2.1. Transverse deflecting cavitiy

The difference between a TDC and an accelerating cavity is that the TDC gives the beam transverse force to change the transverse trajectory and have no contribution to the beam acceleration. The TDC are required in accelerators for a variety of applications (Brut, Reference Brut2012), which need a time-dependent transverse deflection of charged particles, such as RF separators, RF-cavity-based bunch length and longitudinal phase space measurements, transverse and longitudinal emittance exchange, etc.

To provide transverse force to the charged particles, the transverse electric field and transverse magnetic field or both should be given by the Lorentz force equation $ {\bf F} = q \cdot ({\bf E} + {\bf v} \times {\bf B})$ , where q is the particles charge and v is the particles velocity. For vertical force, a vertical electric field and/or a horizontal magnetic field are required. Most dipole modes in cavities have both the transverse electric and transverse magnetic field at the center. However, these two kinds of field can either add constructively or cancel each other. In order to find the transverse kicker, the Panofsky–Wenzel theorem is used to define the transverse voltage V _y to the transverse variation of the longitudinal field, shown in Eq. (1).

(1) $$V_y = \int F_y /q \cdot dz = \int (E_y + c \times B_x ) \cdot dz = - \displaystyle{{ic} \over w} \int (\nabla _t E_z ) \cdot dz.$$

As the transverse voltage is proportional to the gradient of E _z , it is clear that TE modes, which do not have longitudinal electric fields, cannot be used to provide a transverse deflection. For a pillbox cavity the TM dipole mode is used. The 6 × 6 beam transport matrix (Cornacchia and Emmma Reference Cornacchia and Emmma2002) is shown in Eq. (2) for a vertical deflecting cavity, from which the beam parameters rough evolution after TDC can be easily understood. Beam in the x-direction is not effected by TDC, like a drift; in the y-direction the position y and divergence y′ are changed by TDC with respect to beam position z. The momentum spread is determined by beam vertical coordinate y, vertical divergence y′, and longitudinal position z.

(2) $$\left[ {\matrix{ x \cr {\matrix{ {x^{\prime}} \cr {\matrix{ y \cr {\matrix{ {y^{\prime}} \cr z \cr {\rm \delta} {p} \cr}} \cr}} \cr}} \cr}} \right] = \left[ {\matrix{ {\matrix{ 1 & {l_s} & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr}} & {\matrix{ 0 & {\; \; \; \; \; \; \; \; \; 0} & {\; \; \; \; \; \; \; \; \; 0} \cr 0 & {\; \; \; \; \; \; \; \; \; 0} & {\; \; \; \; \; \; \; \; \; 0} \cr {l_s} & {Kl_s /2} & {\; \; \; \; \; \; \; \; \; 0} \cr}} \cr {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & K \cr}} & {\matrix{ 1 & K & \;\;\;\; 0 \cr 0 & 1 & \;\;\;\;0 \cr {Kl_s /2} & {K^2 l_s /6} & \;\;\;\; 1 \cr}} \cr}} \right]\; \left[ {\matrix{ x \cr {\matrix{ {x^{\prime}} \cr {\matrix{ y \cr {\matrix{ {y^{\prime}} \cr z \cr {\rm \delta} {p} \cr}} \cr}} \cr}} \cr}} \right]_0 $$

l _s is the TDC structure length and K = eV ₀ k/p ₀ c, V ₀ is the total defecting voltage, shown in Eq. (1), k is the wave number k = 2πf/c.

The principle of TDC for beam bunches split is shown in Figure 2. When an electron bunch propagates through the traveling wave structure, the electrons always see a constant phase of the deflecting force (assuming the bunch length can be neglected compared with TDC wavelength). So if one bunch starts to propagate through the structure at some TDC RF phase, when all the RF fields are zero, this bunch will not be deflected. But if one bunch comes to the structures at some other TDC RF phase, this bunch will always see the RF fields at this phase and will be deflected by a nearly constant force along the structure.

Fig. 2. The principle of TDC used for beam bunches split.

Figure 2 shows how a vertical deflector separates three bunches with short bunch interval L _B/2 to three directions. The beam bunches vertical mean position Y _B after TDC and drift distance L is defined with Eq. (3), the second bunch is on the zero-cross phase of TDC and the first and third bunches (from right to left) is on the zero-cross phase plus and minus 90° degree phase. So the second bunch is non-deflected, the first and third bunches have maximum vertical deflection in the negative and positive directions.

(3) $$Y_{\rm B} = \displaystyle{{e{\rm \;} V_{y\;} {\rm \pi} \; f_{{\rm RF\;}} L_{\rm B}} \over {{\rm \beta} ^3 \; E\; \; c\;}} \; L\;, $$

where V _y the transverse deflecting voltage, depends on TDC dipole mode fields, f _RF TDC frequency, L is the distance from TDC center to the observation position and E is the beam energy.

2.2. Twin septum magnet

The deflection force is weak in a TDC cavity because of the limited maximum RF power. Due to this small vertical kick, it needs a long drift distance for enough vertical separated distance among the three bunches. This will make the split system very long and also degrade the beam quality. In principle, a vertical dipole magnet can be added to the beam line at certain position after the TDC and it will give additional vertical kick to get enough vertical separated distance with short drift distance for one direction split bunch. Because the beam bunches after the TDC will propagate in three ways and they are still limited by small separated distance among each other, the TSM is chosen instead of the dipole. The TSM was proposed for early multi-axis advanced radiography (Wang et al., Reference Wang, Lund, Caporao, Chen and Poole1999) and beam spreader for multi-beam lines X-ray free-electron laser (Placidi et al., Reference Placidi, Jung, Ratti and Sun2014). There are also some designs and test of the TSM, which prove this method is feasible in technology. In Figure 3, the transverse cross-sectional view of two kinds design of vertical TSM (Wang et al., Reference Wang, Lund, Caporao, Chen and Poole1999; Placidi et al., Reference Placidi, Jung, Ratti and Sun2014) are shown, upper and down parts are the opposite vertical deflection areas and the center is field-free area.

Fig. 3. Two kinds of vertical TSM design and transverse cross-sectional view, the beam transport direction is toward into the paper.

Figure 4 shows the simplified sketch map of the transverse x- direction magnetic fields distribution along the y-direction in TSM. The upper and down beam pipes with uniform magnetic fields in the transverse  x-direction but opposite polarity for increasing the vertical beam deflection and the center beam pipe is field-free for non-deflected beam transport.

Fig. 4. The simplified sketch map of the magnetic field B _x distribution along the y-direction in TSM.

3.1. Beam bunches split with TDC

For beam bunches split simulation study, the 3D fields map of the TDC same as PITZ (Photo Injector Test facility, Zeuthen, DESY) TDC was used. The TDC is a traveling wave 3 GHz RF copper structure with two stabilizing holes to have the RF field mode with vertical deflecting electric field. It operates at 2π/3 modes. More details of the structure parameters and fields distribution can be found in references (Malyutin Reference Malyutin2014; Huck et al., Reference Huck, Asova, Bakr, Boonpornprasert, Donat, Good, Gross, Hernandez-Garcia, Isaev, Jachmann, Kalantaryan, Khoyoyan, Koehler, Kourkafas, Krasilnikov, Malyutin, Melkuyan, Oppelt, Otevrel, Pohl, Renier, Rublack, Schultze, Stephan, Trowitzsch, Vashchenko, Wenndorff, Zhao, Churannov, Kravchuk, Paramonov, Rybakov, Zavadtsev, Zavadtsev, Lalayan, Smirnov, Sobenin, Gerth, Hoffmann, Hunig, Lishilin and Pathak2015). The deflecting voltage is set to 2.55 MV, corresponding to 5 MW in the cavity for the simulation.

A 2.7 m long beam bunches split system is designed for 40 MeV energy beam transport. Firstly, the beam bunches split only with TDC are simulated to get the bunches separated vertically. The distance parameters shown in Figure 5 are as following: L ₀ is 0.5 m, L _s is 0.53 m, L ₁ is 1.17 m, and L is 2.7 m. The first and most important step is to define the zero-cross phase of the TDC, which means at that TDC RF phase, the beam mean vertical position after TDC is zero, which is used for synchronization for other two bunches. The beam mean vertical position as a function of the TDC phase by ASTRA simulation is shown in Figure 6 as a linear fit, by which the zero-cross phase can be calculated. In the simulation set up, the zero-cross phase for the second bunch is at 58.13°. This method is also useful in the experiment to define the zero-cross phase in line.

Fig. 6. Beam mean vertical and horizontal positions as a function of the TDC phase at z = 2.2 m position.

If the bunches 1 (red) and 3 (blue) can be deflected to the opposite side with maximum deflection angle, the time relationship of the bunches interval t and the TDC frequency should satisfy Eq. (4) with respect to the zero-cross phase.

(4) $$2\pi f_{\rm tdc} \cdot t = \left\{\matrix{\left(\displaystyle{2N + 1 \over 2} \right)\cdot \pi, &\hbox{negative maximum deflection} \cr N\cdot \pi, &\hbox{no deflection} \cr \left(\displaystyle{2N - 1 \over 2} \right) \cdot \pi, &\hbox{positive maximum deflection},} \right.$$

where N is the integer.

The time interval of bunches in a bunch train depends on the laser frequency and the LINAC. Therefore the TDC frequency can be selected to match the beam bunches period or change bunches time interval to match the TDC frequency. In the simulation study, the TDC frequency is 3 GHz and the shortest bunches interval is calculated as t = N/4f _tdc, when N is one, t equals to 83.3 ps. So with 3 GHz TDC the beam bunches interval longer than 83.3 ps times N (N is the integer) can be separated into one of the three directions.

The first (red) bunch is in minus 90° with respect to TDC zero-cross phase, and the third (blue) bunch is in plus 90° with respect to TDC zero-cross phase in simulation. The simulation was done by change the TDC phase with respect to the zero-cross phase, which is equivalent to the bunches with corresponding time interval t with respect to the z position for zero-cross phase bunch (second one). The beam vertical mean position at positive maximum deflection is 2.58, 8.68, and 14.78 mm at z position of 1.2, 2.2, and 3.2 m, respectively. The beam parameters after TDC at three positions are shown in Table 1. From the simulation results, it is clearly shown that, with long drift distance the larger vertical separated distance can be achieved, but the beam energy spread and emittance are also increased. For the non-deflected beam, the y-direction emittance is changed because of the the beam y–z distribution exchange due to deflection (the non-deflection means after TDC the vertical beam mean position is zero, but the bunch head and tail are still deflected).

Table 1. Beam parameters after TDC at three z-positions without TSM

3.2. Beam bunches split with TDC and TSM

From the above simulation, at z = 3.2 m, the separated bunch from the positive or negative directions to center is 14.78 mm. For further beam transport to the target, the space is not enough to put other focusing or bending magnets. Hence, the TSM is added after 1.17 m distance of the TDC, at z = 2.2 m, shown in Figure 5.

The magnetic fields distribution of TSM for simulation is shown in Figure 7, with uniform fields and fringe fields but opposite polarity field in upper and down bending area. The amplitude is 0.1 T. The gap distance of the field-free area is 10 mm, which is constrained by the upper and down beam vertical position at the entrance of the TSM at 2.2 m and the space for the septum plate (thickness is about 2 mm with or without some magnetic shielding material). The same simulation procedure as only with TDC was done, and the vertical mean beam positions along the beam line is shown in Figure 8. At z = 3.2 m position, the vertical separated distance from the positive or the negative to the center is 85.32 mm.

Fig. 7. The magnetic fields B _x distribution of the TSM in the z–y-plane used for simulation.

Fig. 8. Simulated beam vertical mean position along the beam line with TDC and TSM for three TDC phases.

The overall beam distribution and phase space of the upper split beam with TDC and TSM at z = 3.2 m is shown in Figure 9 and the beam parameters are in shown Table 2, compared with the beam split without TSM. It is indicated that the separated distance is drastically increased with TSM. The TSM will induce energy dispersion in the beam line, so more attention should be paid for the subsequent beam transport and beam matching line (BT&ML 1, 2, and 3 in Fig. 1) design for eRad, which is not included in this paper. In the simulation, the non-deflected beam parameters are almost same with TSM and without TSM.

Fig. 9. The overall phase space and particle distribution of the positive vertical deflected bunch with TDC and TSM at z = 3.2 m in local cordinate.

Table 2. The beam bunch parameters at z = 3.2 m position with and without TSM

From particle-tracking simulation, the beam parameters can be gotten at arbitrary positions. This will be useful for subsequent beam transport design in reference coordinate with Courant–Snyder parameters and also proved that this design and simulation method is helpful for the whole eRad system development. The emittance of non-deflected beam in the y-direction is increased a lot, which can be restrained by other y-plane focusing quadrupoles in further beam transport beam line.