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Saturation and pulsed FEL dynamics
Nuclear Instruments
2s__
and Methods
in Physics
Research
A 375 (1996)
198-201
33 I!i!d
NUCLEAR INSTRUMENTS & METHODS IN PHVSICS
ELSEVIER
Saturation and pulsed FEL dynamics L. Giannessi”,
L. Mezi
ENEA, Dip. Innovazione, Setiore Fisica Applicata, Centro Ricerche Frascati. C.P. 65, 00044 Frascati, Rome. Italy
Abstract The behavior of a FEL operating in the pulsed saturated regime, is reproduced by the FEL integral equation, suitably modified to include non-linear effects. The method allows to evaluate the FEL dynamics by means of a relatively negligible computer time. The relevant predictions are compared with available experimental and numerical data.
1. Introduction In most cases, FELs operate with short electron bunches provided by rf accelerators. The pulsed nature of the electron source affects the FEL spectral behavior and the interplay between pulse propagation effects and gain saturation determines the FEL linewidth and power spectrum. The theoretical treatment of the pulsed FEL dynamics has been achieved with different methods, either analytical and numerical [1.2]. The pulsed FEL dynamics is governed by the following set of differential equations [3]:
2. Saturation
%{u(z - AT, T) e’Er’.r’},
~
= -rrg,(z
dr
+ Ar)(e-‘*),Z+.T,
,
specifying the electrons and field coupled evolution respectively, A is the slippage length, r is a dimensionless time ranging from zero to one, g, is the gain coefficient depending on the longitudinal coordinate and proportional to the e-beam current. The average in the second of Eq. (I) is taken over the electron phases and energies in a wavelength (see Table 1 for the meaning of the other symbols). In the small signal regime, Eqs. (I ) can be reduced to the following integral equation (see e.g. Ref. [41), dak, 7) = mg,(z + AT) dr T
X
I
dr’ r’e -‘“oT -i’nP/‘2
*Corresponding 94005334.
author.
Tel.
0168.9002/96/$15.00 Copyright SSDI 0168.9002(95)01203-6
a(Z_ ATI, 7 _ T)) ,
+39
which is obtained after expanding .$ up to the lower order in the field amplitude. The term exp{ - i(.sr~~r)‘} takes into account the effects of a Gaussian electron beam energy distribution. The above technique may be applied without the assumption of the small signal regime by extending to higher orders the expansion in the field amplitude [5], but this approach becomes computationally prohibitive for intensities exceeding a few Is, (where I, is the saturation intensity defined in Table I ). In the next section, a procedure to overcome this limitation is presented.
6 94005180,
fax
(2)
+39
6
and the FEL integral equation
The saturation process induces e-beam bunching along with an energy spread proportional to the field amplitude [6,7]. This energy spread contributes to the reduction of the gain and thus to saturation. A simple mechanism of gain saturation may be that of combining quadratically the natural e-beam energy spread with that induced by saturation, by replacing in Eq. (2) the coefficient
Saturation is not only characterized by an increase of the e-beam energy spread, but it is also accompanied by a variation of the average electron energy and by the growth of higher order momenta in the energy distribution. In strong saturation conditions, the electrons energy distribution is not reproduced by a Gaussian any more. It should be therefore understood that the term pf(III,) is not exactly an energy spread, but may be thought of as a coefficient providing a gain depression due to saturation. We have calculated this coefficient, by comparing the solution of the system of Eqs. (I), in the single mode, small g, limit, with the solution of the modified integral equation
0 1996 Elsevier Science B.V. All rights reserved
L. Gianessi,
L. Me,-i
/ Nucl. Imtr.
and Meth.
in Phys. Res. A 375 (1996)
199
198-201
Table 1 List of the symbols Resonance condition Bessel factor correction Slippage length rms bunch length rms rel. energy spread e-beam current e-beam cross section Alfven current Small signal gain coefficient Saturation intensity
3. Comparisons
da(z, r) = mg,,(z + AT) dr
where we have neglected any field envelope variation during a single round trip. According to this procedure, a parametrization for the function ~2 has been obtained,
where the parameters (Y and /? depend on the initial beam energy spread. This functional form has been tested for intensities up to x = 20. In Fig. 1 the coefficients (Y and p as functions of the initial energy spread are shown. The maximum relative deviation of the gain per pass evaluated from integral (4) and from the system of equations (1) is around 5%. With Eq. (4), which includes the :-dependence, we have reduced the pulse propagation problem to the solution of an integro-differential equation depending on the optical field only.
The results provided by the integration of Eq. (4) have been compared with the experimental data from the LANL experiment [8] and from data available from the SPECTRE simulation code [9]. In Table 2 are shown the main parameters concerning the two configurations considered for the comparison. These are two substantially different configurations, in which the expected round trip net gain is below 100%. The LANL data are relevant to an FEL in which there is a significant reduction of the gain per pass due to energy spread dominated inhomogeneous broadening. At the same time the gain reduction due to pulse propagation effects is quite negligible. We have the opposite situation with the SPECTRE parameters, where we have zero e-beam energy spread, and where the gain is strongly affected by pulse propagation effects. In Fig. 2 we have shown a comparison between a cavity tuning curve published in Ref. [7] (LANL) and the numerical data obtained by integrating Eq. (4). The peak and average values are understood over the macropulse, i.e. peak power is the maximum average power attained on an rf macropulse. In Fig. 3 we have shown a comparison between typical spectra obtained in conditions of high efficiency operation (Figs. 3a and 3b), and with detuned (shortened) cavity
/L /
0.8
/
-
\
\
/
Table 2 Configurations
/ / 0.6
\
/
-
/ / / /
0.4
p 0
I 0.5
I 1
\
\
I 1.5
po3 \\ \ 2 CL&
Fig. 1. Fitting coefficients (Y (continuous line) and p (/I X IO’dashed line) as functions of the initial energy spread p,..
parameters
4 Mm1 g,1 & A [mm1 i-5 Expected gain [%] Active losses [%] Passive losses [%] I, [MWI
LANL
SPECTRE
9.76 1.1 0.88 0.36 0.08 31 5 0.5 13.4
214 3.6 0.0 3.42 7.35 85 10
IV LONG WAVELENGTH
F?3Ls
L. Gianessi. L. Mezi I Nucl. Instr. and Meth. in Phys. Res. A 375 (1996)
200
198-201
optimized conditions (Figs. 3a and 3b). A more quantitative comparison between the spectra is not possible because not all the operating parameters relevant to the measured spectra were given in Ref. [8]. The comparison with the output of the CODE SPECTRE is shown in Fig. 4. On the left column we have the conversion efficiency from electron beam power to laser power as a function of the round trip number. On the right column the relative rms linewidth is given vs. the round trip number. The continuous line represents the integration of Eq. (4), while the dashed line is the output of SPECTRE. The plots labeled (a)-(e) have been obtained at different cavity lengths. By indicating with 6L = 0 the cavity length at which the round trip time (at zero gain) -20 o
- 11.25 -2.5 Output
6.25
15
23.15
32.5
41.25
50
Power (peak - kW)
0 Output Power (average - kW) -
LANLData
Fig. 2. Output power (kW) vs. relative cavity length (pm). The dashed line is obtained from the LANL experiment squares are the result of the integration of Eq. (4).
data,
the
(Figs. 3c and 3d). The qualitative agreement is good, we have a narrow spectrum when the cavity is detuned (Figs. 3c and 3d), and a broad band spectrum in efficiency
b
a 0.02 0.01 -
Oil -ib$4 Wavelength
:
0.03
1 Relative Intensity (A.U.)
‘\-_ 6810 Shift (o/o)
’
d
9.7 9.8 9.9
10
10.1
h@JN
0.1
Ob’
100
200
i Ofi 0.4-
100
200
0;
300
i7 :
0
2-
0.2 ‘~ i o /_~ ~_ 0 100 200
200
300
100
200
300
200
300
7~
1;
‘I
300
Electron beam -> Laser efficiency SPECTRE
((c) and (d)).
100
4
0.8, 0.6.
Fig. 3. Typical spectra from the LANL experiment (left column) and from Eq. (4) (right column) for high efficiency operation ((a) and (b)) and low efficiency operation (detuned = shortened cavity
00;
’
300
I
O() (7~) -
100
Linewidth (RMS. %) From eq. (4)
Fig. 4. Comparison with SPECTRE. Continuous line, integration of Eq. (4). dashed line SPECTRE output. On the left column, efficiency vs. round trip number, on the right column, relative linewidth. The plots (a)-(e) have been obtained at the relative cavity lengths given in Table 3.
L. Giunessi,
Table 3 Cavity lengths
relative
L. Mezi
I Nucl. Instr.
to the plots in Figs. 4a-4e
(larger
and Meth.
6L
in Phvs. Res. A 375 (1996)
201
198-201
considerations, we consider the agreement fairly remarkable.
shown in Fig. 4
means shorter cavity) Plot label
6L [pm1
(a) (b) (Cl (d) (e)
100 200 300 400 500
4. Conclusions
with the accelerating radio frequency period (i.e. distance between e-beam bunches), the plots have been obtained at the cavity lengths shown in Table 3. The parameters considered in the comparison with the SPECTRE output, represent an extreme case in which the e-beam bunch length is of the same order of magnitude of the resonant wavelength. It is a condition where we can observe differences between a model substantially based on the FEL pendulum equation, from which Eq. (4) originates, which is derived in the slow wave approximation, and SPECTRE, which is based on a Hamiltonian model where this approximation is not necessary. These differences are more evident when the cavity length approaches the nominal value and the optical pulse shape attempts to reproduce the e-bunch shape (Fig. 4a. 6L = IO0 coincides
pm). Furthermore we have to stress that the presented method holds in the small gain limit. There are two distinct small g(, assumptions: a) We have
neglected
the explicit
time
dependence
in
the rhs of the integral equation (4), i.e. we have neglected any field envelope modification on a single round trip. b) The fitting procedure used to evaluate the function pt. has been performed in the small R,, limit to eliminate any dependence of cy and /I on gr, The first assumption allows a fast numerical implementation of the integration algorithm, while the second is only a matter of spending some effort to tabulate the LY and /3 coefficients at different gr,, In Figs. 4b-4c we have almost reproduced the linewidth. while we have a slight difference in the output efficiency. This suggest that the approximation a), which directly involves the pulse shape is still holding, while the fitting procedure, approximation b), begins to show its limits. On the light of these
We have proposed a technique to reproduce the FEL pulse propagation dynamics in the strong signal regime. The extension of the method to an intermediate range of R~, values by extending the tables of LYand p coefficients is possible, and would allow to reproduce the dynamical behaviour of most FELs operating in an oscillator configuration.
Acknowledgements The authors owe their gratitude to Daniel Iracane, Pascal Chaix, Philippe Guimbal, Daniel Touati and Nicola Piovella for providing the output of the code SPECTRE, and for their assistance in running it. It is also a pleasure to acknowledge the contribution of Giuseppe (Pino) Dattoli for helpful and enlightening discussions.
References W.B. Colson and S.K. Ride, in: Physics of Quantum Electronics (Addison-Wesley. 1982) p, 377. G. Dattoli and A. Renieri, Nuovo Cimento B 61 ( I981 ) 1.53. W.B. Colson, m: Laser Handbook, 6, eds. W.B. Colson. C. Pellegrini and A. Renieri (Elsevier, 1989) p, 301, G. Dattoli. A. Renieri and A. Torre. in: Lectures on the Free Electron Laser Theory and Related Topics (World Scientific. 1993) p. 475. [51 J.C. Gallardo. L. Elias, G. Dattoli and A. Renieri. Phys. Rev. A 36 (1987) 3222. [61 G. Dattoli. L. Giannessi, P.L. Ottaviani and A. Terre, J. Appl. Phys. 76 ( 1994) 55. I71 G. Dattoli, L. Giannessi, P.L. Ottaviani and A. Segreto, Phys. Plasma, to be published. I81 BE. Newnam. R.W. Warren. R.L. Sheffield, J.C. Goldstein and C.A. Brau. Nucl. Instr. and Meth. A 237 (1985) 187. I91 D. Iracane, P. Chaix and J.L. Ferrer. Phys. Rev. E 49 (1993) 800.
IV. LONG WAVELENGTH
FELs
2s__
and Methods
in Physics
Research
A 375 (1996)
198-201
33 I!i!d
NUCLEAR INSTRUMENTS & METHODS IN PHVSICS
ELSEVIER
Saturation and pulsed FEL dynamics L. Giannessi”,
L. Mezi
ENEA, Dip. Innovazione, Setiore Fisica Applicata, Centro Ricerche Frascati. C.P. 65, 00044 Frascati, Rome. Italy
Abstract The behavior of a FEL operating in the pulsed saturated regime, is reproduced by the FEL integral equation, suitably modified to include non-linear effects. The method allows to evaluate the FEL dynamics by means of a relatively negligible computer time. The relevant predictions are compared with available experimental and numerical data.
1. Introduction In most cases, FELs operate with short electron bunches provided by rf accelerators. The pulsed nature of the electron source affects the FEL spectral behavior and the interplay between pulse propagation effects and gain saturation determines the FEL linewidth and power spectrum. The theoretical treatment of the pulsed FEL dynamics has been achieved with different methods, either analytical and numerical [1.2]. The pulsed FEL dynamics is governed by the following set of differential equations [3]:
2. Saturation
%{u(z - AT, T) e’Er’.r’},
~
= -rrg,(z
dr
+ Ar)(e-‘*),Z+.T,
,
specifying the electrons and field coupled evolution respectively, A is the slippage length, r is a dimensionless time ranging from zero to one, g, is the gain coefficient depending on the longitudinal coordinate and proportional to the e-beam current. The average in the second of Eq. (I) is taken over the electron phases and energies in a wavelength (see Table 1 for the meaning of the other symbols). In the small signal regime, Eqs. (I ) can be reduced to the following integral equation (see e.g. Ref. [41), dak, 7) = mg,(z + AT) dr T
X
I
dr’ r’e -‘“oT -i’nP/‘2
*Corresponding 94005334.
author.
Tel.
0168.9002/96/$15.00 Copyright SSDI 0168.9002(95)01203-6
a(Z_ ATI, 7 _ T)) ,
+39
which is obtained after expanding .$ up to the lower order in the field amplitude. The term exp{ - i(.sr~~r)‘} takes into account the effects of a Gaussian electron beam energy distribution. The above technique may be applied without the assumption of the small signal regime by extending to higher orders the expansion in the field amplitude [5], but this approach becomes computationally prohibitive for intensities exceeding a few Is, (where I, is the saturation intensity defined in Table I ). In the next section, a procedure to overcome this limitation is presented.
6 94005180,
fax
(2)
+39
6
and the FEL integral equation
The saturation process induces e-beam bunching along with an energy spread proportional to the field amplitude [6,7]. This energy spread contributes to the reduction of the gain and thus to saturation. A simple mechanism of gain saturation may be that of combining quadratically the natural e-beam energy spread with that induced by saturation, by replacing in Eq. (2) the coefficient
Saturation is not only characterized by an increase of the e-beam energy spread, but it is also accompanied by a variation of the average electron energy and by the growth of higher order momenta in the energy distribution. In strong saturation conditions, the electrons energy distribution is not reproduced by a Gaussian any more. It should be therefore understood that the term pf(III,) is not exactly an energy spread, but may be thought of as a coefficient providing a gain depression due to saturation. We have calculated this coefficient, by comparing the solution of the system of Eqs. (I), in the single mode, small g, limit, with the solution of the modified integral equation
0 1996 Elsevier Science B.V. All rights reserved
L. Gianessi,
L. Me,-i
/ Nucl. Imtr.
and Meth.
in Phys. Res. A 375 (1996)
199
198-201
Table 1 List of the symbols Resonance condition Bessel factor correction Slippage length rms bunch length rms rel. energy spread e-beam current e-beam cross section Alfven current Small signal gain coefficient Saturation intensity
3. Comparisons
da(z, r) = mg,,(z + AT) dr
where we have neglected any field envelope variation during a single round trip. According to this procedure, a parametrization for the function ~2 has been obtained,
where the parameters (Y and /? depend on the initial beam energy spread. This functional form has been tested for intensities up to x = 20. In Fig. 1 the coefficients (Y and p as functions of the initial energy spread are shown. The maximum relative deviation of the gain per pass evaluated from integral (4) and from the system of equations (1) is around 5%. With Eq. (4), which includes the :-dependence, we have reduced the pulse propagation problem to the solution of an integro-differential equation depending on the optical field only.
The results provided by the integration of Eq. (4) have been compared with the experimental data from the LANL experiment [8] and from data available from the SPECTRE simulation code [9]. In Table 2 are shown the main parameters concerning the two configurations considered for the comparison. These are two substantially different configurations, in which the expected round trip net gain is below 100%. The LANL data are relevant to an FEL in which there is a significant reduction of the gain per pass due to energy spread dominated inhomogeneous broadening. At the same time the gain reduction due to pulse propagation effects is quite negligible. We have the opposite situation with the SPECTRE parameters, where we have zero e-beam energy spread, and where the gain is strongly affected by pulse propagation effects. In Fig. 2 we have shown a comparison between a cavity tuning curve published in Ref. [7] (LANL) and the numerical data obtained by integrating Eq. (4). The peak and average values are understood over the macropulse, i.e. peak power is the maximum average power attained on an rf macropulse. In Fig. 3 we have shown a comparison between typical spectra obtained in conditions of high efficiency operation (Figs. 3a and 3b), and with detuned (shortened) cavity
/L /
0.8
/
-
\
\
/
Table 2 Configurations
/ / 0.6
\
/
-
/ / / /
0.4
p 0
I 0.5
I 1
\
\
I 1.5
po3 \\ \ 2 CL&
Fig. 1. Fitting coefficients (Y (continuous line) and p (/I X IO’dashed line) as functions of the initial energy spread p,..
parameters
4 Mm1 g,1 & A [mm1 i-5 Expected gain [%] Active losses [%] Passive losses [%] I, [MWI
LANL
SPECTRE
9.76 1.1 0.88 0.36 0.08 31 5 0.5 13.4
214 3.6 0.0 3.42 7.35 85 10
IV LONG WAVELENGTH
F?3Ls
L. Gianessi. L. Mezi I Nucl. Instr. and Meth. in Phys. Res. A 375 (1996)
200
198-201
optimized conditions (Figs. 3a and 3b). A more quantitative comparison between the spectra is not possible because not all the operating parameters relevant to the measured spectra were given in Ref. [8]. The comparison with the output of the CODE SPECTRE is shown in Fig. 4. On the left column we have the conversion efficiency from electron beam power to laser power as a function of the round trip number. On the right column the relative rms linewidth is given vs. the round trip number. The continuous line represents the integration of Eq. (4), while the dashed line is the output of SPECTRE. The plots labeled (a)-(e) have been obtained at different cavity lengths. By indicating with 6L = 0 the cavity length at which the round trip time (at zero gain) -20 o
- 11.25 -2.5 Output
6.25
15
23.15
32.5
41.25
50
Power (peak - kW)
0 Output Power (average - kW) -
LANLData
Fig. 2. Output power (kW) vs. relative cavity length (pm). The dashed line is obtained from the LANL experiment squares are the result of the integration of Eq. (4).
data,
the
(Figs. 3c and 3d). The qualitative agreement is good, we have a narrow spectrum when the cavity is detuned (Figs. 3c and 3d), and a broad band spectrum in efficiency
b
a 0.02 0.01 -
Oil -ib$4 Wavelength
:
0.03
1 Relative Intensity (A.U.)
‘\-_ 6810 Shift (o/o)
’
d
9.7 9.8 9.9
10
10.1
h@JN
0.1
Ob’
100
200
i Ofi 0.4-
100
200
0;
300
i7 :
0
2-
0.2 ‘~ i o /_~ ~_ 0 100 200
200
300
100
200
300
200
300
7~
1;
‘I
300
Electron beam -> Laser efficiency SPECTRE
((c) and (d)).
100
4
0.8, 0.6.
Fig. 3. Typical spectra from the LANL experiment (left column) and from Eq. (4) (right column) for high efficiency operation ((a) and (b)) and low efficiency operation (detuned = shortened cavity
00;
’
300
I
O() (7~) -
100
Linewidth (RMS. %) From eq. (4)
Fig. 4. Comparison with SPECTRE. Continuous line, integration of Eq. (4). dashed line SPECTRE output. On the left column, efficiency vs. round trip number, on the right column, relative linewidth. The plots (a)-(e) have been obtained at the relative cavity lengths given in Table 3.
L. Giunessi,
Table 3 Cavity lengths
relative
L. Mezi
I Nucl. Instr.
to the plots in Figs. 4a-4e
(larger
and Meth.
6L
in Phvs. Res. A 375 (1996)
201
198-201
considerations, we consider the agreement fairly remarkable.
shown in Fig. 4
means shorter cavity) Plot label
6L [pm1
(a) (b) (Cl (d) (e)
100 200 300 400 500
4. Conclusions
with the accelerating radio frequency period (i.e. distance between e-beam bunches), the plots have been obtained at the cavity lengths shown in Table 3. The parameters considered in the comparison with the SPECTRE output, represent an extreme case in which the e-beam bunch length is of the same order of magnitude of the resonant wavelength. It is a condition where we can observe differences between a model substantially based on the FEL pendulum equation, from which Eq. (4) originates, which is derived in the slow wave approximation, and SPECTRE, which is based on a Hamiltonian model where this approximation is not necessary. These differences are more evident when the cavity length approaches the nominal value and the optical pulse shape attempts to reproduce the e-bunch shape (Fig. 4a. 6L = IO0 coincides
pm). Furthermore we have to stress that the presented method holds in the small gain limit. There are two distinct small g(, assumptions: a) We have
neglected
the explicit
time
dependence
in
the rhs of the integral equation (4), i.e. we have neglected any field envelope modification on a single round trip. b) The fitting procedure used to evaluate the function pt. has been performed in the small R,, limit to eliminate any dependence of cy and /I on gr, The first assumption allows a fast numerical implementation of the integration algorithm, while the second is only a matter of spending some effort to tabulate the LY and /3 coefficients at different gr,, In Figs. 4b-4c we have almost reproduced the linewidth. while we have a slight difference in the output efficiency. This suggest that the approximation a), which directly involves the pulse shape is still holding, while the fitting procedure, approximation b), begins to show its limits. On the light of these
We have proposed a technique to reproduce the FEL pulse propagation dynamics in the strong signal regime. The extension of the method to an intermediate range of R~, values by extending the tables of LYand p coefficients is possible, and would allow to reproduce the dynamical behaviour of most FELs operating in an oscillator configuration.
Acknowledgements The authors owe their gratitude to Daniel Iracane, Pascal Chaix, Philippe Guimbal, Daniel Touati and Nicola Piovella for providing the output of the code SPECTRE, and for their assistance in running it. It is also a pleasure to acknowledge the contribution of Giuseppe (Pino) Dattoli for helpful and enlightening discussions.
References W.B. Colson and S.K. Ride, in: Physics of Quantum Electronics (Addison-Wesley. 1982) p, 377. G. Dattoli and A. Renieri, Nuovo Cimento B 61 ( I981 ) 1.53. W.B. Colson, m: Laser Handbook, 6, eds. W.B. Colson. C. Pellegrini and A. Renieri (Elsevier, 1989) p, 301, G. Dattoli. A. Renieri and A. Torre. in: Lectures on the Free Electron Laser Theory and Related Topics (World Scientific. 1993) p. 475. [51 J.C. Gallardo. L. Elias, G. Dattoli and A. Renieri. Phys. Rev. A 36 (1987) 3222. [61 G. Dattoli. L. Giannessi, P.L. Ottaviani and A. Terre, J. Appl. Phys. 76 ( 1994) 55. I71 G. Dattoli, L. Giannessi, P.L. Ottaviani and A. Segreto, Phys. Plasma, to be published. I81 BE. Newnam. R.W. Warren. R.L. Sheffield, J.C. Goldstein and C.A. Brau. Nucl. Instr. and Meth. A 237 (1985) 187. I91 D. Iracane, P. Chaix and J.L. Ferrer. Phys. Rev. E 49 (1993) 800.
IV. LONG WAVELENGTH
FELs