Boosting the Coherence of X-Ray Free-Electron Lasers
X-ray free-electron lasers (XFELs) have transformed the study of matter by delivering femtosecond and attosecond pulses at angstrom wavelengths, enabling direct observation of ultrafast structural and electronic dynamics. Despite these successes, XFELs have long lacked a capability central to precision optical science: stable temporal phase coherence. Most XFEL facilities operate in the self-amplified spontaneous-emission (SASE) regime, in which radiation originates from microscopic shot noise in an electron beam. This mechanism produces extremely bright pulses, but shot-to-shot fluctuations in their temporal structure limit their use in phase-sensitive experiments useful for metrology, interferometry, and ultrafast spectroscopy [1]. Now Wenxiang Hu of the Paul Scherrer Institute in Switzerland and the Swiss Federal Institute of Technology (ETH) Zurich report a long-awaited step toward controlling the temporal phase of SASE radiation: the first experimental realization of a “mode-locked” XFEL [2]. Their experiment demonstrates that a SASE XFEL can be driven into a regime in which it emits a stable train of phase-locked attosecond pulses that, in the frequency domain, have a spectral comb structure.
In conventional lasers, mode locking enforces a fixed phase relationship among an optical cavity’s longitudinal modes—discrete, equally spaced frequencies that resonate with the cavity. Without mode locking, these modes would oscillate independently, and their random phases would lead to noisy emission in time. When mode locking synchronizes their phases, the modes interfere constructively at regular intervals, producing a periodic train of ultrashort pulses. Mode locking can be enforced, for example, through a device that modulates the amplitude or the phase of the modes at a frequency equal to the mode spacing [3]. This principle underlies many cornerstones of modern optics, from femtosecond spectroscopy to optical frequency combs.
XFELs, however, lack a physical resonator and therefore do not possess longitudinal modes in the usual sense of a cavity-based laser. They operate in a single pass, as electron bunches propagating through a sequence of undulators emit x-ray radiation. Nevertheless, theory has shown that an effective set of longitudinal modes can be synthesized by inserting, between undulator modules, magnetic chicanes that introduce a fixed delay between successive amplification stages [4, 5]. These delays define a comb of allowed frequencies—sometimes described as “synthetic” longitudinal modes—whose spacing is determined by the relative temporal slippage between the emitted x rays and the electrons. Recent experiments have demonstrated that such multichicane schemes enhance temporal coherence in SASE operation [6].
Hu and co-workers bring the control of temporal coherence to the next level by locking the synthetic modes’ frequencies. To achieve mode locking, they combine the chicane architecture with a modulator—an external optical laser that modulates the electron-beam energy before amplification at the mode-spacing frequency (Fig. 1). This modulation selects discrete regions of the electron bunch that drives amplification in the subsequent XFEL stages, so that the emitted x-ray radiation gets reamplified with a fixed phase relation. XFEL mode locking involves no physical resonator, but the process is well captured by the image of a “traveling resonator”—a moving sequence of gain interactions that enforces phase coherence.
The researchers’ measurements of the x ray and of the electron bunch—both in the frequency and time domains—provide compelling evidence of mode locking. The radiation is emitted as a structured train of attosecond pulses separated by a few femtoseconds, while the spectrum displays a high-contrast frequency comb that is reproducible from shot to shot (Fig. 2). The measured temporal structure of the electron beam at the XFEL output directly maps the temporal structure of the emitted radiation, confirming that the underlying XFEL dynamics are well understood.
It is crucial to clarify what kind of coherence this demonstration achieves. Mode locking acts on the first-order coherence, quantified by the field correlation function , which measures the phase coherence of the emitted radiation’s electric field. A high enables interference, pulse trains, and frequency-comb structures. In this sense, the experiment represents a major advance: Phase coherence is actively imposed on an XFEL, even though emission still starts from the amplification of noise.
This achievement, however, does not automatically imply laser-like photon statistics. These are characterized by the second-order correlation function , which quantifies intensity (or photon-number) fluctuations. Of particular importance is , the value at zero time delay, which measures how likely photons are to arrive together in time. For an ideal single-mode laser, , corresponding to Poissonian photon statistics—photons arrive independently with no tendency to bunch or cluster. Ideal thermal, or chaotic, light has , a manifestation of photon bunching. Partially coherent sources fall between these two values—their photon statistics are said to be “super-Poissonian.”
These correlation properties of light can be measured through so-called Hanbury Brown–Twiss measurements. Such measurements were previously carried out at FLASH, an XFEL in Germany [7], and later at PAL-XFEL in South Korea [8]. They showed that SASE XFEL radiation retains super-Poissonian statistics, with significantly larger than 1. Because the present mode-locked scheme remains SASE-based—amplifying initially incoherent noise—its photon statistics are expected to remain non-Poissonian unless demonstrated otherwise by dedicated correlation measurements.
It is instructive to compare SASE-XFELs with other types of FELs that are externally seeded—they amplify a stable optical laser generated through high-gain harmonic generation or echo-enabled harmonic generation [9, 10]. This approach, pursued by facilities such as the FERMI FEL in Italy, imports coherence from the laser seed, producing pulses with exceptional spectral purity and reproducibility. At FERMI’s DiProI beamline, photon-correlation experiments demonstrated , establishing near-laser-like photon statistics [11].
Mode-locked SASE XFELs therefore cannot replace seeded FELs in applications requiring the strict quantum-optical coherence of a laser. However, suitable laser seeds are only available down to vacuum-ultraviolet wavelengths (20–200 nm). XFELs thus offer a complementary approach for producing phase-structured, multicolor waveforms at shorter wavelengths, where external seeding remains challenging. Hu and colleagues’ demonstration was obtained at soft-x-ray wavelengths of about 2.5 nm.
The availability of trains of phase-locked attosecond x-ray pulses offers striking opportunities. Powerful precision spectroscopy techniques, such as Ramsey-type interferometry, that are routine in optical physics, may now become feasible at x-ray wavelengths. By probing the energies of atomic core levels, such experiments could track, with element specificity, charge migration, spin transfer, and electronic correlations on their natural timescales. More broadly, this work signals a conceptual shift. XFELs are no longer limited to producing “the shortest possible pulses,” but are beginning to offer tailored x-ray waveforms designed to drive atoms, solids, and low-density matter through specific quantum pathways.
Just as optical frequency combs reshaped precision spectroscopy, the emergence of x-ray frequency combs may similarly transform how we probe—and ultimately manipulate—matter at its most fundamental scales.
References
- The physical origin of these fluctuations has been understood since the foundational analyses carried out more than four decades ago: A. M. Kondratenko and E. L. Saldin, “Generation of coherent radiation by a relativistic electron beam in an ondulator,” Part. Accel. 10, 207 (1980); R. Bonifacio et al., “Collective instabilities and high-gain regime in a free electron laser,” Opt. Commun. 50, 373 (1984).
- X. Hu et al., “Demonstration of mode-locked frequency comb for an x-ray free-electron laser,” Phys. Rev. Lett. 135, 265001 (2025).
- Mode locking can in general be achieved actively (through a modulator), passively (through a saturable absorber), or through hybrid schemes.
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- A. Singer et al., “Hanbury Brown–Twiss interferometry at a free-electron laser,” Phys. Rev. Lett. 111, 034802 (2013).
- Y. Y. Kim et al., “Statistical analysis of hard X-ray radiation at the PAL-XFEL facility performed by Hanbury Brown and Twiss interferometry,” J. Synchrotron Radiat. 29, 1465 (2022).
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- P. P. Ribič et al., “Coherent soft X-ray pulses from an echo-enabled harmonic generation free-electron laser,” Nat. Photonics 13, 555 (2019); O. Y. Gorobtsov et al., “Seeded X-ray free-electron laser generating radiation with laser statistical properties,” Nat. Commun. 9, 4498 (2018).