An ultracompact brilliant coherent x-ray source, where both the accelerator and the wiggler are provided by intense laser pulses, promises unsurpassed spectral and temporal qualities.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 radiofrequency 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 xray sources, the free-electron laser (FEL) produces femtosecondduration 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 Source 1 in Stanford, CA, the European XFEL 2 in Hamburg, Germany, and the SPring-8 Compact SASE Source 3 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.When an electron emits a photon, the momentum recoil ishk. This is naturally quantized and can assume only the discrete values n(hk). 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) parameter 6 ρ = ρ mcγ r hk ,which is the ratio of the maximum classical momentum spread (of order mcγ r ρ) tohk. Whenρ 1, many momentum levels are involved since the momentum spread is much larger than the
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