Embon, L. et al. Imaging of super-fast dynamics and flow instabilities of superconducting vortices. Nat. Commun. 8, 85 (2017).
Sachkou, Y. P. et al. Coherent vortex dynamics in a strongly interacting superfluid on a silicon chip. Science 366, 1480–1485 (2019).
Bliokh, K. Y. et al. Roadmap on structured waves. J. Opt. 25, 103001 (2023).
Dennis, M. R., King, R. P., Jack, B., O’Holleran, K. & Padgett, M. J. Isolated optical vortex knots. Nat. Phys. 6, 118–121 (2010).
Berry, M. V. Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves. J. Phys. A Math. Gen. 11, 27–38 (1978).
De Angelis, L., Alpeggiani, F., Di Falco, A. & Kuipers, L. Spatial distribution of phase singularities in optical random vector waves. Phys. Rev. Lett. 117, 093901 (2016).
Berry, M. V. & Dennis, M. R. Phase singularities in isotropic random waves. Proc. R. Soc. Lond. A 456, 2059–2079 (2000).
Hansen, J. P. & McDonald, I. R. Theory of Simple Liquids: With Applications to Soft Matter (Academic Press, 2013).
Toulouse, G. & Kléman, M. Principles of a classification of defects in ordered media. J. Physique Lett. 37, 149–151 (1976).
Blatter, G., Feigel’man, M. V., Geshkenbein, V. B., Larkin, A. I. & Vinokur, V. M. Vortices in high-temperature superconductors. Rev. Mod. Phys. 66, 1125–1388 (1994).
Drori, L. et al. Quantum vortices of strongly interacting photons. Science 381, 193–198 (2023).
Sugic, D. et al. Particle-like topologies in light. Nat. Commun. 12, 6785 (2021).
Indebetouw, G. Optical vortices and their propagation. J. Mod. Opt. 40, 73–87 (1993).
Freund, I. Optical vortex trajectories. Opt. Commun. 181, 19–33 (2000).
Li, P. et al. Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing. Nat. Commun. 6, 7507 (2015).
Yoxall, E. et al. Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity. Nat. Photon. 9, 674–678 (2015).
Caldwell, J. D. et al. Photonics with hexagonal boron nitride. Nat. Rev. Mater. 4, 552–567 (2019).
Kurman, Y. et al. Spatiotemporal imaging of 2D polariton wave packet dynamics using free electrons. Science 372, 1181–1186 (2021).
Giles, A. J. et al. Ultralow-loss polaritons in isotopically pure boron nitride. Nat. Mater. 17, 134–139 (2018).
Barwick, B., Flannigan, D. J. & Zewail, A. H. Photon-induced near-field electron microscopy. Nature 462, 902–906 (2009).
Feist, A. et al. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature 521, 200–203 (2015).
Madan, I. et al. Holographic imaging of electromagnetic fields via electron-light quantum interference. Sci. Adv. 5, eaav8358 (2019).
Nabben, D., Kuttruff, J., Stolz, L., Ryabov, A. & Baum, P. Attosecond electron microscopy of sub-cycle optical dynamics. Nature 619, 63–67 (2023).
Gaida, J. H. et al. Attosecond electron microscopy by free-electron homodyne detection. Nat. Photon. 18, 509–515 (2024).
Bucher, T. et al. Coherently amplified ultrafast imaging using a free-electron interferometer. Nat. Photon. 18, 809–815 (2024).
Nye, J. F. & Berry, M. V. Dislocations in wave trains. Proc. R. Soc. Lond. A 336, 165–190 (1974).
Nye, J. F. The motion and structure of dislocations in wavefronts. Proc. R. Soc. Lond. A 378, 219–239 (1981).
Andersen, M. F. et al. Quantized rotation of atoms from photons with orbital angular momentum. Phys. Rev. Lett. 97, 170406 (2006).
Tsesses, S. et al. Tunable photon-induced spatial modulation of free electrons. Nat. Mater. 22, 345–352 (2023).
Török, P. & Munro, P. R. T. 2004. The use of Gauss-Laguerre vector beams in STED microscopy. Opt. Express 12, 3605–3617 (2004).
Berry, M. et al. Roadmap on superoscillations. J. Opt. 21, 053002 (2019).
Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons. Nature 412, 313–316 (2001).
Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12, 5448–5456 (2004).
Davis, T. J. et al. Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution. Science 368, eaba6415 (2020).
Ni, J. et al. Multidimensional phase singularities in nanophotonics. Science 374, eabj0039 (2021).
Berry, M. V. & Dennis, M. R. Knotted and linked phase singularities in monochromatic waves. Proc. R. Soc. Lond. A 457, 2251–2263 (2001).
Ostrovsky, E., Cohen, K., Tsesses, S., Gjonaj, B. & Bartal, G. Nanoscale control over optical singularities. Optica 5, 283–288 (2018).
Eckhardt, B., Dörr, U., Kuhl, U. & Stöckmann, H. J. Correlations of electromagnetic fields in chaotic cavities. Europhys. Lett. 46, 134–140 (1999).
Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride. Science 343, 1125–1129 (2014).
Berry, M. V. Vector fields for monochromatic waves with general dispersion: vortex and stagnation singularities. J. Phys. A Math. Theor. 58, 275201 (2025).
Vasnetsov, M., Pas’ ko, V., Khoroshun, A., Slyusar, V. & Soskin, M. Observation of superluminal wave-front propagation at the shadow area behind an opaque disk. Opt. Lett. 32, 1830–1832 (2007).
Dominici, L. et al. Full-Bloch beams and ultrafast Rabi-rotating vortices. Phys. Rev. Res. 3, 013007 (2021).
Jordan, A. N., Howell, J. C., Vamivakas, N. & Karimi, E. Superoscillations and physical applications. In Operator Theory (eds Alpay, D. et al.) https://doi.org/10.1007/978-3-0348-0692-3_101-1 (Springer, 2025).
Berry, M. V. Waves near zeros. In Proc. Conference on Coherence and Quantum Optics, CMB1, OSA Technical Digest (CD) (Optica Publishing Group, 2007).
Zhao, Z. et al. Experimental demonstration of 16-Gbit/s millimeter-wave communications link using thin metamaterial plates to generate data-carrying orbital-angular-momentum beams. In IEEE International Conference on Communications, 1392–1397 (IEEE, 2015).
Stav, T. et al. Quantum entanglement of the spin and orbital angular momentum of photons using metamaterials. Science 361, 1101–1104 (2018).
Gorlach, A. et al. Photonic quantum state tomography using free electrons. Phys. Rev. Lett. 133, 250801 (2024).
Forbes, A., Nothlawala, F. & Vallés, A. Progress in quantum structured light. Nat. Photon. 19, 1291–1300 (2025).
Shpiro, T. et al. Toward photon-induced near-field electron tomography. Preprint at https://arxiv.org/abs/2510.24648 (2025).
Bliokh, K. Y. et al. Theory and applications of free-electron vortex states. Phys. Rep. 690, 1–70 (2017).
Midgley, P. A. & Dunin-Borkowski, R. E. Electron tomography and holography in materials science. Nat. Mater. 8, 271–280 (2009).
Russo, C. J. & Henderson, R. Charge accumulation in electron cryomicroscopy. Ultramicroscopy 187, 43–49 (2018).
Ferrari, B. M. et al. Realization of a pre-sample photonic-based free-electron modulator in ultrafast transmission electron microscopes. ACS Photon. 12, 5864–5873 (2025).
Wu, Y. et al. Enhanced thermal conduction by surface phonon-polaritons. Sci. Adv. 6, eabb4461 (2020).
Piazza, L. et al. Simultaneous observation of the quantization and the interference pattern of a plasmonic near-field. Nat. Commun. 6, 6407 (2015).
Wang, K. et al. Coherent interaction between free electrons and a photonic cavity. Nature 582, 50–54 (2020).
Kfir, O. et al. Controlling free electrons with optical whispering-gallery modes. Nature 582, 46–49 (2020).
Bucher, T. et al. Free-electron Ramsey-type interferometry for enhanced amplitude and phase imaging of nearfields. Sci. Adv. 9, eadi5729 (2023).
Liu, S. et al. Single crystal growth of millimeter-sized monoisotopic hexagonal boron nitride. Chem. Mater. 30, 6222–6225 (2018).
Kurman, Y. et al. Dynamics of optical vortices in van der Waals materials. Optica 10, 612–618 (2023).
Gibertini, M., Koperski, M., Morpurgo, A. F. & Novoselov, K. S. Magnetic 2D materials and heterostructures. Nat. Nanotechnol. 14, 408–419 (2019).
de Abajo, F. J. G. et al. Roadmap for photonics with 2D materials. ACS Photon. 8, 3961–4095 (2025).
Maleev, I. D. & Swartzlander, G. A. Jr. Composite optical vortices. J. Opt. Soc. Am. B 20, 1169–1176 (2003).
Bekshaev, A., Chernykh, A., Khoroshun, A. & Mikhaylovskaya, L. Singular skeleton evolution and topological reactions in edge-diffracted circular optical-vortex beams. Opt. Commun. 397, 72–83 (2017).
De Angelis, L., Alpeggiani, F., Di Falco, A. & Kuipers, L. Persistence and lifelong fidelity of phase singularities in optical random waves. Phys. Rev. Lett. 119, 203903 (2017).
De Angelis, L. & Kuipers, L. Effective pair-interaction of phase singularities in random waves. Opt. Lett. 46, 2734–2737 (2021).
Berry, M. V. Regular and irregular semiclassical wavefunctions. J. Phys. A Math. Gen. 10, 2083–2092 (1977).
Kourkoulou, I., Landry, M. J., Nicolis, A. & Parmentier, K. Apparently superluminal superfluids. J. High Energy Phys. 2024, 80 (2024).
Seo, S. W., Ko, B., Kim, J. H. & Shin, Y. I. Observation of vortex-antivortex pairing in decaying 2D turbulence of a superfluid gas. Sci. Rep. 7, 4587 (2017).
Green, S. Fluid Vortices (Springer, 2012).
Fiorino, M. & Elsberry, R. L. Some aspects of vortex structure related to tropical cyclone motion. J. Atmos. Sci. 46, 975–990 (1989).
Poincaré, H. Sur les courbes définies par les équations différentielles. J. Math. Pures Appl. 4, 167–244 (1885).
Vanacore, G. M. et al. Author Correction: Attosecond coherent control of free-electron wave functions using semi-infinite light fields. Nat. Commun. 10, 1069 (2019).
Machado, F., Rivera, N., Buljan, H., Soljačić, M. & Kaminer, I. Shaping polaritons to reshape selection rules. ACS Photon. 5, 3064–3072 (2018).
Session, D. et al. Optical pumping of electronic quantum Hall states with vortex light. Nat. Photon. 19, 156–161 (2025).
Spektor, G. et al. and Meyer zu Heringdorf, F.J. Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices. Science 355, 1187–1191 (2017).
Lim, S. W. D., Spaegele, C. M. & Capasso, F. Multidimensional optical singularities and their applications. Preprint at https://arxiv.org/abs/2406.00784 (2024).
Thompson, W. On Vortex Atoms by Lord Kelvin. Proc. R. Soc. Edin. 6, 94–105 (1867).
Zhang, Z. et al. Particlelike behavior of topological defects in linear wave packets in photonic graphene. Phys. Rev. Lett. 122, 233905 (2019).
Rumi, G. et al. Hyperuniform vortex patterns at the surface of type-II superconductors. Phys. Rev. Res. 1, 033057 (2019).
Brandt, E. H. Electric field in superconductors with rectangular cross section. Phys. Rev. B 52, 15442 (1995).
Aharonov, Y., Popescu, S. & Rohrlich D. How can an infra-red photon behave as a gamma ray? Preprint TAUP, 1847–1890 (1991).
Berry, M. V. in Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (eds Anandan, J. S. & Safko, J. L.) 55–64 (World Scientific, 1994).
Rogers, E. T. et al. A super-oscillatory lens optical microscope for subwavelength imaging. Nat. Mater. 11, 432–435 (2012).