Consideration of fringe field overlap of quadrupoles in a compact THz-FEL prototype

Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489

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Consideration of fringe field overlap of quadrupoles in a compact THz-FEL prototype✩ Qushan Chen ∗, Bin Qin State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

ARTICLE Keywords: Compact accelerator Fringe field Beam matching

INFO

ABSTRACT Huazhong University of Science and Technology has built up a compact terahertz free electron laser prototype. In the transport line, three quadrupoles with aperture of 45 mm are densely packed and the iron to iron distance are merely 148 mm (Q1-Q2) and 123 mm (Q2-Q3) respectively. This compact arrangement inevitably causes fringe field overlap between the quadrupoles. As the iron to iron induced fringe field reduction has been studied extensively, here we focus on the change in focusing properties of the quadrupole triplet when the fringe field overlap is presented. The three quadrupoles have been measured independently with a Hall probe system and the measurement data are used to do the slice analysis. Comparison among the conventional hard-edge model, the equivalent hard-edge model and the slice model is addressed to evaluate the influence of the fringe field overlap. It is found that the equivalent model is a much better approximation than the conventional model though it has slight difference compared to the slice model. The equivalent model is preferred for beam matching calculation during a physical design phase and the slice model is beneficial for current adjustment in beam commissioning.

1. Introduction Huazhong University of Science and Technology (HUST) has built up a compact terahertz free electron laser (THz-FEL) prototype [1,2]. All elements in the prototype are densely packed due to the limited space. The compact arrangement is space saving, but it also causes some engineering challenges and physical problems. One important problem is the fringe field interference between the adjacent quadrupoles. The transport line of the HUST THz-FEL prototype is designed to match the beam from the upstream linac to the downstream undulator. It mainly consists of a quadrupole triplet and a double bend achromat (DBA) section. The quadrupole triplet assembly (Q1, Q2 and Q3) is shown in Fig. 1. All the three quadrupoles are the same type. Their aperture, iron length and effective length are 45 mm, 77 mm and 100 mm, respectively, which gives an aspect ratio of 1.71. The distance between the centers of Q1 and Q2 is 225 mm, and that of Q2 and Q3 is 200 mm. During the design phase, all the magnets were considered to be independent of each other and were treated with the conventional hard-edge model, which used the quadrupole field index 𝑘 in the center as the field index 𝑘0 of the whole quadrupole. After the factory manufacture, the magnets were measured off line before installation

and the results were consistent with their design values [3]. But in beam commissioning, the designed working points of the quadrupoles did not give the matching results as predicted. So we intend to figure out possible reasons for the discrepancy. Conventionally, the study of fringe field effect usually refers to dipoles [4]. But recently more and more attention has been focused on the quadrupole related fringe field effect, especially in compact cases [5–8]. The quadrupole related fringe field effect usually involves three aspects. The first is the difference between the conventional hardedge model and the real field distribution. In most cases, this difference is negligible, but we can still solve the equivalent transfer matrix of a quadrupole to achieve better accuracy [9]. The second one, which has been studied extensively, is the field reduction induced by iron-to-iron crosstalk [10–12]. This effect can make a difference in many situations, such as a quadrupole and a corrector, a quadrupole and a quadrupole, or a quadrupole and a sextupole, where two magnets with yokes are close to each other. The third one is the fringe field overlap between two adjacent quadrupoles. The fringe field overlap is also caused by close arrangement of magnets, but it mainly results in distortion of focusing property while has nothing to do with field reduction. As the adjacent iron to iron induced field reduction has been reported in many literatures, this paper mainly focuses on the influence

✩ This work is supported by the National Natural Science Foundation of China (No. 11705062) and the China Postdoctoral Science Foundation (No. 2018T110763). ∗ Corresponding author. E-mail address: [email protected] (Q. Chen).

https://doi.org/10.1016/j.nima.2020.163489 Received 10 November 2019; Received in revised form 31 December 2019; Accepted 20 January 2020 Available online 27 January 2020 0168-9002/© 2020 Elsevier B.V. All rights reserved.

Q. Chen and B. Qin

Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489

Fig. 1. The quadrupole triplet assembly in the HUST THz-FEL prototype transport line. The beam direction is from Q1 to Q3. The distance between the quadrupoles is referred to the effective length.

of fringe field overlap, although the two effects are often concomitant. Actually, prior analysis of our machine has indicated that the iron to iron cross talk would cause variation in the transfer matrix of a single quadrupole on the level of 0.2%, which is too small to be considered [13,14]. Besides, nonlinear effects are not included in our discussion because the transport line is relatively short in the HUST THz-FEL prototype. So the content is arranged as follows. First the equivalent transfer matrix of each quadrupole in the transport line is proposed and calculated, based on which the difference between the conventional hard-edge model and the equivalent hard-edge model is discussed. Then we analyze how the focusing property of the quadrupole assembly is changed due to the fringe field overlap. The conclusions are summarized in the end.

Fig. 2. Measured field distribution of Q1 at different transverse positions when the excitation current is 6.5 A. 𝑥-axis corresponds to the horizontal plane.

placed upstream and downstream Q1 to make up a 0.4 m assembly. The transfer matrix of this assembly in the horizontal plane is [ ] 0.3043 0.2630 𝑀con = . (1) −3.4500 0.3043 As for the slice model, larger slice number results in better accuracy but also more computation time. In our case, 400 slices, which indicates an interval of 1 mm, is enough to accurately characterize the real falloff distribution [6]. With the slice model, the transfer matrix of the Q1 assembly ([−0.2 m, 0.2 m]) in the horizontal plane is [ ] 0.3099 0.2653 𝑀slice = . (2) −3.4077 0.3099

2. Fringe field of a single quadrupole 2.1. Conventional hard-edge model and slice model

In this example, the change in the defocusing strength 𝑚21 is 1.2%, relatively small in common sense. But it should be noted that the change is varied with the excitation current. Furthermore, the change will accumulate along the transport line. So it would be meaningful to find a more accurate model to characterize the real fall-off distribution.

For a real quadrupole, it always has fringe fields, in which region the focusing components decrease to zero smoothly. The conventional hard-edge model transforms a real fall-off distribution into a pulse shape. The conventional hard-edge model is a handy method and has been widely used during physical design phases to estimate system parameters. But there is always difference between the conventional hard-edge model and the real fall-off distribution. The slice model, as long as the slice number is large enough, is an accurate method to characterize the focusing property of a real distribution. As the magnets in the HUST THz-FEL prototype have been tested before installation, the measured magnetic field data are used in our analysis. The magnets were measured with a Hall probe with resolution of 0.6 Gauss and the magnetic field data were sampled at every 0.1 mm along the longitudinal direction (𝑧-axis) [15]. For example, Q1 was measured at different transverse positions (Fig. 2). It can be found from Fig. 2 that the transverse components decrease well down to the noise level where is 0.2 m away from the center. So in the following discussion, all analyses are based on the measurement data and [−0.2 m, 0.2 m] referred to the center is taken as the dominant region of each quadrupole. At each longitudinal point, the field gradient is calculated by linear fitting of the 𝐵𝑦 components at different 𝑥 positions. Then the real falloff distribution in the dominant region can be obtained. For the case in which the excitation current is 4 A and the beam energy is 14 MeV, the conventional hard-edge model gives a quadrupole field index 𝑘0 of 36.088 m−2 and an effective length 𝐿 of 101.8 mm. To make a suitable comparison, two drift tubes with equal length of 149.1 mm are

2.2. Equivalent hard-edge model Assuming there is another quadrupole assembly that has the similar configuration with the conventional hard-edge model, there must exist an equivalent quadrupole characterized by another set of 𝑘0 and 𝐿 that satisfy Eq. (2), i.e. 𝑀equ = 𝑀slice , where 𝑀equ is the consecutive multiplication of the transfer matrices of the upstream drift, the equivalent hard-edge quadrupole in the middle and the downstream drift. Then it comes to √ √ − 𝑘0 sin(𝐿 𝑘0 ) = 𝑚21 √ √ √ 1 cos(𝐿 𝑘0 ) − (𝐿𝑎 − 𝐿) 𝑘0 sin(𝐿 𝑘0 ) = 𝑚11 , (3) 2 where 𝐿𝑎 is the length of the assembly; 𝑚11 and 𝑚21 correspond to the transfer matrix elements obtained from the slice model. Substituting 𝐿𝑎 = 0.4 m, 𝑚11 = 0.3099, 𝑚21 = −3.4077 into Eq. (3), the two parameters can be solved as 𝑘0 = 29.455 m−2 , 𝐿 = 125.1 mm. The difference is obvious between the conventional model and the equivalent model. Fig. 3 displays the comparison between the two models. The equivalent model gives a smaller focusing strength but longer effective length than that of the conventional model. It is important to point out that the real fall-off distribution should be symmetric to its center if the equivalent 2

Q. Chen and B. Qin

Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489 Table 1 Effective length of Q1, Q2 and Q3 under the two hard-edge models. The unit is mm. Quad.

Designed 𝐿

Conventional 𝐿

Equivalent 𝐿

Q1 Q2 Q3

100 100 100

101.8 101.9 102.1

125.1 125.4 126.4

Fig. 3. Quadrupole field index of Q1 under the conventional hard-edge model, the equivalent hard-edge model and the real fall-off distribution when the excitation current is 4 A and the beam energy is 14 MeV.

Fig. 5. Superimposed distribution of 𝑘 of the triplet assembly when excited at (3 A, −6 A, 4 A). The edges under the conventional hard-edge model are also drawn.

matrix calculated from the slice model, we can always solve the corresponding equivalent hard-edge model, which can exactly represent the focusing property of the real fall-off distribution. 3. Fringe field overlap in a multi-quad assembly Following the above discussion, it is reasonable to infer that the equivalent model can also exactly characterize the focusing property of a multi-quadrupole assembly, as long as there is no fringe field overlap. But in compact accelerators, this is usually not the case. Fig. 5 shows the superimposed 𝑘 distribution of the quadrupole triplet in the HUST THz-FEL prototype when excited at (3 A, −6 A, 4 A). The distribution is the linear superposition of individual fields of each quadrupole based on their longitudinal positions in the transport line. Table 2 lists the 𝑘0 and 𝐿 of each quadrupole under the conventional model and the equivalent model respectively. According to the results in Table 2, the quadrupoles have no overlap when described under the two hard-edge models. However, it is easy to find from Fig. 5 that the real falloff distribution overlaps. In this case, it is ambiguous to define the dominant region of each quadrupole. One possible way to keep the equivalent model accurate is dividing the superimposed distribution into three segments according to the two zero points in the middle and then solving the equivalent transfer matrix of each segment. In this case, the three segments are no longer symmetric to their own centers, which are usually set to the centers of the quadrupoles. As stressed in the above section, there would be no simple solutions. To evaluate the fringe field overlap induced distortion in focusing property, our analysis is carried out as follows. First the transfer matrices of the triplet are calculated under the two hard-edge models, in which cases the three quadrupoles are treated independently. Then the slice operation is applied to the superimposed distribution and the transfer matrix is calculated accordingly. It is for sure that the slice model gives the accurate result. The difference in the two hard-edge models reflects the fringe field overlap induced distortion.

Fig. 4. 𝑘0 − 𝐼 curves of Q1 under the conventional model and the equivalent model.

model is applied. The transfer matrix of a symmetric distribution is characterized by 𝑚11 = 𝑚22 . Then the upstream and the downstream drifts in the equivalent model can be set to equal length. Otherwise, there is no simple solution to Eq. (3). During beam commissioning, the 𝑘0 − 𝐼 curve is always needed to find the appropriate current of a quadrupole. The general method is to first calculate the quadrupole field indices under several currents and then do linear fitting. Fig. 4 displays the 𝑘0 − 𝐼 curves of Q1 under the two hard-edge models. It is certain that the curve calculated by the equivalent model gives a more accurate result in focusing properties. The same procedure can be applied to Q2 and Q3 and the results are summarized in Table 1. The uniformity of the three quadrupoles is acceptable. With the simulated or measured fall-off distribution of a quadrupole, the focusing property can be well characterized by the slice model, as long as the slice number is large enough. Then based on the transfer 3

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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489

Table 2 𝑘0 and 𝐿 of Q1, Q2 and Q3 under the conventional model and the equivalent model. The excitation currents are 3 A, −6 A, 4 A respectively. Conventional

Q1 Q2 Q3

Equivalent

𝑘0

𝐿

𝑘0

𝐿

27.066 −53.828 35.935

101.8 101.9 102.1

22.037 −44.057 29.105

125.3 125.0 126.4

Fig. 7. Transverse phase space at the end of the triplet assembly under the three different models. The emittance of both planes is set to be 20 μm rad.

which represents the focusing or defocusing strength of the assembly, for example. The difference between the conventional model and the slice model is 1.9%, while that between the equivalent model and the slice model is reduced to 0.5%. The greatest difference comes from the term 𝑚11 . The difference between the conventional model and the slice model is 28.0%, while that between the equivalent model and the slice model is 3.7%. So the results in Eq. (4) indicates that both the conventional model and the equivalent model lead to a more serious distortion in beam size than that in beam divergence. Please note that this conclusion corresponds to the 𝑥 plane. It should be also noted that the difference among those matrix terms is varied with the excitation currents of the quadrupoles. Whether the beam size or the beam divergence is more sensitive to the choice of the models should be determined individually. To better understand the beam optics in the quadrupole triplet, the transfer matrix-based accelerator design code, MAD [16], is used to simulate the beam parameters under the three models. The initial beam parameters are 𝛽𝑥0 = 50 m, 𝛽𝑦0 = 20 m, 𝛼𝑥0 = 0, 𝛼𝑦0 = 0. The current settings are the same as Fig. 5. Fig. 6 displays the beta (𝛽) and gamma (𝛾) functions in the triplet. The flat portion in the gamma function corresponds to the drift space (𝑘 = 0) under the two hard-edge models. For the 𝑥 plane, there is no obvious difference in the gamma function at the end of the triplet, while the beta function given by the slice model is twice that given by the

Fig. 6. Beta and gamma functions in the triplet assembly. The subscript 𝑥 and 𝑦 indicate the horizontal and the vertical plane. The subscript ℎ1, ℎ2, 𝑠 indicate the conventional model, the equivalent model and the slice model.

Based on the results in Table 2 and the superimposed distribution in Fig. 5, the transfer matrices of the triplet under the three models are calculated. [ ] 0.0847 0.7015 𝑀con = , −1.4655 −0.3315 [ ] 0.1219 0.7231 𝑀equ = , −1.4305 −0.2819 [ ] 0.1176 0.7188 𝑀slice = . (4) −1.4376 −0.2838 Eq. (4) only shows the transfer matrices in the 𝑥 plane and the results in the 𝑦 plane can be easily calculated accordingly. It can be found from Eq. (4) that the equivalent model gives a more accurate result than the conventional model, but there is still slight difference between the equivalent model and the slice model. Take the term 𝑚21 , 4

Q. Chen and B. Qin

Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489

Fig. 8. OPERA3D model of the triplet assembly and the simulation results. From right to left: Q1, Q2, Q3. The current configuration is (3 A, −6 A, 4 A).

conventional model. This is consistent with the results of Eq. (4). For the 𝑦 plane, the situation is just reversed. Fig. 7 displays the transverse phase space at the end of the triplet. The distortion of the conventional model is obvious, especially in the 𝑦 plane. Compared to the conventional model, the equivalent model is a much better approximation to the real distribution, but there is still considerable difference in some parameters. For example, the 𝑥 beam sizes given by the equivalent model and the slice model are 4.09 mm and 3.75 mm respectively and the corresponding difference is 9.1%. This indeed reflects the fringe field overlap induced distortion. 4. Consideration of the iron-to-iron cross talk All the above discussions are based on the data acquired from the Hall probe measurement. Especially, the superimposed 𝑘 distribution of the triplet assembly in Fig. 5 is a linear superposition of the three individual fields measured independently, which excludes the influence of the iron-to-iron cross talk. So, it is necessary to assess the impact of the cross talk effect. OPERA3D [17] is used to model and analyze the triplet assembly. To make a suitable comparison, the current configuration in OPERA is the same as that in Fig. 5 and the corresponding simulation result is shown in Fig. 8. The maximum surface magnetic flux density is 0.5723 T, which is far from saturation. The superimposed 𝑘 distribution of the assembly can also be obtained and it gives the transfer matrix (𝑥 plane) as [ ] 0.1129 0.7199 𝑀OPERA = . (5) −1.4326 −0.2767

Fig. 9. The 𝑘 distribution of Q2 in Case 1 (solid line) and the difference of 𝑘 between Case 1 and Case 2 (the dash line). The excitation current of Q2 is −6 A and the winding number is 90. Please note the 𝑘 is plotted in its absolute value for clear presentation.

of magnitude lower than the central 𝑘. The peak corresponding to the yoke of Q3 is higher than that of Q1, because the distance between Q3 and Q2 is 25 mm shorter than that between Q1 and Q2. The transfer matrices (𝑥 plane) of the two cases can also be calculated as

Compared with the last matrix in Eq. (4), the relative difference of 𝑚11 , 𝑚12 , 𝑚21 and 𝑚22 are 4.5%, 0.15%, 0.35% and 2.5%. The difference may come from the iron-to-iron cross talk, numerical error or measurement error et al.. A more reasonable way to investigate the impact of the iron-to-iron cross talk is to compare the 𝑘 distributions of Q2 with and without the yokes of Q1 and Q3. We model two cases: in Case 1, the yokes of Q1 and Q3 are set to the real ferromagnetic material; in Case 2, the yokes of Q1 and Q3 are set to Air. In both cases only Q2 is powered with current of −6 A. The simulation results are shown in Fig. 9. The two peaks in Fig. 9 indicate the yokes of Q1 and Q3 reduce the fringe field of Q2 in some degree, but they are three orders

[ 𝑀Case1 = [ 𝑀Case2 =

3.4273 6.1106

1.7581 3.4262

3.4324 6.1265

1.7598 3.4324

] , ] .

(6)

It is found that the relative difference is no more than 0.26% for all elements, indicating very small influence from the iron-to-iron cross talk. 5

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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163489

5. Summary and comments

Synchrotron Radiation Laboratory for setting up the magnetic field measurement system and providing professional suggestions on the data analysis.

The conventional hard-edge model, which uses the 𝑘 in the center as the 𝑘0 of that quadrupole, is a handy method and has been widely used in physical design phases. But it may cause distortion in some degree when the fringe field effect cannot be neglected. To overcome this problem, the equivalent hard-edge model is proposed. The equivalent model is an exact representation of an element as long as its fringe field is not affected by others. But in a compact accelerator, such as the HUST THz-FEL prototype, there is usually fringe field overlap between quadrupoles. In this case, the conventional model may result in serious distortion. The equivalent model is a much better approximation to the real distribution but it still has slight distortion. In accelerator engineering, beam matching calculation during physical design and current adjustment during beam commissioning are the two inevitable procedures. For compact accelerators and quadrupoles not operated in saturation, the iron to iron cross-talk is insignificant while the fringe field overlap may still exist. Then it is suggested to do beam matching calculation with the equivalent model for its better accuracy. The slice method is not suitable for beam matching calculation because there are too many unknown 𝑘0 to be determined. But the slice model is a good choice for optics verification and current adjustment later on.

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Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Qushan Chen: Conceptualization, Data curation, Formal analysis, Writing - original draft. Bin Qin: Supervision. Acknowledgments The authors would like to thank the accelerator engineers, N. Chen, L. Shang, X.H. Yang, Y.J. Pei, X.Q. Wang et al. from the National

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