An Adjustable , Short-Focal Length Permanent-Magnet Quadrupole Based Electron Beam Final Focus System

Next generation advanced-accelerators such as the plasma wake-field accelerator, and beamradiation interaction scenarios such as inverse-Compton scattering (ICS), depend on the achieving of ultra-small spot sizes in high current beams. Modern injectors and compressors enable the production of high-brightness beams having needed short bunch lengths and small emittances. Along with these beam properties comes the need to produce tighter foci, using stronger, shorter focal length optics. An approach to creating such strong focusing-systems using high field, small-bore permanent-magnet quadrupoles (PMQs) is reported here. A final focus system employing three PMQs, each composed of 16 neodymium iron boride sectors in a Halbach geometry has been installed in the PLEIADES ICS experiment. The field gradient in these PMQs is 560 T/m, the highest ever reported in a magnetic optics system. As the magnets are of a fixed field-strength, the focusing system is tuned by adjusting the position of the three magnets along the beamline axis, in analogy to familiar camera optics. This paper discusses the details of the focusing system, simulation, design, fabrication and experimental procedure in creating ultra-small beams at PLEIADES.


I. INTRODUCTION
In recent years, there has been a revolution in the design and implementation of electron sources due to the introduction of the rf photoinjector [1], which produces beams that have very high brightness, defined as simultaneous high peak current and low transverse emittance.The applications of high brightness electron beams generated by such devices has also continued apace in this time.These applications, e.g., driving a plasma wakefield accelerator (PWFA) [2,3] or providing an electron beam for an inverse Compton scattering (ICS) xray source [4,5], are strongly dependent on production of very dense beams.It has been widely noted that the scaling of such applications to higher performance levels (higher field in the PWFA, brighter for the ICS source) requires shorter bunch lengths σ z and lower transverse emittances ε x,y , a situation that has strong implications for future development of electron beam sources themselves.These considerations have led to increased concentration on photoinjector optimization [6] and development of pulse compression systems.Much less emphasis has been placed, however, on the need to scale the dimensions of the beam focal systems down with other relevant length scales -σ z , ε x,y .The characteristic length scale of the focal system is the minimum "β-function", β * x , which is related to the transverse beam size at focus by the relation σ * x = β * x ε x .As an example of an application that is critically de- * Electronic address: jlim@physics.ucla.edupendent on achieving small β-functions, and the relationship between β * x and σ z , we examine aspects of the PWFA.The maximum longitudinal field (the "wavebreaking" field) achievable in a relativistic plasma wave scales with the plasma wavenumber k p as E W B = m e c 2 k p /e. Further, in the PWFA it has been noted that if the beam is short (k p σ z <1), then the wake is efficiently excited and the plasma wave's electric field amplitude scales [7] as E ∝ k 2 p N b , where N b is the number of beam electrons.
Thus to obtain high gradients in a PWFA one should operate at larger k p and concomitant shorter bunch length σ z .Such scaling has been the motivation behind the use of compressors, and ever-higher density (n 0 ∝ k 2 p ) plasmas in cutting-edge experiments at FNAL [8] and SLAC [9].It is notable that the beams in these experiments have had σ z ≈0.5 mm, corresponding to plasma densities of n 0 ∼ = 10 14 cm −3 .More recently, the SLAC E164X experiments [10] have been run at bunch lengths below 20 µm, with an associated increase in k p by a factor of at least 25.
The effect of using a large k p in a PWFA on the transverse beam handling is quantified by examining the matched β-function associated with the plasma's ionderived focusing, which is extremely short for low beam energy (γm e c 2 ) and large n 0 .For the example of the PWFA case discussed in Ref. [11], the beam energy is 14 MeV, and the resulting β eq =280 µm (k −1 p ∼ = 40µm).Even in present low energy experiments (FNAL, and previously at ANL [12]), β eq was already in the range of several mm.In fact, it has not yet been possible to obtain such small, matched β * ∼ = β eq in these experiments.
The difficulty in matching is greatly exacerbated by the need to longitudinally compress the beam.In the compression process, employing either chicane [13] or velocity bunching [14], significant momentum spread σ δp /p is introduced.When creating a tight beam focus, one must consider therefore the effects of chromatic aberrations which, in the absence of an elaborate, chromatically compensated final focus system, will limit the final spot size.To illustrate the constraints introduced by these effects, we display the following relation governing the ratio of final spot size σ * to initial σ 0 [15], where the final term obtained in the limit β f p σ δp .When the initial β-function β 0 = σ 2 0 /ε is larger than the effective focal length f of the final focus system, strong compression of the beam size is possible, up until the chromatic aberrations begin to dominate.There is a minimum in the beam size, for a fixed focal length system, when β 0 /f ∼ = p/σ δp .In this condition, the demagnification is √ 2σ δp /p.With larger initial beam size, the chromatic contribution to the emittance grows rapidly.Thus, because the demagnification is limited to the relative rms momentum spread, the final beam size cannot be made smaller by a larger initial beam size.These considerations strongly affected β-matching in previous and current PWFA experiments.
The second example, which is indeed the experimental scenario of the remainder of this work, serves to illustrate the implications of Eq. ( 2) on the choice of focusing scheme.A UCLA/LLNL collaboration is now operating an ICS source experiment termed the Picosecond Laser-Electron Inter-Action for the Dynamic Evaluation of Structures (PLEIADES) [16,17], in which a 10-100 TW, sub-ps, 820 nm laser pulse is collided with a velocity compressed, sub-ps 50-75 MeV electron beam.With the present generation of relatively large aperture quads that UCLA employs, which have a maximum gradient of 15 T/m over ∼10 cm length, the minimum value of f (effective focal length for a triplet) is roughly 50 cm, and thus optimum demagnification for σ δp /p=0.5% is only possible for β 0 =100 m, or (with ε x = ε n /γ=0.14 mmmrad) σ 0 =3.2 mm.We therefore anticipate a minimum achievable spot size (Eq.( 2)) of σ * ≈32 µm.This spot size estimate, which has been verified in simulation, is unacceptably large for the ICS source, that eventually demands a factor of better than 4 smaller spot size for nonlinear ICS interaction studies, as well as for increased x-ray photon production.
In addition, by allowing the beam to expand before final focus, space-charge forces (for beams of energy < 100 MeV) will increase the emittance [6], giving an even larger final spot size.This limitation also precludes more elaborate approaches, based on sextupole correction, to eliminate chromatic aberrations.At low energy, transport and final focusing systems must be compact; short short focal lengths are needed.Similar considerations are valid for many higher energy applications as well, such as linear collider (LC) final focusing.
From this introduction, it is clear that one must look into the use of much shorter focal length lenses in order to create minimum β-functions small enough for many modern high-brightness electron beam applications.One might consider superconducting quadrupoles or solenoids, but these devices are neither inexpensive, nor straightforward to build with the necessary small (cm-scale) dimensions.On the other hand, permanent magnet-based quadrupoles (PMQs) can access high-field gradients, and have been under intensive study recently in two arenas: LC main accelerator focusing -to mitigate the cost of the needed power supplies -and final focus magnets in both circular accelerators [18] and LCs [19][20][21][22].
In the LC main accelerator application, an effort has been made to design a moderate strength, but tunable magnet [19].Much of the international linear collider (ILC) oriented work has been concentrated on tunability, while simultaneously holding the magnetic center stable, as required for use in the beam-based alignment process.In addition, both ILC and CESR groups have studied the influence of their respective expected radiation environments on the magnetization of the permanent magnet (PM) material [23], and found the effects to be small.
On the other hand, the need for extremely large field gradients makes the introduction of significant gradient tuning into a PMQ very challenging.In the present work, we discuss an alternative scheme that sidesteps these difficulties, by tuning the strength of the final focus array (a class of quadrupole triplet) only through the longitudinal (z ) positioning of the individual fixed-strength PMQs.In this way, the issue of tunability is separated from both that of alignment, and that of achieving the highest gradient.Indeed, one may access extremely large field gradients by using an optimum, static intra-magnet configuration.Such a magnet configuration was introduced by Halbach [24,25], having a pure permanent magnet geometry consisting of 16 sections, with magnetization oriented at 0 • , ±45 • , and ±90 • with respect to their transverse symmetry plane.This type of magnet has been studied in the context of the Compact Linear Collider (CLIC) program [20,21], and has been implemented at the Cornell Electron Storage Ring (CESR) in the CLEO detector mini-β system [26].
The details of the field calculation of the Halbach sectioned PMQ are discussed in the following section.This geometry is known to allow field gradients approaching the ratio 2B r /r i , where B r is the remanent field of the PM material, and r i is the pole inner radius.For the material used in the present study, NdFeB, B r =1.22 T, and thus a 2.5 mm inner radius may produce over 600 T/m.This field gradient is appropriate for use in the PLEIADES ICS experiment, which provides the context for the development of the PMQ-based focusing system discussed here.
The PLEIADES project has very strict performance demands; one must have a tightly focused electron beam collide with an oncoming tightly focused laser beam, and thus alignment tolerances are stringent, at the 5 µm level.Further, PLEIADES demands a range of operating energies varying over a factor of two.Because the focal length of the system as a whole should be ∼10 cm, the individual PMQ length is chosen to be 1 cm.If one restricts, as is appropriate for a triplet configuration, the focusing phase advance (for the initial operating energy of 70 MeV) to be less than one-half (k q l q = (B /BR)l q ≤ 0.5) in a single 1 cm lens, then a 600 T/m field gradient is deduced.We shall see in the sections below that this estimate on appropriate field gradient yields effective guidelines for optimizing the beam optics in the PLEIADES final focus system.
In the following detailed treatment, we discuss: the basic theory and simulation for a single PMQ magnet (Section II); the simulation of a focusing lattice (Section III); the design and fabrication of the final focus system (Section IV); the measurement of the completed system (Section V); and finally, implementation, usage, and beam measurements (Section VI).

II. MAGNET THEORY AND SIMULATIONS
A main motivation behind the construction of a permanent magnet-based quadrupole is to achieve extremely high magnetic field gradients and simultaneously reduce the physical size of the magnet.This circumvents the limitations of conventional electromagnets, in particular the need to increase the gradient leads to an very high current density in the coils, and thus an elaborate cooling system.In this section, a 2-D analytical model originally approached by Halbach and a 3-D simulation model are presented to illustrate the dependence of the PMQ performance on the geometric and magnetic design parameters in the Halbach pure PM design.

A. 2-D Halbach PMQ Model
We present first the two dimensional Halbach-type PMQ model, reviewing results that were originally established by Halbach [24].These results are then used to guide initial magnetic and beam optics design decisions.
A PMQ structure can be composed of multiple segments -in the design considered here, 16 wedges -where each wedge is a trapezoid.A complete assembly of all 16 segments gives a cylindrical piece with a small bore about the center of its axis.The magnetization easy-axis in each trapezoidal segment rotates 45 • from one segment to the next.A schematic cross section of a 16-piece r o r i FIG.1: A cross section of the 16 segment PMQ and the magnetic-remanent field orientations shown as arrows.
quadrupole is illustrated in Figure 1.
To estimate the achievable gradients one obtains with a PMQ, we first assume a permanent magnet geometry in which the magnetization varies continuously with an azimuthal angle φ.The total magnetic field gradient for this pure quadrupole consists of Amperian current contributions from the boundaries at inner and outer surfaces, as well as from the bulk of the magnetic material, and is given by With B r =1.22 T, r i =2.5 mm, and r o =7.5 mm, we have B =650 T/m.In practice, the field gradient is most strongly controlled by the inner radius r i .It is not possible to make a PMQ with a continuously varying remanent field, but the use of uniformly magnetized segments can well approximate the pure quadrupole case.This was recognized by Halbach, who studied the approximation to the ideal geometry that can be achieved by segmenting the magnet into M geometrically identical pieces, which have magnetization rotated appropriately in each segment, as shown in Figure 1.In this case, the multipole expansion of the M -segmented magnet can be expressed by [24] The segmentation of the magnet into M uniformly magnetized sections is accounted for by the coefficient a n .This term is slightly less than unity, so the field predicted in Eq. (4a) is slightly diminished compared to the field obtained from the ideal case.For the Halbach geometry considered, where the magnet has M =16 segments, the coefficient of quadrupole term, a 2 , is found to be 0.94the field gradient is diminished by only 6% from the ideal geometry.The model analysis above omits the effect of finite length in the longitudinal dimension of the PMQ.This analysis requires a 3-D model, which is presented in the next section.

B. 3-D Radia Simulation Model
The magnetostatic simulation code Radia [27] was used in 3-D modeling and numerical analysis of a 16-piece Halbach PMQ.Radia, in contrast to most magnetostatic field solvers based on Finite Element Method (FEM), is a calculation tool utilizing the Boundary Integral Method (BIM).The code is written in Object Oriented C++ and possesses interface capability with Mathematica for preand post-simulation processing.The magnetostatic object of interest can be formulated as a composition of a set of smaller geometric components bounded by planar polygons which then are linked to make up the desired magnet model.Furthermore, each component can be segmented into smaller subdivisions for improved accuracy of the simulation result.
In this section, we present the simulation results of the ideal PMQ model and analyze the various causes contributing to demagnetizations in the ideal PMQ.The section finishes with the discussion of the field property errors arising in the imperfect geometrical PMQ model.The desired goal from 3-D simulation analysis is to obtain a set of optimized object dimensions which, consistent with a given remanent field, yield an appropriately high field gradient.Our application demands that the overall dimension of the PMQ be consistent with the beam optics dimensions (few cm focal lengths for each lens), and that the beam be easily passed through the small bore where the strong field gradient is produced.The design analysis also must factor in technical difficulties in the manufacturing process.NdFeB, a rare-earth PM material, was chosen for our simulations over other rare-earth types, e.g., SmCo, because of our demand for a higher remanent field range.
Each component wedge block is bounded by two parallel trapezoidal planes separated by a length of l z and a given remanent field of B r .By implementing a simple loop algorithm routine in the Radia, 16 wedge blocks are sequentially stacked side-by-side where the final model design, as seen in Figure 1, is created.The final optimum PMQ dimensions determined from the simulations in considerations of a design compactness and a peak field gradient are: r i =2.5 mm, r o =7.5 mm and l z =10.0 mm.The field gradient that can be produced with this set of parameters is B =573 T/m.Note that the gradient predicted in Radia simulation is lower than that deduced from Eq. (4a) mainly due three-dimensional (edge) effects.The field gradient and the magnetic field profiles plotted in the bore region are shown in Figure 2. The effective magnetic length determined from the simulation is l ef f =10.4 mm (see Figure 2).

Demagnetization
In this section, two of the most relevant factors contributing to changes in the performance of the NdFeB based PMQ are discussed: relaxation in PMQ due to intra-magnetic interactions, and a partial demagnetization due to temperature variations.The demagnetization studies in Radia are thus used for determining whether the coercivity and remanence in the NdFeB material are adequate.
The close proximity and high remanent fields of the segments can influence the overall magnetic characteristics of the assembled PMQs through mutual magnetic interactions.Relaxation modeling was performed by iterative calculations of the field strength and the magnetization of each wedge block under the influence of the other 15 wedge blocks, until the values reach the specified precision in the stabilized magnetization.
With the NdFeB magnet parameters from the material datasheet and the calculation precision set in Radia, the simulation showed a negligible relaxation in the chosen geometry of the PMQ.However, a permanent magnet material with a high coercivity is still desirable for its high resistance against opposing external fields and its resistance to temperature variation induced demagnetization.
Understanding the effect of temperature variation on the permanent magnet is important in the situation where an accidental exposure of a NdFeB PMQ to a temperature elevation can cause demagnetization as, for instance, during a vacuum bake-out.Demagnetization effects in the simulation studies are introduced through a second order temperature dependence in the magnetization and the coercivity equations as [28]: where T 0 is a reference temperature, and coefficients a i and b i are obtained from the material data sheet.
The relaxation of a PMQ magnetization is performed for 5 different chosen temperatures: 0, 20, 60, 90 and 120 C • .The temperature increase has a negligible effect in the magnetization loss in the PMQ at low temperature elevations, but the effect becomes noticeable with higher temperature.In Figure 4, the irreversible loss of a PMQ magnetization is shown as a function of a temperature change.This loss is small under 100 C • , but it increases dramatically above 120 C • .This behavior is similar to that obtained from simulations reported in Ref. 28.

Geometric and Magnetic Errors
Geometric imperfections in PMQ due to fabrication and assembly errors are investigated in simulations and the evaluation of the magnet performance are discussed here.We concentrated our simulation efforts in three main cases of PMQ imperfections: angular orientation errors in the magnetic easy-axis; sector-to-sector geometric errors; and PMQ bore aperture radius errors.From these simulations, good approximations in allowable tolerances in mechanical errors for the individual modules are obtained.A complete error budget, including alignment errors, is discussed in later sections.
In the first case, the following set of errors are introduced into the magnetization easy-axes: 0.225 • , 0.450 • , 0.675 • and 1.125 • ; corresponding to relative errors of 1%, 2%, 3%, and 5%, respectively.The magnetic field gradient is solved and presented for each angular error cases as shown in the Figure 5; as the magnitude of the angular error increases, the field profile suffers from a shift in the magnetic center and introduction of variation in the field gradient (higher order multipoles) in the bore region.
The wedge geometry is different from that of an ideal trapezoid sector in two different ways: an angular deviation from 22.5 • and a wedge length deviation from |r o −r i |.The following errors in wedge angles (in this particular exercise, the side length error is excluded, as we shall discuss in the magnet fabrication, that final boring out of inner pole surface yields a highly uniform surface) are chosen as the simulation input parameters: 0.225 • , 0.450 • , 0.675 • and 1.125 • ; again corresponding to relative errors of 1%, 2%, 3%, and 5%, respectively, as shown in Figure 6.The error parameters are assigned to an arbitrary number of wedges in different positions which consequently create varying gaps.The field gradient profile shows a significant sloping effect and a reduction in amplitude when the errors exceed 2%.As we shall discuss below, one of the final steps in the manufacturing process is boring out the PMQ hole to a specified value, in our case 2.5 mm.In doing so, it is prudent to understand the effect of an undercut or overcut.As shown in the theoretical analysis, a variation in the PMQ bore radius is inversely proportional to the field.In the final case of our imperfect PMQ model, the following set of PMQ bore radii are considered: 2.45 mm, 2.50 mm and 2.55 mm.As expected, the field gradient predicted in different radius parameters gives 591 T/m and 556 T/m, corresponding to inner radius of 2.45 mm and 2.55 mm, respectively.This variation is ∆B ≈ ±18 T/m.In com- FIG. 6: The field gradient plots with magnet wedge shape errors.Each curve represents wedge shape (mechanical) deviation from the ideal.Again the plots show increasing uneven characteristics with increasing percent error.paring with the 2-D Halbach PMQ model, the variation is deduced from Eq. ( 3) to be |2B r ∆ri r 2 i | ≈19.5 T/m.Three notable error studies are done providing the following results that have been implemented in our magnet design: (1) a magnetization easy-axis angular orientation error is allowable up to 2% before a notable change develops between the magnetic center and the mechanical center; (2) the wedge shape error (the wedge angle) is limited and allowed up to again 2%; and finally (3) bore radius variations of ±0.05 mm from the nominal 2.5 mm radius produced the field gradient variation of ±3% from the ideal B .In Section III, magnet errors that are incorporated into the final focus system have un-desired effects on the beam dynamic, therefore the beam simulation results will be discussed.

III. BEAM DYNAMICS
The high field PMQ allows for short focal lengths, and thus very short β-functions and ultra-small beam sizes from high brightness beams.However, a final focus system that employs these magnets poses a design challenge; magnet errors considered in the previous section pose a severe performance problem in obtaining a small beam spot size.Control of the input beam size is critical to avoid causing excessive beam sizes within the small aperture PMQ magnets.Selection of the final focusing lattice depends on the input beam energy, input beam size, and output sizes desired.Tuning of the final focus system, through adjustment of the magnet spacing, requires iterative simulation for optimization of performance.This section will start with the presentation on the theory of optimum triplet configuration based on thin-lens analysis; then the first-order beam simulation with Trace3D proving the focusability of the final focus system; and finally with the second-order beam simulations with El- egant for magnet errors and beam chromatic aberration effects on the final beam parameters produced.

A. Optimized Triplet Design
In this section the limitations on the final focus spot size using quadrupole triplet focusing are discussed and the optimum configuration of a simple triplet design is found.These design considerations motivate the choices of PMQ configuration and up-stream optics and diagnostics used in an experiment.
Consider here an idealized system composed of three thin lenses separated by drift spaces and alternating between focusing and defocusing in the horizontal (x ) dimension.An illustration of this model problem is shown in Figure 8.The assumed constraints for this design are that the focus be symmetric, and that the incoming beam is symmetric and at a waist, σ x0 = σ y0 and σ x0 = σ y0 = 0.The usual matrix formalism [29] may be employed to describe the particle trajectories in the system.Using M f = 1 0 −1/f 1 for a focusing thin lens (with a change in sign of f in the case of a defocusing lens) and M L = 1 L 0 1 for a drift, the transport matrix of the system is simply For a given maximum obtainable focusing strength, or equivalently minimum focal length, f min , one seeks the parameters f 1 , f 2 , f 3 , L 1 , L 2 , L 3 that maximize the convergence angle, x at the focal point.Ignoring the small initial angle (x 0 = 0), the final angle is given by x f = M 21 x 0 , where (6) This expression can be simplified by requiring a round beam spot at the focus.This forces the matrix element M 21 to be equal for x and y.Setting the righthand side of Eq. ( 6) equal to its counterpart in y leads to the relation: Similarly, requiring x = y at the third lens by equating the M11 elements of the x and y versions of the matrix M = M L2 M f2 M L1 M f1 gives the equation The expected result shown by Eq. ( 8) is that the second lens must be stronger than the first, provided L 1 > 0. Applying Eqs. ( 7) and ( 8) to Eq. ( 6) and solving for variables f 2 , L 1 , and L 2 leads to Now, if it is assumed that the second lens provides the shortest focal length, f 2 = f min , the minimum spot size is achieved by maximizing the expression L 1 L 2 /(L 1 + L 2 ), subject to the constraints given by Eqs. ( 7) and ( 8) and the fact that f 3 ≥ f min .These three conditions lead to the inequality, With back substitution, the optimum triplet configuration has parameters: (10) Note that if Eqs. ( 6)-( 8) were solved for f 3 and the assumption f 3 = f min is made, the maximum convergence angle still occurs with the parameters of Eq. ( 10), showing this (that is, a 2f , −f , +f asymmetric triplet lens configuration) to be the optimal design.

B. Simulations using Trace3D
The beam dynamics simulation codes Elegant [30] and Trace3D [31] were employed to study the initial lattice design, sensitivity to errors, as well as for online tuning of the focusing system during beam runs.
We employ the final electron beam focusing magnet triplet configuration suggested by the thin-lens analysis of the previous section: Focus-Defocus Defocus-Focus Focus (F-DD-FF), where the double strength (half-focal length) lenses are created by concatenating two identical magnets of the type discussed in Section II.The doublelength magnets are moved in tandem, to act as a movable single lens.The asymmetric triplet is first studied using an envelope code based on the first order matrix particle transport simulation code: Trace3D.The optimum lattice configuration was thus studied with this code to establish tuning conditions and performance predictions across a wide range of beam energies.Further, as is discussed below, the code is employed as a tuning guide in the experimental application of the focusing system.
As stated above, the Trace3D code was utilized to demonstrate the focusability of the wide beam energy range through dynamically configuring the drift spaces between PMQ lenses.Two examples of such simulation studies are illustrated in Figure 9 for two different beam energy settings: 72 MeV and 92 MeV.The following characteristic were simulated: rms emittances of ε x =7 mmmrad, ε y =15 mm-mrad and β-functions of β x =2.0 m, β y =0.9 m.Trace3D as in Figure 9, shows that the magnet spacings L 1 , L 2 and L 3 between quadrupole magnets expand with increasing electron beam energy, in order to increase the focal power of the triplet as it is applied to a more rigid beam.Lower beam energy operation, below 60 MeV, is not possible in a system where 560 T/m PMQs are utilized.To access this lower energy range one may employ a weaker focusing lens configuration.The following system has been recently implemented for lower energy (20-35 MeV) ICS experiment mode: weak Focus-strong Defocus-weak Focus (WF-SD-SF).Here the weak PMQ is achieved by boring out an existing PMQ to a larger inner radius, which produces a measured field gradient of 270 T/m.In addition to replacing the first PMQ with a weaker PMQ, the two following lenses consist of a single PMQ as opposed to a double PMQ in the original configuration.Trace3D simulations which show tunable focusability down to 30 MeV have been recently verified experimentally.
Detailed studies, with a sophisticated higher order matrix calculation code, Elegant, of effects on the performance of the PMQ final focus system in the presence of both mechanical and magnetic errors are presented in the following section.

C. Simulations using Elegant
In addition to the magnet manufacturing errors discussed above, there are several types of intra-triplet PMQ alignment errors which may have profound effects upon the performance of the PMQ final focus system.The two most serious are rotated (skewed) PMQs and transverse magnetic center offset error.
Understanding the effects of these errors requires use of a more powerful tool for analysis, the higher-order envelope and tracking code Elegant.The effects of these errors are examined here separately.Solutions for mitigating the effects of the errors in the final focus system's overall mechanical design are discussed below which effectively correct errors to within tolerances allowed.
A systematically rotated field in a quadrupole magnet is expected when the 4-fold symmetry of the quadrupole is broken by an error in manufacturing process, for instance of un-even wedge sector shapes and incorrectly magnetized blocks.In addition, errors in angular placement of the PMQs with respect to the beamline may also produce a skewed magnet.
A simple physical model can be constructed to study the effects of magnet rotation error.The rotation angle error φ around the z -axis is equivalent to a rotation of the beam axes.The transverse space-coordinate rotation is applied through a matrix written as follows [32]: In the simulation code, the skew rotation term is individually introduced in quadrupoles.In our study, we outline a tolerance envelope in rotation by applying a systematic error.A clockwise rotation error is applied to a focusing quadrupole and a counter-clockwise rotation error applied to a defocusing quadrupole.Based on the simplified model constructed, Elegant simulations are generated for three different cases of magnet rotation angle errors: 0, 0.01 and 0.02 radians.The normalized emittances in both transverse dimensions display growth proportional to the rotation error, as shown in Figure 10.The final rms beam spot size grows from 20 to 34 µm when a 0.01 rad systematic rotation error was introduced.The beam spot size produced of course grows progressively worse with increasing rotation; in the case of 0.02 rad rotation error, a 200% growth in final beam spot size is found.
In actuality, all of the PMQs shared the same magnet characteristics, as they are all created by slicing each magnet from a long assembly.Thus, a rotation error due to a segment or magnetization variability found in one particular magnet is equally found in other magnets and therefore an azimuthal rotation error is essentially eliminated from the focusing system.Any other rotation errors can be also introduced through incorrect mounting of the lens assembly.Therefore, considerable emphasis was placed on mechanical tolerances.In particular, magnet holders for enclosing PMQs are specifically machined to comply with less than 0.01 radian angular error.
The electron beam dynamics in the presence of displacement error in PMQ center with respect to beamline axis is also studied in the simulation code.As noted in the magnet simulations of magnet easy-axis orientation and wedge errors, the magnetic center is shifted from the nominal geometric center of PMQ due to the cumulative effect of magnet errors.When such PMQs are installed into the translation rail of the final focus system, magnet centers of the focusing and the defocusing PMQs will be in offset positions due to 90 degrees rotation rela- tion.The effects of quad misalignments are to produce steering; if the vertical midplane of a quadrupole is displaced vertically by δy, it will introduce a dipole field δB x = B δy [32].Furthermore, this steering produces momentum dispersion, which leads to degraded performance.Determining the acceptable tolerance in magnet displacement places demands on the types of adjustments one needs in mechanical design of magnet holders.These tolerances are established through simulation.In practice, with magnetic center measurement data, corrections are applied through adjustments, after centerline measurement of each magnet, of the PMQ holder.
To simulate misalignments, the PMQs are simply specified to have transverse misalignments in the Elegant input deck.The focusing PMQs are assigned a vertical displacement error and the defocusing PMQs the same horizontal displacement.The following values in displacement error are used: 0, 0.1, 0.2 and 0.4 mm.The displacement error's effect on the horizontal normalized emittance is less evident when compared to the vertical emittance as the beam encounters a greater total misalignment in this direction, as shown in Figure 12.The emittance growth due to misalignment is less severe than that due to the skew rotation error, because it is driven mainly by momentum dispersion, which is a small effect for the design momentum spread of 0.5%.The emittance growth in vertical term is near negligible up to 0.1 mm misalignment, growing to 8% at a larger 0.4 mm displacement from the centers.
As noted in the introduction, the use of short focal length magnets in the final focus system is in fact motivated by the need to evade chromatic effects.We now discuss the aberrations expected due to momentum errors.The focusing error from chromatic aberration arises due to the fact that the focusing strength is inversely proportional to the particle momentum.Thus, particles with more momentum will focus downstream of particles with a nominal beam energy, and this chromatic aberration causes a blurring in the beam spot size.A severe case of a beam demagnification is un-desirable at PLEIADES as the x-ray flux is inversely proportional to the (final beam size) 2 at the interaction.The effects of beam momentum spread effect on the final focused beam spot have been simulated with Elegant, and are shown in Figure 11.
As mentioned in the introduction, the chromatic aber- ration becomes the dominant factor contributing to the final beam size as the input beam size σ 0 is expanded.Since Eq. ( 2) was derived as a generalization to a thick triplet lens, as opposed to the original first-order, thin lens treatment found in Ref. 15, a simulation study of its validity was performed using Elegant.The results of this analysis are plotted in Figure 13.In this study, we used the original design emittance of the PLEIADES injector, so the overall scale of the achievable spot sizes is smaller.
As can be seen in Figure 13, chromatic aberration dominates the initial beam emittance above the optimum beam input size σ 0 > σ 0 | opt where The corresponding minimum final beam spot size obtained in the presence of the chromatic aberrations is The simulations shown in Figure 13 assume a round input beam with initial β-functions in the range 2 m≤ β ≤250 m, 0.6% fractional rms momentum spread and 3.6×10 −2 mm-mrad rms geometric emittances.The Elegant results are plotted against the prediction of Eq. ( 2).In this way, the effective focal length of the final focus system in the x -dimension was determined to be f ef f =6.2 cm.The predicted optimum input β-function for the minimum final beam focal spot size of 5 µm is 10 m for this case, which has a relatively small emittance.For the actual experimental parameters with degraded emittance, the initial optimum β is smaller.

A. Magnet Design and Fabrication
Fabrication of an ultra-high field gradient Halbachtype PMQ with good field quality has proven to be x16 x6 14: An illustration of the PMQ manufacturing process showing an individual wedge, the assembled 16 wedge rod, and a single PMQ cut from the rod.
very difficult from an engineering perspective; assembling highly magnetized, small trapezoidal segments into a precisely aligned unit with a small bore poses mechanical challenges.In addition, fabrication of multiple PMQs with both identical magnetic and geometric characteristics increases the constraints.
Nevertheless, a simplified fabrication procedure for producing a high quality, and ultra-high gradient PMQ may be given as follows: a long trapezoidal wedge which is pre-magnetized to a B r =1.22 T is machined to a specified design shape from a larger magnet block through an high precision cutting process, wire electrical discharge machining (EDM).The 16 magnetized wedges are then coupled together carefully into an equally long aluminum keeper tube.In addition to magnet wedges being held inside the keeper, a thin layer of cohesion material is applied, strongly joining neighboring magnets.The tube keeper and a bonding glue together ensure structural stability of the PMQ under extreme magnetic repulsions.The long assembly is then cut into six identical PMQs, with mechanical length of 10 cm, to ensure consistent performance between all PMQs.In particular, construction of a single, long magnet assembly assured that the relative azimuthal (roll) errors of individual units may be essentially eliminated.Finally, the inner diameter of bore is slowly ground to the specified value of 5 mm with a diamond cutter.An illustration which schematically summarizes the PMQ fabrication procedure is shown in Figure 14.

B. Magnet Material Properties
High performance permanent magnet materials are available in large quantities and a variety of formulations such as neodymium iron boride (NdFeB), samarium cobalt (SmCo) and ferrite (Fe 2 O 3 ).NdFeB is particularly suitable for our application due to its low-cost, high remanent field, high intrinsic coercivity (H ci =21 kOe), and good mechanical characteristics.In contrast, SmCo is generally known to be weaker in B r by 25%, extremely brittle and more expensive than NdFeB.On the other hand, the major disadvantage of NdFeB is that its Curie temperature is much lower than that of SmCo.Consequently, a quadrupole fabricated from NdFeB is more sensitive to temperature fluctuations.
For the application at hand, the benefits of NdFeB far outweighed the drawbacks.A particular grade of material was selected based on various parameters, availability and homogeneity.Key parameters of the NdFeB (grade 35SH) used in the PMQs are listed in Table I.It is worth noting that higher remanent fields are available in different grades; however, it was felt that grade 35SH provided sufficient field without compromising coercivity and homogeneity.In applications requiring larger bores and higher field gradients, one might consider other grades.
The manufacturing process required uniform magnet material properties because no sorting process was relied upon when selecting for the individual magnet wedges.Instead, the long magnet wedges were all machined from a single pre-magnetized block, with the field orientation assured by the cut angle rather than a postmachining magnetization.This recipe transferred the possible sources of error and precision to the machining process, and spared us from having to measure very small magnets and sort hundreds of wedges.

C. Mover System
Before implementing the PMQ system on PLEIADES, general problems associated with the application of this focusing system in the experimental environment had to be solved.Addressing the mechanical tolerances and space constraints of the PMQ system, while maintaining an environment consistent with ultra-high vacuum (UHV) and magnetic fields, required careful integration.
Precision machining of all parts eliminated the need for adjustable alignment in the movers that the PMQs were mounted upon.Computer controlled custom-fabricated vacuum linear motion feedthroughs were utilized to control each quadrupole's longitudinal position, which was precisely constrained by a rail system.Motion of the PMQs was set by use of stepper motors and LabVIEW control software.The entire magnet axis of the system was aligned by use of a pulsed wire system and optical techniques.Other features of the assembly include a ceramic ablation disc at the front of the assembly to protect the magnets from misaligned high-intensity Falcon laser damage (during the ICS experiment), and a scintillating disc at the (beam entrance) side of the assembly, for locating mis-steered electron beams.
From a mechanical design approach, the system had to meet the following requirements: independent (bench top, pre-installation) correction of magnetic center to mechanical center errors for each PMQ assembly, independent position control of PMQs to within 100 µm in the beamline direction, magnetic center-to-center transverse placement with 10 µm precision for each PMQ to a center line along the range of motion, insertion of assembly into a standard 4 inner diameter (ID) vacuum chamber with 6 outer diameter (OD) flanges, and compatibility with vacuum at a nominal pressure of < 10 −6 Torr.
A mechanically simple design was chosen to comply with the above stringent requirements.A precision computer numerical control (CNC) machined stainless steel conflat flange was fabricated to serve as a reference mon-FIG.15: A rendered CAD drawing of the final-focus assembly.The three focusing elements are shown on the three-rod alignment system, connected to the main vacuum flange (6 OD), with the linear actuators extending from the backside.The stepper motors and optical encoders are also shown in the back.
ument, mechanical support and vacuum seal.Three precision ground rods were fitted into the flange by means of expansion and shrinkage fits.
Pairs of magnets were supported by CNC machined aluminum disks, each with three precision linear bearings press fit into the outer diameter.The disks and bearings move along the rods, and are controlled by custom fabricated vacuum linear motion feedthroughs.The entire mechanical assembly is supported by the main flange using a massive precision ground stand.The stand affords five axis alignment using the three shimmed or machined legs, Figure 15.
An initial set of disks were machined to place each PMQ into the mechanical center of the assembly (±0.0005 ).Using optical techniques (theodolite and autolevel) along with a pulsed wire system, the magnetic center of each PMQ was found relative to the assembled mechanical center -this measurement included offsets due to the sag of the rods, and machining errors.New discs, if needed, then be machined with the offsets found, thus zeroing out any errors.Fiducials -clear cylinders with etched cross hairs -are placed inside the front and back PMQ for optical alignment.

V. PMQ SYSTEM MEASUREMENTS
Various measurements on the PMQs were performed using pulsed wire and Hall probe.In comparison with the 3-D magnetic simulations and tolerance studies, the quadrupole magnet properties indeed agree well with predicted values.More details of measurement techniques, as well as their usefulness for strength and alignment testing, are discussed in the following sections.
FIG. 16: A single PMQ module is shown set in a test Vblock while on the pulsed-wire metrology system.The wire can faintly be seen through the PMQ.Behind the PMQ, the laser-detector ring is visible.

A. Pulsed-Wire Technique
Measurements of the PMQs were made with the pulsed wire technique, which was pioneered in the context of tuning undulator magnets [33].The test set-up included a single meter-long, 50 µm diameter BeCu wire which was stretched out through the PMQ magnetic field region and anchored at end mounts.One end was fixed, while the other end had a pulley over which the wire was tensioned by a hanging weight.A pulse of 20-50 V, with a duration of a few msec to sec, and a repetition rate of 1 Hz was produced from a high voltage pulser triggered by a function generator.When the current pulse is passed along the wire through the PMQ magnetic region, it interacts with the PMQ B -field and deflects the wire.A detector consisting of a photo-diode sensor and a diode laser -with a small focusing lens to focus the beam on to the center of the wire -mounted in a Plexiglas ring, and arranged so the transmitted light is proportional to the wire displacement, was employed to quantify the deflection.Two sets of laser-sensors were used in order to measure both transverse deflections simultaneously.
The pulsed wire technique was used to detect nonlinearity in the B -field dependence on transverse displacement.The BeCu wire was translated horizontally through the magnetic center from −2.0 mm to +2.0 mm.The deflection amplitude gives a measure of the (longitudinally) integrated vertical component of the B -field.According to the data reported in Figure 17, the transverse B -field shows no apparent higher multipole content; the dependence of B y on x is linear to within measurement error over the scan interval.

B. Alignment Measurements
The final focus system in a F-DD-FF triplet configuration requires a high degree of alignment of magnetic centers between the component PMQs.Determination of the relationship between the geometric and magnetic centerlines of each individual PMQ as well as that of the complete lattice was accomplished using the pulsed wire technique.
The PMQ magnetic center to the mechanical center alignment has been studied using the pulsed wire technique.Each PMQ was mounted on a custom design V-block mount, which in turn is bolted onto two high precision micrometer stages which provide vertical and horizontal position variation.First, the end mounts were adjusted until the wire was level with respect to the vibration isolation optical table.The PMQ was then translated in both transverse directions until the BeCu wire was aligned to the geometric center of the magnet bore as the process was observed with a theodolite (see Figure 16).If pulsed wire deflection was observed on the digital oscilloscope connected to the detector apparatus, the BeCu wire was relocated through translations of micrometer stages to find the PMQ magnetic center (zero field).From the pulsed wire measurement, it was determined that the magnetic center was aligned to the geometric center to within the instrument observation capability which was 25 µm.
A more difficult alignment measurement of the PMQ mover system has been carried out employing the pulsed wire setup.A crucial requirement of the final focus system is that the PMQs riding on the rails maintain transverse alignment when moved longitudinally.A main source of motion deviation can be expected due to the sagging of the rail rods due to gravity.This sagging can introduce a dynamic steering of the electron beam at the IP, as the PMQs are moved.
The alignment of the final mover system was observed with a high precision theodolite for horizontal motion and an auto-level for vertical motion.The auto-level indicated a noticeable sagging in the vertical direction.The alignment in the horizontal direction, however, was good to within 25 µm, like that of the pulsed wire system.The maximum vertical sagging was observed when magnets were moved to the downstream end of the mover system and minimized when they were placed at the fixed end of the rails.The sagging was corrected with a set-screw located on the bottom of the ring-piece which ties together the free ends of the rails.

C. Hall Probe Measurements
Although the pulsed wire technique was well suited to the tasks described above with high spatial resolution, attempts to use it as a means to measure the field gradient were limited by the apparatus conditions: wire sag, dispersion and sensitivity to vibration noise.In particular, the wavelength dependence of the wire response prevented calibration of the 1 cm long magnet measurements with a known 10 cm long electromagnetic quadrupole.As an alternative option for quantitative measure- ments, a miniature magnetic Hall probe was employed to determine the field gradient and the effective magnetic length of the PMQ.With the PMQ mounted on the same translation micrometer stages, the probe scans horizontally reading B y .The field gradient obtained from this measurement in the center of the PMQ bore region was measured to be 560 T/m (see Figure 18) using this method.We note that a Hall probe is limited in spatial resolution, which in turn limits the characterization of non-linear field components in a quadrupole magnet.An effective magnetic length was also determined using a Hall probe by reading the value of B y at each point along a longitudinal line located 0.5 mm diagonally from the magnet axial axis.The Radia simulation predicted an effective magnetic length of 10.4 mm, where the magnetic length is defined as the integral B y dz/B max , with B max indicating the peak field.The measured trace, as shown in Figure 19, shows a good agreement with the simulation.While the entire range of the magnet is not  accessible to measurement, the reproduction of the Radia results allows us to deduce the effective length, which are employed as inputs in the beam dynamics simulations, to be essentially that given by Radia.

VI. BEAM BASED MEASUREMENTS
A. The PLEIADES Experiment The PMQ system was installed in the PLEIADES ICS experiment, in which a relativistic (25-80 MeV), highbrightness electron beam is collided with a 50 femtosecond, up to 1 Joule, 820 nm laser to produce short pulse (0.3-5 psec), high peak brightness, hard x-rays, tunable in energy from 40-140 keV.
The number of x-ray photons, N x , produced by the interaction of the laser and electron beams is determined by the intensities of the beams as they overlap: where n L ( x, t) and n e ( x, t) are the laser photon and electron beam densities, respectively, σ T is the total Thomson cross section, v is the electron beam velocity, and k is the laser propagation wave number.For the PLEIADES experiment, a 180 • scattering geometry is used.The integrated x-ray flux is found, assuming Gaussian spatial and temporal profiles for both beams, and that the laser and electron beam pulse durations are less than the Rayleigh range and β-function of the laser and electron foci, respectively, to be Here σ e and σ L are the electron and laser beam spot sizes at the interaction point, and N e and N L are the total numbers of electrons and laser photons in the pulses.Note that PLEIADES experimental design parameters include a bunch charge approaching 1 nC (N e = 6 × 10 9 ) and a laser photon energy approaching 1 Joule at a wavelength around 1 µm, resulting in N L = 5 × 10 18 .Assuming the incident laser and electron beams are matched such that σ e = σ L , then to obtain a flux of x-rays suitable for a single shot diffraction experiment (about 10 8 ), an rms interaction spot size in the 10's of microns is required.Further, since initial x-ray data was obtained with σ L < σ e , the PLEIADES ICS experiment benefits significantly from decreasing the electron beam size at the interaction point and is therefore an excellent location to initially implement the PMQ final focus system.

B. Installation at PLEIADES
The PLEIADES electron beam is generated using a BNL/SLAC/UCLA style 1.6 cell photo-cathode RF gun [34].Following the gun, the beam is accelerated to energies of 20-100 MeV using as many as four S-band 2.5 m traveling wave sections.For the measurements described below, the gun produced 200-300 pC bunches with 3 ps rms duration, while the final beam energy was varied between 50 to 70 MeV, and the typical normalized, rms emittance was around 10 mm-mrad.
To install the PMQs as the e-beam final focus system, the high energy electron beamline was redesigned both to provide approximate beam matching into the PMQs and to retain diagnostic capabilities.A schematic of this beamline is shown in Figure 20.
The laser-electron IP (interaction point) is located at the center of a 6-inch cube vacuum chamber.As described above, the PMQ mover assembly is mounted directly into the interaction chamber, allowing positioning of the magnets at a range of distances directly upstream of the IP (see Figure 21).The minimum distance of the final PMQ to the IP is determined by the width of a polished aluminum cube positioned at the IP.This cube functions as the spatial and temporal overlap diagnostic by directing the laser light and the optical transition radiation (OTR) produced by the electron beam into a CCD camera and a streak camera.In addition to enabling the overlap of the two beams, the CCD images naturally allow optimization of the focused spot size.This cube is extracted from the beam path for x-ray production but cannot be fully removed from the PMQ path.As a result, the minimum distance from the edge of the final PMQ to the IP is 5 mm.
After the IP, a round pole dipole magnet is used to bend the electron beam onto a trajectory which separates it from the x-ray pulse, allowing the beam to be dumped into a shielded Faraday cup, minimizing the noise reaching the x-ray detectors.Note that a spectrometer magnet was also located just upstream of the final matching quads to measure the energy and energy-spread of the beam, which are essential steps in tuning the linac.

C. Tuning the Final Focus Optics
In accordance with the quad triplet model used to design the PMQ final focus system, the beam must be manipulated up-stream of the PMQs to produce a moderately focused waist at the entrance to the first PMQ.The first step in this process is the determination of the rms Twiss parameters of the beam exiting the accelerator, which is done using the quadrupole scan emittance measurement technique.In the quad scan measurement, the current in the first electromagnetic (EM) quad in Figure 20 is varied while the rms beam size is measured after a known drift length.Taking into account the finite length of the quadrupole magnet (using a "thick lens" analysis) the final beam size in each transverse direction, σ f , follows the well-known relation [35] where l q,d are the EM quad and drift lengths, respectively, K is the focusing strength of the quad given by its magnetic field gradient and the beam momentum, K = eB /p, and the initial Twiss parameters are β i = σ 11,i /ε, α i = −σ 12,i /ε, and γ i = σ 22,i /ε, where ε is the rms emittance.Because quad scans must be performed as part of the routine operating procedure, the process was automated using LabVIEW based control system software.The immediate determination of the emittance provides the linac operator with feedback on the quality of the linac tune, as well as giving the Twiss parameters needed for matching the beam into the PMQs.
Once the Twiss parameters have been determined, they are used as input to a Trace3D model of the beamline.Given the beam parameters at the first element of the model, and those desired at some latter point, Trace3D can perform a matching function, in which a user defined set of model parameters are adjusted in an algorithm to produce the desired final beam.In this case, we adjust the strengths of EM quad doublet and triplet to produce a beam waist (α = 0) in both transverse dimensions at the entrance to the PMQs with an rms size around 350 µm.An example of the output of this matching process is shown in Figure 22.The choice of rms beam size at the final focus entrance is a compromise between emittance and aberration considerations, and will be discussed further below.
Once the Trace3D model has determined the proper settings for the upstream quadrupoles, measuring the beam size at two pop-in beam profile monitors and comparing the results with the model predictions allows a partial check of the model.With good agreement found between model and measurement, the Trace3D model is used again to determine the proper spacing of the PMQs.The result of this matching process is shown in Figure 23.
Here the spacing between 3 PMQ holders is adjusted to produce the smallest β-function possible at the IP.The linac operator uses the Trace3D values of upstream quad strengths and PMQ positions as a starting point.One important practical consideration the operator must take into account is steering the beam onto the axis defined by the PMQ mover system.This is needed both to minimize the effects of chromatic and spherical aberrations in the PMQs and to precisely define the direction of the ICS x-rays generated.There are two steering magnets located between the EM quad triplet and the PMQs that are used for this job.Steering is also made easier by imaging the beam on a plastic scintillator mounted on the upstream face of the first PMQ holder.This scintillator has a 5 mm diameter hole that is mounted concentric to the PMQ aperture.With these tools, proper beam steering is accomplished using the two steering magnets to iteratively move the beam to the center of the scintillator hole and observe the motion (or lack of motion) of the beam at the IP while moving the final PMQ holder.
The process described above has been used to focus the PLEIADES electron beam at the IP using the PMQ system.The spot size was measured using OTR generated by the beam colliding with the interaction cube diagnostic as described above.This light is collected by a CCD camera/lens system and digitized using a computer frame grabber.A typical and highly repeatable beam spot is shown in Figure 24.In this case we focused a 70 MeV, 200 pC beam to less than 20 µm rms size in both transverse dimensions.This is a significant improvement over the previous final focus system, which achieved spot sizes of roughly 50 µm, rms.

D. Emittance Measurements with the PMQs
A variation of the quad scan technique for emittance measurement is the so-called three screen or moving screen measurement, in which there is a focusing lens followed by a drift space and measurement screen.However, in this case the beam size is measured while varying the drift length instead of the lens strength.In the simplified thin lens analysis (where the focal length is f = 1/Kl q ) the equation analogous to Eq. ( 16) is where β * is the minimum β-function at the waist, ε 0 is the beam emittance, and σ res is the image resolution  obtained from the limitations of optical transport and camera performance.In the free drift, the square of the rms beam spot size expands quadratically away from the beam waist position, z 0 .
In the case of the PMQ final focus system, the focal length, f, is fixed and the drift length, z, is changed most easily by moving all three of the PMQ holders in unison.This is not technically a correct application of Eq. ( 17), since by moving the quadrupoles instead of the screen, we are varying the initial Twiss parameters.However, since we know the initial β-function of the input beam to be at least 1 m, and since the scans must only cover several β * , or less than one cm, the initial conditions remain essentially constant over the scan.
Thus, the analysis of the moving screen technique to the "moving quads" measurement is valid in the present scenario.One such dataset and analysis is shown in Figure 25.The effects of limited image resolution can be clearly seen in these measurements.The expected parabolic dependence of σ 2 on z is observed well away from the minimum beam sizes, but the data give a nearly flat dependence in the zone near z 0 .In the case studied in Figure 25, normalized initial emittances measured at 58.2 MeV beam energy using the EM quad triplet are ε x =16.6 mm-mrad and ε y =9.7 mm-mrad, while the normalized emittances obtained with the PMQ scan are ε x =16.3 mm-mrad and ε y =12.9 mm-mrad.During the initial period of running the final focus system, beam spot-sizes as low as σ rms ∼18 µm were measured in both transverse dimensions after optimization.
As noted, this is an upper limit on the actual beam size, due to the resolution limitations in the measurement system.We note from Figure 25 that the value of the σ res obtained from the best fit of the quad scan data to Eq. ( 17) is also 18 µm.With the value of σ res determined by the fit to the data, we deduce that the minimum beam size in each transverse dimension in Figure 25 would have been in the range of 15 µm in the ab-sence of image resolution problems (e.g.geometric and chromatic aberrations).In order to observe such small sizes, a reconfiguring of the optics between the interaction region OTR surface and the viewing CCD camera will be undertaken in the future.One of the most critical needed capabilities the PMQ-based final focus system, that of tunability over a wide range of beam energies, has been demonstrated.Through longitudinal motion of the PMQs, the final focus system was capable of adjusting to consistently produce a small beam spot-size at different input beam energies.This variability, over 40% in beam energy, has produced variation of x-ray photon energies over the range between 40 to 140 keV.
Recently, the final focus system was successfully reconfigured by boring out the initial isolated PMQ's inner diameter to lower its gradient by approximately one-half, and removing one of each of the downstream PMQ pairs.This new system has allowed production of an optimum beam focus at as low as 20 MeV electron beam energy.This new capability has allowed use of PLEIADES to pursue ongoing dynamic pump-probe experiments which investigate dynamic diffraction in different heavy-metal materials.

VII. DISCUSSIONS AND CONCLUSIONS
An adjustable short focal length final focus system based on an asymmetric triplet configuration (a single focus-double defocus-double focus quadrupole array F-DD-FF) composed of ultra-high field gradient (experimental value∼560 T/m) PMQs has been developed.The design issues concerning the PMQs themselves were studied in detail using the 3-D magnetostatic simulation code Radia.The PMQ array was fabricated, measured, and placed on a precision moving system.It was then implemented at the ICS final focus system beamline at LLNL PLEIADES.This effort led to creation of ultrasmall beam spot sizes, and an enhanced performance of a high brightness ICS x-ray source, with adjustability demonstrated over a wide range of input beam energies, and thus produced x-ray energies.
The principles behind the short focal length, adjustable PMQ-based final focus system are quite attractive for advanced applications.At UCLA, a similar system, with few cm focal lengths, is presently being implemented in the final focus of a nonlinear ICS experiment in the Neptune laboratory.As the emittance is expected to be a factor of three lower in the Neptune experiment, this test should allow demonstration of the emittance preservation at a level much closer to the stateof-the-art in high brightness photoinjector systems.Additionally, the current experience in creating very high gradient, small bore quadrupoles is of continuing interest for application to linear collider final focus systems, as the demonstrated gradient achieved here exceeded the CLIC efforts by over 15%.Finally, we note that standard F-DD-F PMQ triplets using longitudinal position tuning, with much larger bore and lower gradients have been used at LANSCE for proton radiographic microscopy [36], an application which does not require large demagnification, but nonetheless demands very high strength quadrupoles.

FIG. 3 :
FIG.3: Vector field plot of the PMQ in the bore region, shown in 2D slice, generated from the 3D code Radia.The length of the arrow reflects the field amplitude.

FIG. 4 :
FIG. 4: The magnetic field lines plotted showing decrease in slope as temperature value increases (top) and the percentage loss of field gradient as a function of temperature (bottom).

FIG. 5 :
FIG.5:The field gradient plots with magnetization orientation errors.As the angular orientation error increases, a less symmetrical magnetic profile results with increasing center offset.

FIG. 10 :FIG. 11 :
FIG.10: Rotation error effect on normalized emittance growth through an otherwise ideal triplet as simulated by Elegant.

FIG. 13 :
FIG.13: Study of chromatic aberration effects in PLEIADES PMQ final focus system, using Elegant simulation (dots).Also, the line shown is the theoretical prediction.

FIG. 17 :
FIG. 17: Field linearity scan over accessible PMQ aperture obtained with the pulsed-wire setup.

FIG. 18 :
FIG.18:A comparison of PMQ field profile measurements using a Hall probe to Radia simulation results.

FIG. 21 :
FIG.21:The adjustable PMQ final focus assembly after installation at PLEIADES.The vacuum cube which serves as the IP is shown on the left, with the bellows, precision flange, and linear actuators extending to the right.

FIG. 22 :
FIG. 22: Trace3D model of the beamline used to compute the focusing strength of the upstream electromagnetic quadrupoles for proper matching into the PMQs.The values encircled in a circle line are calculated quad strengths in the upstream EM doublet and triplet.

FIG. 23 :
FIG. 23: Typical Trace3D output specifying the spacing between PMQ blocks and their position with respect to the beam focus.The values encircled in a circle are calculated drift spacings between PMQs.

FIG. 24 :
FIG. 24: False color CCD image of the PMQ focused electron beam.The RMS beam size is 20 µm in both x and y.

FIG. 25 :
FIG. 25: Quadrupole scan of the final focus beam at LLNL PLEIADES, performed by moving the PMQ assembly.
Trace3D simulations demonstrating the tunability of the final focus system.Beam energy: 72 MeV (top) and 92 MeV (bottom).The horizontal axis corresponds to the longitudinal beamline and the vertical axis for the matching β-functions.The final β-functions is in a 3-5 mm range.