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.

# 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.

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

## Access options

\$32.00

All prices are NET prices.

## 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).

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

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).

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).

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

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

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

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

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

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

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).

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

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).

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).

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

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

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).

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

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

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

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

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

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

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

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

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).

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).

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

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).

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

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).

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).

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

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

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).

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).

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

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).

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).

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).

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

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).

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).

## 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

### 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.

## Ethics declarations

### 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.

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

## 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.

### 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.

### 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.

### 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.

### 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.

## 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.

## 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

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

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41586-022-05239-2