Preamble

Superconducting solenoid optimization and field measurements S. Ma, A. Arnold, P. Michel, P. Murcek, A. Ryzhov, J. Schaber, J. Teichert, and R. Xiang 1 National Key Laboratory of Science and Technology on Advanced Laser and High Power Microwave, Institute of Applied Electronics, China Academy of Engineering Physics, Mianyang 621900, China Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany A solenoid is typically used in normally conducting and superconducting radio frequency (SRF) photoinjec- tors to compensate for the projected transverse beam emittance. In the ELBE SRF Gun-II, a superconducting solenoid is positioned inside the gun cryomodule approximately from the end of the gun cavity. The spherical aberration and multipole field effects caused by offset and tilt limit the reduction in beam emittance for high bunch charges. We designed a novel superconducting (SC) solenoid with a lower spherical aberration coefficient. In the simulation, the beam emittance from the spherical aberration decreased by 47%. Both the longitudinal and transverse fields were measured and analyzed using the formalism fitting method to assess the performance of the SC solenoid within the cryomodule and its influence on the beam transverse emittance.

Keywords

Superconducting solenoid, magnetic field, photoinjector, SRF gun.

INTRODUCTION

Over the past decade, a superconducting radiofrequency (SRF) electron source has demonstrated the capability to pro- duce high-quality electron beams, specifically for the “Elec- tron Linac for Beams with High Brilliance and Low Emit- tance (ELBE),” which is a user facility that generates sec- ondary radiation for a wide range of applications [ ]. The SRF gun, which primarily operates in continuous wave mode at repetition rates of 100 kHz , has been instrumen- tal in producing terahertz radiation between

5 THz

which is characterized by high pulse energy and significant average power. Equipped with a 3.5-cell niobium cavity res- onating at

3 GHz

, the SRF gun achieves an acceleration gradient of

8 MV

, corresponding to an on-axis peak elec- tric field of

5 MV

The SRF gun can deliver elec- tron bunches with up to 300 pC bunch charge utilizing ei- ther magnesium (Mg) or cesium telluride ( ) cathodes in conjunction with a 262 nm laser [ ]. This technology shows significant promise for applications in neutron time- of-flight experiments, high-repetition-rate ultrafast electron diffraction, high-power terahertz experiments, and Thomson backscattering experiments that require high bunch charge and repetition rates [ To control the electron beam size and compensate for pro- jected transverse emittance, a superconducting (SC) solenoid is installed 0.7 m downstream from the cavity end within the SRF gun cryomodule [ ]. This distance represents a balanced compromise between the need for magnetic field shielding of the cavity and minimizing the beam emittance.

However, a solenoid introduces a new source of emittance owing to spherical aberration, which is linear with respect to the spherical aberration coefficient and scales with the fourth power of the beam RMS size [ ]. The spherical aber- rations of the solenoid can significantly increase emittance for beams with relatively large diameters [ ]. Additionally, measurements indicate that beam aberrations increase with higher solenoid currents, primarily owing to multipole mag- netic field effects arising from the tilt and offset position er- rors of the solenoid [ ]. This configuration imposes lim- itations on applications such as X-ray free-electron laser fa- cilities, which require a lower emittance and higher bunch charge [ The current SRF gun in operation at ELBE (SRF Gun-II) is part of a developmental progression in SRF gun research and development, which will soon advance to the third-generation SRF Gun-III [ ]. The primary objectives of this new generation are to increase the available bunch charge for tera- hertz production and reduce the transverse emittance through enhanced emittance compensation, correction of multipole magnetic field effects, and mitigation of solenoid aberrations.

A newly redesigned solenoid was constructed to achieve these goals.

Upon completion of the SRF Gun-III cryomodule, magnetic field measurements were conducted to evaluate the performance of the new solenoid. A formalism fitting method was employed to analyze the multipole fields, and the asso- ciated measurement errors were quantified. We also assessed the impact of these fields on the beam emittance. The sec- ond section of this paper describes the basic concept and field measurement method. The influence of the multipole com- ponents, such as the quadrupole and sextupole, is analyzed in this section. A comparison of the new and old designs of SC solenoids is presented in the third section. The fourth sec- tion presents the results of the field and error analyses. The fifth section analyzes the influence of the multipole fields on the beam transverse emittance and solutions. The conclusions are presented in the final section.

BASIC PRINCIPLE Spherical aberration of solenoid Magnetic lenses can exhibit imaging errors, even when the momentum distribution of the particles is negligible. These errors arise because of spherical aberrations that occur as a result of nonlinear magnetic field regions within the lens and

particle rays that deviate from the paraxial approximation.

Spherical aberration is dependent on the third power of the radial position of the particle within the solenoid and can be represented using two distinct integral expressions [

C s = 1

The transverse emittance resulting from solenoid spherical aberration can be calculated as

ε sph = r 4 0 2 √

C 1 = C s

C 2 = 5

On the right side of Eq. 2 , the higher-order terms are sig- nificantly smaller than the leading term. Thus, the spheri- cal aberration emittance scales with the fourth power of the beam RMS size at the solenoid entrance.

In high-bunch- charge and short-bunch-length injectors, the beam RMS size is typically large, which amplifies the effect of spherical aber- rations. Therefore, decreasing the spherical aberration coef- ficient is crucial to mitigate its effect on beam quality. As indicated by Eq. 1 , reducing the spherical aberration coeffi- cient requires minimizing the integral of the first derivative of the solenoid field at the same field integral.

Solenoid axis and multipole field measurement The pulsed wire method is a basic method for analyzing the alignment of magnets [ ]. In this method, a wire is fixed between two points using an external field. A current pulse is applied to the wire. If the external field has a transverse com- ponent, the local wire will move owing to the Lorentz force, which will propagate to two endpoints as traveling waves and be measured by motion sensors. The amplitude of the motion is related to the transverse magnetic field integration along the z-direction and pulse current magnitude. The axis of the external field can be obtained within resolution by analyzing the motions. For solenoids, the other method uses the properties of the solenoid field, in which the longitudi- nal component on the axis is the extremum along the radial direction and the radial component is zero, as follows:

∂B z ( r, z ) ∂r = 0

B r (0 , z ) = 0 . (4)

This method has a worse resolution than the pulsed wire but is concise for the SC solenoid. This is because only one end face of the tube is open in the cryomodule and constructing the wire is challenging.

An important method for analyzing the multipole magnetic field for multipole component measurement is the harmonic coils and wires approach, which was developed by Carlo et al. at CERN LHC [ ]. The basic concept relies on Fara- day’s law of induction; as the magnetic field changes, it in- duces an electromotive force in the coils, enabling accurate determination of the multipole field components. The Fourier coefficients of the magnetic flux function correspond to the strengths of the multipole components of the field.

Although this method provides highly accurate field mea- surements, it requires a multifaceted control system and ad- vanced electrical analysis tools. This approach is impractical for accelerator facilities that do not have access to dedicated field-measurement laboratories. To streamline the measure- ment of multipole modes without specialized equipment, we use a 2D polynomial fitting technique based on data obtained from 3D Hall probe measurements. Drawing on the defini- tions of the dipole, quadrupole, and sextupole components outlined in [ ], we proceed with the following approach:

B dn = 0 e x + J dn e y B ds = − J ds e x + 0 e y B qn = J qn y e x + J qn x e y B qs = − J qs x e x + J qs y e y

B sn = J sn xy e x + 1

2 J

where the field coefficients are denoted by . The subscripts “d,” “q,” and “s” denote the dipole, quadrupole, and sextupole parts, respectively, whereas “n” and “s” specify whether the mode is of the normal or skew type. In the case of solenoids, the transverse magnetic field can be expressed as

B t = J t x e x + J t y e y . (6)

Thus, the horizontal and vertical fields can be expressed as

B x = − J ds + J qn y + ( J t − J qs ) x +

The Hall-effect sensor can accurately measure the horizontal and vertical components of the transverse magnetic field. The coefficients for various multipole modes (e.g., quadrupole and sextupole) can then be derived by applying a numerical fitting algorithm to the functions provided in Eq.

Influence of multipole fields on beam transverse emittance The multipole fields resulting from misalignment and man- ufacturing errors in the solenoid can affect the beam shape

and emittance. Because the dipole field components primar- ily cause beam deflection, we focus on the effects of the quadrupole and sextupole components. Details on the influ- ence of these fields and the methods for their correction can be found in [ The 4D phase–space transport matrix for an ideal quadrupole lens under the thin-lens approximation followed by a solenoid lens can be expressed as follows:

M sol M quad =

Here, represents the effective length of the solenoid, de- fined as , where is the longitudinal magnetic field as a function of the position , and is the peak mag- netic field along the solenoid axis.

The parameter is the electron charge, is the electron mass, is the speed of light, is the relativistic factor, and is the normalized velocity of the particle.

A normal quadrupole lens positioned at the entrance of the solenoid provides focus with a focal length . After the beam passes through the normal quadrupole and solenoid, the re- sulting beam matrix takes the following form:

σ ( s ) = M sol M quad σ (0) (M sol M quad ) T . (9)

The additional transverse emittance resulting from the com- bined effects of the quadrupole focus and solenoid coupling between transverse planes is expressed as sin 2 Here, is the beam-normalized emittance. If the quadrupole field has a rotating angle , the resulting transfer matrix in- corporates cross-plane coupling terms between the - and directions and is expressed as

M rotquad ( α 1 , f ) =

and Eq. changes to sin 2( To mitigate the emittance growth caused by the quadrupole component parasiting the solenoid, a pair of correction quadrupoles consisting of a normal and skew quadrupole are installed downstream of the solenoid at a distance . These quadrupoles work together and have a rotation angle

ε n, total = βγ ���� σ x, sol σ y, sol f sin 2 ( KL s + α 1 )

sin(2 Equation demonstrates that the effectiveness of the cor- rector is strongly dependent on the beam RMS size at the corrector position. In practical applications, the distance be- tween the solenoid and corrector is a key parameter. In the SRF gun configuration, the corrector is positioned 437 m downstream of the solenoid, directly outside the cryomod- ule. Using ASTRA, a beam dynamics simulation tool [ we performed simulations and compared the results with the theoretical predictions yielded by Eqs. . Figure presents the simulation results for different distances. The simulations revealed that, when the distance parameters were fixed, the insufficient corrector focal strength failed to coun- teract the quadrupole field from the solenoid properly. This indicates that, for identical parasitic quadrupole strengths in both the solenoid and corrector, shorter distances result in more effective emittance compensation. However, if the dis- tance is too long, as shown by the case (blue line) in Fig. , the corrector either fails to cancel the quadrupole field of the solenoid or requires a significantly stronger focal strength to achieve the same effect.

According to Eq. , the beam size at the corrector is al- ways smaller than that at the solenoid because the solenoid focuses the beam, and the corrector is positioned within the focal length of the solenoid. Theoretically, if the corrector is placed beyond the focal length of the solenoid, where the beam RMS size is equal to or exceeds that at the solenoid, the emittance compensation improves for a given quadrupole strength. However, because the focal length of the solenoid varies depending on the specific operational parameters, the corrector cannot compensate for the emittance growth that occurs between the solenoid and corrector.

Therefore, to achieve optimal emittance compensation, the corrector should be positioned as close as possible to the solenoid. , a strength of , and was rotated by 23 de- grees relative to the solenoid principal axis. The beam had a kinetic energy of

4 MeV

. The solenoid had a maximum magnetic field of

171 T

, characterized by an effective length ( . Ad- ditionally, a corrector magnet with an effective length of and a focal length of 16 m was positioned downstream to adjust the beam dynamics.

As discussed in [ ], the sextupole component can in- troduce additional transverse emittance to the beam, which is primarily dependent on the radial second-order derivative of the magnetic field and effective length of the field region.

To simplify the analysis of sextupole effects, we assume that fringe fields are negligible, which is a valid assumption in this context because fringe effects are minor compared to the main field. Additionally, we consider a paraxial approximation in which the transverse momentum of the beam is much smaller than its longitudinal momentum. Under these conditions, the Lorentz force equation yields the force in the x-direction as follows:

F x = dp x dt = e ( v y B z − v z B y ) . (14)

Considering that dz = βcdt , the force becomes

dp x dz = − eB y . (15)

The y-component of the sextupole field is given by [

���� x,y =0

B y ( x, y ) = 1

The sextupole field introduces transverse momentum, as fol- lows:

���� x,y =0

p x dp x = ∆ p x = − e 2

∆ p x ( x, y ) = − e 2 ∂ 2 B y ∂x 2

���� x,y =0 L e f f � x 2 − y 2 � .

The beam emittance is

ε n = σ x σ p x mc , (18)

where are the RMS size and horizontal momen- tum of the beam, respectively: = 0)]

σ 2 p x =

denotes the transverse beam distribution. For example, may take the form of a Gaussian distribution

2 σ 2 x , where σ x is the beam RMS width, or a uniform distribution.

The distribution function influences the extent to which the sextupole component contributes to the emittance growth, as the nonlinear components of the field interact with the par- ticle density profile in different manners, depending on the distribution. The resulting normalized emittance induced by the sextupole field for both the Gaussian and uniform beam distributions is expressed as

����� x,y =0 ,

e f f

ε n , sextupole =

e f f

���� x,y =0 .

ε n , sextupole = 1

From Eq. , the increasing transverse emittance from the parasitic sextupole component is linear with respect to the ef- fective length of the sextupole component and cubic with re- spect to the beam RMS size. This indicates that larger beam sizes or stronger sextupole components can significantly in- crease the emittance. To mitigate this effect, a compensat- ing sextupole lens must be inserted near the solenoid. This compensator can counteract the influence of the sextupole component by introducing an opposing sextupole field com- ponent. Additionally, reducing the second-order derivative of the transverse magnetic field in the solenoid can help to diminish the sextupole component effect because a smaller second-order derivative reduces the strength of the nonlinear component that contributes to emittance growth.

NOVEL DESIGN OF SC SOLENOID To reduce spherical aberration, we optimized the design of the SC solenoid. Magnetic field calculations were performed using Poisson /Superfish, which is widely used to simulate electromagnetic fields in accelerator components [ ]. The results of these calculations are shown in Fig. . The NbTi wire coils remained unchanged in both designs, whereas the pure iron yoke was optimized by increasing its radius from 75 mm = 38 mm and doubling its length = 120 mm . This optimization significantly altered the magnetic field profile of the solenoid.

The normalized longitudinal field and its first derivative for both designs are shown in Fig. , where the new design ex- hibited a substantial reduction in the first derivative at the solenoid field edges. The effective length of the solenoid in- creased from 56 mm 27 mm in the new design.

We conducted simulations using ASTRA to compare the beam dynamics of the old and new solenoid designs. In these simulations, the beam kinetic energy was set to

5 MeV

the space-charge effect was neglected to focus on the im- pact of the solenoid on the beam. At the solenoid entrance, the beam exhibited zero emittance with a Gaussian trans- verse distribution profile. The magnetic field integral identical for both designs at . The maxi- mum magnetic fields were 0.171 and 0.151 T for the old and new designs, respectively. The integration of the first- derivative square was

14 T

for the old design and

62 T

for the new design. Figure shows the spherical aberration emittance at the solenoid exit for both designs. The new design demonstrated a marked improvement, with the fourth-power fitting coef- ficient being approximately half that of the original design.

The coefficient was reduced by approximately 47%, from 0061 mm 0033 mm . For a 600 pC beam with a Gaussian transverse distribution profile and 25 mm RMS size at the solenoid position, the transverse projected emittance at from the cathode decreased 458 mm with the old-design SC solenoid to 768 mm with the new-design SC solenoid at the same focal strength. Additionally, when the solenoid is not perfectly aligned, the newly designed SC solenoid will exhibit superior behavior. For instance, if the alignment has a 1 mm offset in the x-direction, the transverse projected emittance will decrease from 814 mm 902 mm The average slice emittance decreases by approximately 25%.

MEASUREMENT SETUP Multipole components arise when an offset or tilt exists between the magnetic field axis of the solenoid and its me- chanical axis. Misalignments of this type can cause multipole components such as quadrupole or sextupole components to appear. These multipole components distort the beam pro- file and increase the transverse emittance, thereby degrading the overall beam quality.

Identifying and correcting these misalignments is crucial for optimizing the solenoid perfor- mance.

Figure shows a photograph of the magnetic field mea- surement system. The 3D coordinate measurements and me- chanical alignment of the solenoid were performed using a Quantum Max metrology tool (mechanical measuring arm) from FARO. This arm was also employed to establish the transverse coordinate origin and ensure proper alignment of the longitudinal axis. Before sealing the cryomodule, this tool was used to determine the mechanical axis of the module and solenoid position relative to the large front-side flange of the cryomodule, which served as the reference plane.

The magnetic field was mapped using three motorized lin- ear stages enabling precise three-axis movements.

The z- direction stage had a travel range of 0 to 270 mm . Figure illustrates the installation of the SC solenoid in the cryomod- The core, fixed by an aluminum support frame, was placed 135 mm away from the original measurement point.

The beam pipe diameter was 39 mm . The z-direction stage had a travel range of 0 to 270 mm Magnetic field measurements were performed using two Hall sensors. The first sensor (SENIS AG) was employed to measure all three components of the magnetic field. operates within a range of -200 to 200 mT , and the resolu- tion is 001 mT . The active area of this 3D probe measures with a thickness of . To ensure the precise alignment and protection of the probes during field mapping, a custom holder was designed (see Fig. (a)). The second was a Magnet-Physik GmbH probe that measures the longitudinal magnetic field component in the range of 3 mT to 3 T (see Fig. (b)).

For the 1D axial probe, measurements of the longitudinal magnetic field component were obtained over a distance of 0 270 mm along the mechanical axis of the solenoid, with a step size of The 3D probe was used to map the transverse magnetic field components over a 12 mm 11 mm area in the hori- zontal and vertical directions (see Fig. ). The center point was the original rectangular point. The step sizes were set to in the x- and y-directions and in the z-direction.

Each measurement plane required approximately 4.5 min.

1 A

FIELD MEASUREMENT RESULTS The longitudinal field along the solenoid axis was mea- sured using the 1D Hall probe, which was selected for its ease of data processing, allowing for a more efficient and accurate analysis compared with other methods. Owing to the dom- inance of the longitudinal field over the weaker transverse components, any minor misalignment between the measure- ment and solenoid axes was considered negligible.

Figure (a) shows the measured longitudinal field profile for currents ranging from

8 A

. Figure (b) com- pares the measured data with the computed field profile, re- vealing excellent agreement between the two. As anticipated from the behavior of the solenoid, the maximum value of exhibited a linear increase with the solenoid current and the slope coefficient was 02 mT , as shown in (a). The effective magnetic length was determined as 990 mm with a measurement uncertainty of 068 mm The results for different currents, which demonstrate a con- sistent relationship between the field strength and current, are shown in Fig.

B z = 3 5 . 2 3 2 I - 0 . 2 0 8

Misalignment measurement Detailed spatial field distribution mapping was performed to measure the SC solenoid field accurately, as shown in . The 3D Hall probe was used for these measurements, with the maximum current limited to

5 A

owing to the range constraints of the probe. To account for the influence of back- ground magnetic fields, preliminary mapping was conducted at a current of

0 A

. These background readings were subse- quently subtracted from the standard field measurements be- fore the numerical analysis.

The data analysis involved determining the extrema coor- dinates (maximum values outside and minimum values inside the solenoid) for each measurement plane using a parabolic fitting method. Linear regression was applied to the center point coordinates in the z-direction at both the solenoid en- trance and exit, yielding two straight lines. The tilt and offset of the solenoid field axis were calculated by averaging the re- sults of these two lines. The SC solenoid offset was 20 mm in the horizontal direction and 02 mm in the vertical direction. The tilts were 87 mrad the horizontal plane and 23 mrad in the vertical plane.

Although the offset of the solenoid can be adjusted by repo- sitioning the x-y stage during SRF gun operation, any tilt in the field axis of the solenoid, once established, cannot be corrected later. Potential sources of misalignment between the magnetic field axis of the solenoid and the measurement

axis include discrepancies between the solenoid magnetic field axis and mechanical axis, inaccuracies in the solenoid alignment relative to the reference plane, errors in the mea- surement coordinate system, and data analysis inaccuracies.

Furthermore, magnetic hysteresis in the solenoid or adjacent components, as well as thermal contraction effects during cooldown to 4 .

5 K

, may contribute to alignment deviations.

Multipole modes The multipole component analysis focused on the trans- verse magnetic field, measured using the 3D Hall probe, to characterize higher-order magnetic field structures accurately.

Before performing a detailed evaluation of the field mapping data, the background magnetic field measurements were sub- tracted to remove any residual environmental noise or base- line offsets and to ensure the accuracy of the subsequent anal- ysis.

The primary objective of the data processing was to com- pute the coefficients of the multipole modes that describe the strength and nature of the higher-order components of the magnetic field. These coefficients were calculated based on the measured transverse fields for each plane, following the formulation provided in Eq. . Accurate center coordinates, which are essential for determining the symmetry and align- ment of the multipole modes, were obtained from previous axis measurements.

In the final step, a fitting procedure was applied to the data for each measurement plane, allowing for the extraction of multipole coefficients that characterize the transverse compo- nents of the field. For example, Fig. shows the multipole component distribution of the SC solenoid at

5 A

In Fig. (a), the solenoid radial field coefficient tained from the measurement is compared with the corre- sponding curve derived from the first derivative of the mea- sured on the axis. The consistency between these two re- sults confirms the validity and accuracy of the measurement process.

Figure (b) illustrates two components of the dipole field parasite in the solenoid. The normal mode exhibited symme- try around the center of the solenoid, whereas the skew mode was asymmetric, indicating the presence of minor field im- perfections.

The overall dipole field, which accounted for both the nor- mal and skew modes, can be described by the following equa- tion, which characterizes the complete dipole structure of the field:

B d = B dn + B ds = − J ds e x + J dn e y . (21)

By integrating the transverse field components along the z-axis, the resulting values determined the magnitude and di- rection of the dipole kick exerted on the beam. This kick influenced the trajectory of the beam as it passed through the solenoid, with both the strength and orientation depending on the transverse field distribution.

are shown as functions of the longitudinal position z, and a distinct phase change is observed at the central plane of the solenoid, which may indicate field reversal or changes in field symmetry across the solenoid.

As shown in Fig. (b), the integration of the dipole field strength was proportional to the current, confirming the proportional relationship between the current and transverse dipole field.

This linearity is key to predicting the beam- steering effects caused by the solenoid under varying oper- ational conditions.

k = 2 5 . 6 5 6 ± 0 . 5 5 m T m m / A y = - 2 7 . 2 7 8 ± 1 . 1 4 7 m T m m / A

5 A

. (b) Integration of dipole mode strength vs. solenoid current.

The parasitic quadrupole component shown in Fig. demonstrates antisymmetry about the central plane of the solenoid, with the dominant normal quadrupole mode revers- ing its sign across the center. Because of this antisymme- try, the integrated quadrupole gradient derived from the z- integration of the quadrupole components was almost zero.

This suggests that, in the absence of other effects, the net fo- cus or defocus from the quadrupole field is minimal over the entire solenoid.

However, the Larmor rotation of the coordinate frame of the beam as it traverses the solenoid can cause coupling be- tween the transverse motions.

This rotation may lead to observable effects on the beam, even when the integrated quadrupole field is negligible. To evaluate the impact of the total quadrupole field on the beam accurately, both the normal and skew components must be combined, as represented by the following equation:

In Fig. 12 [FIGURE:12] (a), the magnitude and phase of the dipole field

B q = B qn + B qs = ( J qn y − J qs x ) e x +( J qn x + J qs y ) e y . (22)

This representation pro- vides insight into the spatial variation and orientation of the quadrupole field components.

In Fig. 13 [FIGURE:13] (b), the integrated quadrupole gradient is plotted against the solenoid current, revealing the linear dependence of the field strength on the current. Notably, the slopes of the currents with different signs in Fig. 13 (b) differ only slightly.

This discrepancy may be attributed to systematic measure- ment errors, such as probe misalignment, noise in the data acquisition system, or nonideal calibration of the Hall probes.

k = 0 . 0 5 1 ± 0 . 0 1 7 m T / A y = - 0 . 0 3 5 ± 0 . 0 0 1 m T / A

To mitigate the effects of the parasitic quadrupole field gen- erated by the SC solenoid, a set of correctors comprising one normal quadrupole and one skew quadrupole were strategi- cally placed 437 mm downstream of the solenoid center in the SRF gun beamline. Each corrector had an effective length 0672 m and provided a gradient of 12 mT When operating SRF Gun-III at an accelerating gradient of

12 MV

, corresponding to a kinetic energy of

6 MeV

, the SC solenoid current was set to

6 A

to focus the beam and mit- igate the growth of the transverse emittance. Consequently, the solenoid induced an integrated quadrupole field gradient 192 mT with a peak value of 48 mT over an effective length of (Color online) Normalized emittance change with quadrupole field rotation angle in simulation.

The beam with a 500 pC bunch charge had

6 MeV

of kinetic energy. We ignored the space charge when considering the quadrupole field in the solenoid.

The solenoid current was

6 A

and located 55 mm downstream of the cathode. The component focal strength of the parasitic quadrupole , and its effective length was Owing to the uncertainty in the phase of the parasitic quadrupole field, simulations using ASTRA were performed to analyze its impact on the beam dynamics. Under specific phase conditions, the parasitic quadrupole field could reduce the initial beam emittance, potentially compensating for the spherical aberration of the solenoid. However, this effect was contingent on the precise phase alignment.

In most cases, the parasitic quadrupole field exacerbated transverse emittance growth, making it critical to cancel this effect using correction techniques. Assuming the worst-case scenario in which the quadrupole field phase is misaligned by , as shown in Fig. , and applying a current of

1 A

the corrector, Fig. demonstrates that the emittance oscilla- tion was significantly reduced. The simulation results confirm that the correctors, when properly adjusted, effectively nullify the influence of the parasitic quadrupole field of the solenoid, thus preventing additional emittance growth.

B s = B sn + B ss = [ J sn xy − 1 2 J ss ( x 2 − y 2 )] e x

2 J

The sextupole field characteristics are illustrated in Fig. (a), which shows the sextupole field coefficient and its phase along the longitudinal direction at a solenoid current of

5 A

The amplitude of the sextupole field was approximately sym- metric with respect to the original point (the solenoid center), with maximum values of approximately 0025 mT occurring in the mirror planes. Additionally, the phase under- went significant changes near the center, with the maximum and minimum strength values observed at these mirror planes.

At the edges of the solenoid, the phase measurement became unstable because of the small magnitude of the sextupole co- efficient, which resulted in significant noise interference and limited accurate fitting. Although the z-integral of the sex- tupole field coefficient is expected to be proportional to the solenoid current, the measured results do not align with this expectation, as shown in Fig. (b). This discrepancy likely arose because the sextupole component was outside the accu- racy range of the measurement method.

We evaluated the influence of the sextupole field on the transverse emittance in the context of SRF Gun-III using a bunch charge of 500 pC The simulations revealed that the beam RMS size at the solenoid position was approxi- mately . When the solenoid current was fixed at

6 A

the sextupole field exhibited an amplitude of approximately 005 mT and an effective length of . For a beam characterized by a lateral Gaussian distribution of the trans- verse profile, the additional normalized emittance introduced by the sextupole field was approximately contrast, for a uniform beam distribution, the additional emit- tance was reduced to approximately 52 mm . Con- sequently, based on Eq. , the overall effect of the sextupole field on the transverse emittance was estimated to be approx- imately 20%.

Analysis of error sources The measurement uncertainties associated with the SC solenoid stem from four primary sources: 1.

Mechanical alignment of the Hall probe: Inaccuracies in the mechanical alignment of the Hall probe can lead to erroneous magnetic field measurements. Even slight misalignments can result in significant discrepancies in the recorded data. 2. Degauss- ing of the yoke: Incomplete or imperfect degaussing of the yoke can introduce residual magnetic fields that may affect the accuracy of the measurements by masking or distorting the intended signal. 3. Intrinsic errors of the Hall probe: Each Hall probe has inherent measurement errors owing to its con- struction and operating principles, which can contribute to the overall uncertainty in the magnetic field measurements. 4.

Data fitting errors: Errors introduced during the data fitting process can arise from the mathematical models used, as well as the selection of fitting parameters. These factors can affect the accuracy of the derived magnetic field values.

The field measurements were conducted over the course of one week, with each full mapping session requiring approx- imately 4.5 h. Owing to the limitations in the measurement system, such as the z-axis movement range and initialization procedure, the Hall probe required daily manual realignment with a marker. This manual realignment resulted in a longitu- dinal position error of less than and a rotational error of less than 60 mrad . Consequently, the longitudinal field er- ror of the solenoid owing to realignment was estimated to be less than 2%, whereas the uncertainty in the transverse field was less than 6%.

The second source of error arises from incomplete degauss- ing of the soft iron yoke. Although the background field was recorded, it was not measured prior to each mapping session.

Background measurements were performed at three key in- stances: before the measurement period, immediately prior to increasing the solenoid current to

4 A

and then to

5 A

, and finally, after measuring the magnetic field at

3 A

. As il- lustrated in Fig. , the background field variation along the mechanical axis was approximately 125 mT . Although we subtracted this background field before performing the mul- tipole field analysis, its variations introduced additional un- certainties. The average differences in the multipole compo- nent integrals were approximately 18.4% for the dipole fields, 30.4% for the quadrupole field gradients, and 21.4% for the sextupole field coefficients.

In terms of the measurement equipment, the active area of the 1D Hall sensor was a circular area with a diameter , representing the average over a circular area of 126 mm . The calibration uncertainty of the 1D hall probe was 0.25% [ ]. The core size of the 3D Hall sensor was 15 mm 15 mm and the measurement preci-

The observed fitting errors were 2%, 73%, and 90% for the dipole fields, quadrupole field gradients, and sextupole field coefficients, respectively.

The overall fitting error reflects the cumulative impact of these individual errors, underscoring the significant influence of the quadrupole and sextupole components on the accuracy of the multipole field measurements. This highlights the im- portance of addressing these fitting errors to improve the mea- surement reliability.

CONCLUSION

We optimized the SC solenoid design to reduce the spheri- cal aberration, with a focus on modifying the geometry and field distribution.

The effective length and field strength coefficient, as determined from measurements under an op- erational current, were found to be in excellent agreement with the simulation results, confirming the accuracy of our solenoid design.

An analysis of the high-order magnetic components, in- cluding the quadrupole and sextupole components, was con- ducted to assess their impact on the beam transverse emit- tance. Although the formalism fitting method used for multi- pole component analysis did not yield highly precise results for the reasons discussed in the previous section, it provided valuable insights into the solenoid performance in the beam- line and can help to guide further optimization efforts.

The correctors installed in the beamline effectively com- pensated for the adverse impact of the quadrupole field on the beam transverse emittance. However, the sextupole compo- nent remains a critical factor in optimizing the future perfor- mance of the solenoid because it introduces nonlinear field effects that significantly impact the beam quality.

A practical solution to mitigate the effects of the sextupole component is to install a sextupole corrector in close prox- imity to the SC solenoid. This would allow for fine-tuning of the higher-order field components and enhance the overall beam quality. This addition could significantly enhance the performance by further mitigating sextupole effects. sion was better than 0.1% [ ]. Additionally, the intrinsic alignment error of the sensor areas in the 3D Hall sensor was approximately 45 mrad , as shown in Fig.

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