Tapered helical undulator system
for high efficiency energy extraction
from a high brightness electron beam

There are several parameters in the design of the TESSA-266 experiment to be optimized in order to maximize the conversion efficiency. Using time-independent single frequency genesis simulations – i.e. assuming infinitely long electron and laser pulses –, a numerical optimization maximizes the bunching factor at the undulator entrance as a function of seed laser Rayleigh length and waist position as well as the prebuncher chicane R56 setting. The resonant phase and optimum tapering design along the undulators are then determined using the Genesis-Informed-Tapering (GIT) algorithm [20] in order to maximize the radiation power at the exit of the system. Effectively the algorithm maximizes the number of particles trapped in the ponderomotive bucket and the deceleration gradient along the interaction. The phase shifts in between the undulator sections (to compensate diffraction and dephasing effects) are also separately optimized. Fig. 2 shows that the maximum power obtained in these simulation studies is 39.5 GW which translates to 11 % peak conversion efficiency, consistent with the average change in the beam energy (right panel). The resonant energy along the interaction decreases by almost 20 $\%$ as can be seen looking at the undulator normalized parameter $K$, but not all particles participate to the interaction and the bunching factor remains around 0.6 for the entire undulator.

The transverse lattice of the TESSA-266 beamline was optimized to maximize the extraction efficiency in the strongly tapered undulators. Due to the inverse-square dependence of efficiency on transverse beam size [28], solutions on how to minimize the average beam size along the interaction were sought [38]. Assuming a 2 mm-mrad emittance, with no added quadrupoles, solely relying on the undulator focusing the average rms size in the undulator can be calculated to be $\langle\sigma_{x}\rangle=80\mu$m. The standard solution of introducing alternating gradient focusing and defocusing quadrupoles in each drift sections yields a $\langle\sigma_{x}\rangle=68\mu$m. It was then found that using a quadrupole doublet in the break section would make the smallest transverse average rms beam size of 46$~{}\mu$m (Fig. 3). The different focusing lattice solutions were compared using the time-independent tapering optimization algorithm, and the solution with a quadrupole doublet in the break section was found to yield the smallest average transverse beam size in the undulators and the highest output power and was decided as the design choice.

The beam from the linac can be matched to this strong focusing channel using two electromagnetic quadrupoles in between the prebuncher and the first section. A list of all the magnetic elements in the TESSA-266 beamline is reported in Table LABEL:tab:lattice. Besides the undulator and the quadrupole doublet in the break sections, this list also includes the dipoles in the injection and modulation chicanes and matching quadrupoles before and after the prebuncher.

While optimizing the tapering for realistic beam distributions has the potential to improve the results by taking into account the actual energy and temporal profiles of the electrons, time-dependent simulations are computationally intensive and not suited to large parameter scans. For this reason, in order to evaluate finite bunch length effects and provide guidance to the optimization of the beam dynamics through the linac, we employed the tapering solution obtained from the time-independent simulations. Due to the slippage phenomenon, it is found that flat current distributions (Figure 4(b)) yield a significantly higher FEL efficiency compared to Gaussian distributions. In Fig. 4(a) we show the output for two electron bunches having identical rms peak current and charge, but different current profiles. It is immediate to see how the radiation pulse overlaps better and for longer with the e-beam current distribution for the flatter current distribution. The energy conversion efficiency can be obtained by the ratio of the total energy in pulse (integral of the power profile along the bunch coordinate) to the e-beam energy (300 pC 343 MeV corresponding to 100 mJ). The flat current profiles shows a significantly larger conversion efficiency 8 $\%$ with respect to the 5 $\%$ obtained using a gaussian pulse. In the limit of very long flat pulse, the 10 $\%$ conversion efficiency limit of the steady state would be recovered. Including in the simulations resistive wall wake effects produced only a small change in the power output. For example, considering a gaussian profile bunch in a stainless steel pipe, time-dependent genesis simulations show a minimal reduction in efficiency from 5.23% to 5.14%. Surface roughness effects are harder to estimate as they depend on the vacuum pipe quality. Assuming similar order of magnitude effect, we don’t expect the experiment results to be significantly affected by wakefields.

A careful management of the injected longitudinal phase space will be required in order to maximize the efficiency for TESSA-266. The way this is accomplished would entirely depend on the available linac system, but here we briefly discuss the case where the experiment is installed at the end of the APS Linac [39] and driven by the high brightness photocathode RF gun (PC) recently been installed as a secondary APS injector [40, 30].

Taking as a reference Fig. 5, electrons from the PC gun accelerate through four linac structures L1, L2, L4, and L5[41]. A bunch compressor is located just after the L2 linac section [42]. Through the L4 and L5 linac sections, the electrons are accelerated to the optimal energy for TESSA-266 experiment ($\gamma=671$). After the linear accelerator, the beam is deflected in a vertical ramp (R56=0.129mm, T566=30mm, U5666=281.8mm) which has a small effect on the final longitudinal phase space, before being sent thru the LEA beamline to the injection chicane (R56=-0.149mm, T566=0.224mm, U5666=-0.147mm) to create a 10 mm offset to enable the injection of the seed laser. A shortcoming of the current linac configuration is the lack of a high frequency RF linearizer that would greatly help in controlling the longitudinal phase space after compression.

In order to better understand the challenge, we can model the energy-position correlation upstream of the bunch compressor–i.e. at the end of PC gun and linacs L1 and L2– using the following expression

where $A_{\text{gun}}$, $A_{1}$, $A_{2}$ and $\phi_{g}$, $\phi_{1}$, and $\phi_{2}$ are the maximum energy gains and the phases of PC gun, L1, and L2 respectively and $k$ is wave vector of S-band structures.

Assuming a short bunch, we can use a Taylor series expansion of Equation 1, and express the relative energy of each particle $p_{i}$ just upstream of the bunch compressor in a polynomial form. Keeping up to third order coefficients we have:

A closer look to Eq. 1 will show that regardless of the beam chirp, for all accelerating phases (between 0 and $\pi$) the energy position correlation is characterized by a negative second order curvature at the entrance at the bunch compressor. After compression, this term is responsible for a skewed, narrow current spike shown in Fig. 6a which would severely degrade the FEL interaction due to slippage even though the peak current is 1  kA.

In principle, it is possible to correct for the curvature if a non linear compression can be applied. In fact, the final longitudinal coordinates after the bunch compressor can be written as:

Combining the two last equations shows that the second order non linear coefficient would be cancelled if the following condition were satisfied:

Unfortunately in a standard magnetic chicane bunch compressor $R_{56}$ and $T_{566}$ are opposite in sign and the only way to satisfy this condition is to introduce non linear elements in the chicane (changing the sign of $T_{566}$ or adjust the curvature coefficient $d_{2}$ to be positive.

Both options have been evaluated using the start-to-end model developed starting with astra simulation up to L1 exit [43], and then elegant to track the particles to TESSA-266 entrance [44]. We first considered adding sextupoles to control $T_{566}$ in the bunch compressor. Two sextupoles would be sufficient to fully linearize the longitudinal dynamics, but in order to avoid severe transverse emittance growth they should be placed at location with a $\pi$ phase advance between them [45, 46]. However, due to the skewed design of the bunch compressor which was to minimize the CSR effects [42] along with the restrictions associated to satisfying the interleaving lattice matching [30], it proved too difficult to get the required phase advance between the two sextupoles.

The favored solution to resolve the nonlinearity is to use a high frequency passive corrugated structure [47], to correct the second order curvature prior to compression[48, 49]. When a high current electron beam goes through a corrugated structure, the resulting wakefield can be used to modify the longitudinal phase distribution of the beam [50]. We considered the use of rectangular corrugated structures instead of round pipes, as this would allow one to vary the gap using a mechanical jaw adjustment. In addition, the effects on the emittance due to the transverse wakefields can be canceled using two flat corrugated structures rotated by 90 degrees with respect to each other [51].

Accurate expressions for the short range wakefield of a flat dechirper are available in the literature [52, 47]. As an initial design study we considered the fundamental mode of longitudinal wake field in a slab geometry corrugated structure [53]:

where the wave vector $k$ depends on the corrugation period $p$, the height $h$ and spacing $g$ of the corrugation teeth, and the pipe radius $a$ as $k=\sqrt{\frac{p}{ahg}}$. In order to keep compatibility with the linac operation for APS storage ring operation, the linearizer can be located between L1 and L2.

The linearizer parameters were optimized using elegant simulations in order to take into account higher order terms in the longitudinal beam dynamics such as space charge forces, CSR and the small contributions from the vertical ramp and the injection chicane before the TESSA-266 setup. A global optimization selects the optimum aperture and wavevector as well as L2 phase to linearize the compression. The L2 voltage is then adjusted to keep constant energy in the chicane. Figure 6b shows elegant output at the prebuncher entrance, for the case where a 0.2 m long linearizer of 2.6 mm gap, and wave vector 1700 m^-1 is placed after L1 structure. As a quality factor for the linearization process we consider the fraction of e-beam slices that fill an ideal rectangular current distribution of 1 kA and 90 $\mu$m which can be used as a proxy for the FEL efficiency in the system. Figure 6 shows the output longitudinal phase spaces (LPS) from elegant at the prebuncher entrance for the case of linearizer off (left) and linearizer on (right). The first one is characterized by a long talk characteristic of non linear compression. The red dashed line shows the ideal 1 kA rectangular current distribution. In the linearizer on case, more than 80% of the beam slices fit within desired 1 kA rectangular current distribution while only 60% without a linearizer. The double-horn shape of the LPS distribution could be suppressed by applying further compensation to eliminate the third order terms [54], but due to the limited space in the bunch compressor it would be challenging to do this while limiting the transverse emittance growth.

The 6-dimensional distribution elegant output (x,xp,y,yp,t,p) is then converted to 6-dimensional genesis input (X,PX,Y,PY,T,P) to complete the TESSA-266 start-to-end simulations. After adjusting the position of the beam centroid to maximize the overlap with the input seed laser pulse, optimized time-dependent simulations yield up to 3 % efficiency, much higher than the 1 % efficiency obtained from the non-linearized compression case which is characterized by a very long low-current beam tail. The main reason of still large reduction in output power with respect to the ideal case is the large energy spread of the final beam. If this could be reduced to the design levels, efficiency above 5% could be obtained. Future studies could involve optimization of the undulator tapering profile considering the actual temporal profile.

Once the initial design working point has been established and the tapering profile of the undulator defined, we carried out a study on the tolerances to key input parameters for the TESSA-266 system using a series of genesis simulations. Acceptable parameters are defined as the ones that reduce the output power by less than 20 $\%$. It was found that when multiple parameters vary at the same time, at first order the reduction in output efficiency can be obtained simply as the quadrature sum of the contributions from each independent parameter variation off the ideal conditions. In Table 3 we summarize the tolerance intervals obtained for all of the parameters including laser waist position, Rayleigh range, energy spread and transverse emittance.

These results are important not only to give an idea of the beam stability required for the experiment, but also as they allow to specify the level of accuracy of the various diagnostics used to characterize the TESSA-266 input beams.

Fig. 7a shows the relative output power variation as a function of the relative seed laser time of arrival as estimated by performing multiple time-dependent simulations varying the seed pulse injection time compared to the peak of the e-beam temporal distribution. The result is $\pm$0.35 ps interval is attributable to the pulse length of the seed laser. The tolerance interval on the injection energy and the magnetic errors in the undulator (also shown in Fig. 7b,c) is set by the amplitude of the ponderomotive resonant bucket in the longitudinal phase space which is 0.2 $\%$. The energy stability of the APS linac (0.1 %) should comfortably be within this tolerances. More attention will be needed to control the linac phase fluctuations to keep the time-of-arrival within the specified tolerances. Finally, in order to quantify the alignment error tolerances we used the simulation to predict the effects due to changes in electron beam position at each undulator section entrance. The results are shown in Fig. 7d.

The undulator design is an evolution of the first tapered helical undulator built by UCLA for IFEL experiments [59]. The undulator permanent magnets are designed with narrow, trapezoidal shape in order to increase field strength in the center of the gap, and the sides are slanted in order to prevent the magnets from slipping out of the holder. Each magnet is glued by Loctite M-31CL to an aluminum holder. Four brass strongbacks are fixed to the undulator end plates and are machined with v-shaped slots where the magnet holders can be inserted and slide. Each four sets of magnets are attached from the bottom of the holders to a tuning plate with #6-32 screws which can be used to adjust the undulator tapering profile. This operation does require a re-measurement of the undulator trajectory so will not be possible once the undulator are installed on the beamline. Figure 10 shows a cross section view of the undulator technical drawing [60].

The undulator magnets are responsible for a non negligible magnetic force on the undulator structure. Based on radia simulations, there is inward force of up to 100N inward on the holders of the magnets with in-out magnetization and a net 46N after averaging on each undulator tuning plate. The total net inward force imposed on each strongback fixed on both ends is estimated to cause bending by 0.06mm based on Solidworks analysis.

The entrance and exit sections are designed to minimize trajectory offset from the geometric center of the undulator. They use 2 mm, 4 mm, and two 8 mm magnets at each end of the array (see Fig. 11(a)) [61] adding a physical length of 34mm at each end of the undulator. Special holders were used to hold the thinner magnets in the strongback.

The magnetic measurements are performed using a setup designed to precisely hold a narrow three axis Hall probe and guide it along the geometrical center of the assembly. In order to satisfy the tight requirements on the magnetic field errors along the undulator, positioning of the Hall probe should be within 200 $\mu$m of the beam axis with less than 2 degrees rotation error. The 2 mm wide Hall probe and the guiding mechanism are designed to fit inside the small diameter beam pipe. This is important as the relative magnetic permeability of the tube is specified to be less than 1.05, however maximum values of 1.08 were measured close at the weld joints. For this reason, final measurements and tuning of the magnetic field were performed with the pipe installed.

The Hall probe scan yields the variation of the three field components $Bx,By,Bz$ as a function of $z$ and is the basic tool used to tune the magnetic field of the undulator. The lack of soft magnetic material in the THESEUS undulators allows the tuning procedure to rely on the principle of superposition. The change in field strength (transverse and longitudinal) as a function of $z$ when a single magnet is moved using a tuning screw is normalized and defined as the magnet tuning eigenfunctions. The eigenfunctions are computed using radia and benchmarked against Hall probe measurements as shown in Fig. 12. A matlab tuning code optimizes the linear combination of eigenfunctions that, when added to a first measurement scan, would minimize the error with respect to the target magnetic field profile.

The optimization fundamentally relies on the probe accurately measuring the fields along a fixed axis in the undulator. Systematic twisting or bending of the tuning carriage caused by machining errors or slight magnetic/gravitational forces must be understood and taken into account. Fortunately, these errors have predictable effects on measurements. Before tuning starts, a reference Hall probe scan is recorded with all magnets pulled out to their mechanical hardstops. An offset from the axis in the scan will result in a non zero measured $Bz$ field sinusoidal and in phase with the $Bx$ field ($y$ offset) or the $By$ field ($x$ offset). Additionally, a roll twist error will manifest in a shifting of the perceived peak field positions. The reference scan is used to generate a systematic transformation map which is applied to the data before optimization to account for such errors. The ability of the reference Hall probe scan to accurately compute probe position has been confirmed by directly observing the vacuum pipe bend (with one of the undulator arrays removed) under with effect of gravity with pulls in the $+\hat{y}$ direction (Figure 13b). The probe twist was consistent with independent measurements obtained reflecting a laser off a mirror glued above the probe carriage (Figure 13a).

Finally, the fields are fine-tuned to minimize the beam slippage through the interaction as shown in Fig. 14a to ensure that the particles see the same phase when they move through the different periods of the undulator. The magnetic error tolerance from the genesis simulations can be translated into a period-average slippage error of less than 0.5 %. The picture show that the achieved tune satisfies the resonance requirement.

After the desired tapering profile is obtained with the Hall probe scans, entrance and exit magnets are tuned last using the pulsed-wire method to zero the first and second field integrals [62, 63, 64]. The method works by passing a current pulse through a wire tensioned through the undulator. The magnetic fields exert forces on the wire as the pulse passes through, creating wave vibrations that propagate in both directions. For a short pulse, the deflection of the wire is proportional to the first field integral and described the velocity of an electron passing through the undulator. For a long pulse, the deflection is proportional to the second field integral and describes an electrons trajectory. The main advantage of this method is in that it provides a direct measurement of the expected beam trajectory in the undulator and gives instantaneous feedback while tuning. Common limitations to be considered are the sag of the wire and dispersion in the vibration waves [65, 66]. The beam trajectory before and after tuning the entrance and exit section for the prebuncher undulator are shown in Fig. 14b.

The TESSA-266 Hybrid Quadrupole design consists of four steel poles with low carbon content ($<$0.06$\%$), and two NdFeB (1.45 T remanence) wedge sectors. Adjustable shims are used to create a magnetic circuit which can siphon magnetic flux density away from the aperture when the shims are brought in towards the magnet. The design has been chosen over other variable permanent quadrupole options [68, 69] to satisfy the constraints over the available beamline distance. When the shims are pulled away from the magnet, the magnetic circuit is severed, and the pole tips are maximally magnetized, still below the saturation limit. The shims are mounted to the steel poles and their positions are controlled with thumbscrews. A schematic is shown in Fig. 18.

The pole tips are machined to match the shape of the ideal hyperbolic potential surfaces in the transverse plane having 4 mm pole tip radius relative to the symmetry axis. The magnetic design has been simulated in radia yielding a peak integrated gradient of 6.16 T, with an effective length of 30.9 mm with the shims all the way out and a range of tunability of up to 30$\%$ with a good field of region of at least 1.5 mm of radius. In that vicinity, the ratio of the quadrupole moment term to all other moments is small, i.e., $\frac{\int B_{y}(r=1.5mm,\theta)\cos(\theta)d\theta}{\int B_{y}(r=1.5mm,\theta)\cos(n\theta)d\theta}<0.001$. Simulation results also indicate magnetic center motion does remain within 50 $\mu$m of the geometric center when an error smaller than 1 mm is applied to the shim displacement position.

A prototype hybrid quadrupole was constructed at RadiaBeam, and measured at UCLA, using an FW Bell STD18-0404 Hall probe. A Zaber translation stage was used to control the longitudinal motion of the probe. The motion is accurate down to 25 $\mu$m. Manual micrometers were used to control the transverse motion of the probe. We measured longitudinal gradient profiles for several shim displacements. In Fig. 19(a), the measured gradient profile with shims all the way in is shown in comparison to the radia prediction assuming a magnetization of 1.39 T for the permanent magnets. The gradient longitudinal profile can be modeled reasonably well by:

where $G_{0}(s)$ is the peak gradient as a function of the shim displacement ($s$), $b$=350 m^-1 is the field stiffness parameter, and $L_{\textrm{eff}}$ as the effective length. The model fit to the measured data is also shown in Fig 19(a). For this profile, the effective length of the quadrupole was found to be 30.7 mm $\pm$ 0.2 mm in agreement with the simulations. The integrated gradient as a function of shim displacement is shown in Fig. 19(b). The integrated gradient was found to be adjustable from 4.2 T to 6.1 T fully satisfying the demands of the TESSA-266 design. Finally, magnetic field was measured along the symmetry axis to estimate the magnetic center motion with a $<$ 50 $\mu$m resolution. The corresponding field variations were integrated and normalized by beam rigidity to give a 0.1 mrad bound on beam centroid kicks.