Unifying frequency metrology across microwave, optical, and free-electron domains

Nat. Commun. 16, 8369 (2025) · DOI: 10.1038/s41467-025-62808-5 · arXiv: 2403.10486

License: CC BY 4.0.

Abstract

Frequency metrology lies at the heart of precision measurement. Optical frequency combs provide a coherent link uniting the microwave and optical domains in the electromagnetic spectrum, with profound implications in timekeeping, sensing and spectroscopy, fundamental physics tests, exoplanet searches, and light detection and ranging. Here, we extend this frequency link to free electrons by coherent modulation of the electron phase by a continuous-wave laser locked to a fully stabilized optical frequency comb. Microwave frequency standards are transferred to the optical domain via the frequency comb, and are further imprinted in the electron spectrum by optically modulating the electron phase with a photonic chip-based microresonator. As a proof-of-concept demonstration, we apply this frequency link in the calibration of an electron spectrometer and verify its precision by measuring the absolute optical frequency. This approach achieves a 20-fold improvement in the accuracy of electron spectroscopy, relevant for investigating low-energy excitations in quantum materials, two-dimensional materials, nanophotonics, and quantum optics. Our work bridges frequency domains differed by a factor of ~10^13 and carried by different physical objects, establishes a spectroscopic connection between electromagnetic waves and free-electron matter waves, and has direct ramifications in ultrahigh-precision electron spectroscopy.

Figures

14 panels with data across 3 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 2

• panel (b): Microwave power spectrum (dBm) against microwave frequency (MHz) recorded with the electronic spectrum analyzer (RBW 5 kHz) with the optical phase-locked loop off (Unlocked). The printed labels of the repetition-rate peak $f_{\mathrm{rep}}$ and the four beatnotes $f_{\mathrm{beat},1}$ to $f_{\mathrm{beat},4}$ are not drawn. data.csv
• panel (b-inset): Inset of panel (b): microwave power spectrum (dBm) against frequency (MHz) with the optical phase-locked loop on (Locked), with the smoothed beatnote drawn as a black curve. The printed arrows marking the beatnote's 598 kHz 3-dB bandwidth are not drawn. data.csv
• panel (c): Optical power spectrum (dBm) against optical frequency (THz) recorded with the optical spectrum analyzer (RBW 0.02 nm): the narrow CW laser peak at $f_{\mathrm{opt}}$ on the plateau of the bandpass-filtered optical frequency comb. data.csv
• panel (d): Normalized electron counts (arb. u.) against electron energy minus 120 keV (eV) at the nominal EELS dispersion of 50 meV/px: raw spectrum (gray), spectrum smoothed with a Gaussian pre-filter (black), deconvolved spectrum (green), and the spectrum without optical modulation (light green shaded). The shaded window marking the inset region is not drawn. data.csv
• panel (d-inset): Inset of panel (d): the same four spectra in the shaded central window, normalized electron counts (arb. u.) against electron energy minus 120 keV (eV); raw (gray), pre-filtered (black), deconvolved (green) and un-modulated (light green shaded). The printed inset has no tick labels. data.csv

Fig. 3

• panel (a): Deviation (meV) of the calibration markers at multiples of the photon energy from the nominal spectrometer energy, against pixel (px), for Exposures 1 (blue) and 2 (orange). Shaded bands are the error bars. The printed inset schematic is not drawn. data.csv
• panel (b): First-order component (eV) of the polynomial fit of the energy solution (pixel-to-energy mapping) against pixel (px), for Exposures 1 (blue) and 2 (orange); the legend gives the calibrated average dispersion of each. data.csv
• panel (c): Second-order component (meV) of the polynomial fit of the energy solution (pixel-to-energy mapping) against pixel (px), for Exposures 1 (blue) and 2 (orange). data.csv
• panel (d): Third-order component (meV) of the polynomial fit of the energy solution (pixel-to-energy mapping) against pixel (px), for Exposures 1 (blue) and 2 (orange). data.csv
• panel (e): Histogram (counts, 25 bins) of the pixel-wise root-mean-square deviation (meV) between the energy solutions of the two exposures, with the normal distribution fit drawn as a red curve, as labelled in the panel. data.csv

Fig. 4

• panel (a): Deviation (meV) of the calibration markers at multiples of the photon energy from the nominal spectrometer energy, against pixel (px), for electrons modulated at four optical frequencies $f_{\mathrm{opt}}$ (legend). Shaded bands are the error bars. data.csv
• panel (b): First-order component (eV) of the polynomial fit of the energy solution (pixel-to-energy mapping) against pixel (px) for the four calibrations, coloured as in (a); the legend gives the calibrated average dispersion of each. data.csv
• panel (c): Pre-calibration error (THz) in the optical frequency measured from the electron spectra against optical frequency $f_{\mathrm{opt}}$ (THz), before calibrating the spectrometer (nominal pixel-to-energy mapping), one point per calibration coloured as in (a), with error bars. data.csv
• panel (d): Post-calibration error (GHz) in the optical frequency measured from the electron spectra against optical frequency $f_{\mathrm{opt}}$ (THz), after calibrating the spectrometer with electrons modulated at $f_{\mathrm{opt}}$ = 194 567 516 MHz (marked calibration), one point per calibration coloured as in (a), with error bars. data.csv

Cite

Yujia Yang, Paolo Cattaneo, Arslan S. Raja, Bruce Weaver, Rui Ning Wang, Alexey Sapozhnik, Fabrizio Carbone, Thomas LaGrange, Tobias J. Kippenberg. Unifying frequency metrology across microwave, optical, and free-electron domains. Nat. Commun. 16, 8369 (2025). https://doi.org/10.1038/s41467-025-62808-5

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