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Coherent surface plasmon polariton amplification via free-electron pumping


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.


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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



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).

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