The peak brilliance of VUV/X-ray free electron lasers (FEL) is by far the highest.

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1 Free electron lasers The peak brilliance of VUV/X-ray free electron lasers (FEL) is by far the highest. Normal lasers are based on stimulated emission between atomic energy levels, i. e. the radiation is amplified when it interacts with systems of bound electrons. In free electron lasers radiation interacts with the free electrons in an electron beam. The emitted radiation is, however, not stimulated in the same sense as in normal lasers.

2 Key idea: As we know from all the formulas derived so far, starting with the classical expression for emission from N accelerating charged particles, the emitted power is P Ne! If the electrons are forced to move in a coherent fashion, so that all N particles accelerate as one superparticle with charge Ne, the corresponding emitted power is P (Ne)! The principle of self-amplified spontaneous emission (SASE) has paved the way to achieve such superparticles by microbunching. For lower energies there are numerous FEL projects in the world ( many of them use synchrotron rings, others use linear accelerators like in the Fig. below ( Basically, the undulator is placed between two mirrors (in a cavity), so that the primarily emitted undulator radiation is reflected and can interact with the electron beam.

3 In FELs the field of the radiation can make the necessary order in the electron beam: microbunches are created. Using the graphics from the same home page of Institute for Free Electron Laser in Osaka we see how this can be achieved. The wavelength of the emitted radiation, λ is, as we know from before, much shorter than the electron period, λ! : λ = λ! K! (1 + 2γ! 2 ) During the time 0 < t < t! the electron travels the distance λ!, whereas the photon travels λ! + λ. So far it is the same type of reasoning that lead to the undulator equation in the section Insertion devices). From the Fig. it is clear that the electron at z! and z! experiences the same accelerating force. It has the same sign as the velocity. Because there is a phase shift corresponding to one λ, the direction of the E-field at the site of this electron changes sign as the direction of the electron velocity changes sign, so that the force is accelerating also at z!. Similarly, you can convince yourself that the electron at z!, z! and z! is decelerated throughout the whole undulator period, and of course, through the whole undulator. These forces will tend to microbunch the electrons; these microbunches then radiate as one supercharge. To get a large number of electrons in these microbunches, obviously the requirements on the emittance of the electron beam and the adjustment of the mirrors are large. This is all the more true in the VUV and X-ray range where the normal incidence reflectivity is close to zero. The FELs have to work without cavities, and the enhancement has to be achieved in a single path. This puts extremely high demands on the electron beam: high energy, around GeV, and very low emittance, which means that very long linear accelerators are needed. The undulator length should be on the order of 100 m, and as usual the poles must be positioned with micrometer accuracy.

4 The most common attempts are based on the Self Amplified Spontaneous Emission (SASE) principle. This principle is very similar to the one described above. The difference is that it is the photons emitted in the beginning of an undulator, which are used for microbunching. To put this argument on a bit more solid ground we follow the argument in the introduction to the book The physics of free electron lasers, by E. L. Saldin, E. A. Schneidmiller, and M. V. Yukov. The rate of change in electron energy is de dt = ev i E where E is the electric field vector of the wave, and v is the transverse velocity of the electron. If the rate of change is constant throughout the undulator there will loose (or gain) energy in a synchronized way. We will derive a condition for this. Let us assume a helical undulator with wave number k =!!!!. The transverse velocity of the electron is v = Kc γ [e! cos kz e! sin kz] and the circularly polarized electromagnetic wave is: E = E[e! cos ω z c t + e! sin ω z c t ] Remembering that dz = v! dt we have de dz = e v! v! E! v! E! =

5 = KceE [cos kz cos ω z γ c t sin kz sin ω z c t ] = = KceE cos [ kz + ω z γ c KceE t ] = cosψ γ If the phase is constant we have the condition for synchronized energy loss (or gain). The resonance condition becomes Again using that dz = v! dt we have dψ = kdz + ω dz ωdt = 0 c k + ω c ω v! = 0 Thus, synchronization takes place when the electromagnetic wave is ahead of the electron beam by one undulator period. The expression can be rewritten: v z f = c v z λ u c = 1 β 1 2γ 2 λ u l 1 with (see fig) l! =!!! = λ, the wavlength of the radiation. We remember the expression,!! which we used as the starting point for deriving the undulator equation, and we can conclude that just the wavelength that is spontaneously produced by the undulator λ = λ u 2γ 2 gives the necessary phase relation for microbunching.

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7 At FERMI the first seeding is demonstrated, in which the control over the shape of the pulse is controlled via external lasers. This significantly increases the longitudinal coherence, and time- definition of the pulse. There are self- seeding schemes (see above) implemented to this end at higher energies. Note that conventional lasers, based on atomic transitions have the problem at X-ray energies, that there are no highly excited states in bound electron systems which have sufficient life time for stimulating emission; it is difficult to create a population inversion and one would think that spontaneous emission will always dominate. However, it was recently shown that the FEL pulse can create just this kind of inversion in Ne ions which results in atomic X-ray lasing. Atomic X-ray lasing (from Rohringer et al., Nature 481,488(2012)). Although higher harmonic generation (HHG) is used to produce coherent radiation at higher energies, free electron lasers are probably the only route to produce high brilliance at short wavelengths. Today there are a few VUV/X-ray laser working: FLASH in Hamburg, LCLS in Stanford, SACLA in Sendai, Japan, and FERMI in Trieste, Italy. The peak brilliance and the time structure often are better than was expected. Several FELs are planned and commissioned all over the world; e.g. the SwissFEL at PSI, and also the European XFEL in Hamburg. There are on-going discussions about the possibility of a Swedish FEL. Visits to the FEL homepages are recommended. the- future- project xfel.spring8.or.jp/

8 There is a Free Electron Laser Course given, based on: A compendium on beam transport and beam diagnostic methods for Free Electron Lasers, A. Lindblad, K. Tiedtke, and S. Svensson, April 2011, ISBN