Beam distribution reconstruction simulation for electron beam probe Yongchun Feng,1, 2 Ruishi Mao,1, ∗ Peng Li,1 Xincai Kang,1 Yan Yin,1 Tong Liu,1, 2 Yaoyao You,1, 2 Yucong Chen,1 Tiecheng Zhao,1 Zhiguo Xu,1 Yanyu Wang,1 and Youjin Yuan1
arXiv:1610.00870v1 [physics.acc-ph] 4 Oct 2016
1
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China 2 University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China (Dated: October 5, 2016) Electron beam probe (EBP) is a new principle detector, which makes use of a low-intensity and low-energy electron beam to measure the transverse profile, bunch shape, beam neutralization and beam wake field of an intense beam with small dimensions. While can be applied to many aspects, we limit our analysis to beam distribution reconstruction. This kind of detector is almost noninterceptive for all of the beam and does not disturb the machine environment. In this paper, we present the theoretical aspects behind this technique for beam distribution measurement and some simulation results of the detector involved. First, a method to obtain parallel electron beam is introduced and a simulation code is developed. And then, EBP as a profile monitor for dense beam is simulated using fast scan method under various target beam profile, such as KV distribution, waterbag distribution, parabolic distribution, Gaussian distribution and halo distribution. Profile reconstruction from the deflected electron beam trajectory is implemented and compared with the actual one, and an expected agreement is achieved. Furthermore, Instead of fast scan, a slow scan, i.e. step-by-step scan, is considered, which lows the requirement for hardware, i.e. Radio Frequency deflector. we calculate the three dimensional electric field of Gaussian distribution and simulate the electron motion under this field. In addition, fast scan along the target beam direction and slow scan across the beam is also presented, and can provide a measurement of longitudinal distribution as well as transverse profile simultaneously. Final, simulation results for China Accelerator Driven Sub-critical System (CADS) and High Intensity Heavy Ion Accelerator Facility (HIAF) are given to investigate the quantitative behavior of EBP.
I.
INTRODUCTION
Beam profile measurement is of prime importance to all of the accelerators, especially for high intense machine, which can reveal the beam width in some locations with small aperture and further match phase space between different parts of an accelerator facility. Conventional techniques [1] for measuring beam distribution involve a very large variety of devices depending on beam particles, intensity and energy, such as scintillator screen, secondary electron emission grid and wire scanner. These equipments typically need to insert a physical object into the beam path, which themselves are destroyed easily under increasing beam intensity and in turn can result in beam loss. So, some kinds of non-interceptive profile monitors have been launched based on different principles, such as ionization profile monitor[2] (IPM), beam induced fluorescence monitor[3] (BIF) and electron beam probe (EBP). The application of charged particles as a probe beam to determine charge distribution, and therefore, the beam profile, has been raised since 1970’s[4, 5]. In those days, people utilize electron beam to diagnose plasma charge distribution. The development of this idea prompts accelerator scientists’ outlook into the potentials of this emerging technique as an alternate approach toward practically no-invasion profile monitor for accelerator beam, especially for high intensity beam. Since then,
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many laboratories around the world have been begining to study and improve EBP and obtain some very valuable results. Among these labs, Canada TRIUMF[6], Lawrence Berkeley National Laboratory (LBNAL)[7], America Spallation Neutron Source (SNS)[8–10] and Fermi National Accelerator Laboratory (FNAL)[11, 12] use electron beam as a probe to detect ion beam profile. Budker Institute of Nuclear Physics (BINP)[13–16] in Russia utilizes electron beam to measure ultra-relativistic electron bunch length and beam distribution. In addition, some labs employ ion beam as a probe to extract ion beam profile, i.e. European Organization for Nuclear Research (CERN)[17], and even some use ion beam to detect electron beam, i.e. Stanford Linear Accelerator Center (SLAC)[18]. The principle behind EBP is that a low energy, low current electron beam was injected across the target beam perpendicularly and then deflected by the target beam collective field (mainly electric field). A screen and CCD located down stream captures deflected electron beam trace, and then, by some mathematic treatment, i.e. derivative, the beam profile can be reconstructed accurately. Since the measurement should not significantly disturb the field generated by target beam, the current of electron beam must be low compared to target beam. EBP is suitable for both circular and linear accelerators and mainly devoted to high intensity beams. Two next-generation accelerator facilities, High Intensity Heavy Ion Accelerator Facility (HIAF)[19] and China Accelerator Driven Sub-critical System (CADS)[20, 21] Program, have been proposed by the Institute of Modern
2 Physics (IMP), which both have high intensity or high power, i.e. energy 1.2GeV/u with intensity 5 ×1011 ppp (particles per pulse) for HIAF and 10 MW for CADS phase I. Measuring beam parameters of these high power accelerators are challenging and usually depends on noninvasive instrument. Such a detector may provide the capacity for fine accelerator tuning and online control of beam stability. The purpose of this article is to present the study and simulation of EBP under various target beam profile distribution. Some interesting results have been achieved with fast scan and slow scan. An example is also presented to deepen the understand of EBP. This simulation is expected to provide the theory basis for test and construction of EBP in the future. II.
PRINCIPLE
EBP uses the deflection of a low energy probe beam under the target beam electromagnetic field to infer the profile information of target beam. Measuring the deflection angle as a function of different impact, one can reconstruct the beam distribution of x or y direction. Theory of profile reconstruction and validity of this theory are presented below. A.
Theory of profile reconstruction
angle between impactor parameter and x direction, and φ = 0 means incident direction is x, φ = π2 for y. In case of steady beam current, according to Maxwell-Faraday equation, electric field is curl free. ~ =0 ∇×E
(2)
Assume electron beam is injected at −x0 and ended at x0 , so, Eq.(2) actually becomes to Z
x0
−x0
Ek d k= 0
(3)
We can draw a conclusion that the net energy change along electron trajectory is zero if we utilize abovementioned hypothesis. Hence, we can assume electron beam has a constant velocity v, which is important to this theory and also is reasonable in some sense (See II B). Next, we investigate the perpendicular direction. Combining Newton’s second law of motion and some simple mathematic treatment, we obtain Z d⊥ e = (Ex cosφ + Ey sinφ)d k (4) dk mv 2 Where e and m are the electron charge and mass, respectively. For small deflection angle, Z e (Ex cosφ + Ey sinφ)d k (5) θ= mv 2 Letting φ = y direction.
π 2,
we can obtain the deflection angle along e mv 2
Z
Ey dx
(6)
dθy e = dy mv 2
Z
dEy dx dy
(7)
θy = after differentiating,
~ = Using Gauss’s law ∇E
FIG. 1. Schematic of electron beam deflection by the target beam.
Without loss of generality, assume electron beam has a tilted incident angle[22] with an impact parameter ρ, as depicted in Fig. 1. The target beam moves along z direction, centered at x = y = 0. Neglecting magnetic field, the transverse electric field can be divided into perpendicular and parallel one. E⊥ = Ex cosφ + Ey sinφ,
Ek = Ex sinφ − Ey cosφ (1)
where Ex and Ey are the horizontal and vertical components of target beam space charge electric field. φ is the
dθy e = dy mv 2
Z
σ(x,y) ǫ0 ,
we obtain
σ(x, y) d dx − ǫ0 dx
Z
Ex dx
(8)
Using Eq. (3), e dθy = dy ǫ0 mv 2
Z
σ(x, y)dx
(9)
R where σ(x, y)dx is the profile of y direction. Above formula states that the derivative of the probe beam deflection angle with respect to impact parameter gives the projection profile of beam cross-section distribution on y direction, which does the same thing as the wire scanner did.
3 B.
Validity of theory
In deriving the profile reconstruction procedure above, we introduced three important hypotheses. Firstly, we neglect magnetic field of target beam, because magnetic field is around the θ direction, which has no influence on the deflection angle along y direction. In addition, for non-relativistic beam, magnetic field is considerably small compared to electric field. We further consider that electron beam velocity remains constant throughout the scan. However, since the electric field component along x direction exerts a force on the electron, electron beam velocity will be changed slightly during its passage, although the net energy change is zero always according to symmetry. To get rid of the error due to velocity change as much as possible, electron beam energy should be much higher than the target beam potential. We also assume that the deflection angle is small, which can be achieved with high electron beam energy. Inevitable errors could be introduced as described by corresponding hypotheses. Therefore, computer simulation is ugly needed.
III.
PRODUCING PARALLEL ELECTRON BEAM
To obtain the profile of y or x direction, electron beam should be scanned along x or y with varying y or x values. The key point to reconstruct beam profile is that electron beam has to be parallel to either axis and perpendicular to target beam. In general, there exists two kinds of idea to produce parallel electron beam. Lawrence Berkeley National Laboratory adopt four dipole magnets of equal strengths to form a chicane system[7], which is similar to a bump system. By virtue of the arrangement, electron beam can be swept in the y axis while remaining parallel to x axis in the gap between the middle two magnets. Although the technology is successfully applied and ion beam profile also reconstructed. Obviously, it can’t make a fast scan and thus profile can’t be measured automatically and rapidly. This situation will change with the development of other version to produce parallel electron beam, which is shown in Fig. 2. It is an advanced configuration and used by several labs, such as SNS, BINP and FNAL. This system consists of an electron gun for electron beam generating, solenoid for electron focusing, RF deflector for fast scan, two thin quadrupoles for forming parallel electron beam, and optical image system. To separate deflected and undeflected trajectory, electrons are scanned through the target beam at a tilted angle, i.e. 45 degree. If scan is aligned vertically, one has to analyze the density distribution of projected electron beam, and through simulations, gives poor quality results[10]. The mathematic model[23] to simulate parallel electron beam is presented below. For simplicity, we regard electron beam as a point charge and no transverse mo-
mentum. We assume electrons start off at the center of RF deflector with an initial phase space coordinates at y axis, 0 y0 = (10) U y0′ 2V d x0 where U is the voltage of RF deflector, which can be adjusted from 0 to maximum to obtain various initial angle. V is the high voltage of electron gun cathode, which can be changed from 1KV to 20KV according to different target beam intensity. d is the RF deflector gap and x0 the deflector length. To observe the focusing behave at z axis, we let the initial phase space coordinates of z direction be 0 z0 (11) = 0.05 z0′ The transfer matrix along the transfer 1 1 l2 1 0 1 l3 My = K1 0 1 K2 1 0 1
line is given by 1 l1 0 (12) 1 0 1
1 l1 1 0 1 l2 1 0 1 l3 0 1 −K1 1 0 1 −K2 1 0 1 (13) where K1 and K2 are the integrated gradient of the first and second quadrupole, l3 is the distance from focus quadrupole center to screen, l2 is the distance form defocus quadrupole center to focus one and l1 is the distance from center of RF deflector to the defocus quadrupole center. According to the condition of point to parallel transport in xy plane and point to point transport in xz plane, we have Mz =
My (22) = K1 l1 (1 + K2 l2 ) + K2 (l2 + l1 ) + 1 = 0
(14)
Mz (12) = − K2 (l1 l3 + l2 l3 ) − K1 (l1 l3 + l2 l2 ) + K1 K2 l1 l2 l3 + (l1 + l2 + l3 ) = 0
(15)
From Eq.(14) and Eq.(15), K1 and K2 can be expressed as an explicit function of l1 , l2 and l3 . For quadratic equation, there exists exactly two sets of solutions, √ √ l1 + l2 l1 + l2 + 2l3 p K1 = l1 l2 (l2 + 2l3 ) p l1 l2 + l22 − l2 (l1 + l2 )(l2 + 2l3 )(l1 + l2 + 2l3 ) K2 = 2l2 (l2 + l2 )l3 (16) and √ √ l1 + l2 l1 + l2 + 2l3 p K1 = − l1 l2 (l2 + 2l3 ) p l l + l22 + l2 (l1 + l2 )(l2 + 2l3 )(l1 + l2 + 2l3 ) K2 = 1 2 2l2 (l2 + l2 )l3 (17)
4
FIG. 2. Layout of electron beam probe. 1.electron gun, 2.solenoid, 3.Radio Frequency deflector, 4.defocus quadrupole, 5.focus quadrupole, 6.target beam, 7.YaG:Ce screen, 8.CCD.
0.10
0.04 0.03 0.02 0.01 0.00 −0.01 −0.02 −0.03 −0.040.0 0.03 0.02 0.01 0.00 −0.01 −0.02 −0.030.0
0.00
y [m]
y [m]
0.05 −0.05 0.4
0.2
0.4
x [m]
0.6
0.8
1.0
0.6
0.8
1.0
0.2
0.4
0.2
0.4
x [m]
0.6
0.8
1.0
0.6
0.8
1.0
z [m]
0.2
z [m]
−0.100.0 0.008 0.006 0.004 0.002 0.000 −0.002 −0.004 −0.006 −0.0080.0
x [m]
FIG. 3. Top: electron trajectory in xy plane with the first solution. Bottom: electron trajectory in xz plane with the first solution.
FIG. 4. Top: electron trajectory in xy plane with the second solution. Bottom: electron trajectory in xz plane with the second solution.
IV.
which represent the totality of all possible cases of the system. The first solution, given by Eq.(16), indicates that the first quadrupole is the defocus one and the second is focus one in xy plane, and vice verse in xz plane. This can provide a wide range of parallel beam in scanning plane and a focused beam in another plane, as shown in Fig.3. The other solution, given by Eq.(17), also provides a parallel beam in xy plane and a focused beam in xz plane (See Fig.4), but the scan amplitude in the region of interaction is much smaller than the first one. Furthermore, in xz plane, the second solution has a large size in the region of interaction, which should be avoided to improve measurement accuracy. Therefore, the first solution seems to be better to form parallel electron beam. Actually, we focus our attention on the first case and simulation is also based on it.
x [m]
PROFILE RECONSTRUCTION WITH FAST SCAN
From Eq. (9), we know that the derivative of the probe beam deflection angle with respect to impact parameter gives the projection profile of beam cross-section distribution on y direction. When scan period is much shorter than bunch length, we can perform a fast scan, so the fine structure of bunch shape can be achieved along the bunch. However, it is difficult to design a RF deflector with such high frequency, i.e. GHz. As an alternate approach, a slow scan is considered. For details, see V. Next, we will calculate the deflection angle with a traditional approach under various beam distribution, such as KV distribution, waterbag distribution, parabolic distribution, Gaussian distribution and halo distribution. The maximum deflection occurs at the boundary for a clearboundary beam and at the 1.585σ for Gaussian beam, which is verified by simulation. The momentum change under target beam space
5
xf
xi
Ey dx ve
(18)
where xi and xf represents the initial position and final position of electron beam. Ey is the y component of target beam electric field and ve the velocity of electron. Hence, deflection angle is θy ≈
∆py px
(19)
where px is the momentum of electron beam. If we calculate out the y component of target beam electric field, the deflection angle can be easily solved, and also the derivative.
0 y [mm]
5
10
actual reconstructed
0.8 0.6 0.4 0.2 −5
0 y [mm]
5
10
FIG. 5. KV distribution. Top: deflected trajectory of electron beam. Bottom: reconstructed and actual profile of target beam.
n(x, y) =
λ , πR2
x2 + y 2 ≤ R2
(20)
where λ is the particle density per unit length. 1D realspace profile is given by 2λ n(y) = πR
0.5 y2 1− 2 R
y component of electric field is given by y 2, y < R Ze R γλ Ey = y 2πǫ0 , y≥R x2 + y 2
(21)
with
K=
−Ze γλ 4πǫ0 V
(22)
(23)
When xi → ∞, the anti tangent value becomes arctan |xyi | → π2 , so the maximum deflection angle is θmax = |Kπ| =
Ze Zγ ib γλ = 4ǫ0 V 4ǫ0 v V
20 15 10 5 0 −5 −10 −15 −20 −10 1.0
−5
(24)
where v is the velocity of target beam and ib is target beam current. The deflected trajectory of electron beam and reconstructed profile of target beam are illustrated in Fig.5.
0 y [mm]
5
10
actual reconstructed
0.8 0.6 0.4 0.2 0.0 −10
where Z is the target beam charge, ǫ0 is the permittivity of vacuum and γ the relativity factor. The deflection angle due to target beam for y > R region is |xi | , y
deflection angle [mrad]
KV distribution
KV distribution is a well known distribution, which discovered by I. Kapchinskij and V. Vladimirskij[24] in 1959. 2D real-space particle number density is defined as
θy = 2K arctan
−5
0.0 −10
normalized intensity
A.
20 15 10 5 0 −5 −10 −15 −20 −10 1.0
normalized intensity
∆py = Fy ∆t = −e
Z
deflection angle [mrad]
charge field in y direction is given by
−5
0 y [mm]
5
10
FIG. 6. Waterbag distribution. Top: deflected trajectory of electron beam. Bottom: reconstructed and actual profile of target beam. B.
Waterbag distribution
2D real-space particle number density is defined as r2 2λ 1 − 2 , x2 + y 2 ≤ R2 (25) n(x, y) = πR2 R 1D real-space profile is given by 1.5 8λ y2 n(y) = 1− 2 3πR R y component of electric field is given by 2 x2 + y 2 y Ze 1− 1− , γλ Ey = R2 2πǫ0 x2 + y 2 1, y ≥ R
(26)
y
