 1. Motivation
 2. Qualitative description
 3. What causes an exponential intensity increase?
 4. Emission by individual electrons
 5. Factors influencing the gain length and the amplification
 6. Microbunching: electrons and waves traveling together
 7. Saturation
 8. The underlying physics
 9. Limitations
 References
 1. Motivation
 2. Qualitative description
 3. What causes an exponential intensity increase?
 4. Emission by individual electrons
 5. Factors influencing the gain length and the amplification
 6. Microbunching: electrons and waves traveling together
 7. Saturation
 8. The underlying physics
 9. Limitations
 References
research papers
A simplified description of Xray freeelectron lasers
^{a}Faculté des Sciences de Base, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH1015 Lausanne, Switzerland
^{*}Correspondence email: giorgio.margaritondo@epfl.ch
It is shown that an elementary semiquantitative approach explains essential features of the Xray freeelectron laser mechanism, in particular those of the gain and saturation lengths. Using mathematical methods and derivations simpler than complete theories, this treatment reveals the basic physics that dominates the mechanism and makes it difficult to realise freeelectron lasers for short wavelengths. This approach can be specifically useful for teachers at different levels and for colleagues interested in presenting Xray freeelectron lasers to nonspecialized audiences.
Keywords: freeelectron laser; SASE; Xray laser.
1. Motivation
Xray freeelectron lasers (XFELs) are finally a reality: the recent success of the Stanford Coherent Light Source (LCLS) (Emma et al., 2010) is attracting considerable attention worldwide, not limited to the directly involved community nor to physics. This makes it desirable to have a theoretical treatment accessible to nonspecialists and students. Past experience with synchrotron sources (Margaritondo, 1988, 1995, 2002) indicates that an effort in this direction may enhance the use of the new machines, extend it to new research communities and facilitate teaching tasks at different levels.
We present here what is, we believe, the simplest description so far of the XFEL mechanism. Without complicated formalism, we can explain the role of relevant factors. The underlying physical phenomena become easily understandable, in particular what makes it difficult to build lasers for Xrays.
Note that because of the relativistic velocity of the electrons in the XFEL, such phenomena are not intuitive. For example, we shall see that the optical amplification depends on the electrons forming microbunches with a space period close to the emitted wavelength. Why, then, is the effect much more difficult to achieve for short Xray wavelengths than for visible light? On the contrary, one could imagine that microbunching is easier to obtain if the distance between microbunches is shorter! We shall see how relativity explains this apparent paradox.
2. Qualitative description
Fig. 1 schematically explains how an XFEL works (Madey, 1971; Dattoli & Renieri, 1984; Dattoli et al., 1995; Patterson et al., 2010; Bonifacio et al., 1984, 1994; Bonifacio & Casagrande, 1985; Pellegrini, 2000; Murphy & Pellegrini, 1985; Kim, 1986; Huang & Kim, 2007; Kim & Xie, 1993; Brau, 1990; Kondratenko & Saldin, 1980; Milton et al., 2001; Schmueser et al., 2008; Feldhaus et al., 2005; Altarelli, 2010; Shintake, 2007; Shintake et al., 2003; Roberson & Sprangle, 1989; Saldin et al., 2000). The optical amplification takes place within electron bunches traveling inside a linear accelerator (LINAC) at a (longitudinal) speed u ≃ c, the speed of light. The emission and amplification of electromagnetic waves are activated by a periodic magnet array (`undulator') with period L. The undulator magnetic field can be written as B = B_{0}sin(2πx/L) = B_{0}sin(2πut/L). Subject to this field, the electrons slightly undulate with a periodic transverse velocity component v_{T}. These oscillations and the corresponding acceleration cause the electron charges to emit electromagnetic waves.
In a normal undulator source the electrons emit electromagnetic waves without correlation with each other (Fig. 1c) and the total intensity is the sum of the intensities produced by individual electrons, proportional to N/Σ, the number of electrons in the bunch divided by the bunch If i is the electron beam current corresponding to the electron bunch in the accelerator, then N/Σ is proportional to i/Σ.
In an XFEL [Figs. 1(b) and 1(d)] the electrons emit in a correlated way (Emma et al., 2010; Dattoli & Renieri, 1984; Dattoli et al., 1995; Huang & Kim, 2007). Assume that a given electron, after entering the undulator, emits a wave. The (transverse) Bfield of this wave and the transverse velocity of the electrons create a longitudinal Lorentz force that pushes the electrons to form microbunches with a periodicity equal to the emitted wavelength. The electrons within a microbunch oscillate all together under the effect of the undulator, and their wave emission is correlated (Fig. 1d). The Efield (or the Bfield) of the waves emitted by individual electrons are added together, rather than their intensity.
This has two consequences: (i) since the wave intensity is proportional to the square of the Efield, the total emitted intensity is proportional to N^{2} rather than to N; (ii) the total wave intensity is progressively amplified along the undulator (Fig. 1e) according, as we shall see, to an exponential law (Emma et al., 2010; Huang & Kim, 2007).
The amplification does not continue indefinitely: saturation occurs after a distance L_{S} (Fig. 1e). One criterion in designing an XFEL is to reach saturation before the end of the undulator (Emma et al., 2010). In most lasers the path available for amplification is expanded by an external optical cavity. This is not possible for Xrays since normalincidence mirrors are extremely ineffective at the corresponding wavelengths. Hence, a `onepass' strategy is required, with strong amplification and a very long undulator.
Note that the starting wave subsequently amplified could be an external Xray beam injected along with the electron beam (a `seed') rather than the spontaneous initial emission of the electrons (Huang & Kim, 2007). In that case the laser works as an amplifier rather than as a selfcontained source. When spontaneous initial emission is used, the mechanism is called SASE (selfamplified spontaneous emission) (Bonifacio et al., 1984).
3. What causes an exponential intensity increase?
This property can be discussed even before analyzing the details of the XFEL mechanism. The amplification is due to the negative work of the force caused by the wave (transverse) Efield (note that the Bfield cannot do any work).
from the electrons to the previously emitted wave. This requires aThe time rate of E_{W}v_{T}, the wave Efield magnitude times the electron transverse velocity. In turn, E_{W} is proportional to the square root of the wave intensity, thus the rate from each electron is proportional to I^{1/2}v_{T}. Therefore, the uncorrelated combination of the effects of individual electrons would not correspond to an exponential increase of the intensity with the distance but to a quadratic law.
for one electron is proportional to the productMicrobunching changes this by forcing the electrons to emit in a correlated way. What causes microbunching? As we already mentioned, microbunching is caused by the interaction between the electrons oscillating in the transverse direction and the transverse Bfield of the previously emitted waves. Indeed, the transverse velocity and the Bfield produce a longitudinal Lorentz force that, as we shall discuss in detail later, pushes the electrons to form microbunches.
The microbunching Lorentz force is proportional to the transverse electron velocity and to the wave Bfield strength B_{W}. Since B_{W} is proportional to the square root of the wave intensity, the microbunching force is proportional to I^{1/2}.
How does microbunching influence the subsequent wave emission? Let us assume that it enhances the correlated emission by a factor proportional to the microbunching force, an assumption that we will justify later. Multiplied by the I/dt = AI with A = constant, corresponding indeed to an exponential intensity increase along the undulator.
rate for each electron, this factor gives dAssuming A = u/L_{G}, we obtain the commonly used form (Bonifacio et al., 1984; Huang & Kim, 2007) for the exponential intensity law,
The parameter L_{G}, called `gain length', characterizes the amplification and the corresponding requirements to obtain lasing.
The functional form of (1) is verified experimentally (Emma et al., 2010). Therefore, we will use it for the rest of our discussion as an empirical fact.
4. Emission by individual electrons
We now summarize some basic features of the emission of an electron traveling in an undulator (Margaritondo, 2002) that are valid, in particular, for an XFEL, and explain fundamental properties such as the emitted wavelength. Since the electron speed is (almost) the speed of light c, the treatment is based on special relativity.
In the electron reference frame, the undulator transverse Bfield (Fig. 2a), after a Lorentz transformation, becomes the combination of a transverse Bfield plus a transverse Efield (Fig. 2b), traveling together at a speed u ≃ c. These are also the characteristics of an electromagnetic wave. The wavelength of this wave is given, in the electron reference frame, by the undulator period corrected for the relativistic Lorentz contraction. In the longitudinal direction the contracted length is L/γ, where γ is the relativistic γfactor, defined by the equation 1/γ ^{2} = (1 − u^{2}/c^{2}) and proportional to the electron energy γm_{0}c^{2} (m_{0} = electron rest mass).
The electron, therefore, `sees' the undulator as an electromagnetic wave (Fig. 2b). This wave causes the electron to oscillate and to emit waves of equal wavelength. Thus, the emitted wavelength in the electron reference frame is L/γ.
However, seen in the laboratory reference frame (Fig. 2c) the wavelength emitted by the moving electron must be further corrected for the longitudinal Doppler effect. The additional correction factor is ∼2γ, so that the wavelength becomes
According to (2), to obtain Xrays the macroscopic undulator period L must be downscaled by many orders of magnitude using a large γ. Thus, an XFEL requires a highenergy accelerator.
Equation (2) is not entirely correct since it does not take into account the impact on γ of the undulator Bfield that induces the electron transverse velocity. The Lorentz force causing v_{T} cannot do any work: it cannot modify the and the overall velocity magnitude. The presence of v_{T} thus causes a decrease in the longitudinal velocity, to values < u. The effective 1/γ^{2} factor in (2) becomes larger than (1 − u^{2}/c^{2}) and depends on B.
It is easy to demonstrate that the corresponding corrected form of (2) is
where the socalled `undulator parameter' K is proportional to the maximum undulator Bfield strength B_{0} and to L. In fact, owing to conservation, the longitudinal speed squared decreases from u^{2} to (u^{2} − v_{T}^{2}). Thus, in (2), 1/γ^{2} changes to 1 − (u^{2} − v_{T}^{2})/c^{2} = (1/γ^{2})(1 + v_{T}^{2}γ^{2}/c^{2}). This is consistent with (3) since, as we shall see later, v_{T} is proportional to B_{0}L/γ. Note that (3) implies that the emitted wavelength of an XFEL can be controlled by changing the undulator Bfield strength.
In a real undulator, and in an XFEL, the emission occurs not at one wavelength but in a wavelength band of width Δλ around the central value defined by (3) [or, in first approximation, by (2)]. This bandwidth can be estimated by taking into account that each electron going through the undulator emits a wave train consisting of a number of wavelengths equal to the number of undulator periods, N_{u}. The time duration Δt of this pulse is the pulse length divided by the speed of light, N_{u}λ/c.
According to the Fourier transforms, a pulse of duration Δt has a frequency bandwidth Δν = 1/Δt; thus, Δν = c/(N_{u}λ). Wavelength and frequency are related as ν = c/λ, which by differentiation gives Δν = cΔλ/λ^{2}, thus Δλ = Δνλ^{2}/c = λ/N_{u} and
a relative wavelength bandwidth decreasing as the number of undulator periods increases.
5. Factors influencing the gain length and the amplification
We will now discuss in detail the mechanism illustrated in Fig. 1. Note that a rigorous theoretical treatment is intrinsically complicated even in the simplest onedimensional case (Bonifacio et al., 1984). It leads to a thirdorder differential equation whose solution is the combination of three terms. One of them dominates during the exponential amplification and justifies it. The exponential amplification is preceded by a preliminary phase with a slower intensity buildup, and is followed by the saturation phase.
We do not try to tackle all these fine theoretical aspects, but explain with simple arguments their qualitative and quantitative consequences, starting from amplification. Remember that the rate of I^{1/2}v_{T}. Thus, to find the amplification we must evaluate v_{T}. However, the total correlated emission intensity from all electrons also depends on microbunching; thus, to find the amplification we must also evaluate the degree of microbunching.
from an individual electron to the preexisting wave is proportional toWe start with v_{T} that is caused (Fig. 1) by the undulator Bfield. For transversemotion dynamics, the relevant equation is Newton's law with the relativistic mass,
which gives
which is proportional to (B_{0}L/γ). Thus, the rate by a single electron is proportional to I^{1/2}(B_{0}L/γ). We will leave out for now the cosine factor, for reasons that will be clarified later.
As to microbunching, the longitudinal microbunching force is proportional to v_{T} and to the wave Bfield (pictured in Fig. 2). In turn, the wave Bfield is proportional to the square root of the wave intensity, and therefore [see (1)] to I_{0}^{ 1/2}exp[ut/(2L_{G})]. The microbunching force can then be written as
This force induces a small longitudinal electron displacement Δx superimposed on the average motion with speed u. For longitudinal dynamics the relevant relativistic equation is derived from the general law that the time derivative of the longitudinal momentum γm_{0}(dΔx/dt) equals the longitudinal force. The result (neglecting the small transverse oscillations) is
where the factor γ^{3}m_{0} is the socalled relativistic `longitudinal mass'. After integration, the above equation gives a longitudinal displacement towards microbunching,
(note that we assumed a negligibly small initial wave intensity for Δx = 0 m, where the amplification and motion towards microbunching start).
Maximum microbunching means that the electrons are concentrated in narrow slabs separated from each other by a distance equivalent to the wavelength λ. The degree of microbunching, corresponding to the fraction of electrons that emit in a correlated way, can be assumed in a first approximation to be proportional to (Δx/λ). The corresponding number of electrons is proportional to N(Δx/λ). Their contribution to the wave intensity is proportional to (i/Σ)(Δx/λ), in turn proportional [see (2)] to (i/Σ){[(B_{0}LL_{G}^{2}/γ^{4})I^{1/2}]/(L/γ^{2})} = (i/Σ)(B_{0}L_{G}^{2}/γ^{2})I^{1/2}.
These arguments justify our previous assumption that microbunching effects correspond to a factor proportional to the longitudinal microbunching force and therefore to I^{1/2}. In addition, they reveal other important elements in this factor. Multiplying the factor by the rate for one electron, we see that the total transfer rate is proportional to
and we can write
this is, indeed, an equation of the form dI/dt = AI, whose solution is (1) as long as u/L_{G} (≃ c/L_{G}) is proportional to (i/Σ)(B_{0}^{2}LL_{G}^{2}/γ^{3}), or
i.e. a result consistent with those (Bonifacio et al., 1984; Huang & Kim, 2007) of rigorous and complete theories and with their conceptual physics foundations.
This result can be expressed in terms of the `FEL parameter' or `Pierce parameter' ρ, corresponding to
introduced by Bonifacio et al. (1984), and linked to the most important FEL properties. Equation (4) thus implies
in agreement with its rigorous theoretical definition.
Equations (4) and (5) put in evidence essential factors that keep the gain length short, as required for an XFEL. First, the undulator parameters B_{0} and L must be maximized, keeping in mind, however, that L also determines the wavelength. The electron beam current must be high and its transverse small. However, the γfactor cannot be freely decreased if we want to obtain Xray wavelengths [see equations (2) and (3)].
6. Microbunching: electrons and waves traveling together
So far we have not considered the sine and cosine factors in the transverse velocity and in the wave. This can be justified a posteriori, based on the fact that the electron microbunching occurs only because of some subtle effects that merit additional analysis (see Fig. 3). Assume that at a certain time (Fig. 3, top) the Bfield of the already existing wave and the electron transverse velocity v_{T} create a Lorentz force f pushing the electron towards a wave node. This can indeed lead to microbunching.
Imagine, however, that electron and wave travel together with exactly the same speed. After onehalf of the undulator period the electron transverse velocity would be reversed whereas the wave Bfield would keep the same direction. The Lorentz force would be reversed and the microbunching destroyed!
Fortunately this does not happen because the electron and the wave do not travel with the same velocity. The (u − c) difference creates precisely the conditions for the microbunching to continue. In fact (Fig. 3, bottom), as the wave travels over a distance L/2 in a time L/(2c), the electron travels over a smaller distance Lu/(2c). The space shift between wave and electron is
Using (2) and since u ≃ c and (1 + u/c) ≃ 2, we see that this shift is ∼λ/2, onehalf wavelength! Thus, after onehalf undulator period both the electron transverse velocity and the wave Bfield are reversed, the Lorentz force keeps the same direction and microbunching continues.
This argument could be formulated in terms of phases: the difference between the electron oscillation phase and the wave phase stays constant. This is why we could so far neglect such phases (corresponding to the sine and cosine functions in the transverse velocity and in the wave), and analyze the phenomena with simple proportionalities.
7. Saturation
The above description, however, is not entirely realistic (Bonifacio et al., 1984; Huang & Kim, 2007). As an electron gives energy to the wave, its own energy is lowered and its longitudinal speed decreases from u to (u − Δu). Assume that the initial position of the electron with respect to the wave is favorable for the transfer of energy, i.e. that the directions of the electron transverse velocity and of the wave Efield produce negative work. The longitudinal speed decrease to (u − Δu) changes these conditions and makes them increasingly less favorable for the electron → wave.
As Δu becomes bigger, at a certain point the electrons no longer give energy to the wave: instead, the wave gives energy to the electrons. This, in turn, increases u until the conditions for from the electron to the wave are restored. Such a mechanism is repeated over and over: the energy oscillates between the wave and the electrons rather than continuing to increase exponentially for the wave (Dattoli & Ranieri, 1984). This is a key phenomenon underlying the saturation of the wave intensity amplification.
In order to estimate the conditions for saturation and in particular the `saturation length' L_{S} (Bonifacio et al., 1984; Huang & Kim, 2007) over which it occurs, we can start again from the rate for one electron, proportional to E_{W}v_{T}. So far we only considered amplitudes: but E_{W} (see Fig. 2) and v_{T} really are oscillating functions with their phases. We have already seen that
As far as the wave is concerned, we can write
where φ is a constant phase angle. A linear change in speed from u to (u − Δu) would modify the electron position at the time t from ut to approximately (ut − Δut/2), where the wave is proportional to cos[2π(ut/λ − Δut/2λ − ct/λ) + φ]. The difference between the two cosine arguments corresponding to u and to (u − Δu) is πΔut/λ. When this difference becomes too big, the conditions are reversed and saturation begins; this occurs for a difference value πΔut/λ related to 2π, i.e. for Δut ≃ 2λ.
Since Δu << u, for x = L_{S} (the saturation length) t ≃ L_{S}/u, and the same condition can be written,
The speed decrease Δu can be evaluated starting from the relativistic energy of the electron, γm_{0}c^{2} = W. By differentiating γm_{0}c^{2} = (1 − u^{2}/c^{2})^{1/2}m_{0}c^{2} with respect to u, this equation gives
where ΔW is the energy loss, i.e. the energy given by the `average' electron to the wave. Thus, (9) becomes
and therefore
where (ΔW/W) is the fraction of its own energy that the `average' electron gives to the wave. Using (2) we finally obtain
Generalized to all electrons, (11) implies that the ratio L/L_{S} approximately corresponds to the portion of the electron beam energy that is given to the wave before saturation occurs.
A closer look at the energy oscillation between the electrons and the wave enables us to make good use of (11) by calculating (ΔW/W). Consider once more the rate, proportional to the product E_{W}v_{T}. Taking for the wave and the transverse velocity the oscillating functions of (7) and (8), this product is proportional to
Using the elementary trigonometric property 2cos(α)cos(β) = cos(α + β) + cos(α − β), this expression is proportional to
actually corresponding not to one oscillation only but to the superposition of two different oscillations. The argument of the second oscillation can be written as
This is a rather fast oscillation whose effects average to zero and can be neglected in our discussion. With a similar procedure, the argument of the first term in (12) can be written as
that, actually, does not correspond to an oscillation but to a constant.
However, we recover the oscillation by taking into account the speed change from u to (u − Δu), so that the same term becomes
which, since L >> λ, is ≃ −2πΔut/λ + φ. This corresponds to an oscillation with frequency 2πΔu/λ, increasing as Δu increases.
In essence, saturation does not occur initially because this energy oscillation frequency is low and only gain takes place, with the characteristic gain length L_{G}. As the frequency increases, the gain length L_{G} becomes comparable with the electron path during one energy oscillation: there is no longer a steady gain and saturation is reached. This saturation criterion is equivalent to say (Murphy & Pellegrini, 1985) that the oscillation frequency becomes comparable with the gain rate given by (1), u/L_{G}. We can therefore write
and, using for Δu the result of (10),
which gives
In terms of the FEL parameter ρ = , equation (13) implies that
revealing another fundamental meaning of this parameter: it is a measure of the effectiveness of the overall et al., 1984; Huang & Kim, 2007; Murphy & Pellegrini, 1985) is consistent with (13) and (14) although the results have slightly different proportionality constants,
from the electrons to the wave. The conceptual physics background of rigorous theories (BonifacioEquation (15) can also be interpreted with a somewhat different and interesting point of view: the stochastic wave emission changes the energy of each electron with respect to the others. This increases the energy spread until saturation occurs. The spread is related to the average energy loss ΔW, therefore (15) implies that ρ is also a measure (Murphy & Pellegrini, 1985) of the relative energy spread of the electron beam at saturation.
Combining (13) and (11), we finally obtain
another interesting property of XFELs, revealing the relation between the saturation length and the gain. Using (15) instead of (13), we obtain a version (Bonifacio et al., 1984) of (16) with a more accurate proportionality constant,
8. The underlying physics
The above discussion brings to light some of the fundamental physics facts in the XFEL mechanism. In particular, it explains why it is more difficult to build freeelectron lasers for Xrays than for larger wavelengths. Basically, for small wavelengths we need highenergy electrons, but high electron energy also increases the gain length, as shown by equation (4).
This brings us back to the apparent paradox that creating microbunches should be easier when they are spaced by a small wavelength, whereas in reality it is not. The paradox is solved by realising that this factor is more than offset by two others that clearly emerge from the above treatment. First, a large γfactor negatively affects the transverse velocity, which is proportional to (B_{0}L/γ). Second, it impacts even more the longitudinal relativistic mass, proportional to γ^{3}. In essence, the large γfactor required for short wavelengths makes the electrons transversally and longitudinally `heavy' and therefore difficult to move, negatively affecting both the individualelectron emission rate and microbunching.
As far as saturation is concerned, it is clear that the wave intensity amplification could not continue forever since at a certain point the electrons would run out of energy. This, however, is not an important feature: much before the electrons lose a substantial portion of their energy they slow down by emitting electromagnetic energy, change their phase with respect to the wave and start taking energy rather than giving it. Afterwards, the energy oscillates between electrons and wave rather than continuing to accumulate in the wave. Other effects also contribute to the saturation of the amplification (Milton et al., 2001) making a full description more complicated.
9. Limitations
Table 1 summarizes the XFEL properties that could be treated, at least semiqualitatively, with our simple description. We note, however, that this approach is certainly not suitable for designing a real XFEL and should not be applied beyond its limitations. First of all, we explicitly treated a planar undulator and did not consider helical insertion devices that are more effective for freeelectron lasers (Bonifacio et al., 1984; Huang & Kim, 2007). Furthermore, our analysis was performed in one dimension, without taking into account threedimensional effects. Finally, an XFEL requires very high amplification that is affected by several additional factors besides those we discussed. The corresponding treatment must be based (Milton et al., 2001) on numerical solutions obtained with very sophisticated methods.

We can mention here the following additional factors affecting the amplification: electron energy spread, angular divergence, transverse electron beam size and diffraction of the wave. To a certain approximation their effects can be accounted for (Milton et al., 2001) by multiplying the gain length by a `degradation factor' χ > 1, so that the role of the parameters as described for example by equation (4) is still (at least qualitatively) valid.
The electron energy spread affects not only the amplification but also the saturation. In fact, amplification mainly starts with the optimal electron energy, whose γfactor determines the wavelength [equations (2) and (3)]. But as the electrons transfer energy to the wave, their own energy decreases. The wave emission is not the same from all electrons, so that different electrons have different energies, with an increasing energy spread. At a certain point the energy spread is so large that there is no gain anymore. This saturation factor is combined and correlated to the previously discussed mechanism.
Other important issues were not treated at all here. We should mention at least the emission coherence and time structure. The coherence of the Xrays produced by a SASE XFEL is very high laterally but limited longitudinally (Bonifacio et al., 1984; Huang & Kim, 2007) because of the stochastic emission of the initial waves; this problem can be solved by seeding.
The time structure of the emitted beam is very interesting since it can reach the femtosecond and subfemtosecond scale. Indeed, we have seen that the time duration of the emission by a single electron is N_{u}λ/c. Taking typical values N_{u} ≃ 10^{3} and λ ≃ 1 Å = 10^{−10} m, this gives ∼0.3 × 10^{−15} s or 0.3 fs. The actual pulse length for a real XFEL is influenced by several factors (Huang & Kim, 2007) that can also be used to control it. But the above basic time scale gives an idea of why the subfemtosecond scale can be reached.
Acknowledgements
This work was supported by the Fonds National Suisse pour la Recherche Scientifique and by the Center for Biomolecular Imaging (CIBM), in turn supported by the LouisJeantet and Leenaards foundations.
References
Altarelli, M. (2010). From Third to FourthGeneration Light Sources: FreeElectron Lasers in the UV and Xray Range, in Magnetism and Synchrotron Radiation, edited by E. Beaurepaire, H. Bulou, E. Scheurer and J. K. Kappler. Berlin: Springer. Google Scholar
Bonifacio, R. & Casagrande, S. (1985). J. Opt. Soc. Am. B, 2, 250–258. CrossRef CAS Google Scholar
Bonifacio, R., De Salvo, L., Pierini, P., Piovella, N. & Pellegrini, C. (1994). Phys. Rev. Lett. 73, 70–73 CrossRef PubMed CAS Web of Science Google Scholar
Bonifacio, R., Pellegrini, C. & Narducci, L. M. (1984). Opt. Commun. 50, 373–378. CrossRef CAS Web of Science Google Scholar
Brau, C. A. (1990). FreeElectron Lasers. Oxford: Academic Press. Google Scholar
Dattoli, G. & Renieri, A. (1984). Experimental and Theoretical Aspects of FreeElectron Lasers, Laser Handbook, Vol. 4, edited by M. L. Stich and M. S. Bass. Amsterdam: North Holland. Google Scholar
Dattoli, G., Renieri, A. & Torre, A. (1995). Lectures in FreeElectron Laser Theory and Related Topics. Singapore: World Scientific. Google Scholar
Emma, P. et al. (2010). Nat. Photon. 4, 641–647. Web of Science CrossRef CAS Google Scholar
Feldhaus, J., Arthur, J. & Hastings, J. B. (2005). J. Phys. B, 38, S799–S819. Web of Science CrossRef CAS Google Scholar
Huang, Z. & Kim, K. J. (2007). Phys. Rev. Special Topics Accel. Beams, 10, 034801. Web of Science CrossRef Google Scholar
Kim, K. J. (1986). Nucl. Instrum. Methods Phys. Res. A, 250, 396–403. CrossRef Google Scholar
Kim, K. J. & Xie, M. (1993). Nucl. Instrum. Methods Phys. Res. A, 331, 359–364. CrossRef Google Scholar
Kondratenko, K. & Saldin, E. (1980). Part. Accel. 10, 207–216. CAS Google Scholar
Madey, J. (1971). J. Appl. Phys. 42, 1906–1913. CrossRef CAS Web of Science Google Scholar
Margaritondo, G. (1988). Introduction to Synchrotron Radiation. New York: Oxford University Press. Google Scholar
Margaritondo, G. (1995). J. Synchrotron Rad. 2, 148–154. CrossRef CAS Web of Science IUCr Journals Google Scholar
Margaritondo, G. (2002). Elements of Synchrotron Light for Biology, Chemistry and Medical Research. New York: Oxford University Press. Google Scholar
Milton, S. V. et al. (2001). Science, 292, 1059955. Web of Science CrossRef Google Scholar
Murphy, J. B. & Pellegrini, C. (1985). Introduction to the Physics of FreeElectron Lasers, in Laser Handbook, edited by W. B. Colson, C. Pellegrini and A. Renieri. Amsterdam: NorthHolland. Google Scholar
Patterson, B. D., Abela, R., Braun, H. H., Flechsig, U., Ganter, R., Kim, Y., Kirk, E., Oppelt, A., Pedrozzi, M., Reiche, S., Rivkin, L., Schmidt, T., Shmitt, B., Strocov, V. N., Tsujino, S. & Wrulich, A. F. (2010). New J. Phys. 12, 035012. Web of Science CrossRef Google Scholar
Pellegrini, C. (2000). Nucl. Instrum. Methods Phys. Res. A, 445, 124–127. Web of Science CrossRef CAS Google Scholar
Roberson, C. W. & Sprangle, P. (1989). Phys. Fluids, B1, 3–41. CrossRef Google Scholar
Saldin, E., Schneidmiller, E. & Yurkov, M. (2000). The Physics of the Free Electron Laser. Berlin: Springer. Google Scholar
Schmueser, P., Dohlus, M. & Rossbach, J. (2008). Ultraviolet and SoftXray FreeElectron Lasers. Berlin: Springer. Google Scholar
Shintake, T. (2007). Proceedings of the 22nd Particle Accelerator Conference (PAC07), 25–29 June 2007, Albuquerque, NM, USA. Piscataway: IEEE Press. Google Scholar
Shintake, T., Tanaka, H., Hara, T., Togawa, K., Inagaki, T., Kim, Y. J., Ishikawa, T., Kitamura, H., Baba, H., Matsumoto, H., Takeda, S., Yoshida, M. & Takasu, Y. (2003). Nucl. Instrum. Methods Phys. Res. A, 507, 382–387. CrossRef CAS Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.