Flatbands have become a cornerstone of contemporary condensed-matter physics and photonics. In electronics, flatbands entail comparable energy bandwidth and Coulomb interaction, leading to correlated phenomena such as the fractional quantum Hall effect and recently those in magic-angle systems. In photonics, they enable properties including slow light1 and lasing2. Notably, flatbands support supercollimation—diffractionless wavepacket propagation—in both systems3,4. Despite these intense parallel efforts, flatbands have never been shown to affect the core interaction between free electrons and photons. Their interaction, pivotal for free-electron lasers5, microscopy and spectroscopy6,7, and particle accelerators8,9, is, in fact, limited by a dimensionality mismatch between localized electrons and extended photons. Here we reveal theoretically that photonic flatbands can overcome this mismatch and thus remarkably boost their interaction. We design flatband resonances in a silicon-on-insulator photonic crystal slab to control and enhance the associated free-electron radiation by tuning their trajectory and velocity. We observe signatures of flatband enhancement, recording a two-order increase from the conventional diffraction-enabled Smith–Purcell radiation. The enhancement enables polarization shaping of free-electron radiation and characterization of photonic bands through electron-beam measurements. Our results support the use of flatbands as test beds for strong light–electron interaction, particularly relevant for efficient and compact free-electron light sources and accelerators.
This is a preview of subscription content, access via your institution
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Data supporting the findings of this study are provided in the Article and its Supplementary Information. Source data are provided with this paper.
Baba, T. Slow light in photonic crystals. Nat. Photon. 2, 465–473 (2008).
Noda, S., Yokoyama, M., Imada, M., Chutinan, A. & Mochizuki, M. Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design. Science 293, 1123–1125 (2001).
Rakich, P. T. et al. Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal. Nat. Mater. 5, 93–96 (2006).
Park, C.-H., Son, Y.-W., Yang, L., Cohen, M. L. & Louie, S. G. Electron beam supercollimation in graphene superlattices. Nano Lett. 8, 2920–2924 (2008).
Pellegrini, C., Marinelli, A. & Reiche, S. The physics of x-ray free-electron lasers. Rev. Mod. Phys. 88, 015006 (2016).
De Abajo, F. G. Optical excitations in electron microscopy. Rev. Mod. Phys. 82, 209 (2010).
Polman, A., Kociak, M. & García de Abajo, F. J. Electron-beam spectroscopy for nanophotonics. Nat. Mater. 18, 1158–1171 (2019).
Sapra, N. V. et al. On-chip integrated laser-driven particle accelerator. Science 367, 79–83 (2020).
Shiloh, R. et al. Electron phase-space control in photonic chip-based particle acceleration. Nature 597, 498–502 (2021).
Friedman, A., Gover, A., Kurizki, G., Ruschin, S. & Yariv, A. Spontaneous and stimulated emission from quasifree electrons. Rev. Mod. Phys. 60, 471 (1988).
Schächter, L. Beam-Wave Interaction in Periodic and Quasi-Periodic Structures (Springer, 2011).
de Abajo, F. G. et al. Cherenkov effect as a probe of photonic nanostructures. Phys. Rev. Lett. 91, 143902 (2003).
Lin, X. et al. Controlling Cherenkov angles with resonance transition radiation. Nat. Phys. 14, 816–821 (2018).
Roques-Carmes, C. et al. Towards integrated tunable all-silicon free-electron light sources. Nature Commun. 10, 3176 (2019).
Haeusler, U., Seidling, M., Yousefi, P. & Hommelhoff, P. Boosting the efficiency of Smith–Purcell radiators using nanophotonic inverse design. ACS Photon. 9, 664–671 (2022).
Yang, Y. et al. Maximal spontaneous photon emission and energy loss from free electrons. Nat. Phys. 14, 894–899 (2018).
Yamaguti, S., Inoue, J.-i, Haeberlé, O. & Ohtaka, K. Photonic crystals versus diffraction gratings in Smith-Purcell radiation. Phys. Rev. B 66, 195202 (2002).
Ochiai, T. & Ohtaka, K. Electron energy loss and Smith-Purcell radiation in two-and three-dimensional photonic crystals. Opt. Express 13, 7683–7698 (2005).
Bendana, X., Polman, A. & de Abajo, F. J. G. Single-photon generation by electron beams. Nano Lett. 11, 5099–5103 (2011).
Fernandes, D. E., Maslovski, S. I. & Silveirinha, M. G. Cherenkov emission in a nanowire material. Phys. Rev. B 85, 155107 (2012).
Adamo, G. et al. Light well: a tunable free-electron light source on a chip. Phys. Rev. Lett. 103, 113901 (2009).
Pendry, J. & Martin-Moreno, L. Energy loss by charged particles in complex media. Phys. Rev. B 50, 5062 (1994).
So, J.-K. et al. Cerenkov radiation in metallic metamaterials. Appl. Phys. Lett. 97, 151107 (2010).
Kaminer, I. et al. Spectrally and spatially resolved Smith-Purcell radiation in plasmonic crystals with short-range disorder. Phys. Rev. X 7, 011003 (2017).
Liu, F. et al. Integrated Cherenkov radiation emitter eliminating the electron velocity threshold. Nat. Photon. 11, 289–292 (2017).
Kfir, O., Di Giulio, V., de Abajo, F. J. G. & Ropers, C. Optical coherence transfer mediated by free electrons. Sci. Adv. 7, eabf6380 (2021).
Hayun, A. B. et al. Shaping quantum photonic states using free electrons. Sci. Adv. 7, eabe4270 (2021).
Dahan, R. et al. Resonant phase-matching between a light wave and a free-electron wavefunction. Nat. Phys. 16, 1123–1131 (2020).
Adiv, Y. et al. Quantum nature of dielectric laser accelerators. Phys. Rev. X 11, 041042 (2021).
Kremers, C., Chigrin, D. N. & Kroha, J. Theory of Cherenkov radiation in periodic dielectric media: emission spectrum. Phys. Rev. A 79, 013829 (2009).
Chiu, C.-K. & Schnyder, A. P. Classification of reflection-symmetry-protected topological semimetals and nodal superconductors. Phys. Rev. B 90, 205136 (2014).
Roques-Carmes, C. et al. A framework for scintillation in nanophotonics. Science 375, eabm9293 (2022).
Brenny, B., Coenen, T. & Polman, A. Quantifying coherent and incoherent cathodoluminescence in semiconductors and metals. J. Appl. Phys. 115, 244307 (2014).
Wang, Z., Yao, K., Chen, M., Chen, H. & Liu, Y. Manipulating Smith-Purcell emission with Babinet metasurfaces. Phys. Rev. Lett. 117, 157401 (2016).
Jing, L. et al. Polarization shaping of free-electron radiation by gradient bianisotropic metasurfaces. Laser Photon. Rev. 15, 2000426 (2021).
Tang, H. et al. Modeling the optical properties of twisted bilayer photonic crystals. Light Sci. Appl. 10, 157 (2021).
Cerjan, A., Hsu, C. W. & Rechtsman, M. C. Bound states in the continuum through environmental design. Phys. Rev. Lett. 123, 023902 (2019).
Cerjan, A. et al. Observation of bound states in the continuum embedded in symmetry bandgaps. Sci. Adv. 7, eabk1117 (2021).
Guerrera, S. & Akinwande, A. I. Nanofabrication of arrays of silicon field emitters with vertical silicon nanowire current limiters and self-aligned gates. Nanotechnology 27, 295302 (2016).
Adiv, Y. et al. Observation of 2D Cherenkov radiation. Preprint at https://arXiv.org/abs/2203.01698 (2022).
Feist, A. et al. Cavity-mediated electron-photon pairs. Science 377, 777–780 (2022).
Varkentina, N. et al. Cathodoluminescence excitation spectroscopy: nanoscale imaging of excitation pathways. Preprint at https://arXiv.org/abs/2202.12520 (2022).
Black, D. S. et al. Net acceleration and direct measurement of attosecond electron pulses in a silicon dielectric laser accelerator. Phys. Rev. Lett. 123, 264802 (2019).
Schönenberger, N. et al. Generation and characterization of attosecond microbunched electron pulse trains via dielectric laser acceleration. Phys. Rev. Lett. 123, 264803 (2019).
Niedermayer, U. et al. Low-energy-spread attosecond bunching and coherent electron acceleration in dielectric nanostructures. Phys. Rev. Appl. 15, L021002 (2021).
Fallah, A., Kiasat, Y., Silveirinha, M. G. & Engheta, N. Nonreciprocal guided waves in the presence of swift electron beams. Phys. Rev. B 103, 214303 (2021).
Peng, S. et al. Probing the band structure of topological silicon photonic lattices in the visible spectrum. Phys. Rev. Lett. 122, 117401 (2019).
Yu, Y. et al. Transition radiation in photonic topological crystals: quasiresonant excitation of robust edge states by a moving charge. Phys. Rev. Lett. 123, 057402 (2019).
Mukherjee, S. et al. Observation of a localized flat-band state in a photonic Lieb lattice. Phys. Rev. Lett. 114, 245504 (2015).
Vicencio, R. A. et al. Observation of localized states in Lieb photonic lattices. Phys. Rev. Lett. 114, 245503 (2015).
Slot, M. R. et al. Experimental realization and characterization of an electronic Lieb lattice. Nat. Phys. 13, 672–676 (2017).
Kang, M. et al. Dirac fermions and flat bands in the ideal kagome metal FeSn. Nat. Mater. 19, 163–169 (2020).
Kollár, A. J., Fitzpatrick, M., Sarnak, P. & Houck, A. A. Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Commun. Math. Phys. 376, 1909–1956 (2019).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Wang, P. et al. Localization and delocalization of light in photonic moiré lattices. Nature 577, 42–46 (2020).
Leykam, D., Andreanov, A. & Flach, S. Artificial flat band systems: from lattice models to experiments. Adv. Phys. X 3, 1473052 (2018).
Leykam, D. & Flach, S. Perspective: photonic flatbands. APL Photonics 3, 070901 (2018).
Tang, L. et al. Photonic flat-band lattices and unconventional light localization. Nanophotonics 9, 1161–1176 (2020).
Li, J., White, T. P., O’Faolain, L., Gomez-Iglesias, A. & Krauss, T. F. Systematic design of flat band slow light in photonic crystal waveguides. Opt. Express 16, 6227–6232 (2008).
Lou, B. et al. Theory for twisted bilayer photonic crystal slabs. Phys. Rev. Lett. 126, 136101 (2021).
Dong, K. et al. Flat bands in magic-angle bilayer photonic crystals at small twists. Phys. Rev. Lett. 126, 223601 (2021).
Nguyen, D. X. et al. Magic configurations in moiré superlattice of bilayer photonic crystal: almost-perfect flatbands and unconventional localization. Preprint at https://arXiv.org/abs/2104.12774 (2021).
Leykam, D., Flach, S. & Chong, Y. D. Flat bands in lattices with non-Hermitian coupling. Phys. Rev. B 96, 064305 (2017).
Pan, M., Zhao, H., Miao, P., Longhi, S. & Feng, L. Photonic zero mode in a non-Hermitian photonic lattice. Nat. Commun. 9, 1308 (2018).
Noda, S., Kitamura, K., Okino, T., Yasuda, D. & Tanaka, Y. Photonic-crystal surface-emitting lasers: review and introduction of modulated-photonic crystals. IEEE J. Sel. Top. Quantum Electron. 23, 1–7 (2017).
Longhi, S. Photonic flat-band laser. Opt. Lett. 44, 287–290 (2019).
Xia, S. et al. Unconventional flatband line states in photonic Lieb lattices. Phys. Rev. Lett. 121, 263902 (2018).
Schächter, L. & Ron, A. Smith-Purcell free-electron laser. Phys. Rev. A 40, 876 (1989).
Luo, C., Ibanescu, M., Johnson, S. G. & Joannopoulos, J. Cerenkov radiation in photonic crystals. Science 299, 368–371 (2003).
Andrews, H. & Brau, C. Gain of a Smith-Purcell free-electron laser. Phys. Rev. Accel. Beams 7, 070701 (2004).
Kumar, V. & Kim, K.-J. Analysis of Smith-Purcell free-electron lasers. Phys. Rev. E 73, 026501 (2006).
Freund, H. & Abu-Elfadl, T. Linearized field theory of a Smith-Purcell traveling wave tube. IEEE Trans. Plasma Sci. 32, 1015–1027 (2004).
Brinkmann, R., Derbenev, Y. & Flöttmann, K. A low emittance, flat-beam electron source for linear colliders. Phys. Rev. Accel. Beams 4, 053501 (2001).
Piot, P., Sun, Y.-E. & Kim, K.-J. Photoinjector generation of a flat electron beam with transverse emittance ratio of 100. Phys. Rev. Accel. Beams 9, 031001 (2006).
Nguyen, K. T. et al. Intense sheet electron beam transport in a uniform solenoidal magnetic field. IEEE Trans. Electron Devices 56, 744–752 (2009).
Wang, Z. et al. High-power millimeter-wave BWO driven by sheet electron beam. IEEE Trans. Electron Devices 60, 471–477 (2012).
Yang, Y. et al. Apparatus and methods for generating and enhancing Smith-Purcell radiation. US patent 10,505,334 (2019).
We thank T. Savas for fabricating the samples; A. Massuda and J. Sloan for contributions to the building of the setup; D. Zhu and M. Lončar for sharing equipment; and C. Mao, O. D. Miller and N. Rivera for stimulating conversations. This material is based on work supported in part by the US Army Research Office through the Institute for Soldier Nanotechnologies under contract number W911NF-18-2-0048, the Air Force Office of Scientific Research under the award numbers FA9550-20-1-0115 and FA9550-21-1-0299, the US Office of Naval Research Multidisciplinary University Research Initiative grant N00014-20-1-2325 on Robust Photonic Materials with High-Order Topological Protection, and the U.S.-Israel Binational Science Foundation grant 2018288. Y.Y. acknowledges the support from the start-up fund of the University of Hong Kong and the National Natural Science Foundation of China Excellent Young Scientists Fund (HKU 12222417). C.R.-C. acknowledges funding from the MathWorks Engineering Fellowship Fund by MathWorks Inc.
Y.Y., C.R.-C., S.E.K., I.K., J.D.J. and M.S. declare the following patent: US patent 10,505,334 (ref. 77).
Peer review information
Nature thanks Fang Liu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This file contains Supplementary Sections 1–15, Figs. 1–15 and References. The sections are entitled Radiation experimental setup, Band structure measurements, Beam diameter and divergence, Collection optics, Experimental analysis, Physical mechanism of enhanced interaction, Numerical methods, Distinction between sheet electrons and point electrons, Point and line degeneracies between electron surface and photonic band, Angles of enhanced radiation, Flatband-induced localization, Numerical comparisons with diffractive gratings, Experimental comparisons with diffractive gratings, Generality of the flatband scheme, and Flatband resonances for strong coupling and accelerators.
About this article
Cite this article
Yang, Y., Roques-Carmes, C., Kooi, S.E. et al. Photonic flatband resonances for free-electron radiation. Nature 613, 42–47 (2023). https://doi.org/10.1038/s41586-022-05387-5