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FEL SASE and wave undulators
Optics Communications 285 (2012) 5341–5346
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FEL SASE and wave undulators Giuseppe Dattoli a,n, Vittoria Petrillo b, Julietta V. Rau c a b c
ENEA C.R. Frascati, Via E. Fermi, 45- 00044 Frascati (RM), Italy Universita degli Studi di Milano and INFN-MI, Via Celoria, 16- 20129 Milano, Italy Istituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, Via del Fosso del Cavaliere, 100-00133 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 25 January 2012 Received in revised form 6 July 2012 Accepted 30 July 2012 Available online 23 August 2012
We investigate the working conditions of Free Electron Lasers operating in the SASE regime with wave undulators. We provide general scaling criteria and corroborate them with appropriate numerical simulations. & 2012 Elsevier B.V. All rights reserved.
Keywords: Free Electron Laser Self-Amplified Spontaneous Emission Undulators Wave-undulator Compton backscattering Non linear processes
1. Introduction Free Electron Lasers (FEL) operating with undulators provided by electromagnetic waves were considered in the past [1,2] and recently discussed again [3,4] because of the development of laser technology and the availability of the electron beams with unprecedented large brightness [5].1 High power lasers have undergone a spectacular evolution, the record intensity of 2 1026 ðW=m2 Þ has been obtained [6], and the upcoming petawatt lasers aim at intensities of the order of 1027 ðW=m2 Þ [7]. In the near future, intensities of the order of 1028 1030 ðW=m2 Þ are foreseen at the Extreme-Light-Infrastructure (ELI) [8]. The advantages offered by such an equipped FEL are evident. Electron beam with modest energies could in principle be used to reach the hard X-ray region, as well as the construction of long undulators could be avoided. In contrast, high power lasers, challenging electron beam qualities and handlings, are necessary to provide a set of parameters, supporting the growth of a FEL SASE up to the saturation. In this paper, we will not introduce new concepts and/or different experimental solutions, but we will derive a set of scaling formulae useful for a first design of a linearly polarized wave undulator based FEL SASE. We will study the conditions
n
Corresponding author. Tel: þ39-06-94005421. E-mail address: [email protected] (G. Dattoli). 1 The electron beam brightness is defined as the ratio between the peak current and the product of the normalized emittances. A high brightness beam reaches values of about 1015 A/(mm mrad). 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.07.128
allowing such an operating regime and check its relevant validity using analytical and numerical analyses. In an ordinary SASE FEL, electrons propagate in magnetic undulators characterized by a period lu, by an on axis field B0 and by a strength parameter K. The fact that a FEL can operate with a wave undulator is not surprising. Pantell, Soncini and Puthoff pointed out the possibility of obtaining a laser-like emission by means of the stimulated electron scattering of a CO2 laser (see Ref. [9a]), while Madey described the electron– undulator interaction using the Weiszacker-Williams method of virtual quanta (see Ref. [9b]). Accordingly the radiation emission by the electrons is treated as a Compton backscattering of the pseudo photons of the undulator, which is viewed as an electromagnetic wave with wavelength l* twice the period lu of the undulator itself. Within such a point of view the selection of the wavelength ls (the resonance condition) of the emitted radiation is, accordingly, the result of a double Doppler shift of the undulator ‘‘photons’’ wavelength, according to the prescriptions:
ls ¼
1b n 1b l ¼2 lu , 1þb 1þb
ð1Þ
where g is the Lorentz factor and b the normalized velocity of the electron beam. Including in the analysis the transverse motion induced by the interaction and associated with the number of pseudo-photon density of the undulator field [10], one obtains ! ln K2 , ð2Þ ls ffi 2 1 þ 2 4g
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that combined with (4) yields:
where eB0 lu K¼ 2pm0 c2
ð3Þ
The strength parameter K plays a role of paramount importance in the case of laser undulators. It can, indeed, be understood as the electron mass intensity dependence shift, discussed by Sengupta [11] and later developed by Brown and Kibble [12] in their treatment of the Compton scattering by an intense electromagnetic wave.2 The dynamics of FEL operation with magnetic or wave undulators is essentially the same, if we limit ourselves to modest electron energies and not too short laser wavelengths, in order to avoid quantum corrections. The scaling laws reported in Ref. [13] will, therefore, be used to derive practical formulae yielding the conditions to be satisfied by the wave laser intensity and by the electron beam qualities to guarantee a reliable operation. The paper consists of four sections. Section 2 is devoted to general considerations on the feasibility criteria of the device and on some scaling predictions on output power and saturation length. The relevant numerical test is reported in Section 3, and, finally, Section 4 contains critical remarks on the validity limits of the treatment and on the experimental feasibility of a wave undulator FEL.
2. FEL SASE wave undulator: basic design formulae
ð4Þ
Accordingly, a CO2 laser, with a power density of IðW=m2 Þ ffi 1019 , would be enough to provide a wave undulator with sufficiently large K to support the FEL SASE operation. The FEL intensity evolution is ruled by the so-called Pierce parameter, which, for the present purposes, can be cast in the following form [13]3:
rffi
i1=3 n 8:36 103 h U J A=m2 Uðl ðmÞKf b ðxÞÞ2 ,
g
ð7Þ
This last condition links the various parameters and can be exploited to evaluate the ranges of the laser wave intensity and beam current density which are necessary to achieve FEL operation. The operating conditions become more demanding with the decrease of the laser undulator wavelength due to the dependence on l* 4. The saturation length can be expressed in terms of a certain number of gain lengths4: Lg ðmÞ ffi
ln ðmÞ
pffiffiffi 8p 3r
ð8Þ
The ‘‘full saturation’’ condition, with an output FEL power of pffiffiffi ð9Þ PF ffi 2r PE is usually achieved in about 10 gain length, leading to a saturation length given by: LS ðmÞ ffi
ln ðmÞ
ð10Þ
4r
Ls can be exploited to fix the laser wave pulse length, and, together with the laser transverse section S, allows to specify the pulse energy: n E½J ffi 8:2 108 l rðmÞ I W=m2 Sðm2 Þ,
S ¼ ln Z R ,
In this section, we will merge well-known formulae, developed for magnetic undulator SASE FEL [13] and those worked out in Ref. [14] in the analysis of the electron back-scattering of intense laser, to obtain a set of reference quantities, useful to understand the feasibility of SASE FEL wave undulator device. The strength parameter, associated with a wave undulator of intensity I and wavelength l*, in practical units reads as [13,14]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n I W=m2 K 0:85 105 l ðmÞ
2 ðg rÞ3 J A=m2 I W=m2 f b ffi 2:35 1016 n l ðmÞ4
ð11Þ
Z R RayleighLength We assume, moreover, that the e-beam is matched to the laser beam section and, therefore, its current IE ¼JS is provided by: IE ½A ffi
2:35 1016 2 fb
ðgrÞ3 Sðm2 Þ n I W=m2 l ðmÞ4
ð12Þ
and, in turn, the electron beam power can be written as PE ðMWÞ ffi
2:35 1016 2 fb
ðgrÞ3 S ðm2 ÞEðMeVÞ n I W=m2 l ðmÞ4
ð13Þ
Finally, as a consequence of Eq. (9), the FEL saturated power reads as pffiffiffi 1:2U 2 1016 ðgrÞ4 Sðm2 Þ ð14Þ PF ðMWÞ ffi n 2 I W=m2 l ðmÞ4 fb A more complete description of the SASE intensity growth versus the ‘‘undulator’’ longitudinal coordinate can be specified by the following logistic-like function [13]: P0 BðzÞ , P0 9 1 þ 9P BðzÞ F " pffiffiffi ! pffiffiffi !# z p p 3z 3z z=2Lg z=2Lg þ e cos cos e , BðzÞ ¼ 2 cosh Lg Lg Lg 3 3
PðzÞ ¼ f b ðxÞ ¼ J0 ðxÞJ 1 ðxÞ,
x¼
1 K2 , 4 1 þ K2
ð5Þ
2
where J is the electron beam density current. From the first of Eq. (5) we obtain ðgrÞ3 J A=m2 ffi 1:7 106 2 n Kf b ðKÞl ðmÞ
ð6Þ
2 The strength parameter is indeed associated with the average value (over an optical cycle) of /AmAmS, where Am is the four-potential associated with the laser undulator field. 3 The definition of Pierce parameter given in Eq. (3) does not contain the contributions due to diffraction, it can indeed be included through the following
2 redefinition:r ffi FðDÞ r, FðDÞ ¼ ð1 þ DÞ1=3 , D ¼ 2 ð4 pÞl2sðlb eÞ r , ls ¼ 4 lg2 1 þ K2 , T
ð15Þ where P0 the initial power. Eq. (15) has been shown to be extremely useful for the analytical modeling of the FEL SASE power evolution. The previous identities can be exploited to understand whether the requirements to operate a wave undulator FEL SASE are within the present technological capabilities. Let us consider, for instance, the case of a wave undulator, provided by a CO2 laser, with a power density of I[W/m2]ffi8 1019 (corresponding to K¼0.76). The use of an electron beam with an 4 It must be emphasized that in the case of wave undulators the period usually denoted by lu should be replaced with half the laser wavelength ðl=2Þ.
G. Dattoli et al. / Optics Communications 285 (2012) 5341–5346
energy of 25 MeV yields a FEL operating wavelength around 1.34 nm. A current density of J¼1011(A/m2) ensures the value of r ffi3 10 4, corresponding to a gain length of about 0.77 mm. If we further assume a Rayleigh length of ZR ffi5 10 3 m and an optimum matching between laser and electrons with the laser waist occurring in the middle exponential growth, the useful laser region is about 1–2 cm, permitting the FEL process to reach the onset of saturation. With these parameters, we get a beam current exceeding 5 kA, for a beam with a transverse section around 100 mm. The prevision of the logistic formula (15) for this case is shown in Fig. 1. The ratio between the FEL and the ‘‘undulator’’ power densities is about 10 9. If we use a saturation length of 2 cm, we obtain a corresponding energy of the CO2 laser of 260 J, which is beyond the present state of the art. But, if we consider a laser with an energy per pulse of 32 J, corresponding to IðW=m2 Þ ffi1019 (for the same spot as above), and the same characteristics of the electron beam, we obtain the graph shown in Fig. 2, which refers to a FEL operating at 1.08 nm. These parameters do not guarantee that the system reaches the onset of saturation (r ffi 1.5 10 4), however the system exhibits the exponential growth and is well above the noise region. Before concluding this section, we discuss further points, which need to be clarified (a) The effect of the inhomogeneous broadening has not been taken into account. A FEL operating with a laser undulator demands for an electron (e-) beam with particularly good energy spread and emittances. Otherwise, an increase of the
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Fig. 3. Analogous of Fig. 1 for different values of the e-beam relative energy spread: se ¼ 0 continous line; se ¼ 1,5 104 dot line; se ¼ 2 104 dash line:
saturation length is unavoidable. In the case of the energy spread, such an increase can be quantified as
pffiffi LS ffi 1þ 0:185U 23 m~ 2e LS ,
m~ e ¼
2se
r
ð16Þ
where se is the e-beam relative energy spread. Values around 0.01% (hardly achievable for the range of the e-beam energies and the current densities considered so far) would not create any significant increase for r (around 5 10 4). On the contrary, for the parameters used in Fig. 2, the induced increase of saturation length becomes problematic. In Fig. 3, we report the analogous of Fig. 1, but for different values of the energy spread. (b) The transverse shape of the laser wave should be as flat as possible (possibly super-gaussian) to allow a proper matching with the e-beam and avoid problems of line broadening due to the intensity inhomogeneity. (c) A further source of inhomogeneous broadening is associated with the relative bandwidth of the wave laser, which should satisfy the condition of ðDo=oÞ r r. Since Do ffi 2pðc=LS Þ ffi ð4pcr=lÞ, we obtain ðDo=oÞ ffi 2r and, therefore, the previous condition is barely satisfied, unless we do not consider longer pulses, exceeding the saturation length. (d) Larger laser intensities and electron beam current densities can be obtained by considering smaller transverse sections, as we will discuss in the forthcoming sections.
Fig. 1. FEL intensity growth versus the laser-undulator pulse length (in m), corresponding to a laser intensity of IðW=m2 Þffi 8 1019 .
3. Comparison with a numerical analysis
Fig. 2. FEL intensity growth versus the laser-undulator pulse length (in m), corresponding to a laser intensity of IðW=m2 Þffi 1019 .
In the previous section, we have established general criteria for the feasibility of the device. Here, we will specify the operating parameters and make reference to actual experimental configurations. The power growth function reported in Ref. [6] has been extensively checked with numerical codes (and with the experimental data as well) for FEL operating with magnetic undulators. We have made a comparison with the results from the code developed in Ref. [3], where the specific case of a CO2 laser wave undulator has been discussed. The parameters of the simulation are close to those considered in the previous section, and in Fig. 4 we provide a comparison between the power growth curve, obtained by applying the code developed in [3], and the analytical formulae of the previous section. The agreement is satisfactory, thus confirming the validity of the scaling relations developed here. Some other comments are now necessary. The oscillating behavior of the FEL intensity after saturation is not contained in Eq. (15). It reproduces, indeed, the growth up to the saturation and then yields a
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In this paper, we have dealt with the FEL SASE based on the stimulated Compton back-scattering in the classical regime. Quantum corrections may arise when the induced electron recoil is not completely negligible with respect to the electron energy. Quantum corrections can be treated as a kind of inhomogeneous broadening [15] and can be neglected if 6:
mQ ¼
8 p2 |c
g r ls
r1,
|c reduced Compton Wavelength
ð22Þ
For the examples discussed in this paper the above parameter ranges between mQ ffi 0.2 0.3 and, therefore, the adopted classical treatment is adequate. Fig. 4. FEL SASE power (W) growth versus the laser-undulator longitudinal coordinate (in m), analytical (upper curve) and numerical (lower curve).
flat top. The oscillations have been included in the analytical model (see the third of Ref. [13]), however, they will not be further discussed, being not essential in the present context. In Ref. [3], the laser and e-beam transverse sections have been chosen particularly small to enhance the associated densities, and this assumption allows to get reasonable values of the r parameter. Such a strategy, similar to that followed in Ref. [2], is conceptually consistent with the relations given above. It allows to overcome the drawback of the l 4 scaling in Eq. (12) and yields the possibility of relaxing the request on the power and current densities. In contrast, these conditions are certainly more demanding, as concerns the design of the laser and beam optics. As was already stressed above, the electron and laser beam cross sections should be matched along the interaction region, and this requires that
en bT ffi g l Z R
ð17Þ
Furthermore, the inhomogeneous broadening effects, induced by the e-beam divergence, are not harmful if [13]:
2 g en
ffir 2 1þ K2 bT
ð18Þ
While those associated with the transverse field inhomogeneity are under control if [13]:
2 g en
2
1þ K2
2
gT
kb ffi r,
ð19Þ
with kb being the wave number associated with the electron betatron motion. We can ignore this last effect if the transverse shape of the laser wave undulator is flat. By combining Eqs. (17) and (18), we end up with the following condition on the normalized emittance: sffiffiffiffiffi g ls ZR , en ffi pffiffiffiffiffiffiffiffi 2 3 p Lg qffiffiffiffiffiffiffiffiffiffi ð20Þ bT ffi3 Z R Lg The diffraction correction parameter D given in Eq. (3) can be explicitly calculated from the above relations as5: pffiffiffi 3 ls Lg , ð21Þ Dffi 4 p l ZR which is negligible for the most practical cases. pffiffi 2 The diffraction parameter can also be cast in the form D ¼ 8p32 ðegnles DÞ , eD ¼ where eD is the three dimensional emittance. 5
b ls g Lg 2p
4. Concluding comments As already mentioned, we have considered here a linearly polarized wave undulator. In a linearly polarized magnetic undulator, electrons radiate in the forward direction at odd multiples of the fundamental frequency (namely, on ¼ n os , os ¼ 2 p=ls , n ¼ 2 h þ 1), as a consequence of the modulation of the longitudinal motion, induced by the Lorentz force. The mechanism of non-linear harmonic generation in FEL SASE devices is associated with the coherent emission at higher harmonics, induced by the bunching of the electron beam, caused by the fundamental. The non-linear Compton process is the back-scattering of higher order harmonics in the intense linearly polarized laser fields [16]. In the case of a SASE wave undulator, we can speculate on the possible occurrence of non-linear harmonics generation. We can estimate the maximum power of the harmonics using the scaling relation [13]: 2 1 f Pn ¼ pffiffiffi b,n PF f b,n ¼ J n1 ðnxÞJn þ 1 ðnxÞ, f b,1 ¼ f b ð23Þ 2 2 n nf b The use of Eq. (14) yields pffiffiffi 1:2U 2 1016 f b,n 2 gr 4 Pn MW=m2 ¼ pffiffiffi n f b lðmÞ nI MW=m2
ð24Þ
The possibility of observing an appreciable coherent signal from non-linear harmonics generation is associated with sufficiently large K values, which should range on the order of unity (or even larger). By keeping the value of the laser undulator intensity corresponding to K¼1, we obtain for n ¼3: P3 MW=m2 =PF ffi 3:5 103 ð25Þ The amount of power predicted by Eq. (25) for the third harmonic is significantly below that of the fundamental, but it is still significant at short wavelength. The relatively large K values required by the observation of the non-linear effects demand for a higher laser energy. However, since the first peak of the higher harmonics occurs before that of the fundamental (in the case of the third harmonic it is between 1/2 and 2/3 of the full saturation length), it will be possible to relax the request on the energy by using shorter laser pulses. Although the system does not reach full saturation at the fundamental because of the shortness of the ‘‘undulator’’, a significant amount of power can be observed at higher harmonics, as a consequence of the bunching induced by the power of the first. It might be also argued that the quantum corrections might modify 6 Eq. (22) can be written in terms of the normalized emittance by noting that if en ¼ M gðls =4pÞ we get mQ ¼ Mð2p|c =ren Þ r 1, M is a kind of quality factor.
G. Dattoli et al. / Optics Communications 285 (2012) 5341–5346
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Eqs. (24) and (25) and the conclusions we have drawn. However, there are two reasons to believe that the previous results are essentially correct (a) The classical analysis of non-linear Compton process is generally accepted as satisfactory [17], for the range of parameters we have considered so far. In this region quantum effects do not play any significant role, since the work done by the laser field on the electrons is negligible with respect to the electron energy (namely ð_=4Þos ðK=ð1þ K 2 =2ÞÞ 5 E), (see Ref. [17]). (b) The generation of non-linear harmonics is the result of a complex mechanism, associated with the bunching and the induction of a relatively large energy spread on the electron beam [13]. These effects occur at large intensities of the FEL field, so that any quantum effect is negligible with respect to the classical contributions. Other physical mechanisms could modify the previous conclusions. It is worth to remark that the motion of electrons is modified by radiation under the influence of an external field. The associated effects, currently referred as radiation reaction contributions, can be treated using the Dirac–Abraham–Lorentz equation (plagued by the unwanted runaway solutions and causality problems) or by the Landau–Lifshitz equation (see Refs. [18,19]). Both treatments predict that this effect can be neglected, if the following identity is satisfied: 1 K2 a_os 5E, 2 3 1þ K 2
e2 a¼ 4pe0 _c 1
ð26Þ
(which is less restrictive than the condition given in the previous point a). As mentioned above, the present day technology is mature enough to conceive the experimental programs aiming at the realization of a laser undulator FEL SASE. We work out an example, which is within the present capabilities [4,20], the parameters we are referring to are reported in Table 1. In this proposal, the electrons interact with the undulator, after focusing the laser beam and the electron beam on a small spot. The values chosen are compatible with the following FEL parameters: K ffi 0:346, r ffi1:13 103 . In Fig. 5, we report the FEL SASE intensity evolution, and our analysis confirms the prediction of a saturation length of about 5 mm. The reduction of the laser pulse by a factor 3 might allow a K value around 1 and the possibility of observing a significant amount of non-linear signal at 30 KeV. In all the previous calculations, we have assumed that the longitudinal profile of the laser wave undulator is constant. However, a dependence on z accounts for the effect of the diffraction. In general, the laser spot size is a function of z ¼ ðz=Z R Þ. The variation of the spot size along z induces a variation of the laser intensity and a consequent variation of the Table 1 Laser and e-beam parameters for a FEL wave undulator compatible with the present technology. Electron beam E(MeV) I(A) E EX r,
77.3 1.5 103 8.6
EX ðkeVÞ ¼ _ os Laser energy (J) Laser pulse duration (ps)
10 30 30
Fig. 5. FEL intensity (ðW=m2 Þ) growth versus the laser-undulator longitudinal coordinate (in mm), for the parameters reported in Table 1.
strength parameter, which may be a further source of inhomogeneous broadening. Therefore, the condition to be fulfilled by the waist is: WðzMAX Þ2 Wð0Þ2
ffi r,
zMAX ¼
LS 2Z R
ð27Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 If we consider a Gaussian mode with WðzÞ ¼ W 0 1 þ z , the pffiffiffiffi previous condition yields ðLS =2Z R Þ ffi r. The previous argument can be enforced by considering it from the point of view of the hourglass effect [20], responsible for the luminosity reduction in linear colliders. The rigorous inclusion of this effect within the present context is quite delicate. It requires a full three dimensional analysis, in which the evolution of the transverse mode of the wave undulator is considered along with the relevant contribution on the FEL SASE field evolution. A feeling of the importance of this effect can however be obtained by making an average, along the interaction region, on the longitudinal dependence of the transverse dimensions of the wave undulator. Our estimation includes the longitudinal diffraction, but keeps the electron bunch section constant inside the wave undulator. Accordingly we obtain a r reduction factor provided by ½ðsT =W 0 ÞðZ R =LS Þ1=3 , where sT is the transverse electron beam section. The effect has been overestimated, but for the present parameters we may expect a reduction of 60% of the Pierce parameters. This point has to be taken in serious account and the optimization of a FEL SASE wave undulator device should be considered accordingly, along the lines discussed in this paper. In the absence of a focusing effect of the laser radiation wave front, the above quoted reduction of the r parameter reflects on an analogous request of increase both the laser intensity and the beam current in order to fulfill the conditions for FEL operation deriving from Eq. (7). We have developed general considerations regarding the possibility of operating a FEL SASE, using a wave undulator. The concept of free electron coherent devices of this kind dates back to the late sixties of the last century, before the Madey’s proposal of 1971 (see Ref. [9]). Now, after more than 40 years, the state of the art seems to be appropriate for beginning experiments on this subject. The analytical results contained in this paper, and validated by a numerical analysis, offer simple means for a quick evaluation of the feasibility of such a device. W ðzMAX Þ
2
Acknowledgements The Authors are deeply grateful to Dr. M. Quattromini for the significant contributions, the constructive criticisms, the constant
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G. Dattoli et al. / Optics Communications 285 (2012) 5341–5346
interest and his invaluable presence. G. D. expresses his sincere appreciation to Prof. F. Minniti for kind remarks and encouragement.
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[9] (a) R.H. Pantell, G. Soncini, H.E. Puthoff, IEEE Journal of Quantum Electronics 4 (1968) 905; (b) J.M. Madey, Journal of Applied Physics 42 (1971) 1906. [10] G. Dattoli, A. Renieri, In: M.L. Stitch, M.S. Bass (Eds.), Laser Handbook IV, North Holland, Amsterdam, 1985. [11] N.D. Sengupta, Bulletin of Mathematical Society (Calcutta) 44 (1952) 175. [12] L.S. Brown, T.W.B. Kibble, Physical Review 133 (1964) A705. [13] G. Dattoli, P.L. Ottaviani, S. Pagnutti, Journal of Applied Physics 97 (2005) 113102; G. Dattoli, P.L. Ottaviani, S. Pagnutti, Booklet for FEL design, published by ENEA-Edizioni Scientifiche, 2007. [14] Y.Y. Lau, Fei He, D.P. Umstadter, R. Kowalczyk, Physics of Plasmas 10 (2003) 2155. [15] G. Dattoli, A. Renieri, In: W.B. Colson, C. Pellegrini, A. Renieri (Eds.) Laser Handbook, V, North Holland, Amsterdam, 1990. [16] S. Chen, A. Maksimchuk, D. Umstadter, Nature 396 (1998) 653; M. Babzien, I. Ben-Zvi, K. Kusche, I.V. Pavlishin, I.V. Pogorelsky, D.P. Siddons, V. Yakimenko, D. Cline, F. Zhou, T. Hirose, Y. Kamiya, T. Kumita, T. Omori, J. Urakawa, K. Yokoya, Physical Review Letters 96 (2006) 054802. [17] T. Heinzl, Journal of Physics: Conference Series 198 (2009) 012005. [18] H. Spohn, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004; F. Rohrlich, Classical Charged Particles, World Scientific, Singapore, 2007; J. Koga, T. Zh, S.V. Esirkepov, Bulanov, Physics of Plasmas 12 (2005) 093106. [19] A. Di Piazza, K.Z. Hatsagortsyan, C.H. Keitel, arXiv:08110.1703v2 [physics.class-ph], 26 (2009). [20] M.A. Furman, Hourglass effect for Asymmetric Colliders, LBL 30833 (1991).
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FEL SASE and wave undulators Giuseppe Dattoli a,n, Vittoria Petrillo b, Julietta V. Rau c a b c
ENEA C.R. Frascati, Via E. Fermi, 45- 00044 Frascati (RM), Italy Universita degli Studi di Milano and INFN-MI, Via Celoria, 16- 20129 Milano, Italy Istituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, Via del Fosso del Cavaliere, 100-00133 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 25 January 2012 Received in revised form 6 July 2012 Accepted 30 July 2012 Available online 23 August 2012
We investigate the working conditions of Free Electron Lasers operating in the SASE regime with wave undulators. We provide general scaling criteria and corroborate them with appropriate numerical simulations. & 2012 Elsevier B.V. All rights reserved.
Keywords: Free Electron Laser Self-Amplified Spontaneous Emission Undulators Wave-undulator Compton backscattering Non linear processes
1. Introduction Free Electron Lasers (FEL) operating with undulators provided by electromagnetic waves were considered in the past [1,2] and recently discussed again [3,4] because of the development of laser technology and the availability of the electron beams with unprecedented large brightness [5].1 High power lasers have undergone a spectacular evolution, the record intensity of 2 1026 ðW=m2 Þ has been obtained [6], and the upcoming petawatt lasers aim at intensities of the order of 1027 ðW=m2 Þ [7]. In the near future, intensities of the order of 1028 1030 ðW=m2 Þ are foreseen at the Extreme-Light-Infrastructure (ELI) [8]. The advantages offered by such an equipped FEL are evident. Electron beam with modest energies could in principle be used to reach the hard X-ray region, as well as the construction of long undulators could be avoided. In contrast, high power lasers, challenging electron beam qualities and handlings, are necessary to provide a set of parameters, supporting the growth of a FEL SASE up to the saturation. In this paper, we will not introduce new concepts and/or different experimental solutions, but we will derive a set of scaling formulae useful for a first design of a linearly polarized wave undulator based FEL SASE. We will study the conditions
n
Corresponding author. Tel: þ39-06-94005421. E-mail address: [email protected] (G. Dattoli). 1 The electron beam brightness is defined as the ratio between the peak current and the product of the normalized emittances. A high brightness beam reaches values of about 1015 A/(mm mrad). 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.07.128
allowing such an operating regime and check its relevant validity using analytical and numerical analyses. In an ordinary SASE FEL, electrons propagate in magnetic undulators characterized by a period lu, by an on axis field B0 and by a strength parameter K. The fact that a FEL can operate with a wave undulator is not surprising. Pantell, Soncini and Puthoff pointed out the possibility of obtaining a laser-like emission by means of the stimulated electron scattering of a CO2 laser (see Ref. [9a]), while Madey described the electron– undulator interaction using the Weiszacker-Williams method of virtual quanta (see Ref. [9b]). Accordingly the radiation emission by the electrons is treated as a Compton backscattering of the pseudo photons of the undulator, which is viewed as an electromagnetic wave with wavelength l* twice the period lu of the undulator itself. Within such a point of view the selection of the wavelength ls (the resonance condition) of the emitted radiation is, accordingly, the result of a double Doppler shift of the undulator ‘‘photons’’ wavelength, according to the prescriptions:
ls ¼
1b n 1b l ¼2 lu , 1þb 1þb
ð1Þ
where g is the Lorentz factor and b the normalized velocity of the electron beam. Including in the analysis the transverse motion induced by the interaction and associated with the number of pseudo-photon density of the undulator field [10], one obtains ! ln K2 , ð2Þ ls ffi 2 1 þ 2 4g
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that combined with (4) yields:
where eB0 lu K¼ 2pm0 c2
ð3Þ
The strength parameter K plays a role of paramount importance in the case of laser undulators. It can, indeed, be understood as the electron mass intensity dependence shift, discussed by Sengupta [11] and later developed by Brown and Kibble [12] in their treatment of the Compton scattering by an intense electromagnetic wave.2 The dynamics of FEL operation with magnetic or wave undulators is essentially the same, if we limit ourselves to modest electron energies and not too short laser wavelengths, in order to avoid quantum corrections. The scaling laws reported in Ref. [13] will, therefore, be used to derive practical formulae yielding the conditions to be satisfied by the wave laser intensity and by the electron beam qualities to guarantee a reliable operation. The paper consists of four sections. Section 2 is devoted to general considerations on the feasibility criteria of the device and on some scaling predictions on output power and saturation length. The relevant numerical test is reported in Section 3, and, finally, Section 4 contains critical remarks on the validity limits of the treatment and on the experimental feasibility of a wave undulator FEL.
2. FEL SASE wave undulator: basic design formulae
ð4Þ
Accordingly, a CO2 laser, with a power density of IðW=m2 Þ ffi 1019 , would be enough to provide a wave undulator with sufficiently large K to support the FEL SASE operation. The FEL intensity evolution is ruled by the so-called Pierce parameter, which, for the present purposes, can be cast in the following form [13]3:
rffi
i1=3 n 8:36 103 h U J A=m2 Uðl ðmÞKf b ðxÞÞ2 ,
g
ð7Þ
This last condition links the various parameters and can be exploited to evaluate the ranges of the laser wave intensity and beam current density which are necessary to achieve FEL operation. The operating conditions become more demanding with the decrease of the laser undulator wavelength due to the dependence on l* 4. The saturation length can be expressed in terms of a certain number of gain lengths4: Lg ðmÞ ffi
ln ðmÞ
pffiffiffi 8p 3r
ð8Þ
The ‘‘full saturation’’ condition, with an output FEL power of pffiffiffi ð9Þ PF ffi 2r PE is usually achieved in about 10 gain length, leading to a saturation length given by: LS ðmÞ ffi
ln ðmÞ
ð10Þ
4r
Ls can be exploited to fix the laser wave pulse length, and, together with the laser transverse section S, allows to specify the pulse energy: n E½J ffi 8:2 108 l rðmÞ I W=m2 Sðm2 Þ,
S ¼ ln Z R ,
In this section, we will merge well-known formulae, developed for magnetic undulator SASE FEL [13] and those worked out in Ref. [14] in the analysis of the electron back-scattering of intense laser, to obtain a set of reference quantities, useful to understand the feasibility of SASE FEL wave undulator device. The strength parameter, associated with a wave undulator of intensity I and wavelength l*, in practical units reads as [13,14]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n I W=m2 K 0:85 105 l ðmÞ
2 ðg rÞ3 J A=m2 I W=m2 f b ffi 2:35 1016 n l ðmÞ4
ð11Þ
Z R RayleighLength We assume, moreover, that the e-beam is matched to the laser beam section and, therefore, its current IE ¼JS is provided by: IE ½A ffi
2:35 1016 2 fb
ðgrÞ3 Sðm2 Þ n I W=m2 l ðmÞ4
ð12Þ
and, in turn, the electron beam power can be written as PE ðMWÞ ffi
2:35 1016 2 fb
ðgrÞ3 S ðm2 ÞEðMeVÞ n I W=m2 l ðmÞ4
ð13Þ
Finally, as a consequence of Eq. (9), the FEL saturated power reads as pffiffiffi 1:2U 2 1016 ðgrÞ4 Sðm2 Þ ð14Þ PF ðMWÞ ffi n 2 I W=m2 l ðmÞ4 fb A more complete description of the SASE intensity growth versus the ‘‘undulator’’ longitudinal coordinate can be specified by the following logistic-like function [13]: P0 BðzÞ , P0 9 1 þ 9P BðzÞ F " pffiffiffi ! pffiffiffi !# z p p 3z 3z z=2Lg z=2Lg þ e cos cos e , BðzÞ ¼ 2 cosh Lg Lg Lg 3 3
PðzÞ ¼ f b ðxÞ ¼ J0 ðxÞJ 1 ðxÞ,
x¼
1 K2 , 4 1 þ K2
ð5Þ
2
where J is the electron beam density current. From the first of Eq. (5) we obtain ðgrÞ3 J A=m2 ffi 1:7 106 2 n Kf b ðKÞl ðmÞ
ð6Þ
2 The strength parameter is indeed associated with the average value (over an optical cycle) of /AmAmS, where Am is the four-potential associated with the laser undulator field. 3 The definition of Pierce parameter given in Eq. (3) does not contain the contributions due to diffraction, it can indeed be included through the following
2 redefinition:r ffi FðDÞ r, FðDÞ ¼ ð1 þ DÞ1=3 , D ¼ 2 ð4 pÞl2sðlb eÞ r , ls ¼ 4 lg2 1 þ K2 , T
ð15Þ where P0 the initial power. Eq. (15) has been shown to be extremely useful for the analytical modeling of the FEL SASE power evolution. The previous identities can be exploited to understand whether the requirements to operate a wave undulator FEL SASE are within the present technological capabilities. Let us consider, for instance, the case of a wave undulator, provided by a CO2 laser, with a power density of I[W/m2]ffi8 1019 (corresponding to K¼0.76). The use of an electron beam with an 4 It must be emphasized that in the case of wave undulators the period usually denoted by lu should be replaced with half the laser wavelength ðl=2Þ.
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energy of 25 MeV yields a FEL operating wavelength around 1.34 nm. A current density of J¼1011(A/m2) ensures the value of r ffi3 10 4, corresponding to a gain length of about 0.77 mm. If we further assume a Rayleigh length of ZR ffi5 10 3 m and an optimum matching between laser and electrons with the laser waist occurring in the middle exponential growth, the useful laser region is about 1–2 cm, permitting the FEL process to reach the onset of saturation. With these parameters, we get a beam current exceeding 5 kA, for a beam with a transverse section around 100 mm. The prevision of the logistic formula (15) for this case is shown in Fig. 1. The ratio between the FEL and the ‘‘undulator’’ power densities is about 10 9. If we use a saturation length of 2 cm, we obtain a corresponding energy of the CO2 laser of 260 J, which is beyond the present state of the art. But, if we consider a laser with an energy per pulse of 32 J, corresponding to IðW=m2 Þ ffi1019 (for the same spot as above), and the same characteristics of the electron beam, we obtain the graph shown in Fig. 2, which refers to a FEL operating at 1.08 nm. These parameters do not guarantee that the system reaches the onset of saturation (r ffi 1.5 10 4), however the system exhibits the exponential growth and is well above the noise region. Before concluding this section, we discuss further points, which need to be clarified (a) The effect of the inhomogeneous broadening has not been taken into account. A FEL operating with a laser undulator demands for an electron (e-) beam with particularly good energy spread and emittances. Otherwise, an increase of the
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Fig. 3. Analogous of Fig. 1 for different values of the e-beam relative energy spread: se ¼ 0 continous line; se ¼ 1,5 104 dot line; se ¼ 2 104 dash line:
saturation length is unavoidable. In the case of the energy spread, such an increase can be quantified as
pffiffi LS ffi 1þ 0:185U 23 m~ 2e LS ,
m~ e ¼
2se
r
ð16Þ
where se is the e-beam relative energy spread. Values around 0.01% (hardly achievable for the range of the e-beam energies and the current densities considered so far) would not create any significant increase for r (around 5 10 4). On the contrary, for the parameters used in Fig. 2, the induced increase of saturation length becomes problematic. In Fig. 3, we report the analogous of Fig. 1, but for different values of the energy spread. (b) The transverse shape of the laser wave should be as flat as possible (possibly super-gaussian) to allow a proper matching with the e-beam and avoid problems of line broadening due to the intensity inhomogeneity. (c) A further source of inhomogeneous broadening is associated with the relative bandwidth of the wave laser, which should satisfy the condition of ðDo=oÞ r r. Since Do ffi 2pðc=LS Þ ffi ð4pcr=lÞ, we obtain ðDo=oÞ ffi 2r and, therefore, the previous condition is barely satisfied, unless we do not consider longer pulses, exceeding the saturation length. (d) Larger laser intensities and electron beam current densities can be obtained by considering smaller transverse sections, as we will discuss in the forthcoming sections.
Fig. 1. FEL intensity growth versus the laser-undulator pulse length (in m), corresponding to a laser intensity of IðW=m2 Þffi 8 1019 .
3. Comparison with a numerical analysis
Fig. 2. FEL intensity growth versus the laser-undulator pulse length (in m), corresponding to a laser intensity of IðW=m2 Þffi 1019 .
In the previous section, we have established general criteria for the feasibility of the device. Here, we will specify the operating parameters and make reference to actual experimental configurations. The power growth function reported in Ref. [6] has been extensively checked with numerical codes (and with the experimental data as well) for FEL operating with magnetic undulators. We have made a comparison with the results from the code developed in Ref. [3], where the specific case of a CO2 laser wave undulator has been discussed. The parameters of the simulation are close to those considered in the previous section, and in Fig. 4 we provide a comparison between the power growth curve, obtained by applying the code developed in [3], and the analytical formulae of the previous section. The agreement is satisfactory, thus confirming the validity of the scaling relations developed here. Some other comments are now necessary. The oscillating behavior of the FEL intensity after saturation is not contained in Eq. (15). It reproduces, indeed, the growth up to the saturation and then yields a
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In this paper, we have dealt with the FEL SASE based on the stimulated Compton back-scattering in the classical regime. Quantum corrections may arise when the induced electron recoil is not completely negligible with respect to the electron energy. Quantum corrections can be treated as a kind of inhomogeneous broadening [15] and can be neglected if 6:
mQ ¼
8 p2 |c
g r ls
r1,
|c reduced Compton Wavelength
ð22Þ
For the examples discussed in this paper the above parameter ranges between mQ ffi 0.2 0.3 and, therefore, the adopted classical treatment is adequate. Fig. 4. FEL SASE power (W) growth versus the laser-undulator longitudinal coordinate (in m), analytical (upper curve) and numerical (lower curve).
flat top. The oscillations have been included in the analytical model (see the third of Ref. [13]), however, they will not be further discussed, being not essential in the present context. In Ref. [3], the laser and e-beam transverse sections have been chosen particularly small to enhance the associated densities, and this assumption allows to get reasonable values of the r parameter. Such a strategy, similar to that followed in Ref. [2], is conceptually consistent with the relations given above. It allows to overcome the drawback of the l 4 scaling in Eq. (12) and yields the possibility of relaxing the request on the power and current densities. In contrast, these conditions are certainly more demanding, as concerns the design of the laser and beam optics. As was already stressed above, the electron and laser beam cross sections should be matched along the interaction region, and this requires that
en bT ffi g l Z R
ð17Þ
Furthermore, the inhomogeneous broadening effects, induced by the e-beam divergence, are not harmful if [13]:
2 g en
ffir 2 1þ K2 bT
ð18Þ
While those associated with the transverse field inhomogeneity are under control if [13]:
2 g en
2
1þ K2
2
gT
kb ffi r,
ð19Þ
with kb being the wave number associated with the electron betatron motion. We can ignore this last effect if the transverse shape of the laser wave undulator is flat. By combining Eqs. (17) and (18), we end up with the following condition on the normalized emittance: sffiffiffiffiffi g ls ZR , en ffi pffiffiffiffiffiffiffiffi 2 3 p Lg qffiffiffiffiffiffiffiffiffiffi ð20Þ bT ffi3 Z R Lg The diffraction correction parameter D given in Eq. (3) can be explicitly calculated from the above relations as5: pffiffiffi 3 ls Lg , ð21Þ Dffi 4 p l ZR which is negligible for the most practical cases. pffiffi 2 The diffraction parameter can also be cast in the form D ¼ 8p32 ðegnles DÞ , eD ¼ where eD is the three dimensional emittance. 5
b ls g Lg 2p
4. Concluding comments As already mentioned, we have considered here a linearly polarized wave undulator. In a linearly polarized magnetic undulator, electrons radiate in the forward direction at odd multiples of the fundamental frequency (namely, on ¼ n os , os ¼ 2 p=ls , n ¼ 2 h þ 1), as a consequence of the modulation of the longitudinal motion, induced by the Lorentz force. The mechanism of non-linear harmonic generation in FEL SASE devices is associated with the coherent emission at higher harmonics, induced by the bunching of the electron beam, caused by the fundamental. The non-linear Compton process is the back-scattering of higher order harmonics in the intense linearly polarized laser fields [16]. In the case of a SASE wave undulator, we can speculate on the possible occurrence of non-linear harmonics generation. We can estimate the maximum power of the harmonics using the scaling relation [13]: 2 1 f Pn ¼ pffiffiffi b,n PF f b,n ¼ J n1 ðnxÞJn þ 1 ðnxÞ, f b,1 ¼ f b ð23Þ 2 2 n nf b The use of Eq. (14) yields pffiffiffi 1:2U 2 1016 f b,n 2 gr 4 Pn MW=m2 ¼ pffiffiffi n f b lðmÞ nI MW=m2
ð24Þ
The possibility of observing an appreciable coherent signal from non-linear harmonics generation is associated with sufficiently large K values, which should range on the order of unity (or even larger). By keeping the value of the laser undulator intensity corresponding to K¼1, we obtain for n ¼3: P3 MW=m2 =PF ffi 3:5 103 ð25Þ The amount of power predicted by Eq. (25) for the third harmonic is significantly below that of the fundamental, but it is still significant at short wavelength. The relatively large K values required by the observation of the non-linear effects demand for a higher laser energy. However, since the first peak of the higher harmonics occurs before that of the fundamental (in the case of the third harmonic it is between 1/2 and 2/3 of the full saturation length), it will be possible to relax the request on the energy by using shorter laser pulses. Although the system does not reach full saturation at the fundamental because of the shortness of the ‘‘undulator’’, a significant amount of power can be observed at higher harmonics, as a consequence of the bunching induced by the power of the first. It might be also argued that the quantum corrections might modify 6 Eq. (22) can be written in terms of the normalized emittance by noting that if en ¼ M gðls =4pÞ we get mQ ¼ Mð2p|c =ren Þ r 1, M is a kind of quality factor.
G. Dattoli et al. / Optics Communications 285 (2012) 5341–5346
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Eqs. (24) and (25) and the conclusions we have drawn. However, there are two reasons to believe that the previous results are essentially correct (a) The classical analysis of non-linear Compton process is generally accepted as satisfactory [17], for the range of parameters we have considered so far. In this region quantum effects do not play any significant role, since the work done by the laser field on the electrons is negligible with respect to the electron energy (namely ð_=4Þos ðK=ð1þ K 2 =2ÞÞ 5 E), (see Ref. [17]). (b) The generation of non-linear harmonics is the result of a complex mechanism, associated with the bunching and the induction of a relatively large energy spread on the electron beam [13]. These effects occur at large intensities of the FEL field, so that any quantum effect is negligible with respect to the classical contributions. Other physical mechanisms could modify the previous conclusions. It is worth to remark that the motion of electrons is modified by radiation under the influence of an external field. The associated effects, currently referred as radiation reaction contributions, can be treated using the Dirac–Abraham–Lorentz equation (plagued by the unwanted runaway solutions and causality problems) or by the Landau–Lifshitz equation (see Refs. [18,19]). Both treatments predict that this effect can be neglected, if the following identity is satisfied: 1 K2 a_os 5E, 2 3 1þ K 2
e2 a¼ 4pe0 _c 1
ð26Þ
(which is less restrictive than the condition given in the previous point a). As mentioned above, the present day technology is mature enough to conceive the experimental programs aiming at the realization of a laser undulator FEL SASE. We work out an example, which is within the present capabilities [4,20], the parameters we are referring to are reported in Table 1. In this proposal, the electrons interact with the undulator, after focusing the laser beam and the electron beam on a small spot. The values chosen are compatible with the following FEL parameters: K ffi 0:346, r ffi1:13 103 . In Fig. 5, we report the FEL SASE intensity evolution, and our analysis confirms the prediction of a saturation length of about 5 mm. The reduction of the laser pulse by a factor 3 might allow a K value around 1 and the possibility of observing a significant amount of non-linear signal at 30 KeV. In all the previous calculations, we have assumed that the longitudinal profile of the laser wave undulator is constant. However, a dependence on z accounts for the effect of the diffraction. In general, the laser spot size is a function of z ¼ ðz=Z R Þ. The variation of the spot size along z induces a variation of the laser intensity and a consequent variation of the Table 1 Laser and e-beam parameters for a FEL wave undulator compatible with the present technology. Electron beam E(MeV) I(A) E EX r,
77.3 1.5 103 8.6
EX ðkeVÞ ¼ _ os Laser energy (J) Laser pulse duration (ps)
10 30 30
Fig. 5. FEL intensity (ðW=m2 Þ) growth versus the laser-undulator longitudinal coordinate (in mm), for the parameters reported in Table 1.
strength parameter, which may be a further source of inhomogeneous broadening. Therefore, the condition to be fulfilled by the waist is: WðzMAX Þ2 Wð0Þ2
ffi r,
zMAX ¼
LS 2Z R
ð27Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 If we consider a Gaussian mode with WðzÞ ¼ W 0 1 þ z , the pffiffiffiffi previous condition yields ðLS =2Z R Þ ffi r. The previous argument can be enforced by considering it from the point of view of the hourglass effect [20], responsible for the luminosity reduction in linear colliders. The rigorous inclusion of this effect within the present context is quite delicate. It requires a full three dimensional analysis, in which the evolution of the transverse mode of the wave undulator is considered along with the relevant contribution on the FEL SASE field evolution. A feeling of the importance of this effect can however be obtained by making an average, along the interaction region, on the longitudinal dependence of the transverse dimensions of the wave undulator. Our estimation includes the longitudinal diffraction, but keeps the electron bunch section constant inside the wave undulator. Accordingly we obtain a r reduction factor provided by ½ðsT =W 0 ÞðZ R =LS Þ1=3 , where sT is the transverse electron beam section. The effect has been overestimated, but for the present parameters we may expect a reduction of 60% of the Pierce parameters. This point has to be taken in serious account and the optimization of a FEL SASE wave undulator device should be considered accordingly, along the lines discussed in this paper. In the absence of a focusing effect of the laser radiation wave front, the above quoted reduction of the r parameter reflects on an analogous request of increase both the laser intensity and the beam current in order to fulfill the conditions for FEL operation deriving from Eq. (7). We have developed general considerations regarding the possibility of operating a FEL SASE, using a wave undulator. The concept of free electron coherent devices of this kind dates back to the late sixties of the last century, before the Madey’s proposal of 1971 (see Ref. [9]). Now, after more than 40 years, the state of the art seems to be appropriate for beginning experiments on this subject. The analytical results contained in this paper, and validated by a numerical analysis, offer simple means for a quick evaluation of the feasibility of such a device. W ðzMAX Þ
2
Acknowledgements The Authors are deeply grateful to Dr. M. Quattromini for the significant contributions, the constructive criticisms, the constant
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interest and his invaluable presence. G. D. expresses his sincere appreciation to Prof. F. Minniti for kind remarks and encouragement.
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