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  • Open Access

Energy estimates for wave amplification in quasiperiodic Fibonacci time-modulated media

Ilya Ruchiev and Bryn Davies*

  • *Contact author: bryn.davies@warwick.ac.uk
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Phys. Rev. A 114, 023502 – Published 3 August, 2026

DOI: https://doi.org/10.1103/w9bj-gmml

Abstract

We show that Fibonacci time quasicrystals can be approximated by temporal supercells, which capture the nonperiodic but self-similar modulation in an analytically tractable form. This reveals a fractal hierarchy of instability gaps, indexed by the wave number k, in which wave energy is amplified exponentially. These estimates are validated by the observation of super k gaps: instability regions that are independent of the duration of the temporal supercell. This approach predicts the regions of parametric amplification and provides arbitrarily accurate estimates of the energy growth rate.

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References (56)

  1. F. R. Morgenthaler, Velocity modulation of electromagnetic waves, IEEE Trans. Microwave Theory Techn. 6, 167 (1958).
  2. D. Holberg and K. Kunz, Parametric properties of fields in a slab of time-varying permittivity, IEEE Trans. Antennas Propag. 14, 183 (1966).
  3. A. A. Oliner and A. Hessel, Wave propagation in a medium with a progressive sinusoidal disturbance, IEEE Trans. Microwave Theory Techn. 9, 337 (1961).
  4. J. C. Simon, Action of a progressive disturbance on a guided electromagnetic wave, IEEE Trans. Microwave Theory Techn. 8, 18 (1960).
  5. E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Alù, and J. B. Pendry, Photonics of time-varying media, Adv. Photonics 4, 014002 (2022).
  6. S. Yin, E. Galiffi, and A. Alù, Floquet metamaterials, eLight 2, 8 (2022).
  7. P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, and J. B. Pendry, Fresnel drag in space–time-modulated metamaterials, Proc. Natl. Acad. Sci. USA 116, 24943 (2019).
  8. A. Cullen, A travelling-wave parametric amplifier, Nature (London) 181, 332 (1958).
  9. M. Raiford, Degenerate parametric amplification with time-dependent pump amplitude and phase, Phys. Rev. A 9, 2060 (1974).
  10. A. Nikzamir, K. Rouhi, A. Figotin, and F. Capolino, How to achieve exceptional points in coupled resonators using a gyrator or PT-symmetry, and in a time-modulated single resonator: High sensitivity to perturbations, EPJ Appl. Metamat. 9, 14 (2022).
  11. K. Rouhi, H. Kazemi, A. Figotin, and F. Capolino, Exceptional points of degeneracy directly induced by space–time modulation of a single transmission line, Antennas Wirel. Propag. Lett. 19, 1906 (2020).
  12. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature (London) 496, 196 (2013).
  13. R. Fleury, A. B. Khanikaev, and A. Alu, Floquet topological insulators for sound, Nat. Commun. 7, 11744 (2016).
  14. T. T. Koutserimpas and C. Valagiannopoulos, Multiharmonic resonances of coupled time-modulated resistive metasurfaces, Phys. Rev. Appl. 19, 064072 (2023).
  15. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-Gonzalez, Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ɛ(t), Phys. Rev. A 79, 053821 (2009).
  16. E. O. Hiltunen and B. Davies, Coupled harmonics due to time-modulated point scatterers, Phys. Rev. B 110, 184102 (2024).
  17. R. Tirole, S. Vezzoli, E. Galiffi, I. Robertson, D. Maurice, B. Tilmann, S. A. Maier, J. B. Pendry, and R. Sapienza, Double-slit time diffraction at optical frequencies, Nat. Phys. 19, 999 (2023).
  18. Y.-G. Peng, C.-Z. Qin, D.-G. Zhao, Y.-X. Shen, X.-Y. Xu, M. Bao, H. Jia, and X.-F. Zhu, Experimental demonstration of anomalous Floquet topological insulator for sound, Nat. Commun. 7, 13368 (2016).
  19. M. Fink, Time reversed acoustics, Phys. Today 50(3), 34 (1997).
  20. R. Pierrat, J. Rocha, and R. Carminati, Causality and instability in wave propagation in random time-varying media, Phys. Rev. Lett. 134, 233801 (2025).
  21. T. T. Koutserimpas and R. Fleury, Electromagnetic waves in a time periodic medium with step-varying refractive index, IEEE T. Antenn. Prpag. 66, 5300 (2018).
  22. R. Carminati, H. Chen, R. Pierrat, and B. Shapiro, Universal statistics of waves in a random time-varying medium, Phys. Rev. Lett. 127, 094101 (2021).
  23. S. F. Koufidis, T. T. Koutserimpas, F. Monticone, and M. W. McCall, Enhanced scattering from an almost-periodic optical temporal slab, Phys. Rev. A 110, L041501 (2024).
  24. G. He, B. Ye, R. Gong, C. Yao, Z. Liu, K. W. Murch, N. Y. Yao, and C. Zu, Experimental realization of discrete time quasicrystals, Phys. Rev. X 15, 011055 (2025).
  25. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951 (1984).
  26. B. Simon, Almost periodic Schrödinger operators: A review, Adv. Appl. Math. 3, 463 (1982).
  27. Y. Liu, L. F. Santos, and E. Prodan, Topological gaps in quasiperiodic spin chains: A numerical and K-theoretic analysis, Phys. Rev. B 105, 035115 (2022).
  28. H. J. Putley, B. Davies, F. J. Rodríguez-Fortuño, A. Y. Bykov, and A. V. Zayats, Mixing skyrmions and merons in topological quasicrystals of evanescent optical field, Optica 12, 614 (2025).
  29. A. Agazzi, J.-P. Eckmann, and G. M. Graf, The colored Hofstadter butterfly for the honeycomb lattice, J. Stat. Phys. 156, 417 (2014).
  30. S. Y. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. Math. 150, 1159 (1999).
  31. D. Morison, N. B. Murphy, E. Cherkaev, and K. M. Golden, Order to disorder in quasiperiodic composites, Commun. Phys. 5, 148 (2022).
  32. A. Jagannathan, The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality, Rev. Mod. Phys. 93, 045001 (2021).
  33. M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett. 50, 1870 (1983).
  34. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
  35. M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological photonic quasicrystals: Fractal topological spectrum and protected transport, Phys. Rev. X 6, 011016 (2016).
  36. B. Davies and R. V. Craster, Symmetry-induced quasicrystalline waveguides, Wave Motion 115, 103068 (2022).
  37. D. Beli, M. I. N. Rosa, L. Lomazzi, C. De Marqui, Jr., and M. Ruzzene, Interface states in two-dimensional quasicrystals with broken inversion symmetry, Phys. Rev. Appl. 23, 024039 (2025).
  38. B. Davies, G. J. Chaplain, T. A. Starkey, and R. V. Craster, Graded quasiperiodic metamaterials perform fractal rainbow trapping, Phys. Rev. Lett. 131, 177001 (2023).
  39. B. Davies et al., Roadmap on metamaterial theory, modelling and design, J. Phys. D: Appl. Phys. 58, 203002 (2025).
  40. M. A. Shubin, Almost periodic functions and partial differential operators, Russ. Math. Surv. 33, 1 (1978).
  41. B. Davies and C. Thalhammer, Convergence of supercell and superspace methods for computing spectra of quasiperiodic operators, Proc. R. Soc. A 481, 20240894 (2025).
  42. D. Damanik, M. Embree, and A. Gorodetski, Spectral properties of Schrödinger operators arising in the study of quasicrystals, in Mathematics of Aperiodic Order, edited by J. Kellendonk, D. Lenz, and J. Savinien (Birkhäuser, Basel, 2015), Vol. 309, pp. 307–370.
  43. N. Wellander, S. Guenneau, and E. Cherkaev, Two-scale cut-and-projection convergence; homogenization of quasiperiodic structures, Math. Methods Appl. Sci. 41, 1101 (2018).
  44. E. Lustig, Y. Sharabi, and M. Segev, Topological aspects of photonic time crystals, Optica 5, 1390 (2018).
  45. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, Dynamics of light propagation in spatiotemporal dielectric structures, Phys. Rev. E 75, 046607 (2007).
  46. A. Naweed and M. S. Akhtar, Unconventional optical response of time-domain Fibonacci quasicrystals, Opt. Lett. 50, 4746 (2025).
  47. K. A. Lurie, An Introduction to the Mathematical Theory of Dynamic Materials (Springer, Cham, 2017).
  48. G. W. Milton and O. Mattei, Field patterns: A new mathematical object, Proc. R. Soc. A 473, 20160819 (2017).
  49. B. Davies and L. Morini, Super band gaps and periodic approximants of generalised Fibonacci tilings, Proc. R. Soc. A 480, 20230663 (2024).
  50. L. Morini and M. Gei, Waves in one-dimensional quasicrystalline structures: Dynamical trace mapping, scaling and self-similarity of the spectrum, J. Mech. Phys. Solids 119, 83 (2018).
  51. If either |x0|<2 or |x1|<2, then we can apply corollary 3.10 of Ref. [41]. If not, then we are in the standard case of combining two unit cells with gaps, see, e.g., Ref. [56]. Once {xn} is outside [−2,2] and is growing, a simple argument can be used to show it will keep growing forever, as in Theorem 3.1 of Ref. [49].
  52. D. Marripour and J. Abouie, From time crystals to time quasicrystals: Exploring quasiperiodic phases in transverse field Ising chains, Phys. Rev. B 112, 174307 (2025).
  53. B. Vial and R. V. Craster, Quasinormal modes of Floquet media slabs, Phys. Rev. B 112, 214201 (2025).
  54. C. M. Hooper, I. R. Hooper, and S. A. R. Horsley, Quasinormal modes in dispersive photonic time crystals, Phys. Rev. A 113, 043526 (2026).
  55. I. Ruchiev, Fibonacci modulation in time-modulated wave systems [Computer software], Zenodo, 2025, doi: 10.5281/zenodo.17191922.
  56. J.-I. Hori and H. Matsuda, Structure of the spectra of disordered systems I: Fundamental theorems, Prog. Theor. Phys. 32, 183 (1964).

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Phys. Rev. A 114, 023502– Published 3 August, 2026
  • Received 23 November 2025
  • Accepted 17 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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