Passive longitudinal phase space linearizer

We report on the possibility to passively linearize the bunch compression process in electron linacs for the next generation x-ray free electron lasers. This can be done by using the monopole wakefields in a dielectric-lined waveguide. The optimum longitudinal voltage loss over the length of the bunch is calculated in order to compensate both the second-order rf time curvature and the second-order momentum compaction terms. Thus, the longitudinal phase space after the compression process is linearized up to a fourth-order term introduced by the convolution between the bunch and the monopole wake function.


I. INTRODUCTION
In an x-ray free electron laser (FEL), the electron bunch is usually compressed to increase the peak current [1] and thus increase the overall efficiency. This compression process is strongly affected by nonlinear effects such as the sinusoidal rf time curvature and the second-order pathlength dependence on particle energy in the magnetic chicane. In [2] and references contained therein, it is proposed to use a short section of radio-frequency accelerating fields at a higher harmonic with respect to the main accelerating linac to linearize the longitudinal phase space of the overall compression process. This technique linearizes the compression transformation through second order. In this note we describe the use of a passive dielectriclined waveguide to linearize the longitudinal phase space through the compression process instead of an active higher harmonic system. But use of the passive linearizer is not perfect and, after the convolution with the charge distribution, it introduces a fourth-order term in the position-energy behavior that changes the initial bunch profile.
In Sec. II and using the concepts presented in [3], we summarize the longitudinal single-mode wakes in a dielectric-lined waveguide considering only the two first characteristic waves. In Sec. III we give an explicit expression for the energy change within a rectangular bunch distribution upon passage through a dielectric-lined waveguide when only the first characteristic wave is considered for the wake function. Thus, as suggested in [2], we work out the voltage needed to completely cancel the secondorder compression factor. Furthermore, we give a condition to limit the residual nonlinear effects due to the fourthorder term introduced by the convolution between the bunch distribution and the induced wake function of the linearizer. In Sec. IV we present the tracking results considering two cases for a rectangular bunch distribution and the linear compression factor. It is worthwhile noting that the energy change within the bunch due to passage through the dielectric-lined waveguide strongly depends on the charge and bunch distribution. For this reason we also look at the sensitivity of the process as a function of the bunch distribution, in particular the bunch length, and charge variations. In Sec. V we consider a linear ramped bunch distribution such as that used in [1] and, as in the previous section, we work out the voltage needed to cancel the second-order compression terms. For this case we also give a condition to limit the residual nonlinear effects. In Sec. VI we present the tracking results when a passive linearizer is used together with an active fourth-harmonic system in the one-stage compression option of the FERMI@Elettra project [4].

II. LONGITUDINAL WAKE FUNCTION
Following [3], we summarize in this section relevant aspects of the longitudinal single-mode wakes generated in a dielectric-lined waveguide via passage of a charged particle. Let us consider a charged particle traveling with a speed close to that of light through a pipe of circular cross section and radius r 2 . Let us also assume that the wall of the pipe have infinite conductivity and that it is lined with dielectric material between radii r 1 < r < r 2 . In the case of such a dielectric-lined waveguide, the longitudinal wake function for the monopole mode, considering only the two first characteristic waves, is w jj ðzÞ % A 0 cosðk s zÞ þ A 1 cosðk s1 zÞ; (1) where the amplitudes are where Z 0 is the impedance of free space, ¼ r 1 =r 2 , and PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 13, 034401 (2010) The American Physical Society is the relative permittivity. The wave numbers of the characteristic waves are defined by where x 0 and x 1 are the first two positive zeros of the analytical function: and where J 0 and Y 0 are, respectively, the Bessel function of order zero and the Neumann function of order zero. We assume that the higher-order modes of wave numbers k sn are suppressed due to the practical consideration that there will be a finite rms bunch length . However, in Sec. VI we give the particle tracking result when the second characteristic wave of wave number k s1 is also considered. Figures 1 and 2 show the amplitudes and wave numbers of the two first characteristic waves as a function of the parameter for r 2 ¼ 2:5 mm and r 2 ¼ 3 mm, respectively.

III. RECTANGULAR BUNCH DISTRIBUTION
For a rectangular bunch distribution R ðzÞ with fullwidth L b , total charge Q, and distributed symmetrically about z ¼ 0, the energy change within the bunch due to its passage through a dielectric-loaded vacuum chamber of length L and subject to the wake of the first characteristic monopole mode is given by Let us consider that the electron beam passes through one linac segment and then through a round pipe partially filled with dielectric material as described above. If the beam has an initial mean energy E i , then following the linac and after the round pipe the electron energy for particles with jz 0 j L b =2 is where k 1 is the rf wave number, V 1 is the amplitude of the voltage of the linac segment, z 0 is the longitudinal position of the particle with respect to the bunch center, and the bunch head is at z 0 < 0, A s ¼ ðQLA 0 Þ=ðk s L b Þ, and s ¼ ðk s L b Þ=2. The rf phase 1 is defined to be zero at accelerating crest and a phase in the interval À90 < 1 < 0 will accelerate the bunch head less than the bunch tail. The synchronous mode can provide a decelerating voltage at the bunch center when its phase is s $ =2 and this condition can be obtained by choosing a dielectriclined waveguide with k s $ =L b . The relative energy deviation can be expanded in a power series about the reference particle and becomes where E 0 is the chicane energy. In Eq. (11) we choose to expand up to the fourth order in bunch length coordinate z 0 for the energy change due to the dielectric-loaded vacuum chamber, while assuming that k 1 L b ( 1 we choose to expand only up to the second order in bunch length for the energy gain from the linac. Let us write the relative energy deviation in (11) as A magnetic chicane is next used to transform energy deviations into path-length deviations. The chicane transformation can be written to third order in relative energy deviation as where T 566 % À3R 56 =2 and U 5666 % 2R 56 for a typical chicane, and R 56 , T 566 , and U 5666 are the linear, secondorder, and third-order transport matrix elements relating energy offset to longitudinal displacements following transport through the chicane. Putting Eq. (12) into Eq. (13) we obtain Fixing , an order analysis is performed and Eq. (14) can be simplified as z % ð1 þ aR 56 Þz 0 þ ðbR 56 þ a 2 T 566 Þz 2 0 þ dR 56 z 4 0 : (15) In order to evaluate the quality of the analytical treatment and the degree of confidence in the assumption concerning the approximations, we perform a comparison between the results obtained with Eq. (13) without approximations and how this deviates from the approximation of Eq. (15). For this purpose, we consider a realistic set of parameters reported in Table I and also used in the following paragraphs. These parameters are close to the design parame-ters of the FERMI@Elettra project [1]. Figure 3 shows that there is very good agreement between the two results.
To compensate the quadratic term in the transformation in Eq. (15), the second term must be set to zero: bR 56 þ a 2 T 566 ¼ 0; (16) Choosing to hold the energy at the chicane constant and at E 0 and assuming that the synchronous mode is decelerating, then it is necessary to increase the voltage of the linac: A second requirement is to hold the compression factor, C ¼ 1=ð1 þ aR 56 Þ, constant. If the R 56 is fixed, then the requirement is to hold the parameter a constant: Using (18) to eliminate the eV 1 terms from Eq. (17), we can calculate the amplitude of the voltage A s needed to completely cancel the second-order compression terms: If k s ) k 1 and sinð s Þ $ 1 then the amplitude A 0 of the synchronous mode can be written In Eq. (15) the residual nonlinear effects are small if the fourth-order term is maintained small compared with the linear compression term, that is, considering the worst condition at the tail of the bunch where z 0 ¼ L b =2. The condition (23) can be rearranged as follows: Here we have assumed k s L b $ and sinð s Þ $ 1.
Basically, there are two possibilities to limit the residual nonlinear effects: the first one is to reduce the bunch length L b from the photoinjector and the second one is to limit the linear compression factor C ¼ 1=ð1 þ aR 56 Þ. When the charge and the length of the electron bunch are fixed, we can choose a dielectric-lined vacuum chamber with k s $ =L b and A 0 from Eq. (21). In order to accommodate a range of bunch charges and current distributions, one possible idea would be implement a set of passive linearizers on a movable stage. Each linearizer would be ''tuned'' to different values of k s and A 0 and so could be used for a range of bunch length and charge. Another possible idea would be to use a parallel plate waveguide configuration where changing the gap could be used to vary the synchronous mode. In the following paragraphs, we present the results from the tracking analysis considering two bunch length cases both with a total charge of 1 nC. For both cases we also consider relative bunch length and relative charge variations up to 20% with respect to the nominal parameters at the injector in order to gauge the method's sensitivity to fluctuations.

IV. TRACKING RESULTS FOR A RECTANGULAR BUNCH DISTRIBUTION
In this section we present the LITRACK [5] tracking results for two bunch length cases, L b ¼ 3:5 mm and L b ¼ 2 mm. In the former case the linear compression factor is limited to C ¼ 3:5, while in the latter the linear compression factor is limited to C ¼ 6. The nominal energy at the chicane is E 0 ¼ 230 MeV, while the initial energy is E i ¼ 96 MeV. The wave number of the main accelerating linac is k 1 ¼ 62:9 m À1 . A rectangular bunch distribution with total charge Q ¼ 1 nC is used for both cases.
A. Case 1: L b ¼ 3:5 mm and C ¼ 3:5 Figure 4 shows the simulated energy profile, longitudinal phase space, and bunch current distribution after the bunch compressor chicane without second-order compensation (V 1 ¼ 149:4 MeV, 1 ¼ À26:3 deg , and R 56 ¼ À3:95 cm). A current spike develops at the bunch head due to the combined rf and compression nonlinearities. Figure 5 shows the phase space and the bunch current distribution after the bunch compressor chicane when an active fourth-harmonic rf system is used to compensate the quadratic term introduced by the accelerating rf system and the compressor (in this case the harmonic voltage is À12:7 MV while V 1 ¼ 161:0 MeV, 1 ¼ À24:3 deg , and R 56 ¼ À3:95 cm). The compressed current distribution is nearly unchanged with respect to the initial profile (i.e. it is still rectangular) and the energy-z correlation is linear. Table I lists the rf and bunch compression parameters when a passive linearizer is used. The harmonic voltage A s was calculated from Eq. (20) and k s ¼ =L b % 890 m À1 . The voltage and phase of the linac were adjusted in order to be used with the passive linearizer. Figure 6 shows the phase space after the chicane when the passive linearizer is used to compensate the quadratic term. If the remaining fourth-order term remains limited, then the phase space is quite linear and the current distribution does not develop a significant sharp current spike at the head of the bunch. On the other hand, there is a distortion of the current distribution due to the fourth-order term introduced by the passive linearizer. Table I  lined waveguide that would meet the requirements necessary to produce the results of Fig. 6. This dielectric-lined waveguide would have radii r 1 ¼ 2:48 mm and r 2 ¼ 3:5 mm, would be 1-m long, and would be lined with a dielectric material with ¼ 2:8. We next look at the sensitivity of the process as a function of the bunch length and charge variations. Figure 7 plots the relative peak current ÁI pk =I pk , relative mean energy ÁE=E, and bunch arrival time variation Át f after the bunch compression as a function of the bunch length variation ÁL b =L b (upper plots) and the relative charge variations ÁQ=Q (lower plots) at the injector. As an example, the upper middle plot in Fig. 7 indicates that a AE19% bunch length variation at the injector causes an approximately to Ç8% relative peak current variation after the bunch compression. Table II lists the sensitivities of the electron beam parameters to the relative bunch length and charge variations. Each sensitivity number quoted in the table independently causes an 8% relative peak current increase, a 0.1% relative mean energy increase, and a 150 fs bunch arrival time increase. Figure 8 shows the bunch current distributions after the bunch compression chicane for different bunch length variations at the injector and for different relative bunch charge variations. We can  (Color) Relative mean energy ÁE=E, relative peak current ÁI pk =I pk , and bunch arrival time variations Át f as a function of the relative initial bunch length variations ÁL b =L b (upper plots) and relative initial bunch charge variations ÁQ=Q (lower plots). TABLE II. Individual sensitivities. Each parameter variation causes a ÁI pk =I pk ¼ þ8% relative peak current change, or a ÁE=E 0 ¼ þ0:1% relative energy change or a Át f ¼ 150 fs bunch arrival time change after the bunch compression.

Parameters
ÁI pk =I pk ¼ þ8% still see that the current distributions do not develop a significant current spike at the bunch head for any of the simulated cases.
B. Case 2: L b ¼ 2 mm, and C ¼ 6 If the bunch length is reduced, then the compression factor can be increased as seen from the condition (24). Table III lists the rf and compression parameters for the second-order compensation when the compression factor C ¼ 6. In this case the wave number of the linearizer is Figure 9 shows the phase space after the magnetic chicane. As in case 1, the phase space remains quite linear and the current distribution does not develop a significant current spike at the head. However, as in case 1 the current distribution displays a distortion due to the fourth-order term if compared with the situation in Fig. 5 when an x-band system is used. In this case the harmonic voltage A s , calculated from Eq. (20), is approximately 3 times less than the previous case 1 (0.304 MV versus 0.94 MV). Thus, if the bunch length L b is reduced approximately 40% with respect to case 1, then the linear compression factor can be increased without increasing the nonlinear effects, i.e. C ¼ 6. Table III lists the parameters for a possible dielectric-lined waveguide that would meet the requirements necessary to produce the results of Fig. 9. In this case such a dielectric-lined waveguide would be constructed with the same radius r 2 and dielectric material as in the previous case but it would be 325 mm long, i.e., considerably shorter than in the previous case.
As in the previous case a sensitivity analysis was performed and Fig. 10 plots the relative peak current ÁI pk =I pk , relative mean energy ÁE=E, and bunch arrival time variation Át f after bunch compression as a function of the bunch length variations ÁL b =L b (upper plots) and the relative charge variations ÁQ=Q (lower plots) at the injector. As an example, the upper middle plot in Fig. 10    indicates that a AE20% bunch length variation at the injector causes an approximately 5% to À8% relative peak current variation after bunch compression. Table IV lists the sensitivities of the electron beam parameters to the relative bunch length and charge variations. As before each sensitivity number quoted in the table independently causes an 8% relative peak current increase, or a 0.1% relative mean energy increase, or a 150 fs bunch arrival time increase. Figure 11 shows the bunch current distributions after the bunch compression chicane for different bunch length variations at the injector and for different relative bunch charge variations. The current distributions do not develop a significant sharp current spike at the bunch head in any cases studied.

V. RAMPED BUNCH DISTRIBUTION
The operation of x-ray free electron lasers (FELs) relies on extremely high quality electron beams. In the presence of wakefields, the electron density distribution plays an important role in the formation of the electron bunch distribution at the end of the accelerator. The electron density distribution with a linear ramped peak current, obtained through a quadratic ramp in the laser intensity, can in principle compensate the nonlinear time dependent variations due to the linac structural wakefields [6]. For this reason we consider a linear ramped charge distribution that can be defined as   FIG. 10. (Color) Relative mean energy ÁE=E, relative peak current ÁI pk =I pk , and bunch arrival time variation Át f as a function of the relative initial bunch length variations ÁL b =L b (upper plots) and relative initial bunch charge variations ÁQ=Q (lower plots). TABLE IV. Individual sensitivities. Each parameter variation causes a ÁI pk =I pk ¼ þ8% relative peak current change, or a ÁE=E 0 ¼ þ0:1% relative energy change, or a Át f ¼ 150 fs bunch arrival time change after the bunch compression.

Parameters
ÁI pk =I pk ¼ þ8% 8 < À 20 >20 P. CRAIEVICH Phys. Rev. ST Accel. Beams 13, 034401 (2010) 034401-8 The energy change within the bunch itself in a dielectriclined vacuum chamber of length L is given by In this case the synchronous mode can provide a decelerating voltage at the bunch center when its phase is s $ and this condition can be obtained by choosing a dielectric-lined waveguide with k s $ 2=L b . The energy change in Eq. (26) can be written as As in Sec. III for the rectangular bunch distribution, we can calculate the amplitude A s of the voltage needed to completely cancel the second-order compression terms: If k s ) k 1 then the amplitude A 0 of the synchronous mode can be written where A s0 is given in Eq. (22). Following the same consideration as in the rectangular bunch distribution, the residual nonlinear effects are small if the following condition holds: Here we have assumed k s L b $ 2 and k s ) k 1 . If we compare the conditions described by Eqs. (24) and (30), then we observe that the conditions required to achieve a similar level of the beam quality as for the rectangular beam case are 4 times more stringent.

VI. TRACKING RESULTS USING A PASSIVE LINEARIZER AND FOURTH-HARMONIC SYSTEM
From the previous sections we have seen that the passive linearizer could work well with a ramped current distribution if the needed harmonic voltage is not too high. An idea could be to use the passive linearizer together with an active fourth-harmonic system for the linearization of the longitudinal phase space. A possible advantage of this option is that both needed voltages, passive and active, could be reduced. In this section we present the tracking results when both systems are used in the one-stage compression option of the FERMI@Elettra project [4]. This option foresees a ramped current distribution with total charge Q ¼ 0:8 nC. The single bunch beam is produced by a photoinjector, then accelerated up to 250 MeV and compressed with a single magnetic chicane. After the compression the beam is still accelerated approximately to 1.1 GeV (linac end) by a linear accelerator, and finally transported to the undulators, where the free electron lasing occurs. Figure 12 shows the simulated energy profile, longitudinal phase space, and bunch current distribution at the linac end with parameters given in [4] and using only the fourth-harmonic rf system to compensate the nonlinearities. In this case the required fourth-harmonic voltage is 19 MV with an rf phase of À195 . Figure 13 shows the simulated energy profile, longitudinal phase space, and bunch current distribution at the linac end using a combined passive and active system. In this case the fourthharmonic voltage is 8.5 MV and the synchronous mode has an amplitude of 4:4 MV=ðnC mÞ and a wave number k s ¼ 1440 m À1 . Assuming the fourth-harmonic accelerating structure presented in [7], we need an rf power of 4.2 MW if we use the passive linearizer, and 21 MW if we only use the fourth-harmonic system. This becomes a practical issue as x-band rf power systems below 5 MW are commercially available, but at present above that value they are not.
The required voltage for the passive linearizer and the relative wave number can be reached with one dielectriclined waveguide 1-m long with radii r 1 ¼ 2:4 mm and The dielectric material should have a relative dielectric constant ¼ 2:2 with a thickness of 0.6 mm. The second characteristic wave of the waveguide has the following parameters: A 1 ¼ 0:5 MV=ðnC mÞ and k s1 ¼ 5300 m À1 . Figure 14 shows the simulated energy profile, longitudinal phase space, and bunch current distribution at the linac end considering also the second characteristic wave. In such a case the fourth-harmonic voltage is 9 MV and the rf phase À200 . This is needed to compensate the spurious effect due to the second characteristic wave.

VII. CONCLUSION
In this paper a passive linearizer is implemented instead of an active fourth-harmonic linearizer system to compensate both for the second-order rf time curvature and for the second-order momentum compaction terms. Using the passive linearizer, the current distribution does not develop a significant sharp peak current spike at the head of the beam. On the other hand, there is a distortion of the current distribution due to the fourth-order term introduced by the convolution between the wake function of the dielectriclined waveguide and the charge distribution. A condition to limit the nonlinear effects due to the fourth-order term is given. Basically, there are two possibilities in keeping the spurious effects small: the first one is to reduce the bunch length from the photoinjector and the second one is to use a small linear compression factor. We have presented the LITRACK tracking results considering two cases of a rectangular bunch distribution with different bunch length and different linear compression factors. For both cases the results have shown that the linearizer can be successfully used in place of the active fourth-harmonic system. Since the energy change within the bunch due to passage through the dielectric-lined waveguide strongly depends on the bunch distribution and charge, we have carried out a sensitivity study for both the case of charge variations and for variations in bunch length of the considered bunch distributions. We have seen that the current distributions do not develop a significant sharp peak current spike at the bunch head for any of the simulated cases. On the other hand, the linearizer is, however, tailored to a particular bunch charge and current distribution. In this case in order to accommodate a range of bunch charges and current distributions, a possible idea is to implement the passive linearizer involving multiple waveguides with separate properties on a movable stage. Each of the dielectric-lined waveguides tailored would be tailored with a different wave number k s and amplitude A 0 . Furthermore, the results for the linear ramped bunch distribution, such as that used in [1], were presented. Finally, we have successfully studied the passive linearizer together with an active fourth-harmonic system for the one-stage compression option of the FERMI@Elettra project. As a conclusion, the linearizer can, in principle, be used instead of an active higher harmonic system if the compression factor or the bunch length from the photoinjector is limited. In addition, another result from this work is the suggestion to use the passive linearizer together with an active higher harmonic system. In this way the required higher harmonic voltage could be reduced.
As a final point we must consider the transverse wakefields that can be generated in the dielectric-lined waveguide. The wake functions can be calculated as shown in [3]. For the passive linearizers proposed in this paper, the amplitude of the transverse wake functions are of the same order of magnitude, approximately a factor 2-3 larger when compared to the active fourth-harmonic cavity foreseen for the FERMI layout. Thus, in principle, the transverse wake effects due to the passive linearizer may result in a limited but acceptable amount of emittance growth.