Permanent Magnet-Based Quadrupoles for Plasma Acceleration Sources

The laser plasma accelerator has shown a great promise where it uses plasma wakefields achieving gradients as high as GeV/cm. With such properties, one would be able to build much more compact accelerators, compared to the conventional RF ones, that could be used for a wide range of fundamental research and applied applications. However, the electron beam properties are quite different, in particular, the high divergence, leading to a significant growth of the emittance along the transport line. It is, thus, essential to mitigate it via a strong focusing of the electron beam to enable beam transport. High-gradient quadrupoles achieving a gradient greater than 100 T/m are key components for handling laser plasma accelerator beams. Permanent magnet technology can be used to build very compact quadrupoles capable of providing a very large gradient up to 500 T/m. We present different designs, modeled with a 3D magnetostatic code, of fixed and variable systems. We also review different quadrupoles that have already been built and one design is compared to measurements.


Introduction
Ever since the discovery of the electron using a cathode-ray tube at the end of the 19th century, particle accelerators have seen tremendous progress where they became widely used tools for basic research, industry, medicine, material science etc. For the past decades, innovative and compact accelerators have been developed that address the increasing requirements in different fields. Laser Plasma Acceleration (LPA) was proposed in 1979 [1], where ultra-high accelerating fields can be produced to generate electron beams up to several GeV energy, short beam size (fs), very small transverse sizes (µm) and high peak current (1-10 kA) [2,3]. The concept of LPA is based on focusing an intense laser pulse onto a gas to drive large-amplitude plasma waves that then act as accelerating structures for particles. The LPA accelerator surpasses the conventional RF one in terms of energy gradient and compactness. However the characteristics of the beam phase space are very different. In the longitudinal phase space, where the conventional accelerators typically operate at ultra-low energy spread (~0.01-0.1%) and small bunch duration (few picoseconds), the LPA exhibits ultra-short electron bunch (few femtoseconds) but with large energy spread (few %). As for the transverse phase space, the LPA appears as a diverging (few mrads) point source of few µm size, whereas the conventional accelerator operates with typically tens of µm divergence and tens to hundreds of µm size. Consequently, the chromatic effects associated with the divergence and the energy spread, that are generally negligible on conventional accelerators, appear to play a major role and complicate the transport [4][5][6][7][8]. Thus, it is of great importance to mitigate these effects, transport and control the beam addition eliminating a large element of infrastructure for the water cooling system. Permanent magnet based quadrupoles can be reduced in size without losing the magnetic field strength making them suitable for future compact accelerators including LPA.
In this paper, we review compact designs of PMQs suitable for different applications. Whereas present synchrotron radiation sources operate with gradients of 10-20 T/m, future diffraction limited light sources at an intermediate energy will require compact quadrupoles of gradients around 100 T/m. In colliders, the beam energy is much larger and strong quadrupoles of gradients 100-300 T/m are needed for the final focusing. Regarding plasma wakefield accelerators, where the electron beam is generated with initial large divergence, quadrupoles of gradient 100-500 T/m and above are required with low multipole contents to ensure a good handling of the beam transport. We present here solutions offered by permanent magnet quadrupoles and compare the performance of different designs of fixed and variable gradient with numerical simulations using Radia code [36]. We then review the quadrupoles that are designed for the mentioned applications. Measurements using a stretched wire conducted for one design are compared to simulations.

Quadrupole Characteristics
Quadrupoles consist mainly of four magnets with magnetization vectors directed as shown Figure 1, in which the dipole terms at the core of the system are canceled and the field magnitude increases with the radial distance, so particles that are farther from the center experience a larger force. In the scheme, the electron experiences an inward force towards the center in the horizontal plane and outwards in the vertical plane, Thus, a quadrupole focuses in one plane and defocuses in the other.

Gradient
In the case of permanent magnets, the current density is zero, so starting with Maxwell equations: where B is the magnetic field. Considering that the field along the longitudinal direction is zero (B s = 0) and does not vary ( ∂B ∂s = 0), and neglecting higher order multipoles for small transverse displacement: where G = ∂B z ∂x = ∂B x ∂z is the field gradient along the horizontal axis x and vertical axis z. Equation (1) is represented in Figure 1 where the field increases for larger distance from the center.

Multipoles
The magnetic field of realistic quadrupoles, even though designed to generate only one multipolar order, contains higher order multipoles called harmonics resulting from the structure and magnetic or mechanical assembly imperfections. In case of an infinitely long magnet, the local field B inside a quadrupole can be expressed as: where n is the multipolar order, B n and A n the normal and skew multipolar coefficients respectively and r the radius for which multipolar coefficients are computed or measured. A perfect quadrupolar field consists only of B 2 term, as for the field generated by a ideal quadrupole contains systematic multipole terms (B 6 , B 10 , B 14 , ...) due to the device geometry. Mechanical and magnetic impurities give rise to the non-systematic terms including the skew components. The vertical field component is negative for x < 0, positive for x > 0, and null on the axis. This evolution is the same for the horizontal field along the z. In a 3D model, the following components a n and b n defined as a n = A n .ds and b n = B n .ds are introduced. The quadrupole field gradient (2D and 3D) can be expressed as:

Halbach Structure
A Halbach structure is an array of magnets with specific arrangements generating a concentrated strong magnetic field on one side whilst creating a weak stray field on the opposite side [37]. Different multipole systems can be build using such structure as displayed in Figure 2 where 12 segmented magnets are used to design a dipole, quadrupole and a sextupole. Concerning the quadrupole, and considering a long magnetic length (L) the gradient can be expressed as follows [37]: where B r is the remanence field, M the number of segments, R i the inner radius and R o the outer radius. As M tends to infinity K = 1, implying that the quadrupole gradient is enhanced for larger number of segments, but the magnet gets more complicated and expensive to manufacture. The quadrupoles are characterized using Radia magnetostatic code based on boundary integral method. Nd 2 Fe 14 B magnets are most commonly used due to their high remanent field and coercivity at room temperature [38][39][40]. The materials and quadrupole parameters are presented in Table 1.

Hybrid Quadrupole
This section presents different quadrupole designs following a Halbach structure with the addition of poles to enhance the gradient. Figure 5 shows three designs: Type (I) composed of four magnets and four poles, Type (II) composed of eight magnets and four half magnets half poles, Type (III) using 12 segmented structure in which four of them are half poles half magnets and the rest are magnets surrounded by pole ring shaped to divert the magnetic field lines back into the core of the system.  Figure 6 shows the gradient computed for the three designs of Figure 5 and are compared to the Halbach rings. Type (I) achieves a gradient higher by a factor of~2 compared to the four segmented Halbach ring. Indeed, the addition of poles achieve a higher gradient and a less costly system. Type (II) design has the same gradient as the 8 segments Halbach ring with a slight 2% increase. The gradient achieved with the 12 segmented Halbach ring is quite remarkable as compared to type (III) with a much more complex design to generate a similar gradient.

Variable Permanent Magnet Quadrupoles
In conventional electro magnets, the field, and so the gradient can be changed by varying the electric current. As for permanent magnets, it becomes much more challenging due to the requirement of mechanical motion to adjust the positions of the magnets taking into account the large magnetic forces induced by the system. Figure 7 shows examples of permanent magnet based quadrupoles with variable gradient: Type (IV) is composed of a Halbach hybrid ring at the center surrounded by a Halbach ring of 12 sections, and by rotating the ring the gradient is varied. Type (V) also constitutes of a Halbach hybrid ring at the center with cylindrical magnets capable of rotating around the longitudinal axis to provide gradient tunability. Type (VI) uses four magnetic rods built into a structure that can be moved to vary the gradient.

Fixed Gradient
Several Permanent Magnet based Quadrupoles (PMQs) with fixed gradient were built for particular applications: • A 12-modified PM Halbach ring system in which four of them are half permanent magnet and half iron poles (II) is proposed at Kyoto University in collaboration with SLAC as a final focus lens for a positron-electron linear collider accelerator [41]. The system has a bore radius of 7 mm with a magnetic length of 100 mm generating a maximum gradient of 289 T/m. At the department für Physik, in München for a  [43]. Three PMQs of the same design, each made up of a Halbach ring of 16 segments, have been installed in the PLEIADES ICS experiment as a final focus system [4]. The bore radius is very small (2.5 mm) with a magnetic length of 10.4 mm providing the largest gradient recorded using PM technology that is around 560 T/m alongside a focusing tunability through longitudinal movement of the magnets. • A compact PMQ with a hybrid type design of interest for ultimate storage rings has been designed at ESRF [44]. The magnetic structure includes rectangular PM blocks and soft iron poles. A prototype with a bore radius of 12 mm and a magnetic length of 226 mm has been built (see Figure 9) provides a gradient of 85 T/m.

Variable Gradient
Various original designs were proposed and developed for the permanent magnet quadrupole to provide a variable gradient: • A double Halbach ring structure of type (IV) was fabricated at Kyoto U. / SLAC collaboration for a final focus in a linear collider. The system consists of an inner hybrid ring with 8 sectors Halbach system and the outer ring being a pure 12-magnet Halbach structure rotating around the first one resulting in gradient variation from 17 T/m up to 120 T/m for a 230 mm magnetic length and 10 mm bore radius [45]. • A super hybrid quadrupole combining permanent magnetic material, coils and soft magnetic material has been built (see Figure 10) and tested for the new Brazilian Synchrotron Light Source (Sirius) [46]. The system has a bore radius of 27.5 mm and a magnetic length of 288 mm providing a maximum gradient of 28 T/m with a 30% tunability.

•
An adjustable strength PM system has been built and tested, in collaboration between SLAC and Fermilab, for the Next Linear Collider. The system is based on four PM blocks and four soft iron poles of type (VI), with the possibility to retract linearly the permanent magnet blocks enabling an integrated gradient variation between 7 T and 68.7 T (gradient between 13 T/m and 115 T/m) for a 6.5 mm bore radius [47]. • A modified the 12-sector Halbach design composed of eight magnets and four poles surrounded by air-cooled electromagnetic coils as presented in Figure 11 has been proposed in the framework of CLIC/CERN collaboration [48]. The bore radius is 4.125 mm with a magnetic length of 300 mm achieving a maximum gradient of 610 T/m with 20% tunability. • Two variable systems have been built in a collaboration of STFC Daresbury Laboratory and CERN for the Compact LInear Collider (CLIC) project [49,50] with an objective to collide electron-positron at an energy of 3 TeV. The first design, shown in Figure 12-left, provides a gradient from 15 to 60.4 T/m with a magnetic length of 241 mm. The second design, shown in Figure 12-right, provides a gradient from 2.9 to 43.8 T/m with a magnetic length of 194 mm length. The strength is adjusted by moving the PMs vertically away from the center, and by creating an air gap the gradient is reduced.

•
Two variable quadrupoles are proposed for an interdigital H-mode drift tube linear accelerator using KONUS beam dynamics [51]: The first using an external adjustable electromagnets as shown in Figure 13-(left), with a bore radius of 12.5 mm providing a gradient from 50 to 100 T/m. The second using internal adjustable permanent magnets as shown in Figure 13-(right) that is similar to type (IV) but with additional number of segmented magnets. For a bore radius of 12.5 mm, the achieved gradient can be varied from 50 to 102 T/m. • Seven systems (QUAPEVA) of type (V) have been built in collaboration between SOLEIL and SigmaPhi [52,53] for the COXINEL project with an objective of FEL amplification using an LPA source. The QUAPEVA cosists of Nd 2 Fe 14 B magnets and permendur poles achieving a maximum gradient of 202 T/m and 45% tunability. The systems have a bore radius of 6 mm but with different magnetic lengths. Figure 14 presents the three particular configurations of the tuning magnets. The maximum and minimum gradient are obtained by orienting the tuning magnets easy axis towards the central magnetic poles. In these cases, the magnetic flux is either added (Figure 14a) or removed (Figure 14c) from the central poles. The average gradient is achieved when the tuning magnets are in the reference position, i.e. their easy axis is perpendicular to the central magnetic poles (Figure 14b). The QUAPEVAs are supported by translation tables (horizontal and vertical displacement) used to compensate any residual magnetic axis shift when varying the gradient, to perform electron beam based alignment [54,55]. The resulting mechanical design is shown in Figure 15 (left), also an assembled QUAPEVA on the translation table in Figure 15 (right).

Measurement
The QUAPEVAs are characterized using the stretched wire measurement, in which the multipoles are determined at a radius of 4 mm that corresponds to the good field region. The stretched-wire bench [56] has been used for magnetic field integral measurements. Table 2 presents the maximum gradient for the seven systems (QUAPEVAs) measured with the stretched wire. The difference with respect to the simulations reduces for longer magnetic length due to an insufficient number of mesh in the longitudinal axis that is time consuming.  (1)) with the stretched wire for the seven systems at a radius of 4 mm, and ∆ sim the deviation with respect to the simulated one.  Table 3 compares the measured and simulated multipoles for the QUAPEVA of magnetic length 100 mm. The dipolar term (b 1 ) in the simulation is not equal to zero due to computational errors, and the one measured is probably due to a slight orientation of one of the cylindrical magnets or the precision of the bench. The discrepancy with respect to simulations increases for multipoles with weaker signal.

Summary
The models presented previously are summarized in Table 4 with their acheivable gradient and tunability. Other characterizations of a quadrupole are the systematic multipoles (b 6 , b 10 , b 14 , ...) that derive from the structure geometry and the non-systematic multipoles that emerge from magnets/poles impurities or mechanical errors. The beam quality can quickly deteriorate if these terms are not kept low; however their effect highly depends on the beam size in the quadrupole aperture and the gradient homogeneity also known as good field region. In the case of the LPA, the specifications in terms of multipoles are less stringent than a recirculating acclerator, typically by one order of magnitude. On the other hand, the gradient tuneability should be larger. Table 5 presents a summary of the PMQs already built that are presented in the previous section.

Conclusions
A permanent magnet-based quadrupole is quite a compact device achieving a very high gradient of the orders of hundreds of T/m making it essential for future compact linear accelerators including laser plasma accelerators. Conventional electromagnet technology, with copper conductors wrapped around soft iron poles, enables to tune the gradient by simply changing the current but with significant continuous power consumption to maintain the magnetic field. Furthermore, superconducting magnets produce very large magnetic field but requires an expensive Helium infrastructure to keep the magnets in superconducting state. Thus, with zero power consumption, permanent magnet-based systems pave the way towards a green society and with new innovative designs. Also the challenges of wide tunability and high field homogeneity are presentely mitigated with the various examples of built systems.
So far, PMQs are intalled and commissioned for storage rings, LPA and collider applications, but the high multipole contents set a drawback for some applications like diffraction limited light sources. A proposed design to cross that barrier is to design hyperbolic-shaped magnets and poles to reduce the multipole contents in compromising on the gradient variability.