## 1. 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 (

$\mathsf{\mu}$m) 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

$\mathsf{\mu}$m size, whereas the conventional accelerator operates with typically tens of

$\mathsf{\mu}$m divergence and tens to hundreds of

$\mathsf{\mu}$m 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 to enable the utilization of such accelerator for applications such as Free Electron Laser (FEL) [

9,

10,

11] that allows for time-resolved atomic scale imaging.

Undulator radiation has been successively observed using LPA source [

12,

13,

14], while FEL based applications remain very challenging due to the large energy spread and divergence. In the high gain FEL [

15,

16], the high density electron beam interacts with the radiation and gets micro-bunched where a longitudinal coherence is achieved and then the radiation is exponentially amplified until saturation. Large energy spread deteriorates the micro-bunching efficiency and smears out the electron bunch, which prevents the transfer of energy from the electron beam to the resonant mode of wavelength

${\lambda}_{r}$ [

17]. The energy spread rms (

${\sigma}_{\gamma}$) has to satisfy the following relation:

${\sigma}_{\gamma}<\gamma \rho $, where

$\gamma $ is the relativistic factor and

$\rho $ the Pierce parameter

$\propto {(I/{\sigma}_{x}{\sigma}_{z})}^{1/3}$ with

I the peak current and

${\sigma}_{x,z}$ the transverse beam sizes rms.

$\rho $ is normally of the orders of 10

${}^{-3}$, so this condition is not acomplished by typical LPA beams since energy spreads have been measured to be ~10

${}^{-2}$ for MeV-GeV [

18]. In order to use these beams for FEL, the energy spread should be reduced, for example, by cutting a part of the initial distribution using a demixing chicane to select a smaller range of energies [

19] or take advantage of the energy spread by opting for a different approach with the use of transverse gradient undulators (TGUs) [

20,

21]. The FEL amplification highly depends on the overlapping of the electron beam and the wave. Electrons tend to oscillate around the undulator axis with a period larger than the undulator period (betatron oscillations) that reduces the overlapping efficiency. So another FEL condition, the well known Pellegrini criterion [

22], has to be satisfied:

${\u03f5}_{n}<\frac{\gamma {\lambda}_{r}}{4\pi}$, where

${\u03f5}_{n}$ is the normalized emittance. Even though the beam at the plasma-vacuum interface has a quite low normalized emittance (1 mm·mrad), a problem arises regarding the transport where the chromatic emittance increases quadratically with the divergence along a distance

s [

5]:

${\u03f5}_{n}\left(s\right)\propto {\sigma}_{\gamma}{\sigma}_{x}^{{}^{\prime}2}s$, making the beam transport a big issue if not quickly mitigated.

After the emergence of the concept of magnetically self-focusing electron beam of density

${n}_{e}$ by ions from a residual gas [

23] or more generally by a plasma of density

${n}_{p}$, two regimes of plasma lens can be considered. In the over-dense regime (

${n}_{e}<<{n}_{p}$); the electron beam moves away due to the plasma and self generates an azimuthal magnetic field, of focusing strength

$K=2\pi {r}_{e}{n}_{e}/\gamma $ with

${r}_{e}$ the classical radius of the electron [

24]. In the under-dense regime (

${n}_{e}>>{n}_{p}$), the electron beam pass induces a strong wave in the plasma background and can be focused by the ions uniformly distributed, with a strength given by

$K=2\pi {r}_{e}{n}_{p}/\gamma $. Afterwards, Passive Plasma Lens (PPL) have been proposed and developed [

25], which can provide high gradients, but the focusing properties depend on the electron beam itself and can present aberrations. PPL is further developed theoretically [

26] and used for an LPA experiment [

27]. Active Plasma Lens (APL), where the azimuthal magnetic field is controlled by a discharge in the plasma, has been proposed [

28,

29]. APL has been applied to ion beams [

28,

30] and to LPA applications [

31,

32,

33,

34,

35]. APL provide high gradient of the order of kT/m, tunability and radially symmetric focusing, but are subjected to emittance degradation and charge reduction due to highly non-linear focusing arising from current discharge nonuniformity. Furthermore, their use in experiments that run for couple of weeks adds an additional level of risk. Plasma lenses are still under development and the use of conventional magnet can still appear to be more robust.

A FODO high gradient quadrupoles lattice placed very close to the plasma-vacuum interface allows for handling and controling the LPA beam divergence. However, their focusing is not symmetric, unlike APL, and at least three systems are required to provide a round beam. Conventional accelerators, operating at an intermediate energy, use electro-magnet technology for quadrupoles to transport the beam where the rigidity is limited to few tens of T·m. High energy accelerators require larger gradient quadrupoles to transport the beam, where the electro-magnet technology reaches a limit it cannot surpass. Superconducting magnets come in handy for such applications but they are much more expensive than the conventional electro-magnets due to the the cryogenic cost (installation and operation) and the possibility of a quench due to synchrotron radiation and image charges. Thus, permanent magnet advantages come into play with the absence of power supplies and cables, and in 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.