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Coherent surface plasmon polariton amplification via free-electron pumping

Abstract

Surface plasmonics with its unique confinement of light1,2 is expected to be a cornerstone for future compact radiation sources and integrated photonics devices. The energy transfer between light and matter is a defining aspect that underlies recent studies on optical surface-wave-mediated spontaneous emissions3,4,5. However, coherent stimulated emission of free electrons, which is essential for free-electron light sources, and its dynamical amplification process remain to be disclosed in a clear, unambiguous and calibrated manner. Here we present the coherent amplification of terahertz surface plasmon polaritons via free-electron-stimulated emission: a femtosecond optical pulse creates an in-phase free-electron pulse with an initial terahertz surface wave, and their ensuing interactions intensify the terahertz surface wave coherently. The underlying dynamics of the amplification, including a twofold redshift in the radiation frequency over a one-millimetre interaction length, are resolved as electromagnetic-field-profile evolutions using an optical pump–probe method. By extending the approach to a properly phase-matched electron bunch, our theoretical analysis predicts a super-radiant surface-wave growth, which lays the ground for a stimulated surface-wave light source and may facilitate capable means for matter manipulation, especially in the terahertz band.

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Fig. 1: Coherent SPP amplification by stimulated emission.
Fig. 2: Spatiotemporal dynamics of the electromagnetic field of the THz SPPs.
Fig. 3: Spatial and spectral properties of the THz SPPs during the amplification process.
Fig. 4: Simulated electron phase evolution inside the longitudinal electric-field component of the THz SPPs.
Fig. 5: Towards a stimulated SPP light source.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

References

  1. Barnes, W. L., Dereux, A. & Ebbesen, T. W. Surface plasmon subwavelength optics. Nature 424, 824–830 (2003).

    Article  ADS  CAS  PubMed  Google Scholar 

  2. Torma, P. & Barnes, W. L. Strong coupling between surface plasmon polaritons and emitters: a review. Rep. Prog. Phys. 78, 013901 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Wong, L. J., Kaminer, I., Ilic, O., Joannopoulos, J. D. & Soljačić, M. Towards graphene plasmon-based free-electron infrared to X-ray sources. Nat. Photon. 10, 46–52 (2015).

    Article  ADS  Google Scholar 

  4. Rivera, N., Kaminer, I., Zhen, B., Joannopoulos, J. D. & Soljacic, M. Shrinking light to allow forbidden transitions on the atomic scale. Science 353, 263–269 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  5. Pizzi, A. et al. Graphene metamaterials for intense, tunable, and compact extreme ultraviolet and X-ray sources. Adv. Sci. 7, 1901609 (2020).

    Article  CAS  Google Scholar 

  6. Rivera, N. & Kaminer, I. Light–matter interactions with photonic quasiparticles. Nat. Rev. Phys. 2, 538–561 (2020).

    Article  Google Scholar 

  7. Berini, P. & De Leon, I. Surface plasmon-polariton amplifiers and lasers. Nat. Photon. 6, 16–24 (2012).

    Article  ADS  CAS  Google Scholar 

  8. Hu, H. et al. Surface Dyakonov–Cherenkov radiation. eLight 2, 2 (2022).

    Article  Google Scholar 

  9. Dahan, R. et al. Imprinting the quantum statistics of photons on free electrons. Science 373, eabj7128 (2021).

    Article  CAS  PubMed  Google Scholar 

  10. Barwick, B., Flannigan, D. J. & Zewail, A. H. Photon-induced near-field electron microscopy. Nature 462, 902–906 (2009).

    Article  ADS  CAS  PubMed  Google Scholar 

  11. Park, S. T., Lin, M. & Zewail, A. H. Photon-induced near-field electron microscopy (PINEM): theoretical and experimental. New J. Phys. 12, 123028 (2010).

    Article  ADS  CAS  Google Scholar 

  12. Ritchie, R. H. Plasma losses by fast electrons in thin films. Phys. Rev. 106, 874–881 (1957).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  13. Kent, A. J. 11th International Conference on Phonon Scattering on Phonon Scattering in Condensed Matter (Phonons 2004), St. Petersburg, Russia, 25–30 July 2004. Phys. Status Solidi B 241, 2651–2654 (2005).

  14. Giles, A. J. et al. Ultralow-loss polaritons in isotopically pure boron nitride. Nat. Mater. 17, 134–139 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Ma, W. et al. In-plane anisotropic and ultra-low-loss polaritons in a natural van der Waals crystal. Nature 562, 557–562 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  16. Cai, X. et al. Plasmon-enhanced terahertz photodetection in graphene. Nano Lett. 15, 4295–4302 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  17. Seo, M. & Park, H.-R. Terahertz biochemical molecule-specific sensors. Adv. Opt. Mater. 8, 1900662 (2020).

    Article  CAS  Google Scholar 

  18. Deng, X., Li, L., Enomoto, M. & Kawano, Y. Continuously frequency-tuneable plasmonic structures for terahertz bio-sensing and spectroscopy. Sci. Rep. 9, 3498 (2019).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  19. Dahan, R. et al. Resonant phase-matching between a light wave and a free-electron wavefunction. Nat. Phys. 16, 1123–1131 (2020).

    Article  CAS  Google Scholar 

  20. Feist, A. et al. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature 521, 200–203 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  21. Morimoto, Y. & Baum, P. Diffraction and microscopy with attosecond electron pulse trains. Nat. Phys. 14, 252–256 (2017).

    Article  Google Scholar 

  22. Kozak, M., Schonenberger, N. & Hommelhoff, P. Ponderomotive generation and detection of attosecond free-electron pulse trains. Phys. Rev. Lett. 120, 103203 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  23. Zhou, C. et al. Direct mapping of attosecond electron dynamics. Nat. Photon. 15, 216–221 (2021).

    Article  ADS  CAS  Google Scholar 

  24. Caldwell, J. D. et al. Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons. Nanophotonics 4, 44–68 (2015).

    Article  CAS  Google Scholar 

  25. Zhang, X. et al. Terahertz surface plasmonic waves: a review. Adv. Photon. 2, 014001 (2020).

    Article  ADS  CAS  Google Scholar 

  26. Urata, J. et al. Superradiant Smith–Purcell emission. Phys. Rev. Lett. 80, 516–519 (1998).

    Article  ADS  CAS  Google Scholar 

  27. Avetissian, H. K., Avchyan, B. R., Matevosyan, H. H. & Mkrtchian, G. F. Free-electron nanolaser based on graphene plasmons. Laser Phys. 31, 055801 (2021).

    Article  ADS  CAS  Google Scholar 

  28. Zheng, Z., Kanda, N., Konishi, K. & Kuwata-Gonokami, M. Efficient coupling of propagating broadband terahertz radial beams to metal wires. Opt. Express 21, 10642–10650 (2013).

    Article  ADS  PubMed  Google Scholar 

  29. Tian, Y. et al. Femtosecond-laser-driven wire-guided helical undulator for intense terahertz radiation. Nat. Photon. 11, 242–246 (2017).

    Article  ADS  CAS  Google Scholar 

  30. Zeng, Y. et al. Guiding and emission of milijoule single-cycle THz pulse from laser-driven wire-like targets. Opt. Express 28, 15258–15267 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  31. Wang, K. L. & Mittleman, D. M. Metal wires for terahertz wave guiding. Nature 432, 376–379 (2004).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Wang, K. & Mittleman, D. M. Dispersion of surface plasmon polaritons on metal wires in the terahertz frequency range. Phys. Rev. Lett. 96, 157401 (2006).

    Article  ADS  PubMed  Google Scholar 

  33. Nakajima, H., Tokita, S., Inoue, S., Hashida, M. & Sakabe, S. Divergence-free transport of laser-produced fast electrons along a meter-long wire target. Phys. Rev. Lett. 110, 155001 (2013).

    Article  ADS  PubMed  Google Scholar 

  34. Tian, Y. et al. Electron emission at locked phases from the laser-driven surface plasma wave. Phys. Rev. Lett. 109, 115002 (2012).

    Article  ADS  PubMed  Google Scholar 

  35. Bocoum, M. et al. Anticorrelated emission of high harmonics and fast electron beams from plasma mirrors. Phys. Rev. Lett. 116, 185001 (2016).

    Article  ADS  PubMed  Google Scholar 

  36. Malka, G. & Miquel, J. L. Experimental confirmation of ponderomotive-force electrons produced by an ultrarelativistic laser pulse on a solid target. Phys. Rev. Lett. 77, 75–78 (1996).

    Article  ADS  CAS  PubMed  Google Scholar 

  37. Jiang, Z. P. & Zhang, X. C. Electro-optic measurement of THz field pulses with a chirped optical beam. Appl. Phys. Lett. 72, 1945–1947 (1998).

    Article  ADS  CAS  Google Scholar 

  38. Sarkisov, G. S. et al. Fountain effect of laser-driven relativistic electrons inside a solid dielectric. Appl. Phys. Lett. 99, 131501 (2011).

    Article  ADS  Google Scholar 

  39. Systemes, D. CST Studio Suite 2020 https://www.3ds.com/products-services/simulia/products/cst-studio-suite/ (2020).

  40. Tokita, S. et al. Collimated fast electron emission from long wires irradiated by intense femtosecond laser pulses. Phys. Rev. Lett. 106, 255001 (2011).

    Article  ADS  PubMed  Google Scholar 

  41. Feng, C. Theoretical and Experimental Studies on Novel High-Gain Seeded Free-Electron Laser Schemes (Springer, 2015).

  42. Schlauderer, S. et al. Temporal and spectral fingerprints of ultrafast all-coherent spin switching. Nature 569, 383–387 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  43. Suzuki, F., Fujita, N. & Sato, F. In Optical Components and Materials XVI 189–195 (SPIE, 2019).

  44. Teramoto, K. et al. Half-cycle terahertz surface waves with MV/cm field strengths generated on metal wires. Appl. Phys. Lett. 113, 051101 (2018).

    Article  ADS  Google Scholar 

  45. Smith, G. S. On the interpretation for radiation from simple current distributions. IEEE Antennas Propag. Mag. 40, 39–44 (1998).

    Article  Google Scholar 

  46. Sommerfeld, A. Electrodynamics: Lectures on Theoretical Physics Vol. 3 (Academic Press, 2013).

  47. Goubau, G. Surface waves and their application to transmission lines. J. Appl. Phys. 21, 1119–1128 (1950).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. King, M. & Wiltse, J. Surface-wave propagation on coated or uncoated metal wires at millimeter wavelengths. IRE Trans. Anntenas Propag. 10, 246–254 (1962).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (grant numbers 11922412, 11874372, 12104473 and 12104471), the Strategic Priority Research Program (B) (grant number XDB16), the Shanghai Pilot Program for Basic Research – Chinese Academy of Science, Shanghai Branch, the Key Research Program of Frontier Sciences, CAS., the Youth Innovation Promotion Association of Chinese Academy of Sciences, and the Shanghai Sailing Program (grant numbers 20YF1454900 and 21YF1453900). We thank J. Zi and L. Shi from Fudan University for discussions regarding the SPP properties.

Author information

Authors and Affiliations

Authors

Contributions

Y.T. and R.L. conceived and supervised the project. D.Z., Z.L. and Y.B. conducted the experimental measurements. Y.B. and Y.T. developed the theory. D.Z., Y.B. and Y.Z. performed the data analyses. Y.Z., D.Z. and Y.B. wrote the manuscript. All authors reviewed and discussed the manuscript and made substantial contribution to it.

Corresponding authors

Correspondence to Yafeng Bai, Ye Tian or Ruxin Li.

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Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Masaki Hashida and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Magnetic-field evolutions of the THz SPPs on the 500 μm-distant TGG.

The metal-wire borders are marked by the dashed lines. The times are in units of ps.

Extended Data Fig. 2 Magnetic-field evolutions of the THz SPPs on the 750 μm-distant TGG.

The metal-wire borders are marked by the dashed lines. The times are in units of ps.

Extended Data Fig. 3 Magnetic-field evolutions of the THz SPPs on the 1000 μm-distant TGG.

The metal-wire borders are marked by the dashed lines. The times are in units of ps.

Extended Data Fig. 4

SPPs’ magnetic-field dynamics analysis. a, Illustration of the magnetic field of the TM01 mode on a metallic wire structure. The wire is aligned along the z axis (perpendicular to the paper plane). Hence, only the \({B}_{x}=Bcos\varphi =Bcos(arctan(d/h))\) component of the magnetic field can be imprinted on the polarization changes of the probe light. b, The propagation velocity of the magnetic profiles on different TGG distances when compared to a spherical wavefront (solid line, predicted by \(z=\sqrt{{(0.3{\tau }_{d})}^{2}-{d}^{2}-{h}^{2})}\). Noteworthy, the superluminal propagation speeds result from the different projection angles of the SPPs when it encounters the TGG surface, giving rise to the large phase velocity on the TGG.

Source data

Extended Data Fig. 5 Demonstration of the dominant mode of the THz SPPs.

a, Illustration of \({c}_{01}\) (magenta line) as a function of z at a specific moment. The black dashed line marks the position where \({c}_{01}\) is maximal in the longitudinal cross-section of the electric-field profile (Ey). (b and c–h) Several snapshots of transverse cross-section of the electric-field profile (Ey) transected at the z where \({c}_{01}\) is maximal.

Source data

Extended Data Fig. 6 Time evolution of the TM01 weight of the THz SPPs.

The TM01 weight \({c}_{01}\) (taken at the transverse cross-section where it is maximal) gains from an initial value of 0.35 to a dominant value of 0.95 at 5 ps.

Source data

Extended Data Fig. 7 Time evolution of the electrons’ axial momentum \({\hat{p}}_{z}=\gamma {\beta }_{z}\).

The electrons experience a deceleration process, in which the lost energy underlies the THz SPPs amplification during their interaction.

Source data

Extended Data Fig. 8 SPP field strength dependence on the energy of excited electrons.

The simulation is carried out using electrons respectively with 100 keV, 200 keV, and 300 keV kinetic energy (50% FWHM energy spread), whose results indicate that higher electron kinetic energy may lead to higher strengths of the SPPs.

Source data

Supplementary information

Supplementary Video 1

Video showing the evolution dynamics of the THz SPPs at 500-μm TGG distance. The video is composed of the p- (a) and s-polarized (b) snapshots of the magnetic-field profiles, and the magnetic-field evolutions (c,d) with varying time delays. The contours in (d) mark both the profile and the strength variances of the magnetic field that are acquired from Video 1 (c). Note that the stationary branch (branch II) is almost indiscernible in the s-polarized snapshots, indicating its origin irrelevant to magnetic-field-induced polarization changes by the THz SPPs. When the laser pump illuminates the wire (whose edge is marked by the grey dotted lines) from the left fringe of each frame, the THz SPPs are excited and propagate along the wire towards the right side. Here, the first THz SPPs appear around τd = 2.47 ps and keeps growing until it reaches a relatively slow growth rate and stable profile during 3.60 ps < τd < 4.40 ps. The further increasing of the THz SPPs is impeded by the dephasing between the electron and the THz SPPs.

Peer Review File

Source data

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Zhang, D., Zeng, Y., Bai, Y. et al. Coherent surface plasmon polariton amplification via free-electron pumping. Nature 611, 55–60 (2022). https://doi.org/10.1038/s41586-022-05239-2

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