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Effects of beam loss on the stability of a recirculating electrostatic accelerator for a long-pulse free-electron laser
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
Effects of beam loss on the stability of a recirculating electrostatic accelerator for a long-pulse free-electron laser Sung Oh Choa,*, Byung Cheol Leeb, Young Uk Jeongb a
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guesong-dong Yuseong-gu, Daejon 305-701, South Korea b Laboratory for Quantum Optics, Korea Atomic Energy Research Institute, P.O. Box 105, Yusong, Daejon 305-600, South Korea Received 21 January 2003; received in revised form 1 July 2003; accepted 21 July 2003
Abstract For a long-pulse operation of a free-electron laser driven by a recirculating electrostatic accelerator, the effects of beam loss on the operational stability of the accelerator are theoretically and experimentally analyzed. If a beam loss occurs mainly in the deceleration tube, the terminal voltage can keep constant even though the lost beam current is much higher than the charging current of the accelerator. However, the beam loss can induce an unstable operation of the accelerator such as a voltage breakdown or an increase of the beam loss with time. The accelerator instability can be the main problem disturbing the long-pulse operation of the FEL. From the analysis, a method to increase the available pulse width of the FEL is presented, which is to enlarge the capacitances between the neighboring electrodes of the deceleration tube. r 2003 Elsevier B.V. All rights reserved. PACS: 29.17.+w; 29.27.Bd; 41.60.Cr; 41.75.Fr Keywords: Recirculating electrostatic accelerator; Beam loss; Stability; Free-electron laser
1. Introduction Recirculating electrostatic accelerators are widely used for a long-pulse or a continuous-wave (CW) free electron laser (FEL) in millimeter or far infrared region [1–7]. FELs generally require a beam current more than several amperes. However, an electrostatic accelerator can supply a continuous beam current less than several tens mA because of the current capability of a high voltage generator (HVG). One solution to overcome the *Corresponding author. Tel.: +82-42-869-3823; fax: +8242-869-3810. E-mail address: [email protected] (S.O. Cho).
limited capability of the HVG is to use a recirculating electrostatic accelerator. The recirculating accelerator recovers both an energy and a charge of the electron beam that is spent for a wave generation at an undulator. Due to the beam recovery, a beam current much higher than the supplying current of the HVG can be used for the FEL generation. A FEL driven by a recirculating accelerator is composed of an electron gun, an acceleration tube, a high voltage terminal, a deceleration tube, and a collector. While the beam transports from the electron gun to the collector, a part of the beam can be lost because of a degradation of the beam quality at the undulator [8], bad beam optics, and a non-perfect beam collection of a collector [9,10].
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.07.028
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409 Table 1 Main parameters of the accelerator Beam energy, Eb Gun voltage, Vg Beam current, Ib Charging current, Ich Number of electrodes, N Resistance of terminal, RT Capacitance of terminal, CT Resistance of electrodes, Rd Capacitance of electrodes, Cd Breakdown voltage, VBD
Max. 430 keV Max. 30 keV Max. 2 A 7.5 mA 10 1 GO 170 pF 100 MO 1.97 nF 65 kV
The beam loss reduces the beam-recovery efficiency [1,11]. Generally the FEL requires the beam-recovery efficiency higher than 99% for the CW operation [1,5]. However, it has been reported that the maximum recovery efficiency obtained in the lasing experiments of the FELs is about 96% and thus the FELs should be operated in pulse with the maximum pulse width less than 50 ms [1–5,12]. We have developed a mm-wave FEL driven by a recirculating electrostatic accelerator [5], which has the main parameters shown in Table 1. The FEL is designed to operate in CW mode with the output power of about 20 kW. In the preliminary experiments of the FEL we have obtained 1 kW millimeter wave with the maximum pulse width of 20 ms, which is limited due to the beam loss [5]. For a successful CW operation or for an increase of the pulse width of the FEL, the amount of the beam loss should be minimized. Furthermore, analyses of the beam loss effects on the FEL operation are needed. Several groups analyzed the effects of the beam loss on the terminal voltage that determines the beam energy in the electrostatic accelerator [7,11,13]. In this paper, we have analyzed theoretically and experimentally the effects of the beam loss on the operational stability of the FEL.
2. Analysis of the effects of beam loss on the FEL operation 2.1. Terminal voltage change In a FEL driven by a recirculating accelerator, some of the beam can be lost during the transport
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from an electron gun to a collector while the other is recovered at the collector. The beam loss changes the energy of the beam entering the undulator because it reduces the terminal voltage and thus it affects the operation of the FEL. The maximum allowable terminal-voltage drop, DVT ; for a single-mode operation of a FEL can be derived from the requirement of the beam energy entering the undulator [14]: DVT DEb 1 ¼ p VT Eb 2Nw
ð1Þ
where VT is the terminal voltage, Eb is the beam energy, DEb is the energy spread of the beam, and Nw is the number of an undulator period. The effects of the beam loss on the terminal voltage are determined by the position where the beam is lost [11]. We consider a FEL with a positive-terminal configuration [3,5] in which an undulator is located inside a high voltage terminal. The analysis results are also applied to a FEL with a negative-terminal configuration [1]. In the FEL, most of the beam loss occurs after the undulator, that is, inside the terminal or in the deceleration tube because the beam quality degrades and the energy spread increases after the FEL interaction at the undulator [1,5]. The electrons may hit several electrodes of the deceleration simultaneously, however for simplicity of the analysis, it is assumed that the beam loss with the current of Iloss occurs only at a single electrode located n sections before the end of the deceleration tube. Then, the accelerator is depicted by an equivalent circuit of Fig. 1. As shown in Fig. 1, the electrodes of the deceleration tube are connected in series, whereas the terminal is combined with the deceleration tube in parallel. Accordingly, the equivalent circuit of the accelerator can be simplified by introducing the total capacitance CTot ð¼ CT þ Cd =NÞ and the total resistance RTot ð¼ NRd RT =ðNRd þ RT ÞÞ of the accelerator. The subscripts T and d denote the terminal and the electrode of the deceleration tube, respectively, which has the total number of the electrodes, N: As discussed in the Ref. [11], not all the lost current Iloss contributes to a terminal-voltage drop. The lost current Iloss shown in Fig. 1 is divided into I1 ; which flows through the left ðN nÞ electrodes
ARTICLE IN PRESS 404
S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
Fig. 1. An equivalent circuit of the accelerator with a beam loss inside a deceleration tube.
and the terminal, and I2 ; which flows through the right n electrodes and the ground. Assuming that Rd Cd ERT CT ; which is justified in our FEL as shown in Table 1, the currents I1 and I2 are given by [11] nCT nRd I1 ¼ Iloss ¼ Iloss NCTot RT þ NRd ðN nÞCT þ Cd I2 ¼ Iloss NCTot ðN nÞRd þ RT ¼ Iloss : RT þ NRd
ð2Þ
The terminal voltage is changed only by the current I1 while I2 does not induce the terminalvoltage drop. Then, the transient terminal voltage VT and the rate of terminal voltage change are given by ðt=RTot CTot Þ
VT ðtÞ ¼ VT0 e n þ Ich Iloss RTot ð1 eðt=RTot CTot Þ Þ ð3Þ N dVT Ich IR ðn=NÞIloss ðt=RTot CTot Þ ¼ e dt CTot
ð4Þ
where Ich is the charging current of the HVG and IR ð¼ VT0 =RTot Þ is the initial current that flows through the resistor RTot : If the beam loss occurs inside a high voltage terminal, the behavior of the terminal voltage can be found by substituting n ¼ N in the previous equations. Eqs. (3) and (4) show that the amount of the terminal-voltage drop depends on the position of the beam loss. If the right-hand side in Eq. (4) is
higher than zero or if Iloss ðN=nÞðIch IR Þ; the FEL satisfy the requirement of the terminal voltage, Eq. (1), for a CW operation. As an example, the requirement for the CW operation of our FEL is given below. The accelerator has the beam current of 2 A, Ich of 7.5 mA, and N of 10. The current IR ; for the maximum terminal voltage of 400 kV, is 0.8 mA. If the beam loss occurs only inside the terminal, the requirement for the CW operation comes to the lost beam current Iloss p6:7 mA or the recovery efficiency RX99:7%. However, in case beam loss occurs at the first electrode of the deceleration tube, the requirement is greatly reduced to Iloss p67 mA or RX96:7%. This shows that if the beam loss occurs mainly in the deceleration tube, the terminal voltage requirement for the CW operation can be satisfied although the lost beam current is much higher than the charging current of the HVG. 2.2. Operational stability of the accelerator The beam current lost in the deceleration tube changes the voltage distribution of the electrodes and can lead to an unstable operation of the accelerator. As shown in Fig. 1, the current I1 charges the electrodes of the deceleration tube while I2 discharges the electrodes. Then, the voltage differences between two adjacent electrodes, DVd1 and DVd2 ; are derived as DVd1 ðtÞ ¼ DVd0 eðt=Rd Cd Þ þ ðI1 þ ich ÞRd ð1 eðt=Rd Cd Þ Þ DVd2 ðtÞ ¼ DVd0 eðt=Rd Cd Þ ðI2 ich ÞRd ð1 eðt=Rd Cd Þ Þ:
ð5Þ
Here, DVd1 ðDVd2 Þ is the voltage difference between two electrodes located at the left (right) of the electrode where the beam loss occurs and DVd0 is the initial value of the voltage difference. Eq. (5) shows that the beam loss increases DVd1 and decreases DVd2 : Note that DVd2 can be a negative value. The modified DVd1 and DVd2 change the voltage distribution of the electrodes of the deceleration tube. The change in the voltage distribution affects the optics of the beam traveling the deceleration tube and can induce a further
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
beam loss. Then a vicious circle of the beam loss, or an increase of the beam loss with time, can arise. Increase in the beam loss rapidly drops the terminal voltage and finally prevents the FEL operation. In addition, the beam loss increases DVd1 and this value can reach the breakdown voltage between the electrodes. The breakdown interrupts an operation of the FEL. Thus, the beam loss occurring in the deceleration tube can make the accelerator unstable and disturb the operation of the FEL. We have calculated the effects of the beam loss on both the modified voltage distribution and the beam trajectory inside the deceleration tube of our FEL system. The modified voltage distribution of the electrodes is calculated by using the PSPICE code [15] and the beam trajectory is calculated by using the EGUN code [16]. Fig. 2 shows the calculated result after 400 mA hits the first electrode of the deceleration tube for 10 ms. The initial terminal voltage 300 kV is equally distributed to 10 sections of the deceleration electrodes and the capaciatance Cd is 267 pF. The terminal voltage drops just by 0.4 kV by the beam loss, which itself makes negligible effect on the FEL operation. Although the lost beam current 400 mA is much higher than the charging current 7.5 mA of the HVG, the change in the terminal voltage is not so large. This is because the position of the beam loss is the first electrode of the deceleration tube that is close to the ground. However, DVd1 increases from 30 kV to about 36 kV and DVd2 changes from 30 to 23.7 kV. The modification of the voltage distribution, as shown in Fig. 2, causes a part of the beam to reflect and hit several electrodes of the deceleration tube. Fig. 2 shows that an initial beam loss can induce a further beam Deceleration Tube
RADIUS (cm)
4 299.6
227.8
156.3
84.1
(unit : kV) 12 -23.7 0
Collector
3 2 1 0
0
5
10 15 AXIAL DISTANCE (cm)
20
25
Fig. 2. Calculation results of the beam loss effects on both the modified voltage distribution and the beam trajectory inside the deceleration tube.
405
loss. If such a phenomenon occurs in the FEL, the amount of the beam loss increases with time and then the FEL cannot be operated at a certain time after the beam loss occurs. Fig. 2 and the analysis in the previous section explain that the accelerator instability can be more serious than the requirement of the terminal voltage for a long-pulse or a CW operation of the FEL. This is because the terminal voltage requirement can be satisfied even for a large amount of the beam loss, but the beam loss can lead to the accelerator instability that prevents the FEL operation. 2.3. Maximum available pulse width of the FEL The accelerator instability caused by the beam loss limits the operating pulse width of the FEL. The voltage differences, DVd1 and DVd2 in Eq. (5), are approximated as DVd1 ðtÞEDVd0 þ
ðI1 þ ich ir Þt ; Cd
DVd2 ðtÞEDVd0
ðI2 ich ir Þt for t{Rd Cd : ð6Þ Cd
Here, ir ð¼ DVd0 =Rd Þ is the current that flows through the voltage dividing resistors of the electrodes. DVd1 is increased by a beam loss and it should be less than the breakdown voltage VBD between two neighboring electrodes for a stable operation of the accelerator. This limits the maximum available pulse width of the FEL, Ts;max ; which can be calculated by Ts;max E
Cd ðVBD DVd0 Þ : I1 þ ich ir
ð7Þ
For stable operation of the FEL, one should also avoid the process of a vicious circle of beam loss that is caused by electrons repelled back into the high voltage terminal before exiting the deceleration tube. In order to quantify the maximum allowable lost beam current satisfying this requirement, we consider the energy of an electron beam reaching the nth electrode, En : It is assumed that DVd2 ðtÞ has become negative and therefore the voltage is minimal at nth electrode, where incoming electrons may be repelled. The energy En
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
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is given by
299.6
! DVdi þ Vg
DEb
ð8Þ
i
where Vg is the gun voltage and DEb is the beam energy that is converted to wave energy at the undulator; DEb has the maximum value of Eb =2Nw according to Eq. (1). If En is negative, the electron beam cannot transport through the deceleration tube but reflects at the nth electrode. Then the reflected electrons are lost in the deceleration tube. This produces the vicious circle of the beam loss. To prevent this, En should be positive, which gives us the available pulse width of the FEL, TE;max ; as TE;max E
Cd ðn DVd0 þ Vg Eb =2eNw Þ : nðI2 ich ir Þ
ð9Þ
The terminal-voltage drop induced by the beam loss also limits the available pulse width of the single-mode FEL. The maximum available pulse width, Tt;max ; is obtained from Eqs. (1) and (3) as Tt;max E
CTot VT0 2Nw ðIR þ ðn=NÞIloss Ich Þ for Tt;max 5RTot CTot :
ð10Þ
The maximum available pulse width of the FEL is the minimum value of Ts;max ; TE;max ; and Tt;max : As described earlier, the instability of the accelerator can be more serious problem for a long-pulse operation of the FEL than the terminal voltage requirement. Considering the accelerator instability, we present a method to increase the available pulse width of the FEL, which is to enlarge the capacitance between the electrodes, Cd : If Cd is small, DVd1 in Eq. (6) reaches rapidly the breakdown voltage of the electrodes and DVd2 becomes rapidly negative. These can induce an unstable operation of the accelerator and shorten the available pulse width. By enlarging Cd ; we can reduce the changing rate of DVd1 and DVd2 : This improves the stability of the accelerator operation and thus increase Ts;max and TE;max : In addition, an enlarged Cd increases the total terminal capacitance CTot ð¼ CT þ Cd =NÞ and reduces the rate of the terminal-voltage drop. As a result Tt;max increases as can be found in Eq. (10). Therefore,
RADIUS (cm)
En ¼ e
n X
Deceleration Tube
4
239.3
178.9
(unit : kV)
118.5
58.1
0
Collector
3 2 1 0
0
5
10 15 20 AXIAL DISTANCE (cm)
25
Fig. 3. Calculation results of the voltage distribution and the beam trajectory for an increased capacitance Cd :
an enlarging Cd increases the available pulse width of the FEL. In order to investigate the effects of the Cd on the accelerator stability, the voltage distribution and the beam dynamics are calculated for an increased Cd of our FEL. The amount and the position of the beam loss are the same with those in Fig. 2. The capacitance Cd is increased by 1.7 nF compared to Fig. 2. Fig. 3 shows the calculated result of the accelerator behavior. DVd1 slightly increases to 30.2 kV while DVd2 decreases to 27.9 kV. The changes in DVd1 and DVd2 are much lower than those in Fig. 2 because of the increased Cd : Due to the small change in the voltage distribution, the beam passes through the deceleration tube without an additional loss. Comparing Fig. 3 to Fig. 2, we can find that the accelerator stability can be improved by increasing Cd :
3. Experimental results In order to investigate the effects of beam loss on the accelerator, we have slightly modified the FEL system. The undulator is replaced by a drift tube with an inside diameter of 40 mm and a single-stage Faraday cup has been used for a collector instead of a multi-stage-depressed collector. Beam-recovery experiments have been made for this configuration and Fig. 4 is the typical result of the experiments. The accelerator is operated in pulse with the maximum pulse width of 50 ms, which is limited by the present gun power supply. Fig. 4 shows that about 99.9% of the current emitted from the gun arrives at the collector and the terminal voltage does not change during the pulse. If we could get such
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
Ib Ic
407
Ib Ic
-20
0
20
VT
VT
Vg
Vg
40
60
80
Time (µs)
-20
0
20
40 Time (µs)
60
80
Fig. 4. A typical result of the beam-recovery experiment. Ib is the emission current (0.5 A/div), Ic is the collection recovered (0.5 A/div), VT is the terminal voltage (1 kV/div, VT0 ¼ 300 kV), and Vg is the gun voltage (5 kV/div).
Fig. 5. The accelerator behavior when 52 mA is lost at the first electrode of the deceleration tube with Cd of 267 pF. The notations and the corresponding vertical scales are the same with those in Fig. 4.
a high recovery efficiency when the undulator is installed, we can operate the FEL in CW mode. Although the accelerator can be operated with practically no beam loss, we have intentionally induced a beam loss to investigate the effects of the beam loss. Both the position and the amount of the beam loss are controlled by the magnetic fields of two deflection magnets that are installed after the acceleration tube and before the deceleration tube, respectively. Fig. 5 shows the temporal waveforms of the current emitted from the gun, the current arriving at the collector, and the terminal voltage when a part of the beam current is forced to hit the first electrode of the deceleration tube. The amount of the beam loss, which corresponds to the difference between the emission current and the collection current, is initially 52 mA. It increases with time and becomes 330 mA at the end of the pulse. The increase of the beam loss can be explained by the previous analysis result: a beam loss changes the voltage distribution of the deceleration tube and the modification can lead to a further beam loss. Note that the terminal voltage keeps nearly constant even though the amount of the beam loss is dramatically increased during the pulse. Fig. 5 demonstrates that the vicious circle of the beam loss can occur although the terminal voltage keeps nearly constant during the pulse.
To investigate the behavior of the accelerator after the pulse shown in Fig. 5, a beam current of 330 mA is controlled to hit the first electrode of the deceleration tube. Fig. 6 is the experimental result. The collection current decreases with time and ultimately becomes ‘‘0’’ at about 35 ms. Initially only 330 mA out of the total beam current of 0.8 A is lost. However, the amount of the beam loss increases further due to the voltage redistribution of the deceleration tube and finally no beam current reaches the collector after 35 ms. The voltage of the first electrode drops rapidly and becomes negative. A small breakdown occurs at 35 ms and then a large breakdown follows. After the small breakdown the accelerator shows an unstable operation as shown in Fig. 5 and practically the accelerator cannot be operated. The terminal voltage drops just by 1 kV before the breakdown. Figs. 5 and 6 show that the main factor limiting the operating pulse width of the FEL may not be the change in the terminal voltage but the unstable operation of the FEL. We proposed that the accelerator instability could be improved by increasing Cd : In order to demonstrate the effects of the Cd ; ceramic capacitors with 1.7 nF are installed between the electrodes of the deceleration tube. Cd is increased from 267 pF to 1.967 nF. Fig. 7 shows the behavior of the accelerator with the increased Cd when the beam loss occurs at the first electrode of the
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enlarging Cd can improve the accelerator stability and thus it can increase the operating pulse width of the FEL.
Ib Ic VT V1
-20
0
20
40 Time (µs)
4. Conclusions
60
80
Fig. 6. An experimental result of the accelerator instability. V1 is the voltage of the first electrode of the deceleration tube (10 kV/div) and other notations and the corresponding vertical scales are the same with those in Fig. 4.
Ib Ic
VT
Vg
-20
0
20 40 Time (µs)
60
80
Fig. 7. An experimental demonstration for the improvement of the accelerator stability. The notations and the corresponding vertical scales are the same with those in Fig. 4.
deceleration tube. The amount of the initial beam loss is 400 mA, which is more than that of Fig. 6. On the contrary to Fig. 6, the amount of the beam loss does not increase but keeps constant during the pulse. This is because the voltages of the electrodes do not change seriously due to the enlarged Cd ; and so the beam can arrive at the collector without an additional loss inside the deceleration tube. Moreover, no voltage breakdown has been observed during the pulse. The terminal voltage changes only by 0.3 kV for such a large beam loss and it satisfies the Eq. (1) for the FEL operation. Therefore, it is confirmed that an
In the operation of the FEL driven by a recirculating electrostatic accelerator, some of the beam can be lost inside the high voltage terminal or in the deceleration tube. If a beam is lost in the deceleration tube, only a part of the lost beam current contributes to the terminal-voltage drop. So the terminal voltage can keep nearly constant although the amount of the beam loss is much higher than the supplying current of the accelerator. However, the beam loss can make the accelerator unstable and this limits the operating pulse width of the FEL. Fortunately the instability of the accelerator can be improved by increasing the capacitance between the electrodes of the deceleration tube. Consequently, if the FEL is designed that the beam loss occurs mainly in the deceleration tube, it can be operated with a long pulse width even for a large amount of the beam loss.
References [1] L.R. Elias, G. Ramian, J. Hu, G. Ramian, Nucl. Instr. and Meth. A 237 (1985) 203. [2] B.L. Militsyn, et al., Nucl. Instr. and Meth. A 483 (2002) 259. [3] W.H. Urbanus, et al., Fus. Eng. Des. 53 (2001) 423. [4] A. Abramovich, et al., Nucl. Instr. and Meth. A 407 (1998) 16. [5] B.C. Lee, S.O. Cho, Y.U. Jeong, S.K. Kim, B.H. Choi, J. Lee, K.H. Chung, Nucl. Instr. and Meth. A 375 (1996) 28. [6] I. Kimel, L.R. Elias, Nucl. Instr. and Meth. A 358 (1985) 186. [7] C.C. Wright, A.I. Al-Shamma’a, J. Lucas, R.A. Stuart, Nucl. Instr. and Meth. A 445 (2000) 187. [8] I. Boscolo, V. Stagno, V. Variale, R. Calabrese, G. Bottoni, D. Candolfo, L. Tecchio, Appl. Phys. B 50 (1990) 71. [9] T.J.P. Ellison, Nucl. Instr. and Meth. A 364 (1995) 27. [10] A.N. Sharapa, A.V. Shemyakin, Nucl. Instr. and Meth. A 351 (1994) 295. [11] S.O. Cho, et al., Rev. Sci. Instr. 67 (1996) 3491.
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409 [12] A. Amir, L.R. Elias, D.J. Gregoire, R.J. Hu, J.P. Kotthaus, G. Ramian, A. Stern, Appl. Phys. Lett. 47 (1985) 1251. [13] A. Gover, et al., Nucl. Instr. and Meth. A 296 (1990) 720.
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[14] H.P. Freund, T.M. Antonsen Jr., Principles of Freeelectron Lasers, Chapman & Hall, London, 1992, p. 11. [15] MicroSim Corporation, PSpice Circuit Analysis, 1989. [16] W.B. Herrmannsfeldt, SLAC-331, 1988.
Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
Effects of beam loss on the stability of a recirculating electrostatic accelerator for a long-pulse free-electron laser Sung Oh Choa,*, Byung Cheol Leeb, Young Uk Jeongb a
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guesong-dong Yuseong-gu, Daejon 305-701, South Korea b Laboratory for Quantum Optics, Korea Atomic Energy Research Institute, P.O. Box 105, Yusong, Daejon 305-600, South Korea Received 21 January 2003; received in revised form 1 July 2003; accepted 21 July 2003
Abstract For a long-pulse operation of a free-electron laser driven by a recirculating electrostatic accelerator, the effects of beam loss on the operational stability of the accelerator are theoretically and experimentally analyzed. If a beam loss occurs mainly in the deceleration tube, the terminal voltage can keep constant even though the lost beam current is much higher than the charging current of the accelerator. However, the beam loss can induce an unstable operation of the accelerator such as a voltage breakdown or an increase of the beam loss with time. The accelerator instability can be the main problem disturbing the long-pulse operation of the FEL. From the analysis, a method to increase the available pulse width of the FEL is presented, which is to enlarge the capacitances between the neighboring electrodes of the deceleration tube. r 2003 Elsevier B.V. All rights reserved. PACS: 29.17.+w; 29.27.Bd; 41.60.Cr; 41.75.Fr Keywords: Recirculating electrostatic accelerator; Beam loss; Stability; Free-electron laser
1. Introduction Recirculating electrostatic accelerators are widely used for a long-pulse or a continuous-wave (CW) free electron laser (FEL) in millimeter or far infrared region [1–7]. FELs generally require a beam current more than several amperes. However, an electrostatic accelerator can supply a continuous beam current less than several tens mA because of the current capability of a high voltage generator (HVG). One solution to overcome the *Corresponding author. Tel.: +82-42-869-3823; fax: +8242-869-3810. E-mail address: [email protected] (S.O. Cho).
limited capability of the HVG is to use a recirculating electrostatic accelerator. The recirculating accelerator recovers both an energy and a charge of the electron beam that is spent for a wave generation at an undulator. Due to the beam recovery, a beam current much higher than the supplying current of the HVG can be used for the FEL generation. A FEL driven by a recirculating accelerator is composed of an electron gun, an acceleration tube, a high voltage terminal, a deceleration tube, and a collector. While the beam transports from the electron gun to the collector, a part of the beam can be lost because of a degradation of the beam quality at the undulator [8], bad beam optics, and a non-perfect beam collection of a collector [9,10].
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.07.028
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409 Table 1 Main parameters of the accelerator Beam energy, Eb Gun voltage, Vg Beam current, Ib Charging current, Ich Number of electrodes, N Resistance of terminal, RT Capacitance of terminal, CT Resistance of electrodes, Rd Capacitance of electrodes, Cd Breakdown voltage, VBD
Max. 430 keV Max. 30 keV Max. 2 A 7.5 mA 10 1 GO 170 pF 100 MO 1.97 nF 65 kV
The beam loss reduces the beam-recovery efficiency [1,11]. Generally the FEL requires the beam-recovery efficiency higher than 99% for the CW operation [1,5]. However, it has been reported that the maximum recovery efficiency obtained in the lasing experiments of the FELs is about 96% and thus the FELs should be operated in pulse with the maximum pulse width less than 50 ms [1–5,12]. We have developed a mm-wave FEL driven by a recirculating electrostatic accelerator [5], which has the main parameters shown in Table 1. The FEL is designed to operate in CW mode with the output power of about 20 kW. In the preliminary experiments of the FEL we have obtained 1 kW millimeter wave with the maximum pulse width of 20 ms, which is limited due to the beam loss [5]. For a successful CW operation or for an increase of the pulse width of the FEL, the amount of the beam loss should be minimized. Furthermore, analyses of the beam loss effects on the FEL operation are needed. Several groups analyzed the effects of the beam loss on the terminal voltage that determines the beam energy in the electrostatic accelerator [7,11,13]. In this paper, we have analyzed theoretically and experimentally the effects of the beam loss on the operational stability of the FEL.
2. Analysis of the effects of beam loss on the FEL operation 2.1. Terminal voltage change In a FEL driven by a recirculating accelerator, some of the beam can be lost during the transport
403
from an electron gun to a collector while the other is recovered at the collector. The beam loss changes the energy of the beam entering the undulator because it reduces the terminal voltage and thus it affects the operation of the FEL. The maximum allowable terminal-voltage drop, DVT ; for a single-mode operation of a FEL can be derived from the requirement of the beam energy entering the undulator [14]: DVT DEb 1 ¼ p VT Eb 2Nw
ð1Þ
where VT is the terminal voltage, Eb is the beam energy, DEb is the energy spread of the beam, and Nw is the number of an undulator period. The effects of the beam loss on the terminal voltage are determined by the position where the beam is lost [11]. We consider a FEL with a positive-terminal configuration [3,5] in which an undulator is located inside a high voltage terminal. The analysis results are also applied to a FEL with a negative-terminal configuration [1]. In the FEL, most of the beam loss occurs after the undulator, that is, inside the terminal or in the deceleration tube because the beam quality degrades and the energy spread increases after the FEL interaction at the undulator [1,5]. The electrons may hit several electrodes of the deceleration simultaneously, however for simplicity of the analysis, it is assumed that the beam loss with the current of Iloss occurs only at a single electrode located n sections before the end of the deceleration tube. Then, the accelerator is depicted by an equivalent circuit of Fig. 1. As shown in Fig. 1, the electrodes of the deceleration tube are connected in series, whereas the terminal is combined with the deceleration tube in parallel. Accordingly, the equivalent circuit of the accelerator can be simplified by introducing the total capacitance CTot ð¼ CT þ Cd =NÞ and the total resistance RTot ð¼ NRd RT =ðNRd þ RT ÞÞ of the accelerator. The subscripts T and d denote the terminal and the electrode of the deceleration tube, respectively, which has the total number of the electrodes, N: As discussed in the Ref. [11], not all the lost current Iloss contributes to a terminal-voltage drop. The lost current Iloss shown in Fig. 1 is divided into I1 ; which flows through the left ðN nÞ electrodes
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Fig. 1. An equivalent circuit of the accelerator with a beam loss inside a deceleration tube.
and the terminal, and I2 ; which flows through the right n electrodes and the ground. Assuming that Rd Cd ERT CT ; which is justified in our FEL as shown in Table 1, the currents I1 and I2 are given by [11] nCT nRd I1 ¼ Iloss ¼ Iloss NCTot RT þ NRd ðN nÞCT þ Cd I2 ¼ Iloss NCTot ðN nÞRd þ RT ¼ Iloss : RT þ NRd
ð2Þ
The terminal voltage is changed only by the current I1 while I2 does not induce the terminalvoltage drop. Then, the transient terminal voltage VT and the rate of terminal voltage change are given by ðt=RTot CTot Þ
VT ðtÞ ¼ VT0 e n þ Ich Iloss RTot ð1 eðt=RTot CTot Þ Þ ð3Þ N dVT Ich IR ðn=NÞIloss ðt=RTot CTot Þ ¼ e dt CTot
ð4Þ
where Ich is the charging current of the HVG and IR ð¼ VT0 =RTot Þ is the initial current that flows through the resistor RTot : If the beam loss occurs inside a high voltage terminal, the behavior of the terminal voltage can be found by substituting n ¼ N in the previous equations. Eqs. (3) and (4) show that the amount of the terminal-voltage drop depends on the position of the beam loss. If the right-hand side in Eq. (4) is
higher than zero or if Iloss ðN=nÞðIch IR Þ; the FEL satisfy the requirement of the terminal voltage, Eq. (1), for a CW operation. As an example, the requirement for the CW operation of our FEL is given below. The accelerator has the beam current of 2 A, Ich of 7.5 mA, and N of 10. The current IR ; for the maximum terminal voltage of 400 kV, is 0.8 mA. If the beam loss occurs only inside the terminal, the requirement for the CW operation comes to the lost beam current Iloss p6:7 mA or the recovery efficiency RX99:7%. However, in case beam loss occurs at the first electrode of the deceleration tube, the requirement is greatly reduced to Iloss p67 mA or RX96:7%. This shows that if the beam loss occurs mainly in the deceleration tube, the terminal voltage requirement for the CW operation can be satisfied although the lost beam current is much higher than the charging current of the HVG. 2.2. Operational stability of the accelerator The beam current lost in the deceleration tube changes the voltage distribution of the electrodes and can lead to an unstable operation of the accelerator. As shown in Fig. 1, the current I1 charges the electrodes of the deceleration tube while I2 discharges the electrodes. Then, the voltage differences between two adjacent electrodes, DVd1 and DVd2 ; are derived as DVd1 ðtÞ ¼ DVd0 eðt=Rd Cd Þ þ ðI1 þ ich ÞRd ð1 eðt=Rd Cd Þ Þ DVd2 ðtÞ ¼ DVd0 eðt=Rd Cd Þ ðI2 ich ÞRd ð1 eðt=Rd Cd Þ Þ:
ð5Þ
Here, DVd1 ðDVd2 Þ is the voltage difference between two electrodes located at the left (right) of the electrode where the beam loss occurs and DVd0 is the initial value of the voltage difference. Eq. (5) shows that the beam loss increases DVd1 and decreases DVd2 : Note that DVd2 can be a negative value. The modified DVd1 and DVd2 change the voltage distribution of the electrodes of the deceleration tube. The change in the voltage distribution affects the optics of the beam traveling the deceleration tube and can induce a further
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
beam loss. Then a vicious circle of the beam loss, or an increase of the beam loss with time, can arise. Increase in the beam loss rapidly drops the terminal voltage and finally prevents the FEL operation. In addition, the beam loss increases DVd1 and this value can reach the breakdown voltage between the electrodes. The breakdown interrupts an operation of the FEL. Thus, the beam loss occurring in the deceleration tube can make the accelerator unstable and disturb the operation of the FEL. We have calculated the effects of the beam loss on both the modified voltage distribution and the beam trajectory inside the deceleration tube of our FEL system. The modified voltage distribution of the electrodes is calculated by using the PSPICE code [15] and the beam trajectory is calculated by using the EGUN code [16]. Fig. 2 shows the calculated result after 400 mA hits the first electrode of the deceleration tube for 10 ms. The initial terminal voltage 300 kV is equally distributed to 10 sections of the deceleration electrodes and the capaciatance Cd is 267 pF. The terminal voltage drops just by 0.4 kV by the beam loss, which itself makes negligible effect on the FEL operation. Although the lost beam current 400 mA is much higher than the charging current 7.5 mA of the HVG, the change in the terminal voltage is not so large. This is because the position of the beam loss is the first electrode of the deceleration tube that is close to the ground. However, DVd1 increases from 30 kV to about 36 kV and DVd2 changes from 30 to 23.7 kV. The modification of the voltage distribution, as shown in Fig. 2, causes a part of the beam to reflect and hit several electrodes of the deceleration tube. Fig. 2 shows that an initial beam loss can induce a further beam Deceleration Tube
RADIUS (cm)
4 299.6
227.8
156.3
84.1
(unit : kV) 12 -23.7 0
Collector
3 2 1 0
0
5
10 15 AXIAL DISTANCE (cm)
20
25
Fig. 2. Calculation results of the beam loss effects on both the modified voltage distribution and the beam trajectory inside the deceleration tube.
405
loss. If such a phenomenon occurs in the FEL, the amount of the beam loss increases with time and then the FEL cannot be operated at a certain time after the beam loss occurs. Fig. 2 and the analysis in the previous section explain that the accelerator instability can be more serious than the requirement of the terminal voltage for a long-pulse or a CW operation of the FEL. This is because the terminal voltage requirement can be satisfied even for a large amount of the beam loss, but the beam loss can lead to the accelerator instability that prevents the FEL operation. 2.3. Maximum available pulse width of the FEL The accelerator instability caused by the beam loss limits the operating pulse width of the FEL. The voltage differences, DVd1 and DVd2 in Eq. (5), are approximated as DVd1 ðtÞEDVd0 þ
ðI1 þ ich ir Þt ; Cd
DVd2 ðtÞEDVd0
ðI2 ich ir Þt for t{Rd Cd : ð6Þ Cd
Here, ir ð¼ DVd0 =Rd Þ is the current that flows through the voltage dividing resistors of the electrodes. DVd1 is increased by a beam loss and it should be less than the breakdown voltage VBD between two neighboring electrodes for a stable operation of the accelerator. This limits the maximum available pulse width of the FEL, Ts;max ; which can be calculated by Ts;max E
Cd ðVBD DVd0 Þ : I1 þ ich ir
ð7Þ
For stable operation of the FEL, one should also avoid the process of a vicious circle of beam loss that is caused by electrons repelled back into the high voltage terminal before exiting the deceleration tube. In order to quantify the maximum allowable lost beam current satisfying this requirement, we consider the energy of an electron beam reaching the nth electrode, En : It is assumed that DVd2 ðtÞ has become negative and therefore the voltage is minimal at nth electrode, where incoming electrons may be repelled. The energy En
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is given by
299.6
! DVdi þ Vg
DEb
ð8Þ
i
where Vg is the gun voltage and DEb is the beam energy that is converted to wave energy at the undulator; DEb has the maximum value of Eb =2Nw according to Eq. (1). If En is negative, the electron beam cannot transport through the deceleration tube but reflects at the nth electrode. Then the reflected electrons are lost in the deceleration tube. This produces the vicious circle of the beam loss. To prevent this, En should be positive, which gives us the available pulse width of the FEL, TE;max ; as TE;max E
Cd ðn DVd0 þ Vg Eb =2eNw Þ : nðI2 ich ir Þ
ð9Þ
The terminal-voltage drop induced by the beam loss also limits the available pulse width of the single-mode FEL. The maximum available pulse width, Tt;max ; is obtained from Eqs. (1) and (3) as Tt;max E
CTot VT0 2Nw ðIR þ ðn=NÞIloss Ich Þ for Tt;max 5RTot CTot :
ð10Þ
The maximum available pulse width of the FEL is the minimum value of Ts;max ; TE;max ; and Tt;max : As described earlier, the instability of the accelerator can be more serious problem for a long-pulse operation of the FEL than the terminal voltage requirement. Considering the accelerator instability, we present a method to increase the available pulse width of the FEL, which is to enlarge the capacitance between the electrodes, Cd : If Cd is small, DVd1 in Eq. (6) reaches rapidly the breakdown voltage of the electrodes and DVd2 becomes rapidly negative. These can induce an unstable operation of the accelerator and shorten the available pulse width. By enlarging Cd ; we can reduce the changing rate of DVd1 and DVd2 : This improves the stability of the accelerator operation and thus increase Ts;max and TE;max : In addition, an enlarged Cd increases the total terminal capacitance CTot ð¼ CT þ Cd =NÞ and reduces the rate of the terminal-voltage drop. As a result Tt;max increases as can be found in Eq. (10). Therefore,
RADIUS (cm)
En ¼ e
n X
Deceleration Tube
4
239.3
178.9
(unit : kV)
118.5
58.1
0
Collector
3 2 1 0
0
5
10 15 20 AXIAL DISTANCE (cm)
25
Fig. 3. Calculation results of the voltage distribution and the beam trajectory for an increased capacitance Cd :
an enlarging Cd increases the available pulse width of the FEL. In order to investigate the effects of the Cd on the accelerator stability, the voltage distribution and the beam dynamics are calculated for an increased Cd of our FEL. The amount and the position of the beam loss are the same with those in Fig. 2. The capacitance Cd is increased by 1.7 nF compared to Fig. 2. Fig. 3 shows the calculated result of the accelerator behavior. DVd1 slightly increases to 30.2 kV while DVd2 decreases to 27.9 kV. The changes in DVd1 and DVd2 are much lower than those in Fig. 2 because of the increased Cd : Due to the small change in the voltage distribution, the beam passes through the deceleration tube without an additional loss. Comparing Fig. 3 to Fig. 2, we can find that the accelerator stability can be improved by increasing Cd :
3. Experimental results In order to investigate the effects of beam loss on the accelerator, we have slightly modified the FEL system. The undulator is replaced by a drift tube with an inside diameter of 40 mm and a single-stage Faraday cup has been used for a collector instead of a multi-stage-depressed collector. Beam-recovery experiments have been made for this configuration and Fig. 4 is the typical result of the experiments. The accelerator is operated in pulse with the maximum pulse width of 50 ms, which is limited by the present gun power supply. Fig. 4 shows that about 99.9% of the current emitted from the gun arrives at the collector and the terminal voltage does not change during the pulse. If we could get such
ARTICLE IN PRESS S.O. Cho et al. / Nuclear Instruments and Methods in Physics Research A 515 (2003) 402–409
Ib Ic
407
Ib Ic
-20
0
20
VT
VT
Vg
Vg
40
60
80
Time (µs)
-20
0
20
40 Time (µs)
60
80
Fig. 4. A typical result of the beam-recovery experiment. Ib is the emission current (0.5 A/div), Ic is the collection recovered (0.5 A/div), VT is the terminal voltage (1 kV/div, VT0 ¼ 300 kV), and Vg is the gun voltage (5 kV/div).
Fig. 5. The accelerator behavior when 52 mA is lost at the first electrode of the deceleration tube with Cd of 267 pF. The notations and the corresponding vertical scales are the same with those in Fig. 4.
a high recovery efficiency when the undulator is installed, we can operate the FEL in CW mode. Although the accelerator can be operated with practically no beam loss, we have intentionally induced a beam loss to investigate the effects of the beam loss. Both the position and the amount of the beam loss are controlled by the magnetic fields of two deflection magnets that are installed after the acceleration tube and before the deceleration tube, respectively. Fig. 5 shows the temporal waveforms of the current emitted from the gun, the current arriving at the collector, and the terminal voltage when a part of the beam current is forced to hit the first electrode of the deceleration tube. The amount of the beam loss, which corresponds to the difference between the emission current and the collection current, is initially 52 mA. It increases with time and becomes 330 mA at the end of the pulse. The increase of the beam loss can be explained by the previous analysis result: a beam loss changes the voltage distribution of the deceleration tube and the modification can lead to a further beam loss. Note that the terminal voltage keeps nearly constant even though the amount of the beam loss is dramatically increased during the pulse. Fig. 5 demonstrates that the vicious circle of the beam loss can occur although the terminal voltage keeps nearly constant during the pulse.
To investigate the behavior of the accelerator after the pulse shown in Fig. 5, a beam current of 330 mA is controlled to hit the first electrode of the deceleration tube. Fig. 6 is the experimental result. The collection current decreases with time and ultimately becomes ‘‘0’’ at about 35 ms. Initially only 330 mA out of the total beam current of 0.8 A is lost. However, the amount of the beam loss increases further due to the voltage redistribution of the deceleration tube and finally no beam current reaches the collector after 35 ms. The voltage of the first electrode drops rapidly and becomes negative. A small breakdown occurs at 35 ms and then a large breakdown follows. After the small breakdown the accelerator shows an unstable operation as shown in Fig. 5 and practically the accelerator cannot be operated. The terminal voltage drops just by 1 kV before the breakdown. Figs. 5 and 6 show that the main factor limiting the operating pulse width of the FEL may not be the change in the terminal voltage but the unstable operation of the FEL. We proposed that the accelerator instability could be improved by increasing Cd : In order to demonstrate the effects of the Cd ; ceramic capacitors with 1.7 nF are installed between the electrodes of the deceleration tube. Cd is increased from 267 pF to 1.967 nF. Fig. 7 shows the behavior of the accelerator with the increased Cd when the beam loss occurs at the first electrode of the
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enlarging Cd can improve the accelerator stability and thus it can increase the operating pulse width of the FEL.
Ib Ic VT V1
-20
0
20
40 Time (µs)
4. Conclusions
60
80
Fig. 6. An experimental result of the accelerator instability. V1 is the voltage of the first electrode of the deceleration tube (10 kV/div) and other notations and the corresponding vertical scales are the same with those in Fig. 4.
Ib Ic
VT
Vg
-20
0
20 40 Time (µs)
60
80
Fig. 7. An experimental demonstration for the improvement of the accelerator stability. The notations and the corresponding vertical scales are the same with those in Fig. 4.
deceleration tube. The amount of the initial beam loss is 400 mA, which is more than that of Fig. 6. On the contrary to Fig. 6, the amount of the beam loss does not increase but keeps constant during the pulse. This is because the voltages of the electrodes do not change seriously due to the enlarged Cd ; and so the beam can arrive at the collector without an additional loss inside the deceleration tube. Moreover, no voltage breakdown has been observed during the pulse. The terminal voltage changes only by 0.3 kV for such a large beam loss and it satisfies the Eq. (1) for the FEL operation. Therefore, it is confirmed that an
In the operation of the FEL driven by a recirculating electrostatic accelerator, some of the beam can be lost inside the high voltage terminal or in the deceleration tube. If a beam is lost in the deceleration tube, only a part of the lost beam current contributes to the terminal-voltage drop. So the terminal voltage can keep nearly constant although the amount of the beam loss is much higher than the supplying current of the accelerator. However, the beam loss can make the accelerator unstable and this limits the operating pulse width of the FEL. Fortunately the instability of the accelerator can be improved by increasing the capacitance between the electrodes of the deceleration tube. Consequently, if the FEL is designed that the beam loss occurs mainly in the deceleration tube, it can be operated with a long pulse width even for a large amount of the beam loss.
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[14] H.P. Freund, T.M. Antonsen Jr., Principles of Freeelectron Lasers, Chapman & Hall, London, 1992, p. 11. [15] MicroSim Corporation, PSpice Circuit Analysis, 1989. [16] W.B. Herrmannsfeldt, SLAC-331, 1988.