The quantum free-electron laser

Ultrashort pulses of x-ray radiation from synchrotron sources have become ubiquitous tools for investigating the structure of matter. Their immense usefulness has led to the development of large international facilities. These are based on radio-frequency accelerating cavities and magnetic undulators, and provide brief radiation pulses capable of probing and taking ‘snapshots’ of molecules and solid-state matter. However, synchrotron sources produce pulses of incoherent radiation that are limited to relatively low peak brilliance and durations of order a picosecond and longer. As a next significant step in advancing x-ray sources, the free-electron laser (FEL) produces femtosecond-duration pulses with a peak brilliance seven orders of magnitude higher than synchrotrons.

Several large international teams are constructing FELs to produce x-ray radiation through self-amplified spontaneous emission (SASE): the Linac Coherent Light Source¹ in Stanford, CA, the European XFEL² in Hamburg, Germany, and the SPring-8 Compact SASE Source³ in Hyōgo prefecture, Japan. One drawback of such sources is that they produce pulses composed of many random superradiant spikes with a broad noise spectrum.⁴ In the classical picture of the FEL, this spiky x-ray pulse results from the random initial phases of electrons entering the amplifier. However, it is clear from quantum theory that the emission process is discrete. Moreover, it must include quantization of the electron motion, which completely changes both the properties of the emitted radiation and the resulting momentum distribution of the electrons. Accordingly, an FEL operating in the quantum regime should offer improved performance over its classical counterpart, in particular, enhanced spectral brightness and degree of coherence.

Figure 1. Numerical solutions for L_b=40 L_c (L_b: Electron-bunch length. L_c: Cooperation length.) and δ=0 (δ: Frequency detuning), in (a, c) the classical regime ( and ) and (b, d) the quantum regime ( and ). Graphs (a) and (b) show the scaled intensity, and graphs (c) and (d) the corresponding scaled power spectra as a function of scaled frequency , where ω is the resonance frequency and ω ′ the relative frequency with respect to the ω_s and divided by ρ, the free-electron-laser parameter. The dotted line in (a) marks the front edge of the electron pulse. : Normalized vector-field potential of the amplified free-electron-laser radiation. : Scaled wiggler length. , where v is the velocity of the electrons and t a time interval.

When an electron emits a photon, the momentum recoil is ℏk. This is naturally quantized and can assume only the discrete values n(ℏk). In classical FEL theory, the initial spontaneous-radiation field is amplified through the ‘ponderomotive’ force resulting from the interference of the radiation and undulator fields. This leads to electron bunching on a wavelength scale and exponential amplification with a rate governed by ρ, the FEL parameter.⁵ ρ depends on the undulator period, and magnetic-field strength and electron-beam parameters such as, e.g., the Lorentz factor at resonance for a particular wavelength of the amplified light, γ_r, peak current, and emittance. The number of photons emitted depends on ρ, and is given by the quantum-FEL (QFEL) parameter⁶

which is the ratio of the maximum classical momentum spread (of order mcγ_rρ) to ℏk. When , many momentum levels are involved since the momentum spread is much larger than the level spacing. The discreteness of the momentum becomes irrelevant, and one recovers the classical behavior, characterized by a random series of superradiant spikes. The spectrum of the emitted field is broad and chaotic. Conversely, when , an electron emits a single photon and makes a single momentum transition. The result is a single narrow-line spectrum that is Fourier-limited by the electron-beam duration, i.e., Δω/ω ~ λ/L_b.^6,7 This means that a QFEL operating in the  ngstrom region with an electron-bunch length L_b=1mm could generate radiation with a relative linewidth of 10⁻⁷, much smaller than the envelope linewidth 2ρ of the classical SASE spectrum (typically of order 10⁻³).

Hence, the QFEL could be a very promising x-ray source generating quasi-monochromatic radiation (although at a lower power than in a classical SASE FEL) and a formidable tool for ultra-high-resolution process studies. The ‘quantum purification’ of the SASE spectrum can be interpreted by the following simple argument. The maximum induced energy spread in an FEL is δγ/γ ~ ρ, which in terms of momentum spread is . The QFEL parameter yields the ratio between the maximum momentum spread (induced in the classical regime) and the photon recoil momentum ℏk. Quantum effects become important when , since then the discreteness of momentum exchange is relevant. This provides a simple explanation of the origin of the broad and spiky classical spectrum and its reduction to a single line in the quantum regime (see Figure 1 and videos^8,9). Experimental realization of a QFEL requires a laser wiggler instead of the magnetic wiggler usually used in classical SASE experiments.^1–3 In a laser-wiggler configuration, a low-energy electron beam backscatters the photons of a counterpropagating high-power laser into a photon frequency upshifted by a factor 4γ². However, such a choice sets stringent conditions on the electron- and laser-beam parameters.¹⁰

We propose to exploit the new generation of laser-driven wakefield accelerators,¹¹ where electrons are accelerated to high energies by the electrostatic forces of a laser-driven plasma wave. The advantage is that both the electron beam and the laser beam acting as a wiggler are contained in a guiding structure. The electrons are continuously focused by the transverse fields of the ion ‘bubble,’ while a preformed plasma acts as a waveguide to lead the wiggler laser in maintaining perfect overlap over many Rayleigh lengths. Furthermore, because the accelerator and the FEL are ‘all-optical’ (they both use lasers to provide accelerating and wiggler fields, respectively), they can be placed on a very compact footprint, or perhaps one should even say fingerprint. It should be possible to construct a QFEL driven by a wakefield accelerator that is only a few centimeters long. This presents several significant challenges. The first and most stringent is to produce an electron beam with a sufficiently small energy spread, which must be less than the recoil momentum. This sets a limit of σ_γ/γ < 10⁻⁴, which can be alleviated somewhat by going to very short wavelengths, e.g., 0.05Â. However, the peak current of the electron beam should be greater than 10kA and preferably close to 100kA, which prevailing wisdom does not rule out.

Our next efforts will focus on operating a QFEL with harmonics to reach even shorter wavelengths, either in the seeded or in the SASE mode.