A modified round to flat beam transformation lattice for angular dispersion induced microbunching technique

The angular dispersion induced microbunching (ADM) technique has been proposed to generate high brightness coherent radiation in storage rings by taking advantage of it’s very low vertical emittance. To apply a similar scheme in the linac, it is necessary to reduce the vertical emittance of the electron beam. Generally, angular-momentum-dominated round beams can be generated by immersing the cathode into the axial solenoid magnetic field. The angular momentum can be then removed by skew quadrupoles downstream of the solenoid, resulting in a flat beam with low vertical emittance. In this paper, we propose a possible scheme that holds the possibility to improve the performance of round to flat beam transformation technique by chromatic correction. In this scheme, a chicane is used to introduce the transverse dispersion and sextupoles are employed to correct the chromatic effect. Three-dimensional numerical simulations have been carried out and the results show that a flat beam with transverse emittance ratio of ∼840 can be generated for a bunch charge of 0.5 nC.


Introduction
A new complex that combines the advantages of the energy recovery linac (ERL) and the angular dispersion induced microbunching (ADM) [1] technique has been proposed recently to produce high repetition rate and fully coherent EUV/X-ray pulses [2].Electron beams with low vertical emittances are essential for the realization of this proporal.Typically, the electron beam generated from the photocathode gun has a Gaussian or planar distribution.Therefore, some feasible techniques had proposed to realize round to flat beam transformation.It was found early on that an initially round beam with a net angular momentum can be transformed to an asymmetric beam in transverse dimensions [3].This process can realize transversal emittance exchange and reduce the beam vertical emittance.This round beam with a net angular momentum can be generated by immersing the cathode in solenoid fields.The transformation from this round beam to a asymmetric beam in the transverse dimension can be realized through a skew quadrupole triplet [4].Simulations and experiments have been carried out to prove the potential and feasibility of this technique.A proof-of-principle experiment conducted at the Fermilab/NICADD Photoinjector Laboratory (FNPL) has demonstrated how to generate a flat beam with a measured transverse emittance ratio of 50 [5].In 2006, the smaller normalized root-mean-square (RMS) emittance about 0.4 μm•rad is measured at the Fermilab/NICADD Photoinjector Laboratory [6].To satisfy the needs of linear colliders for flat beams, a team at DESY in Germany increased the emittance ratio to 370 [7].These researches have yielded good results, but it is still not enough for the application of the ADM technique in the linac.In 2004, the chromatic effects in the round-to-flat beam transformation (RTFB) along with the associated impact on the flat beam emittances is studied analytically under the thin lens approximation by Y.-E Sun and K.-J.Kim [8].This study demonstrated that the chromatic effects can not be ignored during the beam transfer.
In this paper, we propose an improved lattice that takes chromatic effects into account and explore a method to compensate it.Simulations has been performed with the tracking program ASTRA [9] and the results demonstrate that a transverse emittance ratio of approximately 840 can be achieved by using the proposed method.

Theory of flat beam generation
Considering the cathode is immersed in solenoid fields, at the exit of the solenoid, the beam is coupled transversely and has a net angular momentum.The coupling of horizontal and vertical coordinates and angles are described by [10]: where =2pc/eBs, being p the momentum of particles, c the light velocity, e the electron charge, Bs the longitudinal magnetic field at the cathode, x 0 , y 0 , x 0 ' and y 0 ' are the transverse position and transverse momentum of the electron at the exit of the solenoid respectively.We can write the tranversally asymmetric flat beam state in this form x x' 0 0 . In order to transform the state in formula (1) in x x' 0 0 , we consider an uncoupled matrix： where A,B are 2×2 matrices of horizontal and vertical motion.Then, we rotate the matrix M AB by =45°: For round to flat transformation, we can get the following expression: giving us the relationship between A,B and F: B=A•F (5) A detailed derivation can be found in Refs.[11,12].
We assume that alpha function 0 = 0 =0 and beta function 0 = 0 = before the transformation (4); Eq.( 5) can be represented by the Courant-Snyder parameters and phase advance at the end of the transformation (4): So, if the beam transformation meets condition (6), the round to flat condition can be realized.After the magnetized beam is transformed into a flat beam, the theoretical transverse normalized emittance is given by [13,14]: where ε ± and ε − are the horizontal and vertical emittance of the flat beam respectively, ε th is thermal emittance of the magnetized beam, is the transverse emittance of the magnetized beam, = 2 .

Lattice design and parameter setting
Here we use the photocathode as the electron source, the initial beam parameters are specified in Table 1.We set the initial root mean square (rms) beam size according the laser spot size of 0.97mm on the cathode.The bunch charge is 0.5nC.The initial bunches show Gaussian distribution in both transverse and longitudinal directions.The layout of the transformation beamline is shown in figure 1.The bunch generates from photocathode which is immersed in the solenoid field.The first solenoid is near the cathode and the magnetic field is 975 Gauss, it is important for the generation of beams with net angular momentum.The second solenoid magnetic field is 775 Gauss.Beam is accelerated by a 1.5-cell rf gun, the accelerating gradient is 32MV/m, we used two 9-cell booster cavity to accelerate the beam to approximately 32MeV in order to reduce the effect of space-charge forces.In order to satisfy condition (6), at least three skew quadrupoles are required.The distance between the three quadrupoles has been studied in reference [15].We consider the chicane and the triplet as a whole element.It's horizontal and vertical transfer matrix correspond to A and B in Eq.( 2).As discussed earlier, we need to tilt chicane and triplet by 45 degree to change the transfer matrix of RFBT from Eq(2) to Eq(3), then insert sextupoles S1, S2 and S3 to correct the chromatic effect.Since there is no element of sextupole in ASTRA, the simulation is carried out by using the sextupole components in a quadrupole instead of the sextupole.Thus in the numerical simulation, S1, S2 and S3 in Figure 3 will be replaced by three quadrupoles, the strength of this type of quadrupole is set to Q_K(j)=0.0001m - to ignore the effects of the quadrupole field, by adjusting the normal sextupole coefficients Q_mult_b(i,j) to generate a sextupole magnetic field, where j=4, 5, 6 represents the element number and i=3 represents the sextupole coefficient.The strength of skew quadrupole Q1, Q2 and Q3 used to meet matching condition ( 6) is represented by Q_K(1), Q_K(2) and Q_K(3) respectively, see Table 2.According to the equation set up under condition (6), we can solve the focusing strength of the triplet.The other parameters are adjusted and optimized according to the simulation results.In this paper, we set the bend angle and radius of dipole to 5 degrees and 5.73m, the dogleg is about 0.1m apart from each other and the dispersion section is about 5.8m, the distance between Q1, Q2 and Q3 is 0.6m and 5.0 m, respectively, S1,S2,S3 are 0.1 meters away from the adjacent quadrupole.

Simulation and analysis
We performed numerical simulations with the tracking program ASTRA.The number of particles is set to 200000.In order to visually compare the effect of sextupoles on the correction of chromatic effect.simulations without sextupoles have also been performed.The evolutions of beam transverse emittance along the beamline are shown in Fig. 4.Where the vertical emittance decreases significantly after adding the sextupole components.The vertical emittance of the electron beam decreases from 0.24 µmrad to 0.065 µmrad.A detailed comparison of the simulation results is given in Table 3.The comparison of flat beam phase space is shown in Figure 5, it can be intuitively found that particles with relative momentum deviation in the periphery of phase space move closer to the center under the action of sextupoles.In the above simulations, the beam thermal emittance was set to be 0.5 µmrad.When the momentum deviation effect is not compensated, the production of the final transverse emittances is 13.06 µmrad 2 , which is reduced to 3.54 µmrad 2 after compensation.

Conclusion
In summary, a new lattice for the RFBT has been given for reducing the impact of the particle momentum deviation.Simulation results show that the vertical emittance of the flat beam decrease from 0.24 µmrad to 0.065 mm mrad, and the transverse emittance ratio increase to 840 after the compensation of the chromatic effect.This ratio is comparable with that in a storage ring.At present, only simulations have been done for the proposed method.Specific application needs further experimental demonstration which is under preparation.

Figure 1 .
Figure 1.Overview of the round to flat transformation beam line.L1 and L2 represent solenoids.CAV1 and CAV2 are two superconducting accelerating cavities respectively.The blue rectangle represents the quadrupole that makes up triplet.

Figure 2 .
Figure 2. Particles with different momentum have different trajectories when passing through the same position of the quadrapole (left).With the sextupole correction, the particles can be focused to the same position (right).

Figure 3 .
Figure 3.The layout of round to flat beam transformation(consider momentum dispersion).The yellow rectangles represent dipole magnets and the blue rectangles represent quadrupole magnets, the green ellipses represent sextupole.

Figure 4 .
Figure 4.The rms beam normalized transverse emittance evolute along the beam line from the photocathode to the exit of RFBT with (solid lines)

Figure 5 .
Figure 5.The left column is the phase space without the sextupole components, and the right column is the phase space after adding the sextupole components.The beam transverse distribution (top), y-y ' phase space (bottom)

Table 1 .
Parameter settings for the photocathode, rf gun and booster cavity during the simulation.

Table 2 .
Parameter of the quadrupoles and normal sextupole coefficients.

Table 3 .
The simulation results of transverse emittance (the unit is µmrad) and ratio.