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Photonic time crystals assisted by quasi-bound states in the continuum

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Pith's one-line read Metasurfaces hosting quasi-bound states can open photonic time-crystal bandgaps with modulation amplitudes as low as 1e-7.

desk verdict The qBIC-assisted bandgap reduction is real and well demonstrated, but the Ge Kerr realizability claim lacks a physically possible pump scheme. read the letter →

arxiv 2507.15644 v1 pith:AYEHUEX6 submitted 2025-07-21 physics.optics

P. Garg , E. Almpanis , L. Zimmer , J. D. Fischbach , X. Wang , M. S. Mirmoosa , M. Nyman , N. Stefanou
show 3 more authors
N. Papanikolaou V. Asadchy C. Rockstuhl
This is my paper · ORCID
classification physics.optics
keywords boundstatesinthecontinuumphotonictimecrystalsmomentumbandgapsmetasurfacestime-varyingsystemsquasi-BICKerrmodulationtemporalFloquetbands
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the prohibitively large permittivity modulations required for photonic time crystals at optical frequencies can be made accessible by structuring the medium as a metasurface of resonators that support quasi-bound states in the continuum. Because such resonances trap light inside the time-varying material for an extended time, a tiny periodic modulation of the electron density opens momentum bandgaps that previously demanded modulation amplitudes near unity. The paper shows this for two metasurface designs: lossless multilayered AZO/silica spheres open a 3% momentum bandgap at a modulation amplitude $M_s = 1.35\times10^{-7}$, and germanium cylinders with realistic losses open the same gap at $M_s = 5\times10^{-5}$, sixteen times below the Mie-resonant reference and orders of magnitude below homogeneous photonic time crystals. If correct, this brings observable optical-frequency photonic time crystals within reach of weak all-optical Kerr driving.

What carries the argument

The load-bearing element is the quasi-bound state in the continuum, a resonance with strongly suppressed radiation losses and high quality factor, embedded in each meta-atom of the metasurface. The paper uses two qBIC mechanisms: a symmetry-protected hybrid mode formed by Friedrich-Wintgen destructive interference in germanium cylinders, and an $\varepsilon=0$ zero-coalescence BIC in AZO/silica multilayered spheres. In the lattice, the qBIC yields an almost flat photonic band, and the temporal modulation $N(t)=N_0(1+M_s\cos\omega_m t)$ folds the band structure about $\Re(\omega)=0.5\omega_m$; the long resonance lifetime prolongs the interaction of light with the time-varying material and widens the momentum bandgap for a fixed $M_s$. Band structures are computed by a T-matrix method for time-varying scatterers, using Floquet-Mie theory and the extended boundary condition method, with complex eigenfrequencies located by the AAA rational-approximation algorithm.

What would settle it

Drive the germanium cylinder metasurface with a pump at $\omega_m=2\pi\times300$ THz tuned to produce $M_s=5\times10^{-5}$ from the known Kerr coefficient of germanium, and probe the response at frequencies near $0.5\omega_m$ for every $k_\parallel$ across the predicted 3% momentum bandgap; the central claim requires modes with positive imaginary frequency (exponential growth) in that range. If no such growing modes appear at this $M_s$, or if they appear only at the $M_s=8.3\times10^{-4}$ needed by the Mie-resonant sphere metasurface, the claimed order-of-magnitude benefit of the qBIC would be refuted.

Watch

Extended reading notes

Core claim

The central discovery is that raising the quality factor of the resonance supported by each meta-atom lowers the temporal modulation amplitude needed to open a momentum bandgap, and quasi-bound states in the continuum provide a practical route to very high quality factors. In a static metasurface, a high-Q qBIC produces a nearly flat photonic band; when the electron density is modulated with frequency $\omega_m$ chosen so that $0.5\omega_m$ sits at the flat band, the folded Floquet bands interact strongly because light dwells in the resonator, and even $M_s\sim10^{-7}$ yields a 3% momentum bandgap in the lossless spherical case. With realistic losses in germanium cylinders ($Q \approx 500$), the same 3% gap requires $M_s = 5\times10^{-5}$, which is 16 times lower than for a metasurface of Mie-resonant germanium spheres and far below the nanostructured damage threshold. The same design opens a full bandgap across the whole Brillouin zone at $M_s = 4.4\times10^{-2}$ for the cylinder metasurface, and the threshold modulation for a full gap in the spherical system vanishes as the true BIC is approached.

Load-bearing premise

The practical claim rests on the assumption that the electron density of germanium can be driven sinusoidally at 300 THz with relative amplitude $5\times10^{-5}$ by the all-optical Kerr effect while staying below the damage threshold; the paper cites the threshold but does not compute the pump intensity required or check competing effects such as two-photon absorption and free-carrier generation.

Editorial extensions

If this is right

  • Optical-frequency photonic time crystals become experimentally plausible: a 3% momentum bandgap is predicted in germanium at $M_s=5\times10^{-5}$, orders of magnitude below the requirement for homogeneous PTCs and safely below the damage threshold of nanostructured germanium.
  • The threshold modulation amplitude for a full bandgap in the lossless sphere system tends to zero as the geometry approaches the true BIC, so improving the Q-factor of the meta-atom is a direct, quantitative route to lowering pump power.
  • Full momentum bandgaps covering the entire Brillouin zone are within reach, although for germanium cylinders at near-infrared frequencies the required $M_s=4.4\times10^{-2}$ remains slightly above the damage threshold; the paper points to low-loss, high-permittivity materials in the radio-frequency range as a path to full gaps at vanishingly small $M_s$.
  • The concept separates the design problem: optimize the isolated meta-atom for a high-Q qBIC rather than engineering the lattice, because band flatness is controlled by the single-scatterer Q-factor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper stops short of converting its modulation amplitudes into concrete pump intensities; a testable extension is to derive the Kerr-effect pump fluence for germanium at $M_s=5\times10^{-5}$ and check whether two-photon absorption or free-carrier generation perturbs the bandgap before thermal damage does.
  • The trend in the paper's threshold data implies a general trade-off: an order-of-magnitude reduction in required modulation amplitude costs roughly an order of magnitude in resonance Q, so material absorption sets a practical floor for how weak the modulation can be.
  • The same enhancement mechanism should transfer to other wave platforms, such as acoustic, elastic, or microwave systems, wherever a high-Q resonance coexists with temporal modulation, and it could also boost time-refraction and temporal-interface effects inside resonators, not just Floquet bandgaps.
  • A near-term experiment could test the scaling in the RF domain first: with supercavity modes in low-loss dielectric resonators, full momentum bandgaps should appear at $M_s$ below $10^{-6}$, a regime where temporal modulation is routinely achievable.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes that metasurfaces made of scatterers supporting quasi-bound states in the continuum (qBICs) can open momentum bandgaps in photonic time crystals at modulation amplitudes far below those required for homogeneous PTCs or Mie-resonant metasurfaces. Two geometries are analyzed: lossless AZO/silica multilayered spheres with a qBIC of Q≈5×10^4, and realistic germanium cylinders with a Friedrich–Wintgen qBIC of Q≈500. Using a T-matrix/Floquet formalism, the authors report a 3% finite momentum bandgap at Ms=1.35×10^-7 for the spheres and at Ms=5×10^-5 for the Ge cylinders, with full bandgaps at larger amplitudes. The Ge result is claimed to be experimentally reachable via the all-optical Kerr effect below the damage threshold.

Significance. The theoretical machinery is a genuine strength: the T-matrix formulation with Floquet harmonics, the AAA-based pole search, the explicit separation of lossless idealized results from lossy realistic ones, and the comparison with a Mie-resonant reference (Supplementary S4) are all state-of-the-art and carefully executed. The predicted scaling of the threshold amplitude with the qBIC Q-factor (Fig. 3) is a crisp, testable trend. If the practical claims were supported, this would be an important step toward optical-frequency photonic time crystals. However, the experimental pathway for the Ge example is not established, which substantially limits the significance as written.

major comments (2)
  1. [Results, 'PTCs based on metasurfaces made from germanium cylinders' (Fig. 4)] The claim that Ms=5e-5 is experimentally reachable in Ge via the all-optical Kerr effect is not supported by a physically realizable pump configuration. The paper sets omega_m=2*pi*300 THz and invokes Ref. [27] for a 'virtually unlimited' modulation frequency, but a Kerr nonlinearity modulates the refractive index with the intensity envelope of the pump, not with the instantaneous carrier field. Generating a 300 THz intensity envelope requires either two pump frequencies separated by 300 THz, both inside the Ge transparency window (below about 162 THz), which is impossible, or a sub-3-fs pulse whose spectrum extends above the bandgap, causing absorption and free-carrier generation. The paper provides no estimate of the pump intensity, two-photon absorption, or free-carrier density needed to realize Ms=5e-5, so the statement that this amplitude is 'well reachable ... without causing thermal damage' is unsupported. This is load-bearing because the Ge example is the only realistic demonstration of the central practical claim.
  2. [Results, germanium cylinders, paragraph citing damage thresholds] The paper quotes the damage threshold of bulk Ge as Ms≈1e-2 and of nanostructured Ge as Ms≈1e-3, citing Refs. [37,38,40,41], but those references report laser-induced damage thresholds in terms of fluence or intensity, not in terms of the dimensionless modulation amplitude Ms. The conversion from Ms to the required pump fluence depends on the nonlinear refractive index n2, the pump wavelength, and the pulse duration, none of which are specified. Without this quantitative conversion, the comparison between Ms=5e-5 and the quoted damage threshold is not meaningful. The authors should either provide the missing estimate or explicitly qualify the damage-threshold comparison as order-of-magnitude only.
minor comments (4)
  1. [Eq. (3)] The definition Delta = |kM + kX|/L is confusing because kM and kX have opposite signs along the M-Gamma-X path; please define them explicitly as positive magnitudes or clarify the sign convention in the text.
  2. [Fig. 2(c) caption] The caption states that the transparency of the bands is directly proportional to the radiative losses; since transparency is a plotting/rendering choice, please rephrase to describe what the transparency represents physically.
  3. [Supplementary S2] The comparison between the lattice qBIC (Delta=0.09%) and the meta-atom qBIC (Delta=3%) at the same Ms is useful, but the text should state explicitly that the modulation frequency was chosen to match the lattice BIC in the former case and the meta-atom qBIC in the latter, so that the comparison isolates the effect of the qBIC Q-factor.
  4. [Methods, 'Band structure calculation'] The convergence parameters (J, lmax, lcut) are listed in Supplementary S6, but no convergence study is shown; for the full-bandgap cases at Ms=4.4e-2, a brief convergence check with respect to the number of Floquet harmonics J would strengthen confidence in the results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the qBIC-assisted bandgap predictions are forward computations with modulation amplitude as an input, not a fitted parameter.

full rationale

I walked the derivation chain. The band structures are obtained by solving Eq. (2) with a T-matrix/Floquet-Mie/EBCM method; the modulation amplitude Ms is imposed as an input (e.g., Ms = 1.35e-7 and Ms = 5e-5) and the momentum gap is read off from the computed complex eigenfrequencies via Eq. (3). The qBIC quality factors are independently characterized from scattering cross sections (Q ≈ 5e4 for the lossless sphere, Q ≈ 500 for Ge cylinders), and the threshold modulation amplitude Mth is then computed as a function of geometry and loss. No parameter is fitted to the gap size, and no equation reduces to its own output. The only design choice is setting ωm so that 0.5ωm coincides with the qBIC flat band; this is a standard Floquet-folding condition rather than a hidden fit. The comparison against Mie-resonant metasurfaces is independently recomputed in Supplementary Section S4 rather than taken on faith from the self-cited Ref. [35]. The practical realizability concern about Kerr-driven 300 THz modulation of germanium is a correctness/feasibility issue, not circularity, because the bandgap calculation does not assume that pump scheme as an input. Overall, the central claim has independent computational content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim depends on several design parameters (geometry, modulation frequency, modulation amplitude) and on the validity of the time-varying Drude-Lorentz model and the numerical truncations. No new physical entities are introduced. The main unverified assumption is the experimental feasibility of the assumed temporal modulation.

free parameters (5)
  • Sphere geometry parameter eta = r1/r2 = 0.611 (eta/eta_BIC = 1.018)
    Chosen close to the BIC condition to obtain Q ~ 5e4; the whole enhancement depends on tuning near the BIC.
  • Lattice period a = a = 3*r2 (spheres), a = 3.5*r (cylinders)
    Design parameters for the metasurface; affect band structure and light coupling.
  • Modulation frequency omega_m = 2*pi*459 THz (spheres), 2*pi*300 THz (cylinders)
    Chosen so that omega_m/2 coincides with the qBIC resonance frequency, which is necessary for the bandgap to open.
  • Modulation amplitude Ms for finite/full bandgaps = Spheres: 1.35e-7, 2.8e-4; cylinders: 5e-5, 4.4e-2
    Selected to demonstrate finite and full bandgaps; the thresholds are computed from the model.
  • Germanium cylinder radius and aspect ratio = r = 465 nm, r/h = 0.698
    Optimized to maximize the qBIC quality factor Q ~ 500.
assumptions (5)
  • domain assumption The Drude-Lorentz model with time-varying electron density N(t) = N0(1 + Ms cos(omega_m t)) correctly describes the temporal modulation of AZO and Ge.
    Used throughout the paper (Supplementary S1) to model the time-varying permittivity.
  • domain assumption The Floquet expansion truncated at J = 7 harmonics and multipoles up to lmax/lcut = 4/6/7 is sufficient for accurate band structures.
    The convergence parameters are listed in Supplementary S6 but no convergence study is shown.
  • domain assumption The static qBIC quality factor of the isolated meta-atom controls the flatness of the bands and the interaction time in the time-modulated lattice.
    Supplementary S2 argues that meta-atom qBICs, not lattice qBICs, dominate the flat band behavior.
  • domain assumption The qBIC resonance of the isolated meta-atom remains at the same frequency and with similar Q under time modulation.
    The band structure calculations assume the static resonance properties are preserved when the time modulation is switched on.
  • standard math Standard electromagnetic scattering theory, including T-matrix, lattice translation matrices, and the treams implementation, is valid for these structures.
    These are established methods; the paper relies on them without re-deriving them.

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Cite this review

Pith. "Pith review of Photonic time crystals assisted by quasi-bound states in the continuum." pith.science (2026). https://pith.science/paper/AYEHUEX6

@misc{pith2026250715644,
  author       = {Pith},
  title        = {Pith review of: Photonic time crystals assisted by quasi-bound states in the continuum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYEHUEX6}},
  note         = {Machine review of arXiv:2507.15644}
}
read the original abstract

Photonic time crystals are a class of artificial materials that have only recently been explored. They are characterized by the ultrafast modulation of the material properties in time, causing a momentum bandgap for light that propagates through such novel states of matter. However, the observation of these unique properties at optical frequencies remains elusive, as the necessary modulation amplitudes of the permittivity to show notable momentum bandgaps are relatively high, inaccessible with available materials. While it has been known that structuring photonic time crystals at the sub-wavelength scale can enhance the momentum bandgap, we push this concept to the extreme by leveraging the nanophotonic toolbox. Specifically, we demonstrate that nanophotonic structures composed of scatterers supporting quasi-bound states in the continuum can significantly reduce the required amplitude of temporal permittivity modulation by enhancing the interaction time between light and time-varying matter. This allows us to observe extremely wide momentum bandgaps despite the material properties having tiny modulation amplitudes. Our approach bridges the concepts of bound states in the continuum and time-varying metamaterials, paving the way toward realizable photonic time crystals at optical frequencies.

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Reference graph

Works this paper leans on

67 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [27]

    Hayran, Z., Khurgin, J. B. & Monticone, F. ℏω versus ℏk: dispersion and energy constraints on time-varying photonic materials and time crystals. Opt. Mater. Express 12, 3904 (2022). URL https://doi.org/10.1364/OME.471672

    doi:10.1364/ome.471672 2022

  2. [1]

    Galiffi, E. et al. Photonics of time-varying media. Adv. Photonics 4, 014002 (2022). URL https://doi.org/10.1117/1.AP.4.1.014002

    doi:10.1117/1.ap.4.1.014002 2022

  3. [2]

    & Monticone, F

    Hayran, Z. & Monticone, F. Capturing broadband light in a compact bound state in the continuum. ACS Photonics 8, 813 (2021). URL https://pubs.acs.org/doi/ 10.1021/acsphotonics.0c01696

    work page doi:10.1021/acsphotonics.0c01696 2021

  4. [3]

    Yang, Z. et al. Ultrafast q-boosting in semiconductor metasurfaces. Nanophoton- ics 13, 2173 (2024). URL https://doi.org/10.1515/nanoph-2023-0718

    work page doi:10.1515/nanoph-2023-0718 2024

  5. [4]

    & Segev, M

    Schiller, O., Plotnik, Y., Segal, O., Lyubarov, M. & Segev, M. Time-domain bound states in the continuum. Phys. Rev. Lett. 133, 263802 (2024). URL https://link.aps.org/doi/10.1103/PhysRevLett.133.263802

    work page doi:10.1103/physrevlett.133.263802 2024

  6. [6]

    Metamaterials with high degrees of freedom: space, time, and more, 657 (De Gruyter, Berlin, Boston, 2021)

    Engheta, N. Metamaterials with high degrees of freedom: space, time, and more, 657 (De Gruyter, Berlin, Boston, 2021). URL https://doi.org/10.1515/ 9783110710687-051. 13

    work page 2021

  7. [7]

    Prudˆ encio, F. R. & Silveirinha, M. G. Synthetic axion response with space-time crystals. Phys. Rev. Appl.19, 024031 (2023). URL https://link.aps.org/doi/10. 1103/PhysRevApplied.19.024031

    work page 2023

  8. [8]

    & Pendry, J

    Huidobro, P., Silveirinha, M., Galiffi, E. & Pendry, J. Homogenization theory of space-time metamaterials. Phys. Rev. Appl. 16, 014044 (2021). URL https: //link.aps.org/doi/10.1103/PhysRevApplied.16.014044

    work page doi:10.1103/physrevapplied.16.014044 2021

Show all 67 references
  1. [9]

    Galiffi, E. et al. Optical coherent perfect absorption and amplification in a time- varying medium. arXiv preprint arXiv:2410.16426(2024). URL https://doi.org/ 10.48550/arXiv.2410.16426

    2024 doi arXiv

  2. [10]

    & Mccall, M

    Koufidis, S., Koutserimpas, T., Monticone, F. & Mccall, M. Electromagnetic wave propagation in time-periodic chiral media. Optical Materials Express 14, 3006 (2024). URL https://doi.org/10.1364/OME.543181

    2024 doi

  3. [11]

    & Engheta, N

    Pacheco-Pe˜ na, V. & Engheta, N. Antireflection temporal coatings.Optica 7, 323 (2020). URL https://doi.org/10.1364/OPTICA.381175

    2020 doi

  4. [12]

    S., Set¨ al¨ a, T

    Mirmoosa, M. S., Set¨ al¨ a, T. & Norrman, A. Quantum state engineering and photon statistics at electromagnetic time interfaces. Phys. Rev. Res.7, 013120 (2025). URL https://link.aps.org/doi/10.1103/PhysRevResearch.7.013120

    2025 doi

  5. [13]

    R., Halevi, P

    Zurita-S´ anchez, J. R., Halevi, P. & Cervantes-Gonz´ alez, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t). Phys. Rev. A 79, 053821 (2009). URL https://link.aps.org/doi/10.1103/ PhysRevA.79.053821

    2009

  6. [14]

    Asgari, M. M. et al. Theory and applications of photonic time crystals: a tutorial. Adv. Opt. Photon.16, 958 (2024). URL https://doi.org/10.1364/AOP.525163

    2024 doi

  7. [15]

    & Shapiro, B

    Zhang, S., Dong, J., Li, H., Xu, J. & Shapiro, B. Longitudinal optical phonons in photonic time crystals containing a stationary charge.Phys. Rev. B110, L100306 (2024). URL https://link.aps.org/doi/10.1103/PhysRevB.110.L100306

    2024 doi

  8. [16]

    Strong localization of photons in certain disordered dielectric super- lattices

    John, S. Strong localization of photons in certain disordered dielectric super- lattices. Phys. Rev. Lett. 58, 2486 (1987). URL https://doi.org/10.1103/ PhysRevLett.58.2486

    1987

  9. [17]

    Inhibited spontaneous emission in solid-state physics and electronics

    Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059 (1987). URL https://doi.org/10.1103/ PhysRevLett.58.2059

    1987

  10. [19]

    Dikopoltsev, A. et al. Light emission by free electrons in photonic time-crystals. Proc. Natl. Acad. Sci.119, e2119705119 (2022). URL https://doi.org/10.1073/ pnas.2119705119

    2022

  11. [20]

    & Shapiro, B

    Carminati, R., Chen, H., Pierrat, R. & Shapiro, B. Universal statistics of waves in a random time-varying medium. Phys. Rev. Lett.127, 094101 (2021). URL https://doi.org/10.1103/PhysRevLett.127.094101

    2021 doi

  12. [21]

    & Segev, M

    Sharabi, Y., Lustig, E. & Segev, M. Disordered photonic time crystals. Phys. Rev. Lett. 126, 163902 (2021). URL https://doi.org/10.1103/PhysRevLett.126. 163902

    2021 doi

  13. [22]

    & Fort, E

    Apffel, B., Wildeman, S., Eddi, A. & Fort, E. Time localization of energy in disordered time-modulated systems. arXiv preprint (2021). URL https://doi. org/10.48550/arXiv.2102.07583

    2021 doi arXiv

  14. [23]

    Reyes-Ayona, J. R. & Halevi, P. Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals. Appl. Phys. Lett. 107, 074101 (2015). URL https://doi.org/10.1063/1.4928659

    2015 doi

  15. [24]

    Park, J. et al. Revealing non-hermitian band structure of photonic floquet media. Sci. Advances 8, eabo6220 (2022). URL https://www.science.org/doi/10.1126/ sciadv.abo6220

    2022

  16. [25]

    Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Advances 9, eadg7541 (2023). URL http://science.org/doi/10.1126/sciadv. adg7541

    2023 doi

  17. [26]

    R., Halevi, P

    Zurita-S´ anchez, J. R., Halevi, P. & Cervantes-Gonz´ alez, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t). Phys. Rev. A79, 053821 (2009). URL https://doi.org/10.1103/PhysRevA. 79.053821

    2009 doi

  18. [28]

    Z., Leon, I

    Alam, M. Z., Leon, I. D. & Boyd, R. W. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science 352, 795 (2016). URL https: //science.org/doi/10.1126/science.aae0330

    2016 doi

  19. [29]

    S., Horsley, S

    Bohn, J., Luk, T. S., Horsley, S. & Hendry, E. Spatiotemporal refraction of light in an epsilon-near-zero indium tin oxide layer: frequency shifting effects arising from interfaces. Optica 8, 1532 (2021). URL https://opg.optica.org/optica/abstract. cfm?URI=optica-8-12-1532. 15

    2021

  20. [30]

    Tirole, R. et al. Double-slit time diffraction at optical frequencies. Nat. Physics 19, 999 (2023). URL https://doi.org/10.1038/s41567-023-01993-w

    2023 doi

  21. [31]

    Zhou, Y. et al. Broadband frequency translation through time refraction in an epsilon-near-zero material. Nat. Commun. 11, 2180 (2020). URL https://doi. org/10.1038/s41467-020-15682-2

    2020 doi

  22. [32]

    Lustig, E. et al. Time-refraction optics with single cycle modulation. Nanopho- tonics 12, 2221 (2023). URL https://doi.org/10.1515/nanoph-2023-0126

    2023 doi

  23. [33]

    Khurgin, J. B. Photonic time crystals and parametric amplification: Similarity and distinction. ACS Photonics 11, 2150 (2024). URL https://pubs.acs.org/doi/ 10.1021/acsphotonics.4c00607

    2024 doi

  24. [34]

    & Steinmeyer, G

    Borchers, B., Br´ ee, C., Birkholz, S., Demircan, A. & Steinmeyer, G. Saturation of the all-optical kerr effect in solids. Opt. Lett. 37, 1541 (2012). URL https: //doi.org/10.1364/OL.37.001541

    2012 doi

  25. [35]

    Wang, X. et al. Expanding momentum bandgaps in photonic time crystals through resonances. Nat. Photonics 19, 149 (2025). URL https://doi.org/10. 1038/s41566-024-01563-3

    2025

  26. [36]

    Optical damage threshold of silicon for ultrafast infrared pulsesVol

    Cowan, B. Optical damage threshold of silicon for ultrafast infrared pulsesVol. 877 (AIP Conf. Proc., 2006). URL https://www.slac.stanford.edu/pubs/slacpubs/ 12000/slac-pub-12089.pdf

    2006

  27. [37]

    N., Pogorelsky, I

    Polyanskiy, M. N., Pogorelsky, I. V., Babzien, M., Vodopyanov, K. L. & Palmer, M. A. Nonlinear refraction and absorption properties of optical materials for high-peak-power long-wave-infrared lasers. Opt. Mater. Express14, 696 (2024). URL https://opg.optica.org/ome/abstract.cf...

    2024

  28. [38]

    Agustsson, R. et al. Measuring single-shot, picosecond optical damage threshold in ge, si, and sapphire with a 5.1 µm laser. Opt. Mater. Express 5, 2835–2842 (2015). URL https://opg.optica.org/ome/abstract.cfm?URI=ome-5-12-2835

    2015

  29. [39]

    K., Soref, R

    Hon, N. K., Soref, R. & Jalali, B. The third-order nonlinear optical coefficients of si, ge, and Si 1−xGex in the midwave and longwave infrared. J. Appl. Phys.110, 011301 (2011). URL https://doi.org/10.1063/1.3592270

    2011 doi

  30. [40]

    et al.Comparison of femtosecond laser-induced damage on unstructured vs

    Kern, C. et al.Comparison of femtosecond laser-induced damage on unstructured vs. nano-structured au-targets. Appl. Phys. A104, 15 (2011). URL https://doi. org/10.1007/s00339-011-6449-2

    2011 doi

  31. [41]

    P., Oulton, R

    Grinblat, G., Li, Y., Nielsen, M. P., Oulton, R. F. & Maier, S. A. Efficient third harmonic generation and nonlinear subwavelength imaging at a higher-order anapole mode in a single germanium nanodisk. ACS Nano 11, 953–960 (2017). URL https://pubs.acs.org/doi/10.1021/acsnano.6...

    2017 doi

  32. [42]

    Rybin, M. V. et al. High-q supercavity modes in subwavelength dielectric res- onators. Phys. Rev. Lett.119, 243901 (2017). URL https://link.aps.org/doi/10. 1103/PhysRevLett.119.243901

    2017

  33. [43]

    Koshelev, K. et al. Subwavelength dielectric resonators for nonlinear nanopho- tonics. Science 367, 288 (2020). URL http://science.org/doi/10.1126/science. aaz3985

    2020 doi

  34. [44]

    Bogdanov, A. A. et al. Bound states in the continuum and Fano resonances in the strong mode coupling regime. Adv. Photonics 1, 016001 (2019). URL https://doi.org/10.1117/1.AP.1.1.016001

    2019 doi

  35. [45]

    & Al` u, A

    Monticone, F. & Al` u, A. Embedded photonic eigenvalues in 3d nanostructures. Phys. Rev. Lett. 112, 213903 (2014). URL https://link.aps.org/doi/10.1103/ PhysRevLett.112.213903

    2014

  36. [46]

    Caspani, L. et al. Enhanced nonlinear refractive index in ϵ-near-zero materi- als. Phys. Rev. Lett.116, 233901 (2016). URL https://link.aps.org/doi/10.1103/ PhysRevLett.116.233901

    2016

  37. [47]

    & Lequime, M

    Gao, L., Lemarchand, F. & Lequime, M. Exploitation of multiple incidences spectrometric measurements for thin film reverse engineering. Opt. Express 20, 15734 (2012). URL https://doi.org/10.1364/OE.20.015734

    2012 doi

  38. [48]

    Silveirinha, M. G. Trapping light in open plasmonic nanostructures. Phys. Rev. A 89, 023813 (2014). URL https://link.aps.org/doi/10.1103/PhysRevA.89.023813

    2014 doi

  39. [49]

    Matrix formulation of electromagnetic scattering

    Waterman, P. Matrix formulation of electromagnetic scattering. Proc. of the IEEE 53, 805 (1965). URL https://ieeexplore.ieee.org/document/1445988/

    1965

  40. [50]

    Garg, P. et al. Modeling four-dimensional metamaterials: a t-matrix approach to describe time-varying metasurfaces. Opt. Express 30, 45832–45847 (2022). URL https://opg.optica.org/oe/abstract.cfm?URI=oe-30-25-45832

    2022

  41. [51]

    & Kuznetsova, L

    Kelly, P., Zhang, W., Liu, M. & Kuznetsova, L. Engineering the structural, plasmonic, and optical properties of multilayered aluminum-doped zinc oxide metamaterial grown by pulsed laser deposition. Appl. Opt.58, 5681 (2019). URL https://doi.org/10.1364/AO.58.005681

    2019 doi

  42. [52]

    Nunley, T. N. et al. Optical constants of germanium and thermally grown germa- nium dioxide from 0.5 to 6.6ev via a multisample ellipsometry investigation. J. Vac. Sci. Technol. B34, 061205 (2016). URL https://doi.org/10.1116/1.4963075

    2016 doi

  43. [53]

    Plasmon modes of axisymmetric metallic nanoparticles: A group theory analysis

    Gantzounis, G. Plasmon modes of axisymmetric metallic nanoparticles: A group theory analysis. J. Phys. Chem. C 113, 21560 (2009). URL https://pubs.acs. org/doi/10.1021/jp908019s. 17

    2009 doi

  44. [54]

    Odit, M. et al. Observation of supercavity modes in subwavelength dielectric resonators. Adv. Mater.33, 2003804 (2021). URL https://doi.org/10.1002/adma. 202003804

    2021 doi

  45. [55]

    & Stefanou, N

    Panagiotidis, E., Almpanis, E., Papanikolaou, N. & Stefanou, N. Optical transi- tions and nonreciprocity in spatio-temporally periodic layers of spherical particles. Adv. Opt. Mater.11, 2202812 (2023). URL https://onlinelibrary.wiley.com/doi/ abs/10.1002/adom.202202812

    2023 doi

  46. [56]

    Waterman, P. C. Matrix theory of elastic wave scattering. J. Acoust. Soc. Am. 60, 567 (1976). URL https://doi.org/10.1121/1.381130

    1976 doi

  47. [57]

    Mackowski, D. W. Exact solution for the scattering and absorption properties of sphere clusters on a plane surface. J. Quant. Spectrosc. Radiat. Transfer109, 770 (2008). URL https://doi.org/10.1016/j.jqsrt.2007.08.024

    2008 doi

  48. [58]

    & Fernandez-Corbaton, I

    Beutel, D., Groner, A., Rockstuhl, C. & Fernandez-Corbaton, I. Efficient simula- tion of biperiodic, layered structures based on the t-matrix method. J. Opt. Soc. Am. B 38, 1782 (2021). URL https://doi.org/10.1364/JOSAB.419645

    2021 doi

  49. [59]

    & Rockstuhl, C

    Beutel, D., Fernandez-Corbaton, I. & Rockstuhl, C. treams – a t-matrix-based scattering code for nanophotonics. Comput. Phys. Commun.297, 109076 (2024). URL https://www.sciencedirect.com/science/article/pii/S0010465523004216

    2024

  50. [60]

    Lamprianidis, A. G. Generalized Transition Matrix Methods for the Analysis of Linear Nanophotonic Systems. Ph.D. thesis, Karlsruher Institut f¨ ur Technologie (KIT) (2023). URL http://publikationen.bibliothek.kit.edu/1000171446

    2023

  51. [61]

    Ptitcyn, G. et al. Floquet–mie theory for time-varying dispersive spheres. Laser Photonics Rev. 17, 2100683 (2023). URL https://doi.org/10.1002/lpor. 202100683

    2023 doi

  52. [62]

    Stefanou, N. et al. Light scattering by a periodically time-modulated object of arbitrary shape: the extended boundary condition method. J. Opt. Soc. Am. B 40, 2842 (2023). URL https://opg.optica.org/josab/abstract.cfm?URI= josab-40-11-2842

    2023

  53. [63]

    https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15

    NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15. URL https://dlmf.nist.gov/. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A....

    2025

  54. [64]

    Rahimzadegan, A. et al. A comprehensive multipolar theory for periodic meta- surfaces. Adv. Opt. Mater. 10, 2102059 (2022). URL https://doi.org/10.1002/ adom.202102059. 18

    2022

  55. [65]

    & Trefethen, L

    Nakatsukasa, Y., S` ete, O. & Trefethen, L. N. The AAA Algorithm for Rational Approximation. SIAM J. Sci. Comput.40, A1494 (2018). URL https://epubs. siam.org/doi/10.1137/16M1106122. Publisher: Society for Industrial and Applied Mathematics

    2018 doi

  56. [66]

    & Binkowski, F

    Betz, F., Hammerschmidt, M., Zschiedrich, L., Burger, S. & Binkowski, F. Effi- cient Rational Approximation of Optical Response Functions with the AAA Algorithm. Laser Photonics Rev.18, 2400584 (2024). URL https://onlinelibrary. wiley.com/doi/10.1002/lpor.202400584

    2024 doi

  57. [67]

    Fischbach, J. D. et al. A Framework to Compute Resonances Arising from Multi- ple Scattering. Adv. Theor. Simul.8, 2400989 (2025). URL https://onlinelibrary. wiley.com/doi/abs/10.1002/adts.202400989

    2025 doi

  58. [68]

    & Evlyukhin, A

    Ustimenko, N., Rockstuhl, C. & Evlyukhin, A. B. Resonances in finite-size all- dielectric metasurfaces for light trapping and propagation control. Phys. Rev. B 109, 115436 (2024). URL https://link.aps.org/doi/10.1103/PhysRevB.109. 115436

    2024 doi

  59. [69]

    #)Static metasurface 𝑄≈90 0.5𝜔! BIC 𝑟!𝑟!𝑟

    Waterman, P. T-matrix methods in acoustic scattering. J. Acoust. Soc. Am.125, 42 (2009). URL https://doi.org/10.1121/1.3035839. Supplementary Material S1 Drude-Lorentz model We use the Drude-Lorentz model to consider dispersion in time-varying media. The polarization density o...

    2009 doi

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