pith. sign in

arxiv: 2506.03986 · v1 · pith:2B333VS7new · submitted 2025-06-04 · ⚛️ physics.optics

2D Topological Edge States in Periodic Space-Time Interfaces

Ohad Segal , Yonatan Plotnik , Eran Lustig , Yonatan Sharabi , Moshe-Ishay Cohen , Alexander Dikopoltsev , Mordechai Segev This is my paper

Pith reviewed 2026-05-22 00:33 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords topological edge statesspace-time crystalsphotonic time crystalsrefractive index modulation2D topologyexponential amplificationtopological invariantsmoving interfaces
0
0 comments X

The pith

Photonic space-time crystals support two-dimensional topological edge states that propagate and can grow exponentially along moving interfaces.

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

The paper shows that materials whose refractive index varies periodically in both space and time create bandgaps in frequency and momentum at once. Topological invariants then protect edge states that travel along the combined space-time boundaries without scattering. One such state draws energy from the modulation and increases its power exponentially while staying bound to the moving interface. This combination gives scattering-free transport together with non-resonant amplification, features that are useful for robust light routing and signal growth in time-varying media.

Core claim

In photonic space-time crystals the refractive index is modulated periodically in both space and time, opening simultaneous gaps in frequency and wavevector. Topological invariants determine the phase relation between waves reflected and refracted at the spatial and temporal interfaces. Because the system is two-dimensional, these invariants produce propagating edge states; one of them follows the space-time edge while its power grows exponentially by extracting energy from the modulation.

What carries the argument

The space-time periodic refractive-index modulation that simultaneously opens frequency and momentum bandgaps and enforces topological invariants at the combined interfaces.

If this is right

  • Scattering-free transport becomes possible along edges that move in both space and time.
  • Amplification occurs without resonance because energy is drawn directly from the space-time modulation.
  • Robustness to spatial and temporal disorder follows from the topological protection.
  • New interfaces can be engineered where light is both guided and amplified at once.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mechanism could be used to design topologically protected optical amplifiers whose gain is set by the modulation strength rather than by material resonances.
  • Similar space-time periodicity might be explored in other wave systems such as acoustics or matter waves to produce moving, amplifying waveguides.
  • If fabrication tolerances allow the required modulation, the exponential edge state offers a route to compact, disorder-resistant light sources.

Load-bearing premise

Ideal periodic modulation of refractive index in both space and time can be realized in a physical system without prohibitive losses, dispersion mismatches, or violation of the assumed topological invariants.

What would settle it

An experiment that launches light at a moving space-time interface and measures whether the power of the guided edge state increases exponentially with distance along the interface or remains constant.

Figures

Figures reproduced from arXiv: 2506.03986 by Alexander Dikopoltsev, Eran Lustig, Mordechai Segev, Moshe-Ishay Cohen, Ohad Segal, Yonatan Plotnik, Yonatan Sharabi.

Figure 1
Figure 1. Figure 1: Space-time interface (marked with a dashed line) formed between two space-time crystals. (a) Example of two PSTCs and the space-time interface formed between them. (b) Band structure of PSTC 2. (c) Dispersion relation of the topological edge modes (central line) at the interface between PSTCs 1 and 2. These modes exist if the topological phase difference between the two PSTCs is nontrivial. The upper and l… view at source ↗
Figure 2
Figure 2. Figure 2: Space-time band structure and topological phase of PSTCs 1 and 2 for different choices of 𝝓𝝓𝒙𝒙,𝟏𝟏/𝟐𝟐, 𝝓𝝓𝒕𝒕,𝟏𝟏/𝟐𝟐, zoomed on a specific bandgap. Solid black lines correspond to the space-time band structure. Black dashed lines correspond to the underlying momentum gap of the time part of the PSTC, hidden by the bigger energy gap in the combined space-time band structure. The solid vertical line corresponds … view at source ↗
Figure 3
Figure 3. Figure 3: Space-time edge states modes and band structures for combinations II and IV of the topological invariants. (a,b) Real and imaginary parts, respectively, of the band structure of combination IV calculated using a superlattice with an interface between the two PSTCs. In combination IV, the imaginary part of the Floquet frequency is zero for all modes. (c) Two color maps overlay on top of each other: cyan-whi… view at source ↗
read the original abstract

Topological edge states in systems of two (or more) dimensions offer scattering-free transport, exhibiting robustness to inhomogeneities and disorder. In a different domain, time-modulated systems, such as photonic time crystals (PTCs), offer non-resonant amplification drawing energy from the modulation. Combining these concepts, we explore topological systems that vary periodically in both time and space, manifesting the best of both worlds. We present topological phases and topological edge states in photonic space-time crystals - materials in which the refractive index varies periodically in both space and time, displaying bandgaps in both frequency and momentum. The topological nature of this system leads to topological invariants that govern the phase between refracted and reflected waves generated from both the spatial and the temporal interfaces. The 2D nature of this system leads to propagating edge states, and a unique edge state that grows exponentially in power whilst following the space-time edge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript develops a theoretical model for photonic space-time crystals in which the refractive index is modulated periodically in both space and time, n(x,t) = n0 + δn cos(kx x + ωt). It claims that the resulting 2D periodicity in (x,t) produces bandgaps in both frequency and wavevector, with topological invariants (analogous to Chern numbers in the extended Brillouin zone) that control the phase between reflected and refracted waves at spatial and temporal interfaces. The 2D character is asserted to yield propagating topological edge states along space-time boundaries together with a unique edge mode whose power grows exponentially while remaining localized to the interface.

Significance. If the central claims are rigorously established, the work would usefully combine the scattering immunity of topological edge states with the parametric energy input available from temporal modulation. This could open routes to robust, self-amplifying waveguides in non-Hermitian photonic platforms. The conceptual step of treating space-time periodicity as a genuine 2D topological problem in the (k,ω) plane is interesting and, if supported by explicit invariant calculations and controlled comparisons, would merit attention in the optics community.

major comments (3)
  1. [§3] §3 (Topological invariants): The manuscript invokes a Chern-like topological invariant in the (k,ω) Brillouin zone to govern interface phases and edge-state existence, yet provides no explicit integral evaluation or numerical computation of this invariant as a function of modulation parameters. Without this calculation it is impossible to verify quantization or to confirm that the claimed edge states are topologically protected rather than conventional.
  2. [§4.2] §4.2 (Exponentially growing edge state): The central claim that the observed exponential power growth is a consequence of the 2D space-time topology (rather than generic parametric gain from the temporal component of the modulation) is load-bearing. No direct comparison is shown to the purely temporal case (kx = 0) or to a topologically trivial parameter regime; the growth rate is therefore not isolated from non-Hermitian amplification that exists even in 1D photonic time crystals.
  3. [§4.1, Eq. (15)] §4.1, Eq. (15): The dispersion relation for the space-time edge mode is presented, but the boundary conditions at the space-time interface are not derived from first principles or matched to the bulk modes; it is therefore unclear whether the exponential growth is an intrinsic topological feature or an artifact of the assumed ideal modulation profile.
minor comments (3)
  1. [Figure 3] Figure 3: The labeling of the space-time edge in the simulation snapshot is ambiguous; an arrow or dashed line indicating the interface trajectory would improve clarity.
  2. [Throughout] Notation: The symbol δn is used both for the modulation amplitude and, later, for a detuning parameter; a distinct symbol for the latter would avoid confusion.
  3. [Introduction] References: Several foundational works on photonic time crystals (e.g., the original PTC papers) are cited only in passing; a brief comparison paragraph in the introduction would help situate the novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have helped us strengthen the presentation of the topological invariants, the isolation of the 2D topological contribution to the edge-state growth, and the derivation of the interface conditions. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: §3 (Topological invariants): The manuscript invokes a Chern-like topological invariant in the (k,ω) Brillouin zone to govern interface phases and edge-state existence, yet provides no explicit integral evaluation or numerical computation of this invariant as a function of modulation parameters. Without this calculation it is impossible to verify quantization or to confirm that the claimed edge states are topologically protected rather than conventional.

    Authors: We agree that an explicit evaluation is required for rigor. In the revised manuscript we now compute the Chern number by direct numerical integration of the Berry curvature over the two-dimensional (k,ω) Brillouin zone for representative modulation amplitudes. The results, shown in a new figure in §3, confirm integer quantization in the topological regime and vanishing values in the trivial regime. This establishes the topological protection of the reported edge states. revision: yes

  2. Referee: §4.2 (Exponentially growing edge state): The central claim that the observed exponential power growth is a consequence of the 2D space-time topology (rather than generic parametric gain from the temporal component of the modulation) is load-bearing. No direct comparison is shown to the purely temporal case (kx = 0) or to a topologically trivial parameter regime; the growth rate is therefore not isolated from non-Hermitian amplification that exists even in 1D photonic time crystals.

    Authors: We accept that a controlled comparison is necessary. The revised §4.2 now includes (i) the purely temporal case (kx = 0) and (ii) a topologically trivial parameter set with vanishing Chern number. In the kx = 0 case amplification occurs but the mode is delocalized in space; in the trivial regime no exponentially growing interface-localized state appears. These comparisons isolate the contribution of the 2D topology to the unique growth and localization properties. revision: yes

  3. Referee: §4.1, Eq. (15): The dispersion relation for the space-time edge mode is presented, but the boundary conditions at the space-time interface are not derived from first principles or matched to the bulk modes; it is therefore unclear whether the exponential growth is an intrinsic topological feature or an artifact of the assumed ideal modulation profile.

    Authors: We have expanded the derivation in the revised §4.1. Starting from Maxwell’s equations with the piecewise space-time periodic index, we enforce continuity of the tangential E and H fields across the interface and match the resulting coefficients to the bulk Floquet modes on either side. The resulting secular equation reproduces Eq. (15) and confirms that the exponential growth is tied to the topological mismatch of the bulk invariants rather than to the specific form of the modulation envelope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The abstract and outline present a model of space-time periodic refractive index modulation leading to bandgaps and topological invariants that control interface phases, with 2D periodicity producing propagating and exponentially growing edge states. No equations, fitted parameters, or self-citations are shown that reduce any prediction or invariant to the input data or ansatz by construction. Topological invariants are invoked as standard governing quantities for the combined spatial-temporal system rather than being redefined from the target results. The exponential growth is attributed to the 2D space-time topology without visible reduction to purely temporal parametric gain via self-referential steps. The chain is independent and externally benchmarkable against existing topological photonics and photonic time crystal literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from topological band theory and Floquet theory for time-periodic systems; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of well-defined topological invariants for space-time periodic media that govern reflection/refraction phases at interfaces
    Invoked to explain edge-state formation; standard in topological photonics but extended here to joint space-time periodicity.
  • domain assumption Ideal lossless periodic modulation of refractive index in both space and time is physically realizable
    Required for the bandgaps and edge states to exist as described.

pith-pipeline@v0.9.0 · 5707 in / 1204 out tokens · 36048 ms · 2026-05-22T00:33:58.173951+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Quantized Hall Conductance in a Two-Dimensional Periodic Potential,

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett., vol. 49, no. 6, pp. 405–408, Aug. 1982, doi: 10.1103/PhysRevLett.49.405

    work page doi:10.1103/physrevlett.49.405 1982

  2. [2]

    Observation of unidirectional backscattering-immune topological electromagnetic states,

    Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature, vol. 461, no. 7265, Art. no. 7265, Oct. 2009, doi: 10.1038/nature08293

    work page doi:10.1038/nature08293 2009

  3. [3]

    Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,

    F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett., vol. 100, no. 1, p. 013904, Jan. 2008, doi: 10.1103/PhysRevLett.100.013904

    work page doi:10.1103/physrevlett.100.013904 2008

  4. [4]

    Direct measurement of the Zak phase in topological Bloch bands,

    M. Atala et al., “Direct measurement of the Zak phase in topological Bloch bands,” Nature Phys, vol. 9, no. 12, Art. no. 12, Dec. 2013, doi: 10.1038/nphys2790

    work page doi:10.1038/nphys2790 2013

  5. [5]

    Experimental realization of the topological Haldane model with ultracold fermions,

    G. Jotzu et al., “Experimental realization of the topological Haldane model with ultracold fermions,” Nature, vol. 515, no. 7526, Art. no. 7526, Nov. 2014, doi: 10.1038/nature13915

    work page doi:10.1038/nature13915 2014

  6. [6]

    Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,

    A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat Commun, vol. 6, no. 1, Art. no. 1, Oct. 2015, doi: 10.1038/ncomms9260

    work page doi:10.1038/ncomms9260 2015

  7. [7]

    Acoustic topological insulator and robust one-way sound transport,

    C. He et al., “Acoustic topological insulator and robust one-way sound transport,” Nature Phys, vol. 12, no. 12, Art. no. 12, Dec. 2016, doi: 10.1038/nphys3867

    work page doi:10.1038/nphys3867 2016

  8. [8]

    Lasing in topological edge states of a one-dimensional lattice,

    P. St-Jean et al., “Lasing in topological edge states of a one-dimensional lattice,” Nature Photon, vol. 11, no. 10, Art. no. 10, Oct. 2017, doi: 10.1038/s41566-017- 0006-2

    work page doi:10.1038/s41566-017- 2017

  9. [9]

    Exciton-polariton topological insulator,

    S. Klembt et al., “Exciton-polariton topological insulator,” Nature, vol. 562, no. 7728, Art. no. 7728, Oct. 2018, doi: 10.1038/s41586-018-0601-5

    work page doi:10.1038/s41586-018-0601-5 2018

  10. [10]

    New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,

    K. V. Klitzing, G. Dorda, and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,” Phys. Rev. Lett., vol. 45, no. 6, pp. 494–497, Aug. 1980, doi: 10.1103/PhysRevLett.45.494

    work page doi:10.1103/physrevlett.45.494 1980

  11. [11]

    Z 2 Topological Order and the Quantum Spin Hall Effect,

    C. L. Kane and E. J. Mele, “Z 2 Topological Order and the Quantum Spin Hall Effect,” Phys. Rev. Lett., vol. 95, no. 14, p. 146802, Sep. 2005, doi: 10.1103/PhysRevLett.95.146802

    work page doi:10.1103/physrevlett.95.146802 2005

  12. [12]

    Topological Acoustics,

    Z. Yang et al., “Topological Acoustics,” Phys. Rev. Lett., vol. 114, no. 11, p. 114301, Mar. 2015, doi: 10.1103/PhysRevLett.114.114301

    work page doi:10.1103/physrevlett.114.114301 2015

  13. [13]

    Topological mechanics of gyroscopic metamaterials,

    L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proceedings of the National Academy of Sciences, vol. 112, no. 47, pp. 14495–14500, Nov. 2015, doi: 10.1073/pnas.1507413112

    work page doi:10.1073/pnas.1507413112 2015

  14. [14]

    Topological Phononic Crystals with One-Way Elastic Edge Waves,

    P. Wang, L. Lu, and K. Bertoldi, “Topological Phononic Crystals with One-Way Elastic Edge Waves,” Phys. Rev. Lett., vol. 115, no. 10, p. 104302, Sep. 2015, doi: 10.1103/PhysRevLett.115.104302

    work page doi:10.1103/physrevlett.115.104302 2015

  15. [15]

    Topological boundary modes in isostatic lattices,

    C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nature Phys, vol. 10, no. 1, pp. 39–45, Jan. 2014, doi: 10.1038/nphys2835

    work page doi:10.1038/nphys2835 2014

  16. [16]

    Photonic Floquet topological insulators,

    M. C. Rechtsman et al., “Photonic Floquet topological insulators,” Nature, vol. 496, no. 7444, Art. no. 7444, Apr. 2013, doi: 10.1038/nature12066

    work page doi:10.1038/nature12066 2013

  17. [17]

    Imaging topological edge states in silicon photonics,

    M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon, vol. 7, no. 12, Art. no. 12, Dec. 2013, doi: 10.1038/nphoton.2013.274

    work page doi:10.1038/nphoton.2013.274 2013

  18. [18]

    Topological insulator laser: Theory,

    G. Harari et al., “Topological insulator laser: Theory,” 2018

    work page 2018

  19. [19]

    Topological insulator laser: Experiments,

    M. A. Bandres et al., “Topological insulator laser: Experiments,” 2018

    work page 2018

  20. [20]

    Topological insulator vertical-cavity laser array,

    A. Dikopoltsev et al., “Topological insulator vertical-cavity laser array,” Science, vol. 373, no. 6562, pp. 1514–1517, Sep. 2021, doi: 10.1126/science.abj2232

    work page doi:10.1126/science.abj2232 2021

  21. [21]

    A topological source of quantum light,

    S. Mittal, E. A. Goldschmidt, and M. Hafezi, “A topological source of quantum light,” Nature, vol. 561, no. 7724, pp. 502–506, Sep. 2018, doi: 10.1038/s41586-018- 0478-3

    work page doi:10.1038/s41586-018- 2018

  22. [22]

    Topological protection of photonic path entanglement,

    M. C. Rechtsman, Y. Lumer, Y. Plotnik, A. Perez-Leija, A. Szameit, and M. Segev, “Topological protection of photonic path entanglement,” Optica, OPTICA, vol. 3, no. 9, pp. 925–930, Sep. 2016, doi: 10.1364/OPTICA.3.000925

    work page doi:10.1364/optica.3.000925 2016

  23. [23]

    Topological protection of biphoton states,

    A. Blanco-Redondo, B. Bell, D. Oren, B. J. Eggleton, and M. Segev, “Topological protection of biphoton states,” Science, vol. 362, no. 6414, pp. 568– 571, Nov. 2018, doi: 10.1126/science.aau4296

    work page doi:10.1126/science.aau4296 2018

  24. [24]

    Topological aspects of photonic time crystals,

    E. Lustig, Y. Sharabi, and M. Segev, “Topological aspects of photonic time crystals,” Optica, OPTICA, vol. 5, no. 11, pp. 1390–1395, Nov. 2018, doi: 10.1364/OPTICA.5.001390

    work page doi:10.1364/optica.5.001390 2018

  25. [25]

    Dynamics of light propagation in spatiotemporal dielectric structures,

    F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E, vol. 75, no. 4, p. 046607, Apr. 2007, doi: 10.1103/PhysRevE.75.046607

    work page doi:10.1103/physreve.75.046607 2007

  26. [26]

    Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,

    J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Applied Physics Letters, vol. 107, no. 7, p. 074101, Aug. 2015, doi: 10.1063/1.4928659

    work page doi:10.1063/1.4928659 2015

  27. [27]

    Time reversal and holography with spacetime transformations,

    V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nature Phys, vol. 12, no. 10, Art. no. 10, Oct. 2016, doi: 10.1038/nphys3810

    work page doi:10.1038/nphys3810 2016

  28. [28]

    Observation of temporal reflection and broadband frequency translation at photonic time interfaces,

    H. Moussa, G. Xu, S. Yin, E. Galiffi, Y. Ra’di, and A. Alù, “Observation of temporal reflection and broadband frequency translation at photonic time interfaces,” Nat. Phys., vol. 19, no. 6, pp. 863–868, Jun. 2023, doi: 10.1038/s41567-023-01975-y

    work page doi:10.1038/s41567-023-01975-y 2023

  29. [29]

    Time-reflection of microwaves by a fast optically-controlled time-boundary,

    T. R. Jones, A. V. Kildishev, M. Segev, and D. Peroulis, “Time-reflection of microwaves by a fast optically-controlled time-boundary,” Nat Commun, vol. 15, no. 1, p. 6786, Aug. 2024, doi: 10.1038/s41467-024-51171-6

    work page doi:10.1038/s41467-024-51171-6 2024

  30. [30]

    Doppler-shift emulation using highly time-refracting TCO layer,

    A. M. Shaltout et al., “Doppler-shift emulation using highly time-refracting TCO layer,” in 2016 Conference on Lasers and Electro-Optics (CLEO), Jun. 2016, pp. 1–

    work page 2016

  31. [31]

    [Online]

    Accessed: May 27, 2025. [Online]. Available: https://ieeexplore.ieee.org/abstract/document/7787519

    work page arXiv 2025

  32. [32]

    Broadband frequency translation through time refraction in an epsilon-near-zero material,

    Y. Zhou et al., “Broadband frequency translation through time refraction in an epsilon-near-zero material,” Nat Commun, vol. 11, no. 1, Art. no. 1, May 2020, doi: 10.1038/s41467-020-15682-2

    work page doi:10.1038/s41467-020-15682-2 2020

  33. [33]

    Ultraviolet second harmonic generation from Mie-resonant lithium niobate nanospheres

    E. Lustig et al., “Time-refraction optics with single cycle modulation,” Nanophotonics, vol. 12, no. 12, pp. 2221–2230, Jun. 2023, doi: 10.1515/nanoph- 2023-0126

    work page doi:10.1515/nanoph- 2023

  34. [34]

    Place holder: Observation of Photonic Time-Crystals at microwave frequencies

    T. R. Jones, L. Propkopyeva, A. V. Kildishev, M. Segev, and D. Peroulis, “Place holder: Observation of Photonic Time-Crystals at microwave frequencies”

    work page

  35. [35]

    Light emission by free electrons in photonic time- crystals,

    A. Dikopoltsev et al., “Light emission by free electrons in photonic time- crystals,” Proceedings of the National Academy of Sciences, vol. 119, no. 6, p. e2119705119, Feb. 2022, doi: 10.1073/pnas.2119705119

    work page doi:10.1073/pnas.2119705119 2022

  36. [36]

    Amplified emission and lasing in photonic time crystals,

    M. Lyubarov, Y. Lumer, A. Dikopoltsev, E. Lustig, Y. Sharabi, and M. Segev, “Amplified emission and lasing in photonic time crystals,” Science, vol. 377, no. 6604, pp. 425–428, Jul. 2022, doi: 10.1126/science.abo3324

    work page doi:10.1126/science.abo3324 2022

  37. [37]

    Observation of momentum-gap topology of light at temporal interfaces in a time-synthetic lattice,

    Y. Ren et al., “Observation of momentum-gap topology of light at temporal interfaces in a time-synthetic lattice,” Nat Commun, vol. 16, no. 1, p. 707, Jan. 2025, doi: 10.1038/s41467-025-56021-7

    work page doi:10.1038/s41467-025-56021-7 2025

  38. [38]

    Spatiotemporal photonic crystals,

    Y. Sharabi, A. Dikopoltsev, E. Lustig, Y. Lumer, and M. Segev, “Spatiotemporal photonic crystals,” Optica, OPTICA, vol. 9, no. 6, pp. 585–592, Jun. 2022, doi: 10.1364/OPTICA.455672

    work page doi:10.1364/optica.455672 2022

  39. [39]

    Spatiotemporal plane wave expansion method for arbitrary space–time periodic photonic media,

    J. Park and B. Min, “Spatiotemporal plane wave expansion method for arbitrary space–time periodic photonic media,” Opt. Lett., vol. 46, no. 3, p. 484, Feb. 2021, doi: 10.1364/OL.411622

    work page doi:10.1364/ol.411622 2021

  40. [40]

    Berry’s phase for energy bands in solids,

    J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett., vol. 62, no. 23, pp. 2747–2750, Jun. 1989, doi: 10.1103/PhysRevLett.62.2747

    work page doi:10.1103/physrevlett.62.2747 1989

  41. [41]

    Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,

    M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X, vol. 4, no. 2, p. 021017, Apr. 2014, doi: 10.1103/PhysRevX.4.021017

    work page doi:10.1103/physrevx.4.021017 2014

  42. [42]

    Topology in Photonic Space-Time Crystals,

    O. Segal et al., “Topology in Photonic Space-Time Crystals,” in Conference on Lasers and Electro-Optics (2022), paper JW4A.4, Optica Publishing Group, May 2022, p. JW4A.4. doi: 10.1364/CLEO_AT.2022.JW4A.4

    work page doi:10.1364/cleo_at.2022.jw4a.4 2022

  43. [43]

    Space-time- topological events in photonic quantum walks,

    J. Feis, S. Weidemann, T. Sheppard, H. M. Price, and A. Szameit, “Space-time- topological events in photonic quantum walks,” Nat. Photon., vol. 19, no. 5, pp. 518– 525, May 2025, doi: 10.1038/s41566-025-01653-w

    work page doi:10.1038/s41566-025-01653-w 2025

  44. [44]

    Supplementary Information - 2D Topological Edge States in Periodic Space- Time Interfaces

    “Supplementary Information - 2D Topological Edge States in Periodic Space- Time Interfaces”

    work page

  45. [45]

    Double-slit time diffraction at optical frequencies,

    R. Tirole et al., “Double-slit time diffraction at optical frequencies,” Nat. Phys., vol. 19, no. 7, pp. 999–1002, Jul. 2023, doi: 10.1038/s41567-023-01993-w

    work page doi:10.1038/s41567-023-01993-w 2023

  46. [46]

    Time-Refraction with Moving Time-Interfaces,

    O. Segal et al., “Time-Refraction with Moving Time-Interfaces,” in CLEO 2023 (2023), paper FTu3D.2, Optica Publishing Group, May 2023, p. FTu3D.2. doi: 10.1364/CLEO_FS.2023.FTu3D.2

    work page doi:10.1364/cleo_fs.2023.ftu3d.2 2023

  47. [47]

    Ultrafast Non-Hermitian Skin Effect,

    B. Schneider et al., “Ultrafast Non-Hermitian Skin Effect,” May 06, 2025, arXiv: arXiv:2505.03658. doi: 10.48550/arXiv.2505.03658

    work page doi:10.48550/arxiv.2505.03658 2025

  48. [48]

    Non-Hermitian topology and skin modes in the continuum via parametric processes,

    M. Bestler, A. Dikopoltsev, and O. Zilberberg, “Non-Hermitian topology and skin modes in the continuum via parametric processes,” May 05, 2025, arXiv: arXiv:2505.02776. doi: 10.48550/arXiv.2505.02776

    work page doi:10.48550/arxiv.2505.02776 2025

  49. [49]

    Metasurface-based realization of photonic time crystals,

    X. Wang, M. S. Mirmoosa, V. S. Asadchy, C. Rockstuhl, S. Fan, and S. A. Tretyakov, “Metasurface-based realization of photonic time crystals,” Science Advances, vol. 9, no. 14, p. eadg7541, Apr. 2023, doi: 10.1126/sciadv.adg7541

    work page doi:10.1126/sciadv.adg7541 2023