective eigenvalues coincide. By further increasing $M_\mathrm{s}$, these two eigenvalues leave the unit circle in a pairwise fashion (red). The colorbar to the right of panel e applies to all panels c-e, schematically indicating the respective EPs by a blue-to-red transition (the actual $M_\mathrm{s}$ value at which the EPs occur differs between panels). d, A special situation occurs when the quasifrequency of the wave field obeys the parametric resonance condition, $\omega/\Omega=0.5$. Then, one eigenvalue vanishes (represented by a star symbol) while another eigenvalue diverges (not shown), at which point the Floquet system acts as both a CPA and a laser. f, The absolute value of the minimal eigenvalue $\lambda_\mathrm{min}$ of $S_\mathrm{F}$ is shown here as a function of the modulation strength $M_\mathrm{s}$ and the quasifrequency $\omega$. For small $M_\mathrm{s}$ (weak modulation) or if the quasifrequency is far detuned from the parametric resonance condition, the minimal eigenvalues, and therefore all eigenvalues, are unimodular (white). This unbroken regime is separated by an exceptional line (black) from the broken regime with $\abs{\lambda_\mathrm{min}}<1$ (yellow). When calculating the winding number along the path $\gamma_1$, we find $\mathop{\mathrm{wind}}\nolimits(S_\mathrm{F}) = 2$, verifying that this path encloses a CPA-lasing point (star symbol). For reference, we also calculate the winding number for the path $\gamma_2$, which does not enclose a CPA-lasing point, and find $\mathop{\mathrm{wind}}\nolimits(S_\mathrm{F})=0$.
• Figure 2: Intensities of the incoming and the outgoing CPA and lasing states at the borders of the slab (see text for parameter values). a and d, Temporal variation of the periodic permittivity function of the slab. b, Intensity of the incoming part of the CPA light field at the left border at $x=-L/2$ (solid red) and at the right border at $x=L/2$ (dashed black) of the slab. The light field approaches the slab from the left and right with the same temporal intensity profile (the two lines overlap). Destructive interference of these pulses eliminates spatial reflections. The wave field builds up a large intensity maximum at times, when the permittivity is rising $0\leq t \leq T/4$ (gray shaded region). In this way, all the energy of the light field gets perfectly absorbed by the time-varying medium. c, The corresponding output intensity at the borders of the slab is reduced by a factor $10^{-10}$. e, Same as b but for the lasing state. This light field builds up large intensity maxima at times, when the permittivity is lowered $-T/4 \leq t \leq 0$ (gray shaded region), receiving energy from the time-modulated slab. f, This leads to a huge amplification of the outgoing intensity by a factor of $10^{10}$. Furthermore, we observe the time-reversal symmetry of the CPA and lasing states: The incoming CPA state is the time-reversed of the outgoing lasing state.
• Figure 3: Behavior of the eigenvalues $\lambda$ of the Floquet scattering matrix $S_\mathrm{F}$ for a time-varying isolated sphere (see text for parameter values). a, We consider the properties of light scattering off a sphere with a time-periodic permittivity function. b-d, The complex eigenvalues of $S_\mathrm{F}$ for varying modulation strengths $M_\mathrm{s}$ and for different quasifrequencies $\omega/\Omega = 0.47, \, 0.5, \, 0.53$, respectively. For all three choices of the input quasifrequency, EPs form, and the system undergoes symmetry-breaking transitions. The red arrows indicate the appearance of additional EPs primarily associated with higher Floquet channels $n<-1$ and $n>0$. For increasing modulation strength, they enter the broken regime but quickly recombine on the unit circle again. Furthermore, only in c, where the light field fulfills the parametric resonance condition $\omega/\Omega=0.5$, some eigenvalues vanish. At this operational condition, the system can act as a CPA (star symbol). Since the eigenvalues come in inverse complex conjugate pairs, there simultaneously exist diverging eigenvalues corresponding to the system acting as a laser (not shown). e, The absolute value of the minimal eigenvalue of $S_\mathrm{F}$ is shown here as a function of the quasifrequency $\omega$ and modulation strength $M_\mathrm{s}$. The unbroken regime where all eigenvalues are unimodular (white) is separated from the broken regimes in which $\abs{\lambda_\mathrm{min}}<1$ (yellow) by exceptional lines (black lines). The star symbol indicates the CPA condition. The red arrow in panel e indicates the broken regime associated with the additional EPs visible in panel c.
• Figure 4: Space-integrated incoming and outgoing intensities for the CPA and lasing states for the time-varying sphere (see text for parameter values) as a function of time $t$. We evaluate the electric fields in the far-field at $r=1000 \times Tc$ and spatially integrate them on an imaginary sphere that encloses the time-varying sphere. a, Incoming and b, outgoing intensity corresponding to the CPA state with an eigenvalue $\lambda=2\times 10^{-4}$. An extreme attenuation of the intensity can be observed, and nearly no outgoing field is produced. c, Incoming and d, outgoing intensity of the lasing state corresponding to an eigenvalue $\lambda=5\times 10^{3}$. Here, we observe an extreme amplification of the incoming light field. Furthermore, we see the time-reversal symmetry of the CPA and lasing states: The incoming CPA state is the time-reversed of the outgoing lasing state.
• Figure 5: Behavior of the eigenvalues $\lambda$ of the Floquet scattering matrix $S_\mathrm{F}$ for a time-varying metasurface. a, We consider the properties of light scattering off a metasurface that consists of a periodic arrangement (on a square lattice) of spheres made from a time-varying medium. The time-varying medium is characterized by the permittivity $\epsilon(\mathbf{r},t)=1+\chi_0 [1 + M_\mathrm{s} \cos(\Omega t)]f(\mathbf{r})$. Here, $f(\mathbf{r})=1$ for $\mathbf{r}$ at the spatial domains occupied by the spheres and $0$ otherwise (see text for the other parameter values). b, The reflectivity $\mathcal{R}$ of the metasurface shown in a as a function of $\omega$ and the $x$-component of Bloch wavevector $k_x$ under static conditions (i.e., $M_\mathrm{s}=0$) and $k_y=0$. We note that at $k_x=0$, there exists a BIC that has a symmetry compatible with TE-polarized plane waves. c, The complex eigenvalues of $S_\mathrm{F}$ for varying modulation strengths $M_\mathrm{s}$ and for the quasifrequency $\omega/\Omega = 0.5$. Here, we observe that at a certain $M_\mathrm{s}$, the light field fulfills the parametric resonance condition, leading to vanishing eigenvalues (indicated by the star symbol), signifying CPA and lasing. d, The absolute value of the minimal eigenvalue of $S_\mathrm{F}$ as a function of the quasifrequency $\omega$ and modulation strength $M_\mathrm{s}$. The unbroken regime where all eigenvalues are unimodular (white) is separated from the broken regime in which $\abs{\lambda_\mathrm{min}}<1$ (yellow) by an exceptional line (black). The star symbol indicates the CPA condition. To prove that $\abs{\lambda_\mathrm{min}}$ vanishes entirely at the CPA point, we calculated the winding number over a loop enclosing the CPA point. We find that the winding number is $\mathop{\mathrm{wind}}\nolimits(S_\mathrm{F}) \approx 2-10^{-4}$ , as expected, due to the TE-polarized qBIC mode of the metasurface.