**Author details**

#### K. Zhukovsky

**4.** The electron beam current*I* must be high andits transverse cross section *σ* small. However, the γ-factor cannot be freely decreased if we want to obtain X-ray wavelengths.

**5.** As follows from the expression for *LG* high peak current is requested for it, such as *I* ~ kA

**7.** The energy spread of the electron beam *σ<sup>e</sup>* has to be as small as possible, preferably *σ<sup>e</sup>* ≈ 10<sup>−</sup><sup>4</sup>

Note also that in the light of the above said the following requirements arise for the electron beam in transverse: low emittance of the beam and preservation of this low emittance; for the longitudinal dimension good compression and acceleration are required. The main negative factors, which affect the amplification, are the electron energy spread, the angular diver‐ gence, the transverse electron beam size, the diffraction of the wave and others. The electron energy spread has negative effect on both the amplification and FEL saturation level. Amplification mainly starts with the optimal electron energy, whose γ-factor determines the wavelength. As the energy is transferred from the electrons to the radiated electromagnetic wave, the energy ofthe electrons naturallydecreases. The wave emissions from allthe electrons differ from each other, because different electrons have different energies. After the waveelectron interaction, the electron beam energy spread increases and at a certain point it grows to a level, where no gain occurs. Moreover, well before the electrons loose a substantial portion of their energy, they slow down by emitting electro-magnetic energy and change their phase with respect to the wave. Thus they begin to take the energy from the wave rather than giving

In conclusion, let us state some areas, where the performance of the X-ray sources of coher‐ ent radiation can be further improved. First of all, the temporal coherence of SASE FELs can be improved. The improved temporal coherence would in turn improve the spectral bright‐ ness of the sources, which means the users will have more useful photons. The way to accomplish it could consist, for example, in seeding X-FEL from a radiation source with good

An alternative approach to single-pass high-gain amplifier schemes is to use cavity feedback in a relatively low-gain system. The development of relatively high-reflectivity diamond

Reducing X-ray pulse durations to the attosecond regime will provide spatiotemporal resolution of atomic processes. Two techniques have so far been reported that can take pulse durations significantly below 100 as toward the atomic unit of time 24 as. The first technique employs a variation of an echo-enabled harmonic generation method and produces pulses of ~20 as duration at a wavelength of 1 nm with the power, which in peaks reaches ~200 MW. The second technique is based on mode-locking in conventional cavity lasers—oscillators. It could generate radiation at 1.5 Å wavelength in sequences of pulses with ~150 as intervals

crystal mirrors in the X-ray regime makes them feasible.

**6.** Transverse electron beam size has to be small, possibly *σR* ~ 10 μm or so.

**8.** The electron and the photon beams have to be overlapped properly.

or more.

220 222High Energy and Short Pulse Lasers

or less.

it.

temporal coherence.

Address all correspondence to: zhukovsk@physics.msu.ru

Department of Theoretical Physics, Physical Faculty, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow
