- Email: [email protected]
A deeper analytical insight into the longitudinal dynamics of a storage-ring free-electron laser
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
A deeper analytical insight into the longitudinal dynamics of a storage-ring free-electron laser G. De Ninnoa,*, D. Fanellib b
a Sincrotrone Trieste, Area Science Park, 34012 Basovizza, Trieste, Italy Department of Cell and Molecular Biology, Karolinska Insitute, SE-171 77 Stockholm, Sweden
Abstract In this paper, a deep characterization is provided of the longitudinal dynamics of a storage-ring free-electron laser. Closed analytical expressions are derived for the main statistical parameters of the system (i.e. beam energy spread, intensity, centroid position and rms value of the laser distribution) as a function of the light-electron beam detuning at each pass inside the interaction region. Moreover, the transition between the stable ‘‘cw’’ regime and the unstable pulsed behaviour is shown to be a Hopf bifurcation. Finally, a feedback procedure is introduced which suppresses the bifurcation and significantly improves the system stability. r 2004 Published by Elsevier B.V. PACS: 29.20. Dh; 41.60.Cr Keywords: Storage ring; Free electron laser; Nonlinear dynamics; Hopf bifurcation
1. The initial model The starting point of our analysis is the wellknown theoretical model, introduced in Ref. [1] and further improved in Refs. [2,3], which allows to capture the main features of the longitudinal dynamics of a Storage-Ring Free-Electron Laser (SRFEL). Such a model describes the coupled evolution of the laser temporal profile and laserinduced beam energy spread. The laser profile, yn ; is updated after each interaction according to ynþ1 ðzÞ ¼ R2 yn ðz eÞ½1 þ gn ðzÞ þ is ðzÞ
ð1Þ
*Corresponding author. E-mail address: [email protected] (G. De Ninno). 0168-9002/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.nima.2004.04.014
where z is the temporal position of the electron bunch with respect to the centroid and pdistribution ffiffiffiffiffiffiffiffiffiffiffiffi R ¼ 1 P (where P stands for the cavity losses) is the mirror reflectivity, the detuning parameter e represents the difference between the electrons revolution period (divided by the number of bunches) and the period of the photons inside the cavity. The term gn stands for the optical gain (assumed to have the same Gaussian profile of the electron bunch) and is ðtÞ accounts for the profile of the spontaneous emission of the optical klystron [4]. The cumulative laser-electron beam delay induced by a finite e value is responsible for the behaviour of the laser intensity, experimentally observed on the Super-ACO and UVSOR FELs: a
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
40
‘‘cw’’ regime1 when e is zero or close to zero, and a stable pulsed regime when e exceeds a given threshold.
2. An explicit, simplified formulation Eq. (1) contains the evolution of the main statistical parameters of the laser distribution, namely the intensity In (zero-order moment), the centroid position with respect to the centre of the electron bunch tn (first-order moment) and the rms value sl;n (second-order moment). In order to gain analytical insight, we have assumed the laser to keep a Gaussian profile and we have calculated the first three moments of the distribution. The details of the calculation are given elsewhere [5]. As a result, by approximating finite differences with differentials, the explicit coupled evolution of the FEL statistical parameters can be cast in the form ds a1 1 ¼ ½a2 I þ 1 s2 dt DT 2s dI R2 I P gi a3 ðs2 1Þ=a2 2s2 ¼ 2 þ 3 a4 s2l t# 2 dt DT R 2s a3 Is þ DT dt t t# gi ðs2 1Þ=a2 s2l ¼ þ 1 a3 a4 dt DT DT s s2 3 dsl 1 gi 1 Is 1 ðs2 1Þ=a2 sl ¼ a3 a4 þ DT 2 dt s3 DT I 2sl 2 s 2 þt a3
2DT ; ts
a3 ¼
O s0 a
2 ;
ð2Þ
ð3Þ
Pse a4 ¼ : gi s0
ð4Þ
Here s represents the laser-induced energy spread normalized to laser-off value s0 ; DT is the 1
Formulation (2) opens up the perspective of a deep analytical study of the SRFEL dynamics. The analysis of the fixed points of Eq. (2) allows, in fact, to characterize the functional dependence of the statistical parameters versus the light electron beam detuning. The fixed points ðI;% s; % t% ; sl Þ are found by imposing dI=dt ¼ ds=dt ¼ dt=dt ¼ dsl =dt ¼ 0; and solving the corresponding system. Assume hereon e > 0; the scenario for eo0 completely being equivalent. After some algebraic calculations, one can express the equilibrium values I;% t% ; sl as function of s: % s% 2 1 I% ¼ ð5Þ a2 8 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi391=2 2 2 <1 s2 = s% 4 % þ t% ¼ þ4e2 A5 ð6Þ :2 a3 ; a3
s2e s20 s20
a2 ¼
3. Equilibrium statistical parameters
sl ¼
where t# ¼ t þ e and a1 ¼
bouncing period of the laser pulse inside the optical cavity, gi stands for the small signal gain, O is the synchrotron frequency, ts the syncrotron damping time and a the momentum compaction, and se represents the equilibrium value (i.e. that reached at the laser saturation) of the beam energy spread at the perfect tuning (i.e. e ¼ 0).
The FEL dynamics is naturally pulsed on the temporal scale of the inter-bunch period, while it appears as ‘‘continuouswave’’ on a larger, millisecond, temporal scale.
s% 2 Is ð1s% 2 Þ=a2 a4 a2 2 s% 1 2gi a3 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi391=4 2 2 = s% 2 s% 4 þ þ4e2 A5 ; a3 a3
ð7Þ
where ð1s% 2 Þ=a2
s% 3 ðs% 2 1Þ a4 A¼ a2 I s gi a 3
:
ð8Þ
The equilibrium value of the energy spread s% can be found numerically by solving the second of Eq. (2), after imposing dI=dt ¼ 0; and making explicit use of Eqs. (5–7)( The results are represented with a solid line in Fig. 1 and display an excellent agreement with simulations based on Eq. (2) (symbols in Fig. 1). Alternatively, it is
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43 Laser intensity
1
0.7
and
Electron's energy spread
1.5
se =s0 3 s2e 1 b¼ gi Is a2 a3 a4 2 s20 s2 log a4 s2e s2e þ e2 1 a2 s20 s20 s0
1.3 0
ε (fs)
1
Laser centroid
0.7
4.5
0 0
ε (fs)
0
1
x 10 -2
4.3 0
ε (fs)
1
Laser rms
ε (fs)
1
Fig. 1. The fixed points are plotted as function of the detuning parameter e: Top left panel: normalized laser intensity. Top right panel: normalized electron-beam energy spread. Bottom left panel: Laser centroid. Bottom right panel: rms value of the laser distribution. The symbols refer to the simulations performed using Eqs. (2), the solid line stands for the analytic approach based on the numerical solution of the second of Eqs. (2) (after imposing dI=dt ¼ 0), while the long-dashed lines represent the result obtained using the closed analytical expressions (9)–(13).
possible to derive a closed analytical expression for s% by performing a perturbative calculation. A quite cumbersome calculation leads to 2 1=2 se s% ¼ þ d ð9Þ s20 where se P 1 s20 2 1 þ G a 3 1 a4 gi s0 R2 2 s2e d¼ 2 2 2 log a4 s0 s s2 s2 2 a3 02 G1 a3 02 G2 02 G1 2 a2 se se se se
ð10Þ with G1 ¼
1 1 s2e pffiffiffi þ c 2 a3 s20
! pffiffiffi c 1 s2e 1 2 þ 2 2 þ 2e b G2 ¼ pffiffiffi a3 a3 s 0 2 c
41
ð11Þ
1 s2 s2e s20 c ¼ 2 e2 þ 4 gi I s a2 a3 a4 a3 s0
2 s2e s 1 e2 e2 : s20 s0
ð14Þ
The above solution is represented in Fig. 1 with a long-dashed line, displaying satisfactory agreement with simulations. Calculations have been performed using the case of the Super-ACO FEL as reference. The values of the relevant parameters are: DT ¼ 120 ns; ts ¼ 8:5 ms; s0 ¼ 5 104 ; se =s0 ¼ 1:5; O ¼ 14 kHz; gi ¼ 2%; P ¼ 0:8%; Is ¼ 1:4 108 : To our knowledge, this study represents the first attempt to characterize the analytic dependence of the equilibrium statistical parameters of the SRFEL versus the temporal detuning e; over the whole region of ‘‘cw’’ behaviour. As a straightforward application of the above derivation, let us focus on the expression for s% l at perfect tuning, i.e. e ¼ 0: We obtain2 1=4 g Is 1 i s% l C 1 þ log ð15Þ st;0 2 P P where we made use of the relation se =s0 C1 þ 0:5 log ðgi =PÞ derived in Ref. [6]. Relation (15) was applied to the case of Super ACO and was shown to reproduce quantitatively the experimental value [5]. Further, a comparison between the estimate of sl derived in the context of super-modes theory [7] and the prediction based on Eq. (15) was drawn, the latter resulting systematically in values closer to the ones measured experimentally [5]. The stability of the fixed point % sðeÞ; ½IðeÞ; t% ðeÞ; sl ðeÞ can be determined by studying % the eigenvalues of the Jacobian matrix associated to system (2). The real part of the eigenvalues are reported in Fig. 2. The system is by definition 2
ð12Þ
ð13Þ
It should be remarked that when e ¼ 0; d ¼ 1=R2 1: Thus, formally, sðe % ¼ 0Þase =s0 ; which is in apparent contradiction with the assumption of the model. However, since R2 ¼ 1 PB1; dðe ¼ 0ÞB0; the residual small discrepancy being related to the approximations involved in the calculation
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
42
3
2
Re
3
λ2
Re λ
-2
Re -4
λ4
Re
λ3
-6 -8
0
0.4
0.8
1.2
Feedback off
Laser Intensity
0
λ1 =
Laser Intensity
Re
0 0.0
200.0 Time (ms)
Feedback on
1.6
ε (fs) Fig. 2. Behaviour of the real parts of the eigenvalues of the Jacobian matrix as a function of the detuning amount. The calculation has been done for the case of the Super-ACO FEL. The transition occurs around 1:3 fs; a value which is in good agreement with experiments.
stable when all the real parts of the eigenvalues are negative. The transition to an unstable regime occurs when at least one of those becomes positive. In general, the loss of stability takes place according to different modalities. Consider, for instance, a Jacobian matrix with a pair of complex conjugate eigenvalues and assume the real parts of all the eigenvalues to be negative. A Hopf bifurcation occurs when the real part of the two complex eigenvalues become positive, provided the others keep their sign unchanged [8]. This situation is clearly displayed in Fig. 2, thus allowing to conclude that the transition between the ‘‘cw’’ and the pulsed regime in a SRFEL is a Hopf bifurcation.
4. Stabilization of the pulsed regime Having characterized the transition from the stable to the unstable steady state in terms of Hopf bifurcation opens up interesting perspectives for the improvement of the system performance. In fact, as it has been shown by several authors [9,10], the chaotic behaviour induced in conventional lasers can be stabilized by using a self-controlled (closed-loop) procedure. In particular, in Ref. [11] the dynamics of a conventional laser was stabilized
0 0.0
200.0
Time (ms) Fig. 3. Behaviour of the FEL intensity in absence (see inset) and in presence of the closed-loop feedback. The parameters utilized for the simulations are those of Super ACO.
around the unstable steady state, arising from a Hopf bifurcation. By virtue of the results of the previous sections, this approach can be extended to the case of a SRFEL. For this purpose the constant detuning e is replaced with the timedependent quantity e ¼ e0 þ bDT
dI dt
ð16Þ
which is added to system (2). Here e0 is assumed to be larger than ec : As it is shown in Fig. 3, when the control is switched off, i.e. b ¼ 0; the laser is unstable and displays periodic oscillations. For b larger than a certain threshold, bc ; the oscillations are damped and the laser behaves as if it was in the ‘‘cw’’ region. Preliminary experimental results have been already obtained at Super ACO [12].
5. Conclusions In this paper, we have shown that making explicit the evolution of the statistical parameters of a SRFEL allows to gain a deep insight into the longitudinal dynamics of the system. Exploiting this analytical characterization, we have introduced a suitable feedback procedure to enlarge the
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
region of stable signal. This result opens up the perspective of improving the performance of last generation SRFELs.
Acknowledgements The work of GDN has been partially supported by EUFELE, a Project funded by the European Commission under FP5 Contract No. HPRI-CT2001-50025. DF thanks U. Skoglund for the useful discussions.
References [1] M. Billardon, et al., Phys. Rev. Lett. 69 (1992) 2368.
43
[2] G. De Ninno, D. Fanelli, M.E. Couprie, Nucl. Instr. and Meth. A 483 (2002) 177. [3] G. De Ninno, et al., Europhys. J. D 22 (2003) 269. [4] N.A. Vinokurov, et al., preprint INP77.59 Novosibirsk, unpublished, 1977. [5] G. De Ninno, D. Fanelli, ELETTRA Internal Note ST/ SL-03/03, 2003. [6] G. De Ninno, D. Fanelli, Phys. Rev. Lett. 92 (2004) 094801. [7] G. Dattoli, private communication. [8] N. Berglund, Nonlinearity 13 (2000) 225. [9] V. Petrov, et al., J. Chem. Phys. 96 (1992) 7503. [10] S. Bielawski, et al., Phys. Rev. E 49 (1994) 971. [11] S. Bielawski, et al., Phys. Rev. A 47 (1993) 3276. [12] M.E. Couprie, Nucl. Instr. and Meth. A, (2004) these proceedings.
Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
A deeper analytical insight into the longitudinal dynamics of a storage-ring free-electron laser G. De Ninnoa,*, D. Fanellib b
a Sincrotrone Trieste, Area Science Park, 34012 Basovizza, Trieste, Italy Department of Cell and Molecular Biology, Karolinska Insitute, SE-171 77 Stockholm, Sweden
Abstract In this paper, a deep characterization is provided of the longitudinal dynamics of a storage-ring free-electron laser. Closed analytical expressions are derived for the main statistical parameters of the system (i.e. beam energy spread, intensity, centroid position and rms value of the laser distribution) as a function of the light-electron beam detuning at each pass inside the interaction region. Moreover, the transition between the stable ‘‘cw’’ regime and the unstable pulsed behaviour is shown to be a Hopf bifurcation. Finally, a feedback procedure is introduced which suppresses the bifurcation and significantly improves the system stability. r 2004 Published by Elsevier B.V. PACS: 29.20. Dh; 41.60.Cr Keywords: Storage ring; Free electron laser; Nonlinear dynamics; Hopf bifurcation
1. The initial model The starting point of our analysis is the wellknown theoretical model, introduced in Ref. [1] and further improved in Refs. [2,3], which allows to capture the main features of the longitudinal dynamics of a Storage-Ring Free-Electron Laser (SRFEL). Such a model describes the coupled evolution of the laser temporal profile and laserinduced beam energy spread. The laser profile, yn ; is updated after each interaction according to ynþ1 ðzÞ ¼ R2 yn ðz eÞ½1 þ gn ðzÞ þ is ðzÞ
ð1Þ
*Corresponding author. E-mail address: [email protected] (G. De Ninno). 0168-9002/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.nima.2004.04.014
where z is the temporal position of the electron bunch with respect to the centroid and pdistribution ffiffiffiffiffiffiffiffiffiffiffiffi R ¼ 1 P (where P stands for the cavity losses) is the mirror reflectivity, the detuning parameter e represents the difference between the electrons revolution period (divided by the number of bunches) and the period of the photons inside the cavity. The term gn stands for the optical gain (assumed to have the same Gaussian profile of the electron bunch) and is ðtÞ accounts for the profile of the spontaneous emission of the optical klystron [4]. The cumulative laser-electron beam delay induced by a finite e value is responsible for the behaviour of the laser intensity, experimentally observed on the Super-ACO and UVSOR FELs: a
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
40
‘‘cw’’ regime1 when e is zero or close to zero, and a stable pulsed regime when e exceeds a given threshold.
2. An explicit, simplified formulation Eq. (1) contains the evolution of the main statistical parameters of the laser distribution, namely the intensity In (zero-order moment), the centroid position with respect to the centre of the electron bunch tn (first-order moment) and the rms value sl;n (second-order moment). In order to gain analytical insight, we have assumed the laser to keep a Gaussian profile and we have calculated the first three moments of the distribution. The details of the calculation are given elsewhere [5]. As a result, by approximating finite differences with differentials, the explicit coupled evolution of the FEL statistical parameters can be cast in the form ds a1 1 ¼ ½a2 I þ 1 s2 dt DT 2s dI R2 I P gi a3 ðs2 1Þ=a2 2s2 ¼ 2 þ 3 a4 s2l t# 2 dt DT R 2s a3 Is þ DT dt t t# gi ðs2 1Þ=a2 s2l ¼ þ 1 a3 a4 dt DT DT s s2 3 dsl 1 gi 1 Is 1 ðs2 1Þ=a2 sl ¼ a3 a4 þ DT 2 dt s3 DT I 2sl 2 s 2 þt a3
2DT ; ts
a3 ¼
O s0 a
2 ;
ð2Þ
ð3Þ
Pse a4 ¼ : gi s0
ð4Þ
Here s represents the laser-induced energy spread normalized to laser-off value s0 ; DT is the 1
Formulation (2) opens up the perspective of a deep analytical study of the SRFEL dynamics. The analysis of the fixed points of Eq. (2) allows, in fact, to characterize the functional dependence of the statistical parameters versus the light electron beam detuning. The fixed points ðI;% s; % t% ; sl Þ are found by imposing dI=dt ¼ ds=dt ¼ dt=dt ¼ dsl =dt ¼ 0; and solving the corresponding system. Assume hereon e > 0; the scenario for eo0 completely being equivalent. After some algebraic calculations, one can express the equilibrium values I;% t% ; sl as function of s: % s% 2 1 I% ¼ ð5Þ a2 8 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi391=2 2 2 <1 s2 = s% 4 % þ t% ¼ þ4e2 A5 ð6Þ :2 a3 ; a3
s2e s20 s20
a2 ¼
3. Equilibrium statistical parameters
sl ¼
where t# ¼ t þ e and a1 ¼
bouncing period of the laser pulse inside the optical cavity, gi stands for the small signal gain, O is the synchrotron frequency, ts the syncrotron damping time and a the momentum compaction, and se represents the equilibrium value (i.e. that reached at the laser saturation) of the beam energy spread at the perfect tuning (i.e. e ¼ 0).
The FEL dynamics is naturally pulsed on the temporal scale of the inter-bunch period, while it appears as ‘‘continuouswave’’ on a larger, millisecond, temporal scale.
s% 2 Is ð1s% 2 Þ=a2 a4 a2 2 s% 1 2gi a3 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi391=4 2 2 = s% 2 s% 4 þ þ4e2 A5 ; a3 a3
ð7Þ
where ð1s% 2 Þ=a2
s% 3 ðs% 2 1Þ a4 A¼ a2 I s gi a 3
:
ð8Þ
The equilibrium value of the energy spread s% can be found numerically by solving the second of Eq. (2), after imposing dI=dt ¼ 0; and making explicit use of Eqs. (5–7)( The results are represented with a solid line in Fig. 1 and display an excellent agreement with simulations based on Eq. (2) (symbols in Fig. 1). Alternatively, it is
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43 Laser intensity
1
0.7
and
Electron's energy spread
1.5
se =s0 3 s2e 1 b¼ gi Is a2 a3 a4 2 s20 s2 log a4 s2e s2e þ e2 1 a2 s20 s20 s0
1.3 0
ε (fs)
1
Laser centroid
0.7
4.5
0 0
ε (fs)
0
1
x 10 -2
4.3 0
ε (fs)
1
Laser rms
ε (fs)
1
Fig. 1. The fixed points are plotted as function of the detuning parameter e: Top left panel: normalized laser intensity. Top right panel: normalized electron-beam energy spread. Bottom left panel: Laser centroid. Bottom right panel: rms value of the laser distribution. The symbols refer to the simulations performed using Eqs. (2), the solid line stands for the analytic approach based on the numerical solution of the second of Eqs. (2) (after imposing dI=dt ¼ 0), while the long-dashed lines represent the result obtained using the closed analytical expressions (9)–(13).
possible to derive a closed analytical expression for s% by performing a perturbative calculation. A quite cumbersome calculation leads to 2 1=2 se s% ¼ þ d ð9Þ s20 where se P 1 s20 2 1 þ G a 3 1 a4 gi s0 R2 2 s2e d¼ 2 2 2 log a4 s0 s s2 s2 2 a3 02 G1 a3 02 G2 02 G1 2 a2 se se se se
ð10Þ with G1 ¼
1 1 s2e pffiffiffi þ c 2 a3 s20
! pffiffiffi c 1 s2e 1 2 þ 2 2 þ 2e b G2 ¼ pffiffiffi a3 a3 s 0 2 c
41
ð11Þ
1 s2 s2e s20 c ¼ 2 e2 þ 4 gi I s a2 a3 a4 a3 s0
2 s2e s 1 e2 e2 : s20 s0
ð14Þ
The above solution is represented in Fig. 1 with a long-dashed line, displaying satisfactory agreement with simulations. Calculations have been performed using the case of the Super-ACO FEL as reference. The values of the relevant parameters are: DT ¼ 120 ns; ts ¼ 8:5 ms; s0 ¼ 5 104 ; se =s0 ¼ 1:5; O ¼ 14 kHz; gi ¼ 2%; P ¼ 0:8%; Is ¼ 1:4 108 : To our knowledge, this study represents the first attempt to characterize the analytic dependence of the equilibrium statistical parameters of the SRFEL versus the temporal detuning e; over the whole region of ‘‘cw’’ behaviour. As a straightforward application of the above derivation, let us focus on the expression for s% l at perfect tuning, i.e. e ¼ 0: We obtain2 1=4 g Is 1 i s% l C 1 þ log ð15Þ st;0 2 P P where we made use of the relation se =s0 C1 þ 0:5 log ðgi =PÞ derived in Ref. [6]. Relation (15) was applied to the case of Super ACO and was shown to reproduce quantitatively the experimental value [5]. Further, a comparison between the estimate of sl derived in the context of super-modes theory [7] and the prediction based on Eq. (15) was drawn, the latter resulting systematically in values closer to the ones measured experimentally [5]. The stability of the fixed point % sðeÞ; ½IðeÞ; t% ðeÞ; sl ðeÞ can be determined by studying % the eigenvalues of the Jacobian matrix associated to system (2). The real part of the eigenvalues are reported in Fig. 2. The system is by definition 2
ð12Þ
ð13Þ
It should be remarked that when e ¼ 0; d ¼ 1=R2 1: Thus, formally, sðe % ¼ 0Þase =s0 ; which is in apparent contradiction with the assumption of the model. However, since R2 ¼ 1 PB1; dðe ¼ 0ÞB0; the residual small discrepancy being related to the approximations involved in the calculation
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
42
3
2
Re
3
λ2
Re λ
-2
Re -4
λ4
Re
λ3
-6 -8
0
0.4
0.8
1.2
Feedback off
Laser Intensity
0
λ1 =
Laser Intensity
Re
0 0.0
200.0 Time (ms)
Feedback on
1.6
ε (fs) Fig. 2. Behaviour of the real parts of the eigenvalues of the Jacobian matrix as a function of the detuning amount. The calculation has been done for the case of the Super-ACO FEL. The transition occurs around 1:3 fs; a value which is in good agreement with experiments.
stable when all the real parts of the eigenvalues are negative. The transition to an unstable regime occurs when at least one of those becomes positive. In general, the loss of stability takes place according to different modalities. Consider, for instance, a Jacobian matrix with a pair of complex conjugate eigenvalues and assume the real parts of all the eigenvalues to be negative. A Hopf bifurcation occurs when the real part of the two complex eigenvalues become positive, provided the others keep their sign unchanged [8]. This situation is clearly displayed in Fig. 2, thus allowing to conclude that the transition between the ‘‘cw’’ and the pulsed regime in a SRFEL is a Hopf bifurcation.
4. Stabilization of the pulsed regime Having characterized the transition from the stable to the unstable steady state in terms of Hopf bifurcation opens up interesting perspectives for the improvement of the system performance. In fact, as it has been shown by several authors [9,10], the chaotic behaviour induced in conventional lasers can be stabilized by using a self-controlled (closed-loop) procedure. In particular, in Ref. [11] the dynamics of a conventional laser was stabilized
0 0.0
200.0
Time (ms) Fig. 3. Behaviour of the FEL intensity in absence (see inset) and in presence of the closed-loop feedback. The parameters utilized for the simulations are those of Super ACO.
around the unstable steady state, arising from a Hopf bifurcation. By virtue of the results of the previous sections, this approach can be extended to the case of a SRFEL. For this purpose the constant detuning e is replaced with the timedependent quantity e ¼ e0 þ bDT
dI dt
ð16Þ
which is added to system (2). Here e0 is assumed to be larger than ec : As it is shown in Fig. 3, when the control is switched off, i.e. b ¼ 0; the laser is unstable and displays periodic oscillations. For b larger than a certain threshold, bc ; the oscillations are damped and the laser behaves as if it was in the ‘‘cw’’ region. Preliminary experimental results have been already obtained at Super ACO [12].
5. Conclusions In this paper, we have shown that making explicit the evolution of the statistical parameters of a SRFEL allows to gain a deep insight into the longitudinal dynamics of the system. Exploiting this analytical characterization, we have introduced a suitable feedback procedure to enlarge the
ARTICLE IN PRESS G. De Ninno, D. Fanelli / Nuclear Instruments and Methods in Physics Research A 528 (2004) 39–43
region of stable signal. This result opens up the perspective of improving the performance of last generation SRFELs.
Acknowledgements The work of GDN has been partially supported by EUFELE, a Project funded by the European Commission under FP5 Contract No. HPRI-CT2001-50025. DF thanks U. Skoglund for the useful discussions.
References [1] M. Billardon, et al., Phys. Rev. Lett. 69 (1992) 2368.
43
[2] G. De Ninno, D. Fanelli, M.E. Couprie, Nucl. Instr. and Meth. A 483 (2002) 177. [3] G. De Ninno, et al., Europhys. J. D 22 (2003) 269. [4] N.A. Vinokurov, et al., preprint INP77.59 Novosibirsk, unpublished, 1977. [5] G. De Ninno, D. Fanelli, ELETTRA Internal Note ST/ SL-03/03, 2003. [6] G. De Ninno, D. Fanelli, Phys. Rev. Lett. 92 (2004) 094801. [7] G. Dattoli, private communication. [8] N. Berglund, Nonlinearity 13 (2000) 225. [9] V. Petrov, et al., J. Chem. Phys. 96 (1992) 7503. [10] S. Bielawski, et al., Phys. Rev. E 49 (1994) 971. [11] S. Bielawski, et al., Phys. Rev. A 47 (1993) 3276. [12] M.E. Couprie, Nucl. Instr. and Meth. A, (2004) these proceedings.