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Storage ring free electron laser: operation with two undulators having opposite circular polarizations
Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
Storage ring free electron laser: operation with two undulators having opposite circular polarizations G. Dattoli*, P.L. Ottaviani1 ENEA, Divisione Fisica Applicata, Centro Ricerche Frascati, C.P. 65, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy Received 6 March 2001
Abstract In Storage Ring Free Electron Lasers two undulators, having opposite helical polarizations and arranged as an Optical-Klystron, may be exploited to obtain linearly polarized radiation, without additional problems associated with mirror degradation, due to the higher on axis harmonic emission. In this paper we explore the dynamical behavior of this device and discuss possible configurations allowing such a possibility. r 2002 Elsevier Science B.V. All rights reserved.
1. Introduction We discuss the theory of Storage Ring Free Electron Lasers (SR-FEL) operating with two undulators having opposite circular polarizations. The paper completes a research line (see Ref. [1]) in which we have explored the FEL dynamics in undulators with variable polarizations, including the case of devices employing Optical-Klystron (OK) configurations with undulators having opposite polarizations. The interest for SR-FEL operating with an OK scheme involving a pair of undulators with opposite helical polarizations is motivated by the fact that it is, in principle, possible to obtain
*Corresponding author. Tel.: +39-06-94005421; fax: +3906-94005334. E-mail address: [email protected] (G. Dattoli). 1 Also at ENEA, Divisione Fisica Applicata, Centro Ricerche Bologna, Via Don Fiammelli 2, Bologna, Italy.
linearly polarized output radiation, without the problems associated with mirror degradation due to the higher on-axis harmonic emission. The dynamical behavior of a FEL with such a configuration has been discussed in Ref. [1], in the case of single pass devices and it has been argued that 1. the dynamics of the system is more complicated than the ordinary case, the two fields do not evolve independently and may interfere destructively; 2. in the case of undulators having the same number of periods, the bunching mechanism privileges the intensity evolution in the second undulator, which grows faster and when close to the saturation suppresses the signal of the first; 3. the intensities of the two fields may be identical at the equilibrium only if one introduces a mechanism which increases the gain of the first
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 9 3 4 - 2
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
669
undulator with respect to the second, as e.g. in the case in which the second undulator has a number of periods smaller than the first. The condition that the two fields have equal intensities is necessary to ensure that they combine to provide an equilibrium linearly polarized field. In this paper we extend the considerations of Ref. [1] to the case of a device operating with a SR. We will discuss the dynamics of the system by exploiting both a numerical and analytical procedure, capable of including all the aspects characterizing the evolution of a SR-FEL device. We have developed an ad hoc numerical code which follows the turn-by-turn evolution of the field and accounts for the intrinsic SR effects (damping and diffusion) combined with the FEL dynamics, which induces a turn-by-turn energy spread, responsible for the system saturation. We have used as benchmark of the code a semi-analytical model based on the rate equation [2]
Fig. 1. Storage ring FEL evolution vs. round trip number. Comparison between analytical (dotted) and numerical results. Parameters (g0 ¼ 1 10@1 , se ¼ 10@4 , 2ðT=ts Þ¼ 3 10@4 , Z¼ 6 10@3 ).
xnþ1 ¼ ð1@ZÞ½Gðxn ; sn Þ þ 1xn " 2T 0:433 2 @ðb=2Þxn 2 si; nþ1 ¼ 1@ e s2i; n þ ts N bxn @1 1@ebxn s2n ¼ s2e þ s2i; n ;
p b ¼ 1:0145 : 2
ð1Þ
In the previous set of equations accounting for the evolution of the induced energy spread ðsi Þ and of the dimensionless intracavity power x: (linked to the intracavity power density and to the saturation power density by x ¼ I=Is ), Z denotes the cavity losses, T the machine revolution period, assumed identical to the cavity round trip period (n denotes the round trip number), ts is the damping time, N the number of undulator periods and se the natural energy spread of the ring. The gain function Gðxn ; sn Þ has been modelled to include the effects of gain reduction due to beaminduced heating and to bunch lengthening. The comparison between semi-analytical and fully numerical computation is provided in Fig. 1 and it is evident that the agreement is fairly good.
Fig. 2. Scheme of the undulator arrangement for FEL operating with undulator magnets having opposite polarizations, the first undulator is assumed left helically polarized, the second right. We will consider three configurations: (a) N1 ¼ N2 , (b) N1 > N2 , (c) N1 ¼ N2 with a dispersive section.
In the following, we will make a combined use of semi-analytical and numerical codes to describe the evolution of a FEL device in which the two undulators are arranged as in Fig. 2, with the first undulator being left helically polarized and the second right.
670
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
We assume that the intrinsic gain of each undulator is large enough that the device can reach saturation even with the other section off and since we want to reduce all the bunching effects which, as already remarked, creates the condition for a faster growth of the field in the second part of the device, we will follow different strategies (a)
(b)
we will compensate the bunching effect in the second section by enhancing the gain of the first part, by choosing e.g. the first undulator with a number of periods larger than the second; we disentangle the two undulators by inserting in between a dispersive section totally mismatched with respect to the beam energy spread so that the system is insensitive to the bunching in the second part of the klystron.
In this last hypothesis the role of the dispersive section is that of eliminating rather than enhancing the bunching contributions. The paper consists of three sections: in Section 2, we will present a few preliminary considerations based on a semi-analytical approach and in Section 3, we will discuss the results of the numerical analysis and some concluding remarks.
2. Preliminary considerations To better clarify the point raised in the previous section, according to which the field grows faster in the second part of the device, we remind that the FEL field amplitude during the transit in the undulator writes 2ð1@e@iv0 t Þ@iv0 tð1 þ e@iv0 t Þ aðtÞ ¼ a0 1 þ pg0 v30 ð2Þ where a denotes the Colson amplitude, g0 the small signal gain coefficient and v0 the frequency detuning parameter. In Fig. 3, we report the field intensity ðjaðtÞj2 ¼ 0:8p4 xÞ vs. the dimensionless
Fig. 3. Dimensionless emitted power vs. undulator length same parameters of Fig. 1(a) round trip n ¼ 0, (b) round trip n ¼ 104 , (c) round trip n¼ 2 104 .
time t (linked to the longitudinal coordinate), it is evident that the most substantive part of the emission occurs in the last few periods of the undulator. To be more quantitative, we note that the power emitted in the last 16% of the undulator is equivalent to that emitted in the first 84%. Eq. (2) does not include saturation or beam energy spread contributions, but it is also evident that, even with the inclusion of multi turn effect and thus of the beam degradation due to the induced energy spread and bunch lengthening, the situation does not change (see Fig. 3). The physical reason of such a behavior is just a consequence of the bunching, which becomes more efficient at the end of the undulator. For the same motivation, we expect that a device, exploiting two undulators with opposite helical polarizations, will produce a linearly polarized field only if the second undulator is shorter than the first (see also Ref. [1]). We have, indeed, observed that in the second undulator the field grows faster because its evolution is due to the intrinsic gain mechanism and to the bunching induced in the previous section. The contribution to the intensity coming from the two contributions can be written as sinðv0 t=2Þ 2 jab ðtÞj2 ¼ jaðtÞj2 þ 4p2 g20 jbj2 ð3Þ v0 t=2 where b is the bunching coefficient induced in the first undulator and is proportional to the field
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
intensity of the first field. We have used the above results to modify our rate equations and model a SR-FEL operating with two different fields; thus, getting the results reported in Fig. 4, according to which, to obtain comparable equilibrium fields it is necessary to operate with the intrinsic gain of the first undulator larger than that of the second.
671
On the other side, if the two sections are totally de coupled the two fields grow independently and if the gain of the two sections are identical they will reach the same intracavity power. To make the two sections independent, we can introduce in between a dispersive part whose length is very far from the OK gain optimizing value and is used to switch off the bunching effect. Just to give an idea of the criteria to follow to choose the ‘‘worst’’ dispersive section, we remind that the value of the dispersive section length optimizing the gain of an OK FEL is [3] d* ¼
1 1 þ m2e ; p me
me ¼ 4Nse
ð4Þ
recall that d ¼ LD =LU ; where LD is the length of the dispersive section and LU that of one undulator, N is the number of periods of one undulator section and se is the e-beam energy spread. The gain contribution of the dispersive part can be written as GD ðd; d * Þ ¼ 8g0 d * e@ð1=2Þx
2
d ð5Þ d* where g0 is the small signal gain associated with one undulator section. By assuming e.g. an energy spread of the order of 10@3 and N ¼ 20; we can totally switch off the dispesive part by choosing x ¼ 10. In the forthcoming section, we will exploit the so far obtained results as the starting point of a more complete numerical investigation. x ¼
3. Numerical analysis and concluding remarks
Fig. 4. Evolution of the intracavity field for the device with undulators having opposite polarizations (analytical computation) (a) the undulators are assumed with equal number of periods ðN1 ¼ N2 ¼ 20Þ the continuous line refers to the first field, the dotted to the second, the intrinsic gain coefficient of each undulator is g0 ¼ 5:88 10@2 , the remaining parameters are the same as in Fig. 1. (b) Same as (a) with N1 ¼ 30, N2 ¼ 10, the intrinsic gain coefficient of the two undulator is g1 ¼ ð3=2Þ3 g0 ; g2 ¼ ð1=2Þ3 g0 .
In the following, we will assume that the undulators, labelled by 1 and 2 in Fig. 2, have left and right helical polarizations, respectively, namely B1 B0 ð@sinðku zÞ; cosðku zÞ; 0Þ B2 B0 ðsinðku zÞ; cosðku zÞ; 0Þ:
ð6Þ
The dynamical behavior of the system is followed by integrating the set of pendulum
672
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
Fig. 5. Same as Fig. 4(a) (same parameters but se ¼ 1:2 10@3 and 2ðT=ts Þ ¼ 6 10@3 ) (numerical computation).
Fig. 7. Same as Fig. 4(a) (N1 ¼ N2 ¼ 20) but with dispersive section d ¼ 20, the two fields reach the same equilibrium value and are totally indistinguishable.
Fig. 6. Same as Fig. 4(b) (N1 ¼ 28, N2 ¼ 12, se ¼ 1:2 10@3 , 2ðT=ts Þ ¼ 6 10@3 , Z¼ 7 10@3 ).
equations derived in Ref. [1], accounting for the left–right polarization interplay, coupled to the equations describing the e-beam longitudinal dynamics in SR. The results of the numerical integration are given in Figs. 5–7 and confirm the conclusions of
the previous section, i.e. we obtain comparable intensities when the gain of the first undulator is larger than the second. We have used in the simulation the parameters of the SR-FEL at Elettra [4] in which the total number of undulator periods (NT ¼ N1 þ N2 ) is 40, to increase the gain of the first section and compensate the effect of the bunching in the second section, we have chosen N1 > N2 . By using the above arrangement, we have obtained the results of Fig. 6, indicating that the equilibrium intracavity intensities of the fields with opposite polarizations reaches almost identical values thus allowing their combination to provide linearly polarized output radiation. As to the ‘‘mismatched’’ dispersive section, we note that, by assuming se D1:2 10@3 and the associated d D1:5, we have checked that values of d larger than 10 eliminate any coupling between the undulators. In this case (see Fig. 7), the fields grow totally independently and reach exactly the same equilibrium value. We must underline that (Figs. 5–7) although specific for the laser parameters of Elettra, uses a value of the ratio T=ts around 3 10@3 (while the effective value is of the order of 10@5) the reason
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
of this choice is due to the computer time, necessary to follow the system up to the saturation, prohibitively large using the actual value. We must also underline that the numerical analysis of the mismatched OK configuration requires a large number of macroparticles in the simulation and this is a further element of increase of the computer time.
673
References [1] G. Dattoli, L. Bucci, Opt. Commun. 177 (2000) 323. [2] P. Elleaume, J. Phys. 45 (1984) 997. G. Dattoli, L. Giannessi, P.L. Ottaviani, Nucl. Instr. and Meth. A 365 (1995) 559. [3] G. Dattoli, P.L. Ottaviani, IEEE JQe-35 (1999) 27. [4] R. Rou, et al., in: J. Feldhaus, H. Weise (Eds.), Free Electron Lasers 1999, Elsevier, Amsterdam, 2000, p. II 21.
Storage ring free electron laser: operation with two undulators having opposite circular polarizations G. Dattoli*, P.L. Ottaviani1 ENEA, Divisione Fisica Applicata, Centro Ricerche Frascati, C.P. 65, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy Received 6 March 2001
Abstract In Storage Ring Free Electron Lasers two undulators, having opposite helical polarizations and arranged as an Optical-Klystron, may be exploited to obtain linearly polarized radiation, without additional problems associated with mirror degradation, due to the higher on axis harmonic emission. In this paper we explore the dynamical behavior of this device and discuss possible configurations allowing such a possibility. r 2002 Elsevier Science B.V. All rights reserved.
1. Introduction We discuss the theory of Storage Ring Free Electron Lasers (SR-FEL) operating with two undulators having opposite circular polarizations. The paper completes a research line (see Ref. [1]) in which we have explored the FEL dynamics in undulators with variable polarizations, including the case of devices employing Optical-Klystron (OK) configurations with undulators having opposite polarizations. The interest for SR-FEL operating with an OK scheme involving a pair of undulators with opposite helical polarizations is motivated by the fact that it is, in principle, possible to obtain
*Corresponding author. Tel.: +39-06-94005421; fax: +3906-94005334. E-mail address: [email protected] (G. Dattoli). 1 Also at ENEA, Divisione Fisica Applicata, Centro Ricerche Bologna, Via Don Fiammelli 2, Bologna, Italy.
linearly polarized output radiation, without the problems associated with mirror degradation due to the higher on-axis harmonic emission. The dynamical behavior of a FEL with such a configuration has been discussed in Ref. [1], in the case of single pass devices and it has been argued that 1. the dynamics of the system is more complicated than the ordinary case, the two fields do not evolve independently and may interfere destructively; 2. in the case of undulators having the same number of periods, the bunching mechanism privileges the intensity evolution in the second undulator, which grows faster and when close to the saturation suppresses the signal of the first; 3. the intensities of the two fields may be identical at the equilibrium only if one introduces a mechanism which increases the gain of the first
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 9 3 4 - 2
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
669
undulator with respect to the second, as e.g. in the case in which the second undulator has a number of periods smaller than the first. The condition that the two fields have equal intensities is necessary to ensure that they combine to provide an equilibrium linearly polarized field. In this paper we extend the considerations of Ref. [1] to the case of a device operating with a SR. We will discuss the dynamics of the system by exploiting both a numerical and analytical procedure, capable of including all the aspects characterizing the evolution of a SR-FEL device. We have developed an ad hoc numerical code which follows the turn-by-turn evolution of the field and accounts for the intrinsic SR effects (damping and diffusion) combined with the FEL dynamics, which induces a turn-by-turn energy spread, responsible for the system saturation. We have used as benchmark of the code a semi-analytical model based on the rate equation [2]
Fig. 1. Storage ring FEL evolution vs. round trip number. Comparison between analytical (dotted) and numerical results. Parameters (g0 ¼ 1 10@1 , se ¼ 10@4 , 2ðT=ts Þ¼ 3 10@4 , Z¼ 6 10@3 ).
xnþ1 ¼ ð1@ZÞ½Gðxn ; sn Þ þ 1xn " 2T 0:433 2 @ðb=2Þxn 2 si; nþ1 ¼ 1@ e s2i; n þ ts N bxn @1 1@ebxn s2n ¼ s2e þ s2i; n ;
p b ¼ 1:0145 : 2
ð1Þ
In the previous set of equations accounting for the evolution of the induced energy spread ðsi Þ and of the dimensionless intracavity power x: (linked to the intracavity power density and to the saturation power density by x ¼ I=Is ), Z denotes the cavity losses, T the machine revolution period, assumed identical to the cavity round trip period (n denotes the round trip number), ts is the damping time, N the number of undulator periods and se the natural energy spread of the ring. The gain function Gðxn ; sn Þ has been modelled to include the effects of gain reduction due to beaminduced heating and to bunch lengthening. The comparison between semi-analytical and fully numerical computation is provided in Fig. 1 and it is evident that the agreement is fairly good.
Fig. 2. Scheme of the undulator arrangement for FEL operating with undulator magnets having opposite polarizations, the first undulator is assumed left helically polarized, the second right. We will consider three configurations: (a) N1 ¼ N2 , (b) N1 > N2 , (c) N1 ¼ N2 with a dispersive section.
In the following, we will make a combined use of semi-analytical and numerical codes to describe the evolution of a FEL device in which the two undulators are arranged as in Fig. 2, with the first undulator being left helically polarized and the second right.
670
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
We assume that the intrinsic gain of each undulator is large enough that the device can reach saturation even with the other section off and since we want to reduce all the bunching effects which, as already remarked, creates the condition for a faster growth of the field in the second part of the device, we will follow different strategies (a)
(b)
we will compensate the bunching effect in the second section by enhancing the gain of the first part, by choosing e.g. the first undulator with a number of periods larger than the second; we disentangle the two undulators by inserting in between a dispersive section totally mismatched with respect to the beam energy spread so that the system is insensitive to the bunching in the second part of the klystron.
In this last hypothesis the role of the dispersive section is that of eliminating rather than enhancing the bunching contributions. The paper consists of three sections: in Section 2, we will present a few preliminary considerations based on a semi-analytical approach and in Section 3, we will discuss the results of the numerical analysis and some concluding remarks.
2. Preliminary considerations To better clarify the point raised in the previous section, according to which the field grows faster in the second part of the device, we remind that the FEL field amplitude during the transit in the undulator writes 2ð1@e@iv0 t Þ@iv0 tð1 þ e@iv0 t Þ aðtÞ ¼ a0 1 þ pg0 v30 ð2Þ where a denotes the Colson amplitude, g0 the small signal gain coefficient and v0 the frequency detuning parameter. In Fig. 3, we report the field intensity ðjaðtÞj2 ¼ 0:8p4 xÞ vs. the dimensionless
Fig. 3. Dimensionless emitted power vs. undulator length same parameters of Fig. 1(a) round trip n ¼ 0, (b) round trip n ¼ 104 , (c) round trip n¼ 2 104 .
time t (linked to the longitudinal coordinate), it is evident that the most substantive part of the emission occurs in the last few periods of the undulator. To be more quantitative, we note that the power emitted in the last 16% of the undulator is equivalent to that emitted in the first 84%. Eq. (2) does not include saturation or beam energy spread contributions, but it is also evident that, even with the inclusion of multi turn effect and thus of the beam degradation due to the induced energy spread and bunch lengthening, the situation does not change (see Fig. 3). The physical reason of such a behavior is just a consequence of the bunching, which becomes more efficient at the end of the undulator. For the same motivation, we expect that a device, exploiting two undulators with opposite helical polarizations, will produce a linearly polarized field only if the second undulator is shorter than the first (see also Ref. [1]). We have, indeed, observed that in the second undulator the field grows faster because its evolution is due to the intrinsic gain mechanism and to the bunching induced in the previous section. The contribution to the intensity coming from the two contributions can be written as sinðv0 t=2Þ 2 jab ðtÞj2 ¼ jaðtÞj2 þ 4p2 g20 jbj2 ð3Þ v0 t=2 where b is the bunching coefficient induced in the first undulator and is proportional to the field
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
intensity of the first field. We have used the above results to modify our rate equations and model a SR-FEL operating with two different fields; thus, getting the results reported in Fig. 4, according to which, to obtain comparable equilibrium fields it is necessary to operate with the intrinsic gain of the first undulator larger than that of the second.
671
On the other side, if the two sections are totally de coupled the two fields grow independently and if the gain of the two sections are identical they will reach the same intracavity power. To make the two sections independent, we can introduce in between a dispersive part whose length is very far from the OK gain optimizing value and is used to switch off the bunching effect. Just to give an idea of the criteria to follow to choose the ‘‘worst’’ dispersive section, we remind that the value of the dispersive section length optimizing the gain of an OK FEL is [3] d* ¼
1 1 þ m2e ; p me
me ¼ 4Nse
ð4Þ
recall that d ¼ LD =LU ; where LD is the length of the dispersive section and LU that of one undulator, N is the number of periods of one undulator section and se is the e-beam energy spread. The gain contribution of the dispersive part can be written as GD ðd; d * Þ ¼ 8g0 d * e@ð1=2Þx
2
d ð5Þ d* where g0 is the small signal gain associated with one undulator section. By assuming e.g. an energy spread of the order of 10@3 and N ¼ 20; we can totally switch off the dispesive part by choosing x ¼ 10. In the forthcoming section, we will exploit the so far obtained results as the starting point of a more complete numerical investigation. x ¼
3. Numerical analysis and concluding remarks
Fig. 4. Evolution of the intracavity field for the device with undulators having opposite polarizations (analytical computation) (a) the undulators are assumed with equal number of periods ðN1 ¼ N2 ¼ 20Þ the continuous line refers to the first field, the dotted to the second, the intrinsic gain coefficient of each undulator is g0 ¼ 5:88 10@2 , the remaining parameters are the same as in Fig. 1. (b) Same as (a) with N1 ¼ 30, N2 ¼ 10, the intrinsic gain coefficient of the two undulator is g1 ¼ ð3=2Þ3 g0 ; g2 ¼ ð1=2Þ3 g0 .
In the following, we will assume that the undulators, labelled by 1 and 2 in Fig. 2, have left and right helical polarizations, respectively, namely B1 B0 ð@sinðku zÞ; cosðku zÞ; 0Þ B2 B0 ðsinðku zÞ; cosðku zÞ; 0Þ:
ð6Þ
The dynamical behavior of the system is followed by integrating the set of pendulum
672
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
Fig. 5. Same as Fig. 4(a) (same parameters but se ¼ 1:2 10@3 and 2ðT=ts Þ ¼ 6 10@3 ) (numerical computation).
Fig. 7. Same as Fig. 4(a) (N1 ¼ N2 ¼ 20) but with dispersive section d ¼ 20, the two fields reach the same equilibrium value and are totally indistinguishable.
Fig. 6. Same as Fig. 4(b) (N1 ¼ 28, N2 ¼ 12, se ¼ 1:2 10@3 , 2ðT=ts Þ ¼ 6 10@3 , Z¼ 7 10@3 ).
equations derived in Ref. [1], accounting for the left–right polarization interplay, coupled to the equations describing the e-beam longitudinal dynamics in SR. The results of the numerical integration are given in Figs. 5–7 and confirm the conclusions of
the previous section, i.e. we obtain comparable intensities when the gain of the first undulator is larger than the second. We have used in the simulation the parameters of the SR-FEL at Elettra [4] in which the total number of undulator periods (NT ¼ N1 þ N2 ) is 40, to increase the gain of the first section and compensate the effect of the bunching in the second section, we have chosen N1 > N2 . By using the above arrangement, we have obtained the results of Fig. 6, indicating that the equilibrium intracavity intensities of the fields with opposite polarizations reaches almost identical values thus allowing their combination to provide linearly polarized output radiation. As to the ‘‘mismatched’’ dispersive section, we note that, by assuming se D1:2 10@3 and the associated d D1:5, we have checked that values of d larger than 10 eliminate any coupling between the undulators. In this case (see Fig. 7), the fields grow totally independently and reach exactly the same equilibrium value. We must underline that (Figs. 5–7) although specific for the laser parameters of Elettra, uses a value of the ratio T=ts around 3 10@3 (while the effective value is of the order of 10@5) the reason
G. Dattoli, P.L. Ottaviani / Nuclear Instruments and Methods in Physics Research A 479 (2002) 668–673
of this choice is due to the computer time, necessary to follow the system up to the saturation, prohibitively large using the actual value. We must also underline that the numerical analysis of the mismatched OK configuration requires a large number of macroparticles in the simulation and this is a further element of increase of the computer time.
673
References [1] G. Dattoli, L. Bucci, Opt. Commun. 177 (2000) 323. [2] P. Elleaume, J. Phys. 45 (1984) 997. G. Dattoli, L. Giannessi, P.L. Ottaviani, Nucl. Instr. and Meth. A 365 (1995) 559. [3] G. Dattoli, P.L. Ottaviani, IEEE JQe-35 (1999) 27. [4] R. Rou, et al., in: J. Feldhaus, H. Weise (Eds.), Free Electron Lasers 1999, Elsevier, Amsterdam, 2000, p. II 21.