Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Observation of optical de Broglie–Mackinnon wave packets

Nature Physics volume 19, pages 435–444 (2023)Cite this article

Subjects

05 June 2024 Editor’s Note: Readers are alerted that there is an error in Equation (2) in this paper. Editorial action is being taken to correct this.

Abstract

de Broglie wave packets accompanying moving particles are dispersive and lack an intrinsic length scale solely dictated by the particle mass and velocity. Mackinnon proposed a localized non-dispersive wave packet constructed out of dispersive de Broglie phase waves that possess an intrinsic length scale via an inversion of the roles of particle and observer. So far, the de Broglie–Mackinnon wave packet has remained a theoretical proposal. Here we report the observation of optical de Broglie–Mackinnon wave packets using paraxial space–time-coupled pulsed laser fields in the presence of anomalous group-velocity dispersion. Crucially, the bandwidth of de Broglie–Mackinnon wave packets has an upper limit that is compatible with the wave-packet group velocity and equivalent mass. In contrast to previously observed linear-propagation-invariant wave packets whose spatio-temporal profiles at any axial plane are X-shaped, those for de Broglie–Mackinnon wave packets are uniquely O-shaped (circularly symmetric with respect to space and time). By sculpting their spatio-temporal spectral structure, we produce dispersion-free de Broglie–Mackinnon wave packets in the dispersive medium, observe their circularly symmetric spatio-temporal intensity profiles and closed-trajectory spectra, and tune the field parameters that uniquely determine the wave-packet length scale.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: de Broglie phase waves and wave packets, and dBM wave packets.
Fig. 2: Optical dBM wave packets.
Fig. 3: Synthesizing and characterizing optical dBM wave packets.
Fig. 4: Observation of optical dBM wave packets in the presence of anomalous GVD and tuning their group velocity.
Fig. 5: Tuning the equivalent rest mass of an optical dBM wave packet.
Fig. 6: Changing the spatio-temporal structure of optical dBM wave packets.

Similar content being viewed by others

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Change history

  • 05 June 2024

    Editor’s Note: Readers are alerted that there is an error in Equation (2) in this paper. Editorial action is being taken to correct this.

References

  1. Berry, M. V. & Balazs, N. L. Nonspreading wave packets. Am. J. Phys.47, 264–267 (1979).

    Article  ADS  Google Scholar 

  2. Greenberger, D. M. Comment on ‘Nonspreading wave packets’. Am. J. Phys.48, 256 (1980).

    Article  ADS  Google Scholar 

  3. Siviloglou, G. A. & Christodoulides, D. N. Accelerating finite energy Airy beams. Opt. Lett.32, 979–981 (2007).

    Article  ADS  Google Scholar 

  4. Zhang, P. et al. Generation of acoustic self-bending and bottle beams by phase engineering. Nat. Commun.5, 4316 (2014).

    Article  ADS  Google Scholar 

  5. Fu, S., Tsur, Y., Zhou, J., Shemer, L. & Arie, A. Propagation dynamics of Airy water-wave pulses. Phys. Rev. Lett.115, 034501 (2015).

    Article  ADS  Google Scholar 

  6. Voloch-Bloch, N., Lereah, Y., Lilach, Y., Gover, A. & Arie, A. Generation of electron Airy beams. Nature494, 331–335 (2013).

    Article  ADS  Google Scholar 

  7. Kaminer, I., Nemirovsky, J., Rechtsman, M., Bekenstein, R. & Segev, M. Self-accelerating Dirac particles and prolonging the lifetime of relativistic fermions. Nat. Phys.11, 261–267 (2015).

    Article  Google Scholar 

  8. Mackinnon, L. A nondispersive de Broglie wave packet. Found. Phys.8, 157–176 (1978).

    Article  ADS  Google Scholar 

  9. de Broglie, L. Recherches sur la théorie des quanta. Ann. de Phys.3, 22 (1925).

    Article  ADS  Google Scholar 

  10. Cohen-Tannoudji, C., Diu, B. & Laloe, F. Quantum Mechanics (Wiley, 1991).

  11. Torres, J. P., Hendrych, M. & Valencia, A. Angular dispersion: an enabling tool in nonlinear and quantum optics. Adv. Opt. Photon.2, 319–369 (2010).

    Article  Google Scholar 

  12. Fülöp, J. A. & Hebling, J. in Recent Optical and Photonic Technologies (ed. Kim, K. Y.) Ch. 11 (IntechOpen, 2010).

  13. Saari, P. & Reivelt, K. Generation and classification of localized waves by Lorentz transformations in Fourier space. Phys. Rev. E69, 036612 (2004).

    Article  ADS  Google Scholar 

  14. Zamboni-Rached, M. & Recami, E. Subluminal wave bullets: exact localized subluminal solutions to the wave equations. Phys. Rev. A77, 033824 (2008).

    Article  ADS  Google Scholar 

  15. Yessenov, M., Hall, L. A., Schepler, K. L. & Abouraddy, A. F. Space-time wave packets. Adv. Opt. Photon.14, 455–570 (2022).

    Article  Google Scholar 

  16. Wilczek, F. A Beautiful Question: Finding Nature’s Deep Design (Penguin Press, 2015).

  17. Saleh, B. E. A. & Teich, M. C. Principles of Photonics (Wiley, 2007).

  18. Kondakci, H. E. & Abouraddy, A. F. Diffraction-free pulsed optical beams via space-time correlations. Opt. Express24, 28659–28668 (2016).

    Article  ADS  Google Scholar 

  19. Parker, K. J. & Alonso, M. A. The longitudinal iso-phase condition and needle pulses. Opt. Express24, 28669–28677 (2016).

    Article  ADS  Google Scholar 

  20. Wong, L. J. & Kaminer, I. Ultrashort tilted-pulse-front pulses and nonparaxial tilted-phase-front beams. ACS Photon.4, 2257–2264 (2017).

    Article  Google Scholar 

  21. Porras, M. A. Gaussian beams diffracting in time. Opt. Lett.42, 4679–4682 (2017).

    Article  ADS  Google Scholar 

  22. Efremidis, N. K. Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type. Opt. Lett.42, 5038–5041 (2017).

    Article  ADS  Google Scholar 

  23. Porras, M. A., Trillo, S., Conti, C. & Di Trapani, P. Paraxial envelope X waves. Opt. Lett.28, 1090–1092 (2003).

    Article  ADS  Google Scholar 

  24. Porras, M. A., Valiulis, G. & Di Trapani, P. Unified description of Bessel X waves with cone dispersion and tilted pulses. Phys. Rev. E68, 016613 (2003).

    Article  ADS  Google Scholar 

  25. Longhi, S. Localized subluminal envelope pulses in dispersive media. Opt. Lett.29, 147–149 (2004).

    Article  ADS  Google Scholar 

  26. Porras, M. A. & Di Trapani, P. Localized and stationary light wave modes in dispersive media. Phys. Rev. E69, 066606 (2004).

    Article  ADS  Google Scholar 

  27. Malaguti, S., Bellanca, G. & Trillo, S. Two-dimensional envelope localized waves in the anomalous dispersion regime. Opt. Lett.33, 1117–1119 (2008).

    Article  ADS  Google Scholar 

  28. Malaguti, S. & Trillo, S. Envelope localized waves of the conical type in linear normally dispersive media. Phys. Rev. A79, 063803 (2009).

    Article  ADS  Google Scholar 

  29. Hall, L. A. & Abouraddy, A. F. Canceling and inverting normal and anomalous group-velocity dispersion using space-time wave packets. Preprint at https://arxiv.org/abs/2202.01148 (2022).

  30. Kondakci, H. E. & Abouraddy, A. F. Optical space-time wave packets of arbitrary group velocity in free space. Nat. Commun.10, 929 (2019).

    Article  ADS  Google Scholar 

  31. Bhaduri, B., Yessenov, M. & Abouraddy, A. F. Anomalous refraction of optical spacetime wave packets. Nat. Photon.14, 416–421 (2020).

    Article  ADS  Google Scholar 

  32. Hall, L. A. & Abouraddy, A. F. A universal angular-dispersion synthesizer. Preprint at https://arxiv.org/abs/2109.13987 (2021).

  33. Saari, P. & Reivelt, K. Evidence of X-shaped propagation-invariant localized light waves. Phys. Rev. Lett.79, 4135–4138 (1997).

    Article  ADS  Google Scholar 

  34. Turunen, J. & Friberg, A. T. Propagation-invariant optical fields. Prog. Opt.54, 1–88 (2010).

    Article  ADS  Google Scholar 

  35. Hernández-Figueroa, H. E., Recami, E. & Zamboni-Rached, M. (eds) Non-Diffracting Waves (Wiley-VCH, 2014).

  36. Chong, A., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Airy–Bessel wave packets as versatile linear light bullets. Nat. Photon.4, 103–106 (2010).

    Article  ADS  Google Scholar 

  37. MacKinnon, E. De Broglie’s thesis: a critical retrospective. Am. J. Phys.44, 1047–1055 (1976).

    Article  ADS  Google Scholar 

  38. Espinosa, J. M. Physical properties of de Broglie’s phase waves. Am. J. Phys.50, 357–362 (1982).

    Article  ADS  Google Scholar 

  39. Donnelly, R. & Ziolkowski, R. W. Designing localized waves. Proc. R. Soc. Lond. A440, 541–565 (1993).

    Article  ADS  Google Scholar 

  40. Bélanger, P. A. Lorentz transformation of packetlike solutions of the homogeneous-wave equation. J. Opt. Soc. Am. A3, 541–542 (1986).

    Article  ADS  Google Scholar 

  41. Longhi, S. Gaussian pulsed beams with arbitrary speed. Opt. Express12, 935–940 (2004).

    Article  ADS  Google Scholar 

  42. Kondakci, H. E. & Abouraddy, A. F. Airy wavepackets accelerating in space-time. Phys. Rev. Lett.120, 163901 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  43. Yessenov, M. et al. What is the maximum differential group delay achievable by a space-time wave packet in free space? Opt. Express27, 12443–12457 (2019).

    Article  ADS  Google Scholar 

  44. Kondakci, H. E. & Abouraddy, A. F. Diffraction-free space–time beams. Nat. Photon.11, 733–740 (2017).

    Article  ADS  Google Scholar 

  45. Unnikrishnan, K. & Rau, A. R. P. Uniqueness of the Airy packet in quantum mechanics. Am. J. Phys.64, 1034–1035 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  46. Sezginer, A. A general formulation of focus wave modes. J. Appl. Phys.57, 678–683 (1985).

    Article  ADS  Google Scholar 

  47. Kondakci, H. E., Alonso, M. A. & Abouraddy, A. F. Classical entanglement underpins the invariant propagation of space–time wave packets. Opt. Lett.44, 2645–2648 (2019).

    Article  ADS  Google Scholar 

  48. Forbes, A., de Oliveira, M. & Dennis, M. R. Structured light. Nat. Photon.15, 253–262 (2021).

    Article  ADS  Google Scholar 

  49. Yessenov, M. & Abouraddy, A. F. Accelerating and decelerating space-time optical wave packets in free space. Phys. Rev. Lett.125, 233901 (2020).

    Article  ADS  Google Scholar 

  50. Hall, L. A., Yessenov, M. & Abouraddy, A. F. Arbitrarily accelerating space-time wave packets. Opt. Lett.47, 694–697 (2022).

    Article  ADS  Google Scholar 

  51. Li, Z. & Kawanaka, J. Velocity and acceleration freely tunable straight-line propagation light bullet. Sci. Rep.10, 11481 (2020).

    Article  Google Scholar 

  52. Sloan, J., Rivera, N., Joannopoulos, J. D. & Soljačić, M. Controlling two-photon emission from superluminal and accelerating index perturbations. Nat. Phys.18, 67–73 (2022).

    Article  Google Scholar 

  53. Bliokh, K. Y. & Nori, F. Spatiotemporal vortex beams and angular momentum. Phys. Rev. A86, 033824 (2012).

    Article  ADS  Google Scholar 

  54. Caloz, C. & Deck-Léger, Z.-L. Spacetime metamaterials–part I: general concepts. IEEE Trans. Antennas Propag.68, 1569–1582 (2020).

    Article  ADS  Google Scholar 

  55. Guo, C., Xiao, M., Orenstein, M. & Fan, S. Structured 3D linear space-time light bullets by nonlocal nanophotonics. Light Sci. Appl.10, 160 (2021).

    Article  ADS  Google Scholar 

  56. Pang, K. et al. Synthesis of near-diffraction-free orbital-angular-momentum space-time wave packets having a controllable group velocity using a frequency comb. Opt. Express30, 16712–16724 (2022).

    Article  ADS  Google Scholar 

  57. Yessenov, M. et al. Space-time wave packets localized in all dimensions. Nat. Commun.13, 4573 (2022).

    Article  ADS  Google Scholar 

  58. Yessenov, M., Chen, Z., Lavery, M. P. J. & Abouraddy, A. F. Vector space-time wave packets. Opt. Lett.47, 4131–4134 (2022).

    Article  ADS  Google Scholar 

  59. Schepler, K. L., Yessenov, M., Zhiyenbayev, Y. & Abouraddy, A. F. Space-time surface plasmon polaritons: a new propagation-invariant surface wave packet. ACS Photon.7, 2966–2977 (2020).

    Article  Google Scholar 

Download references

Acknowledgements

We thank M. Yessenov, K. L. Schepler, D. N. Christidoulides and A. Dogariu for helpful discussions. This work was supported by the US Office of Naval Research (ONR) under contracts N00014-17-1-2458 and N00014-20-1-2789.

Author information

Authors and Affiliations

  1. CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, FL, USA

    Layton A. Hall & Ayman F. Abouraddy

Authors
  1. Layton A. Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ayman F. Abouraddy
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

L.A.H. and A.F.A. conceived the idea and designed the experimental approach. L.A.H. carried out the experiment. L.A.H. and A.F.A. analysed the data and wrote the manuscript.

Corresponding author

Correspondence to Ayman F. Abouraddy.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hall, L.A., Abouraddy, A.F. Observation of optical de Broglie–Mackinnon wave packets. Nat. Phys. 19, 435–444 (2023). https://doi.org/10.1038/s41567-022-01876-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01876-6

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing