- Email: [email protected]
Stabilization of the naturally pulsed zones of the Super-ACO Storage Ring Free Electron Laser
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
Stabilization of the naturally pulsed zones of the Super-ACO Storage Ring Free Electron Laser M.E. Coupriea,*, C. Brunia, G.L. Orlandia, D. Garzellaa, S. Bielawskib # 522, 91 191 Gif-sur-Yvette, Service de Photons, Atomes et Mol!ecules, CEA/DSM/DRECAM, bat. and LURE, bat. 209 D, Universit!e de Paris-Sud, BP 34, 91 898 Orsay Cedex, France b Laboratoire de Physique des Lasers, Atomes et Molecules (UMR 8523) and CERLA, Bat P5, Universit!e des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, France a
Abstract A Storage Ring Free Electron Laser presents different regimes versus the synchronization between the optical pulses bouncing in the optical cavity and the electron bunches stored in the ring. Both experiments and simulations show that, on Super-ACO, the so-called detuning curve presents five zones: a ‘‘CW’’ regime at the ms time scale in the central and lateral ones, and a pulsed behavior at intermediate detunings. Apart from a longitudinal feedback allowing the FEL to be maintained in the central ‘‘cw’’ region for ensuring a high level of stability for the users, a recent feedback allowed the pulsed regions to be stabilized. The new detuning curves are presented. r 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Cr; 42.65.Sf; 42.60Rn Keywords: Storage Ring FEL; Feedback control; Stability; Relaxation oscillations
1. Introduction Free Electron Laser (FEL) oscillators can be viewed as a particular non-linear dynamical system. In this perspective, chaos has been studied on different FEL sources, such as Raman devices [1], and LINAC-based infra-red FEL oscillators [2]. On Storage Ring Free Electron Lasers (SRFEL), a macrotemporal structure at the millisecond time*Corresponding author. Tel.: +33-1-64-46-80-44; fax: +331-64-46-41-48. E-mail address: [email protected] (M.E. Couprie).
scale can appear, for particular detunings (i.e. synchronization between the electron bunches stored in the ring and the optical pulses bouncing in the optical resonator) in addition to the microtemporal structure, corresponding to the temporal pattern of the electron bunches (at a high-repetition rate) [3]. The theoretical and experimental SRFEL responses to a detuning modulation have been studied and they can lead to deterministic chaos [4]. We consider here the control of the naturally pulsed regimes that have been performed on the Super-ACO FEL [5], following the approach developed in the dynamical systems fields applied to conventional lasers, such
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.04.060
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M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
as Nd-YAG lasers [6], Nd-doped optical fiber lasers [7], passively mode locked laser [8], and in class B (pump modulated) lasers [9]. The theoretical and experimental control of the pulsed zones of the Super-ACO Free Electron Laser is here presented.
2. The detuning curve of the Super-ACO FEL 2.1. Experimental behavior of the Super-ACO FEL versus detuning In Fig. 1a, the Super-ACO FEL [10] intensity response for a slow sweep of the detuning shows different regimes. Around perfect tuning and for a small detuning interval (zone 3), the FEL is ‘‘cw’’ at the ms time scale, with the shortest widths of the temporal and spectral distributions, the FEL being closed to the Fourier limit [11]. For intermediate values of the detuning (zones 2 and 4), the FEL presents systematically a pulse structure at the ms time scale, with slightly larger temporal and spectral widths. The detuning curve shows the minima and
Fig. 1. Super-ACO FEL intensity (a) and position (b) ( for measurements and squares for LAS simulations) versus detuning. Energy: 800 MeV; mirror losses: 1%; gain: 2%; wavelength: 350 nm; TEM00 transverse mode. Main RF cavity operating at 100 MHz: A modification of 100 Hz corresponds to a round trip mismatch of 120 ns, or to a cavity length detuning of 18 mm:
the maxima of this macrotemporal structure. For even larger detunings (zones 1 and 5), the laser presents again a ‘‘cw’’ temporal structure at the ms time scale, with larger distributions and reduced power. Fig. 1b shows the evolution of the FEL pulse position with respect to the position of the synchronous electron. This ‘‘arctang’’ like function shows rapid changes around perfect detuning. Clearly, zone 3 is the most suitable one for user applications [12]. Such a type of detuning curve represents a rather general situation for the dynamics of SRFELs, even though the noise can mask the ‘‘cw’’ central region [13]. 2.2. Modelling of the detuning curve First theoretical representation of the behavior of a storage ring FEL was performed according to a phenomenological model in the temporal domain [3]. It describes the evolution of the intensity distribution and temporal position of the FEL pulse (with respect to the electron bunch center of mass), the increase of the electron beam energy spread (‘‘heating’’) induced by the FEL interaction. Using the ‘‘LAS’’ code based on this approach, one can reproduce the FEL intensity and position versus detuning, as illustrated in Fig. 1 [14]. Analogous results can be obtained, for example, with the numerical codes ‘‘SRFELn’’ [15] or ‘‘STOK 2D’’ based on FEL equations [14], or with ‘‘SRFEL’’ in the frequency domain [16], or with Ref. [17]. The pass-to-pass phenomenological model can be simplified for small variations of the detuning by rewriting it in a form of partial differential equations of the normalized laser longitudinal profile Y and the electron beam energy spread s; since the pulse shape varies slowly from pass-to-pass, in the case of a low gain and losses system. It leads to the following equations, at the first order of the cavity losses: @T Y DO@y Y ¼ Y þ G½Y þ Z
ð1Þ
ds2 =dT ¼ a½s20 s2 þ ðs2e s20 ÞI
ð2Þ
where T is the continuous time, expressed in units of cavity photon lifetime (usual time multiplied by the cavity losses and divided by the cavity round trip time tR ). y; the new dimensionless fast time,
ARTICLE IN PRESS M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
synchronous with the electron bunch passage, is the sum of the temporal coordinate t and the delay between the electron bunch and the laser pulse at each round trip dtn ; normalized with respect to tR : DO represents the normalized detuning, G is the laser gain, and ZG is the spontaneous emission. a ¼ 2tR =ts is typically much smaller than 1, ts being the synchrotron damping time. s0 ; s; se ; respectively, refer to the initial energy spread, the energy spread at time t; and at equilibrium. I indicates the laser intensity integrated over the longitudinal profile Y : Eq. (1) expresses the laser intensity evolution. Eq. (2) describes the laser heating damped via the synchrotron oscillations. Fig. 2 shows a typical detuning curve obtained using this simplified model. It is in good agreement with the experimental results and with the previous pass-to-pass simulations.
3. The control of the Super-ACO FEL 3.1. A longitudinal feedback for maintaining the Super-ACO FEL in zone 3 Since a Fourier limit powerful stable laser source is required for the user applications, a feedback aiming to keep the FEL in zone 3 was first developed on Super-ACO [18]. With the position of the FEL pulse changing very rapidly near perfect tuning, it provides a good parameter to evaluate the drift in synchronism. It is deduced from the measurement of the FEL pulse distribution, the drift in synchronism is compensated via the frequency of the RF cavity. Such a feedback system allows the intensity fluctuations to be damped to 1%, the spectral drift to be limited to less than 0.001% and the temporal jitter of the FEL pulse to be reduced, and compensated up to 1 ms; as illustrated in Fig. 3. 3.2. The stabilization of the pulsed regimes of the Super-ACO FEL in zones 2 and 4 The FEL can be viewed as a spatio-temporal system, with relevant coordinates T and y; the effect of the detuning appearing in the form of an advection term DO@y Y : The detuning curve (see
265
Fig. 2. Detuning curve derived from the simplified model in the case of the Super-ACO FEL. Synchrotron damping time: 8 ms; cavity losses 1%, gain: 2%, cavity round trip time: 120 ns:
Fig. 3. Streak camera image showing the compensation of the FEL jitter with the longitudinal feedback system.
Fig. 2), plotted using Eqs. (1) and (2), can also be considered as a bifurcation diagram. Stable stationary states exist around perfect tuning and for large detunings. Between these zones, a limit cycle regime leads to the pulsed behavior of the FEL, together with an unstable stationary state. Then, the control of the pulsed macrotemporal structure of the Super-ACO storage ring FEL becomes possible, by forcing the laser to operate onto the unstable stationary state. As in Refs. [7,8], a signal proportional to the derivative of the laser intensity (with a gain b) is used, and applied to the RF frequency pilot, as for the longitudinal feedback. However, the reference position is set in the centre of zone 2 or 4, and no more in the centre of zone 3. A photomultiplier, whose bandpass is smaller than the revolution frequency, and much larger than the macropulse frequency, is used for the detection. Simulations using the simplified model showed that the pulsed regime of the FEL can be stabilized. Experimentally, the control has been achieved by choosing empirically the feedback gain (b ¼ 0:1–0:5 mW1 ). Fig. 4 shows the controlled laser intensity and the feedback signal with the FEL operation in zone 4. The transient regime
ARTICLE IN PRESS 266
M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
Fig. 6. Detuning curve plotted with a slow triangle function of frequency change applied on the RF pilot, without (off) and with control (on). Fig. 4. Super-ACO FEL intensity I and control signal when the feedback is established and then switched off. The black zones correspond to the pulsed regime of the SUPER-ACO FEL (the macropulses are not resolved at this time scale).
starting from original ‘‘cw’’ behavior. In Fig. 6, one observes a stabilization of the pulse region of one half of the detuning curve, and an increase of the pulses amplitude in the other half.
4. Conclusion
Fig. 5. Streak camera image during the transient following the application of the feedback (a) experimental results (b) simulation with the typical values of the Super-ACO FEL, corresponding to the present experimental conditions.
lasts a few tens of ms leading to the stabilization of the pulsed zones. The resulting laser intensity fluctuations depend on the value of the applied gain b: Fig. 5 shows the behavior of the FEL micropulse observed with a double sweep streak camera, while the control is established. The experimental results in (a) are in good agreement with the simulated ones in (b). The sign of the reaction on the RF frequency pilot is opposite for zones 2 and 4, leading either to a stabilization of the pulsed zones or to an induced pulsed regime
Following the work performed on conventional lasers, a control of the Super-ACO FEL has been applied to stabilize the naturally pulsed regimes of operation, allowing the width of ‘‘cw’’ operation of the FEL to be widened. It is of great importance for the user applications, and for SRFELs for which the central ‘‘cw’’ region is extremely tiny, because of the combination of various parameters such as the synchrotron frequency, synchrotron damping time, gain and cavity losses. It could also probably be very useful for storage ring FEL where line instabilities prevent a good stability in zone 3.
References [1] E.H. Park, et al., Nucl. Instr. and Meth. A 358 (1995) 448; S. Kawasaki, et al., Nucl. Instr. and Meth. A 358 (1995) 114. [2] P. Chaix, et al., Phys. Rev. E 48 (1993) 3259; D.A. Jaroszynski, et al., Nucl. Instr. and Meth. A 407 (1998) 407;
ARTICLE IN PRESS M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
[3] [4]
[5] [6] [7] [8] [9]
[10] [11]
R. Hajima, et al., Nucl. Instr. and Meth. A 475 (2001) 270; W.P. Leemans, et al., Nucl. Instr. and Meth. A 358 (1995) 208. M. Billardon, et al., Phys. Rev. Lett. 69 (1992) 2368. M. Billardon, Phys. Rev. Lett. 65 (1990) 713; G. De Ninno, et al., Eur. Phys. J. D 22 (2003) 269; M.E. Couprie, Nucl. Instr. and Meth. A 507 (2003) 1; C. Bruni, et al., Nucl. Instr. and Meth. A, (2004) these proceedings. S. Bielawski, C. Bruni, G.L. Orlandi, D. Garzella, M.E. Couprie, Phys. Rev. E 69 (2004) 045502 (R). R. Roy, et al., Phys. Rev. Lett. 68 (1992) 1259. S. Bielawski, et al., Phys. Rev. A 47 (1993) 3276. N. Joly, S. Bielawski, Opt. Lett. 26 (2001) 692. M. Ciofini, et al., Phys. Rev. E 60 (1999) 398; R. Meucci, R. Mc Allister, R. Roy, Phys. Rev. E 66 (1999) 026216. M.E. Couprie, et al., Nucl. Instr. and Meth. A 358 (1995) 374. M.E. Couprie, et al., Nucl. Instr. and Meth. A 475 (2001) 229.
267
[12] M.E. Couprie, et al., Nucl. Instr. and Meth. A 375 (1996) 639. [13] M.E. Couprie, P. Elleaume, Nucl. Instr. and Meth. A 259 (1987) 77; H. Hama, et al., Nucl. Instr. and Meth. A 375 (1996) 39; N. Sei, et al., Nucl. Instr. and Meth. A 375 (1998) 187; V. Litvinenko, et al., Nucl. Instr. and Meth. A 475 (2001) 240; R.P. Walker, et al., Nucl. Instr. and Meth. A 475 (2001) 20. [14] C. Bruni, G. De Ninno, ‘‘LAS’’, R. Bartolini, ‘‘STOK 2D’’, Proceedings of the Brainstorming Workshop, Europ. TMR network ‘‘Storage ring FEL towards 200 nm’’, Maratea, Italy, 14–19/04/2002. [15] C.A. Thomas, et al., Nucl. Instr. and Meth. A 507 (2003) 281. [16] T. Hara, et al., Nucl. Instr. and Meth. A 375 (1996) 39. [17] V. Litvinenko, et al., Nucl. Instr. and Meth. A 358 (1995) 53. [18] M.E. Couprie, et al., Nucl. Instr. and Meth. A 358 (1995) 374.
Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
Stabilization of the naturally pulsed zones of the Super-ACO Storage Ring Free Electron Laser M.E. Coupriea,*, C. Brunia, G.L. Orlandia, D. Garzellaa, S. Bielawskib # 522, 91 191 Gif-sur-Yvette, Service de Photons, Atomes et Mol!ecules, CEA/DSM/DRECAM, bat. and LURE, bat. 209 D, Universit!e de Paris-Sud, BP 34, 91 898 Orsay Cedex, France b Laboratoire de Physique des Lasers, Atomes et Molecules (UMR 8523) and CERLA, Bat P5, Universit!e des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, France a
Abstract A Storage Ring Free Electron Laser presents different regimes versus the synchronization between the optical pulses bouncing in the optical cavity and the electron bunches stored in the ring. Both experiments and simulations show that, on Super-ACO, the so-called detuning curve presents five zones: a ‘‘CW’’ regime at the ms time scale in the central and lateral ones, and a pulsed behavior at intermediate detunings. Apart from a longitudinal feedback allowing the FEL to be maintained in the central ‘‘cw’’ region for ensuring a high level of stability for the users, a recent feedback allowed the pulsed regions to be stabilized. The new detuning curves are presented. r 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Cr; 42.65.Sf; 42.60Rn Keywords: Storage Ring FEL; Feedback control; Stability; Relaxation oscillations
1. Introduction Free Electron Laser (FEL) oscillators can be viewed as a particular non-linear dynamical system. In this perspective, chaos has been studied on different FEL sources, such as Raman devices [1], and LINAC-based infra-red FEL oscillators [2]. On Storage Ring Free Electron Lasers (SRFEL), a macrotemporal structure at the millisecond time*Corresponding author. Tel.: +33-1-64-46-80-44; fax: +331-64-46-41-48. E-mail address: [email protected] (M.E. Couprie).
scale can appear, for particular detunings (i.e. synchronization between the electron bunches stored in the ring and the optical pulses bouncing in the optical resonator) in addition to the microtemporal structure, corresponding to the temporal pattern of the electron bunches (at a high-repetition rate) [3]. The theoretical and experimental SRFEL responses to a detuning modulation have been studied and they can lead to deterministic chaos [4]. We consider here the control of the naturally pulsed regimes that have been performed on the Super-ACO FEL [5], following the approach developed in the dynamical systems fields applied to conventional lasers, such
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.04.060
ARTICLE IN PRESS 264
M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
as Nd-YAG lasers [6], Nd-doped optical fiber lasers [7], passively mode locked laser [8], and in class B (pump modulated) lasers [9]. The theoretical and experimental control of the pulsed zones of the Super-ACO Free Electron Laser is here presented.
2. The detuning curve of the Super-ACO FEL 2.1. Experimental behavior of the Super-ACO FEL versus detuning In Fig. 1a, the Super-ACO FEL [10] intensity response for a slow sweep of the detuning shows different regimes. Around perfect tuning and for a small detuning interval (zone 3), the FEL is ‘‘cw’’ at the ms time scale, with the shortest widths of the temporal and spectral distributions, the FEL being closed to the Fourier limit [11]. For intermediate values of the detuning (zones 2 and 4), the FEL presents systematically a pulse structure at the ms time scale, with slightly larger temporal and spectral widths. The detuning curve shows the minima and
Fig. 1. Super-ACO FEL intensity (a) and position (b) ( for measurements and squares for LAS simulations) versus detuning. Energy: 800 MeV; mirror losses: 1%; gain: 2%; wavelength: 350 nm; TEM00 transverse mode. Main RF cavity operating at 100 MHz: A modification of 100 Hz corresponds to a round trip mismatch of 120 ns, or to a cavity length detuning of 18 mm:
the maxima of this macrotemporal structure. For even larger detunings (zones 1 and 5), the laser presents again a ‘‘cw’’ temporal structure at the ms time scale, with larger distributions and reduced power. Fig. 1b shows the evolution of the FEL pulse position with respect to the position of the synchronous electron. This ‘‘arctang’’ like function shows rapid changes around perfect detuning. Clearly, zone 3 is the most suitable one for user applications [12]. Such a type of detuning curve represents a rather general situation for the dynamics of SRFELs, even though the noise can mask the ‘‘cw’’ central region [13]. 2.2. Modelling of the detuning curve First theoretical representation of the behavior of a storage ring FEL was performed according to a phenomenological model in the temporal domain [3]. It describes the evolution of the intensity distribution and temporal position of the FEL pulse (with respect to the electron bunch center of mass), the increase of the electron beam energy spread (‘‘heating’’) induced by the FEL interaction. Using the ‘‘LAS’’ code based on this approach, one can reproduce the FEL intensity and position versus detuning, as illustrated in Fig. 1 [14]. Analogous results can be obtained, for example, with the numerical codes ‘‘SRFELn’’ [15] or ‘‘STOK 2D’’ based on FEL equations [14], or with ‘‘SRFEL’’ in the frequency domain [16], or with Ref. [17]. The pass-to-pass phenomenological model can be simplified for small variations of the detuning by rewriting it in a form of partial differential equations of the normalized laser longitudinal profile Y and the electron beam energy spread s; since the pulse shape varies slowly from pass-to-pass, in the case of a low gain and losses system. It leads to the following equations, at the first order of the cavity losses: @T Y DO@y Y ¼ Y þ G½Y þ Z
ð1Þ
ds2 =dT ¼ a½s20 s2 þ ðs2e s20 ÞI
ð2Þ
where T is the continuous time, expressed in units of cavity photon lifetime (usual time multiplied by the cavity losses and divided by the cavity round trip time tR ). y; the new dimensionless fast time,
ARTICLE IN PRESS M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
synchronous with the electron bunch passage, is the sum of the temporal coordinate t and the delay between the electron bunch and the laser pulse at each round trip dtn ; normalized with respect to tR : DO represents the normalized detuning, G is the laser gain, and ZG is the spontaneous emission. a ¼ 2tR =ts is typically much smaller than 1, ts being the synchrotron damping time. s0 ; s; se ; respectively, refer to the initial energy spread, the energy spread at time t; and at equilibrium. I indicates the laser intensity integrated over the longitudinal profile Y : Eq. (1) expresses the laser intensity evolution. Eq. (2) describes the laser heating damped via the synchrotron oscillations. Fig. 2 shows a typical detuning curve obtained using this simplified model. It is in good agreement with the experimental results and with the previous pass-to-pass simulations.
3. The control of the Super-ACO FEL 3.1. A longitudinal feedback for maintaining the Super-ACO FEL in zone 3 Since a Fourier limit powerful stable laser source is required for the user applications, a feedback aiming to keep the FEL in zone 3 was first developed on Super-ACO [18]. With the position of the FEL pulse changing very rapidly near perfect tuning, it provides a good parameter to evaluate the drift in synchronism. It is deduced from the measurement of the FEL pulse distribution, the drift in synchronism is compensated via the frequency of the RF cavity. Such a feedback system allows the intensity fluctuations to be damped to 1%, the spectral drift to be limited to less than 0.001% and the temporal jitter of the FEL pulse to be reduced, and compensated up to 1 ms; as illustrated in Fig. 3. 3.2. The stabilization of the pulsed regimes of the Super-ACO FEL in zones 2 and 4 The FEL can be viewed as a spatio-temporal system, with relevant coordinates T and y; the effect of the detuning appearing in the form of an advection term DO@y Y : The detuning curve (see
265
Fig. 2. Detuning curve derived from the simplified model in the case of the Super-ACO FEL. Synchrotron damping time: 8 ms; cavity losses 1%, gain: 2%, cavity round trip time: 120 ns:
Fig. 3. Streak camera image showing the compensation of the FEL jitter with the longitudinal feedback system.
Fig. 2), plotted using Eqs. (1) and (2), can also be considered as a bifurcation diagram. Stable stationary states exist around perfect tuning and for large detunings. Between these zones, a limit cycle regime leads to the pulsed behavior of the FEL, together with an unstable stationary state. Then, the control of the pulsed macrotemporal structure of the Super-ACO storage ring FEL becomes possible, by forcing the laser to operate onto the unstable stationary state. As in Refs. [7,8], a signal proportional to the derivative of the laser intensity (with a gain b) is used, and applied to the RF frequency pilot, as for the longitudinal feedback. However, the reference position is set in the centre of zone 2 or 4, and no more in the centre of zone 3. A photomultiplier, whose bandpass is smaller than the revolution frequency, and much larger than the macropulse frequency, is used for the detection. Simulations using the simplified model showed that the pulsed regime of the FEL can be stabilized. Experimentally, the control has been achieved by choosing empirically the feedback gain (b ¼ 0:1–0:5 mW1 ). Fig. 4 shows the controlled laser intensity and the feedback signal with the FEL operation in zone 4. The transient regime
ARTICLE IN PRESS 266
M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
Fig. 6. Detuning curve plotted with a slow triangle function of frequency change applied on the RF pilot, without (off) and with control (on). Fig. 4. Super-ACO FEL intensity I and control signal when the feedback is established and then switched off. The black zones correspond to the pulsed regime of the SUPER-ACO FEL (the macropulses are not resolved at this time scale).
starting from original ‘‘cw’’ behavior. In Fig. 6, one observes a stabilization of the pulse region of one half of the detuning curve, and an increase of the pulses amplitude in the other half.
4. Conclusion
Fig. 5. Streak camera image during the transient following the application of the feedback (a) experimental results (b) simulation with the typical values of the Super-ACO FEL, corresponding to the present experimental conditions.
lasts a few tens of ms leading to the stabilization of the pulsed zones. The resulting laser intensity fluctuations depend on the value of the applied gain b: Fig. 5 shows the behavior of the FEL micropulse observed with a double sweep streak camera, while the control is established. The experimental results in (a) are in good agreement with the simulated ones in (b). The sign of the reaction on the RF frequency pilot is opposite for zones 2 and 4, leading either to a stabilization of the pulsed zones or to an induced pulsed regime
Following the work performed on conventional lasers, a control of the Super-ACO FEL has been applied to stabilize the naturally pulsed regimes of operation, allowing the width of ‘‘cw’’ operation of the FEL to be widened. It is of great importance for the user applications, and for SRFELs for which the central ‘‘cw’’ region is extremely tiny, because of the combination of various parameters such as the synchrotron frequency, synchrotron damping time, gain and cavity losses. It could also probably be very useful for storage ring FEL where line instabilities prevent a good stability in zone 3.
References [1] E.H. Park, et al., Nucl. Instr. and Meth. A 358 (1995) 448; S. Kawasaki, et al., Nucl. Instr. and Meth. A 358 (1995) 114. [2] P. Chaix, et al., Phys. Rev. E 48 (1993) 3259; D.A. Jaroszynski, et al., Nucl. Instr. and Meth. A 407 (1998) 407;
ARTICLE IN PRESS M.E. Couprie et al. / Nuclear Instruments and Methods in Physics Research A 528 (2004) 263–267
[3] [4]
[5] [6] [7] [8] [9]
[10] [11]
R. Hajima, et al., Nucl. Instr. and Meth. A 475 (2001) 270; W.P. Leemans, et al., Nucl. Instr. and Meth. A 358 (1995) 208. M. Billardon, et al., Phys. Rev. Lett. 69 (1992) 2368. M. Billardon, Phys. Rev. Lett. 65 (1990) 713; G. De Ninno, et al., Eur. Phys. J. D 22 (2003) 269; M.E. Couprie, Nucl. Instr. and Meth. A 507 (2003) 1; C. Bruni, et al., Nucl. Instr. and Meth. A, (2004) these proceedings. S. Bielawski, C. Bruni, G.L. Orlandi, D. Garzella, M.E. Couprie, Phys. Rev. E 69 (2004) 045502 (R). R. Roy, et al., Phys. Rev. Lett. 68 (1992) 1259. S. Bielawski, et al., Phys. Rev. A 47 (1993) 3276. N. Joly, S. Bielawski, Opt. Lett. 26 (2001) 692. M. Ciofini, et al., Phys. Rev. E 60 (1999) 398; R. Meucci, R. Mc Allister, R. Roy, Phys. Rev. E 66 (1999) 026216. M.E. Couprie, et al., Nucl. Instr. and Meth. A 358 (1995) 374. M.E. Couprie, et al., Nucl. Instr. and Meth. A 475 (2001) 229.
267
[12] M.E. Couprie, et al., Nucl. Instr. and Meth. A 375 (1996) 639. [13] M.E. Couprie, P. Elleaume, Nucl. Instr. and Meth. A 259 (1987) 77; H. Hama, et al., Nucl. Instr. and Meth. A 375 (1996) 39; N. Sei, et al., Nucl. Instr. and Meth. A 375 (1998) 187; V. Litvinenko, et al., Nucl. Instr. and Meth. A 475 (2001) 240; R.P. Walker, et al., Nucl. Instr. and Meth. A 475 (2001) 20. [14] C. Bruni, G. De Ninno, ‘‘LAS’’, R. Bartolini, ‘‘STOK 2D’’, Proceedings of the Brainstorming Workshop, Europ. TMR network ‘‘Storage ring FEL towards 200 nm’’, Maratea, Italy, 14–19/04/2002. [15] C.A. Thomas, et al., Nucl. Instr. and Meth. A 507 (2003) 281. [16] T. Hara, et al., Nucl. Instr. and Meth. A 375 (1996) 39. [17] V. Litvinenko, et al., Nucl. Instr. and Meth. A 358 (1995) 53. [18] M.E. Couprie, et al., Nucl. Instr. and Meth. A 358 (1995) 374.