Compensation of the effects of a detector solenoid on the vertical beam orbit in a linear collider

1098-4402= This paper presents a method for compensating the vertical orbit change through the interaction region that arises when the beam enters the linear collider detector solenoid at a crossing angle. Such compensation is required because any deviation of the vertical orbit causes degradation of the beam size due to synchrotron radiation, and also because the nonzero total vertical angle causes rotation of the polarization vector of the bunch. Compensation is necessary to preserve the luminosity or to guarantee knowledge of the polarization at the interaction point. The most effective compensation is done locally with a special dipole coil arrangement incorporated into the detector (detector integrated dipole). The compensation is effective for both e e and e e beams, and the technique is compatible with transversecoupling compensation either by the standard method, using skew quadrupoles, or by a more effective method using weak antisolenoids.


I. INTRODUCTION
The future electron-positron International Linear Collider (ILC) requires high luminosity which can only be achieved by colliding very small nanometer scale beams.In the earlier linear collider (LC) projects, Next Linear Collider/Global Linear Collider (NLC/GLC) and TESLA, the beam sizes at the interaction point (IP) were x;y 243; 3 nm and z 110 m for NLC/GLC, and x;y 554; 5 nm and z 300 m for TESLA .
The design of the ILC interaction region (IR) is constrained by the often conflicting requirements of providing strong focusing for the incoming beam, acceptable background environment for the experimental detector, and clean extraction of the outgoing beams.The ILC is specified to have two IRs, at least one of which will likely use a small (up to about 20 mrad) crossing angle in the horizontal plane to facilitate extraction of the outgoing disrupted beams.The crossing angle allows separate incoming and outgoing beam lines, which can be optimized independently.The second IR must additionally be able to accommodate collisions, which require a slightly larger crossing angle, up to 20 -35 mrad.
When the horizontal half crossing angle c is larger than x = z , a crab-crossing technique is required in order to preserve the overlap of the beams in collision at the IP.Two rf cavities located several meters upstream of the IP on both beam lines introduce a kick correlated with longitudinal position within the bunch, so that the bunches rotate and fully overlap at the IP.
The horizontal crossing angle means that the beam traverses the magnetic field of the detector at an angle and thus will be deflected into the vertical plane.The change in the vertical orbit causes degradation of the beam size due to synchrotron radiation (SR), and also causes rotation of the polarization vector if the total vertical angle is nonzero.
This paper discusses both these effects in the context of the NLC design with c 10 mrad and presents possible methods for compensating the vertical angle at the IP and minimizing synchrotron radiation effects.Local compensation using a novel dipole coil integrated with the detector solenoid represents an optimal solution, and is effective for both e e ÿ and e ÿ e ÿ beams.The results will scale with crossing angle for the ILC.
The paper is organized as follows.In Sec.II we discuss the effects on the vertical orbit using the approximation of a detector solenoid with sharp edges.Synchrotron radiation effects due to the vertical deflection and the resulting beam size growth are considered in Sec.III.For a realistic case with the silicon detector, a technique for compensating the vertical IP angle and minimizing SR effects using a detector integrated dipole (DID) corrector is presented in Sec.IV, where the compatibility of the vertical orbit compensation method with the beam size compensation by means of the weak antisolenoids suggested in Ref. [1] is also discussed.Finally, design considerations for the detector integrated dipole are given in Sec.V. edges, schematically shown in Fig. 1.In the case without compensation, the vertical deflection is caused by the edge kick c B 0 L=2B, which occurs when the beam enters the solenoid off axis at c L, and also by the kick linearly distributed in the body of the solenoid.Here c is half of the crossing angle, L is the half-length of the detector solenoid, B 0 is the solenoid field, and B pc=e is the magnetic rigidity of the beam.The body kick integrated from the solenoid entrance to the IP is equal to ÿ2, which is twice the edge kick, and since the body kick has half the lever arm, the resulting vertical offset at the IP cancels exactly (see also Refs.[1,2] for a rigorous proof).The remaining vertical angle at the IP is nonzero and equals ÿ.The maximal deviation of the vertical orbit before the collision is L=4.The vertical angle of the extracted beam, which passes through the entire solenoid, is ÿ2 and the vertical offset at the exit is ÿ3L.
Let us first discuss the impact of the vertical orbit on luminosity.In the case of e e ÿ collisions, which is expected to be the primary mode of operation of the Future Linear Collider, the vertical angles of the opposite beams are antisymmetric, so the beams collide head on and do not experience any loss of luminosity.In the e ÿ e ÿ option, the trajectories are symmetric and the vertical crossing angle must be compensated to preserve the luminosity.Such compensation can be done either with rf cavities to provide vertical crab-crossing, or with the DID corrector method discussed below.In both cases, the vertical deflection will cause growth of the beam size due to synchrotron radiation.The detector integrated dipole can be used to minimize this beam size growth as well.
In addition to luminosity considerations, it may be desirable that the IR optics preserve the beam polarization (the e ÿ or possibly both beams will be longitudinally polarized), as discussed in Ref. [3].A change of the beam orbit by an angle causes the orientation of the polarization vector to rotate by g=2 ÿ 1 due to the anomalous magnetic moment of the electron.In the example shown in Fig. 1, with 45 rad, the polarization vector rotates by about 1.5 , producing a difference between the polarization at the IP and that measured at an upstream polarimeter.Although this spin rotation could be predicted rather accurately, in practice for certain precise physics measurements with either e e ÿ or e ÿ e ÿ , one would benefit if the vertical angle at the IP were compensated to ensure accurate knowledge of the beam polarization.Crab-crossing compensation is not adequate in this case, and local compensation of the vertical orbit angle is needed.
Compensation of the vertical orbit deviation produced by the detector solenoid field can be done locally using a special dipole field incorporated into the detector (detector integrated dipole) and with one additional external corrector on each side.Their combination can correct the vertical IP angle without changing the IP offset.This technique is illustrated in Fig. 1 (bottom) for the idealized case of a solenoid field with sharp edges.A similar technique can be used to flatten the orbit in order to minimize the effects of synchrotron radiation.
Since the direction of the transverse field seen by a particle and the direction of the required compensation field do not depend on the particle charge, the same compensating field works for both e e ÿ and e ÿ e ÿ beams.Also, the inner kick acts on both the incoming and outgoing beams, but the outer kick acts only on the incoming beam.Therefore, the vertical angle of the extracted beam increases only by the value of one outer kick.To facilitate extraction of the beam, and allow downstream polarization diagnostics, the vertical angle of the outgoing beam can be corrected by additional correctors in the extraction line.Inside the detector, the transverse field acting on the outgoing disrupted beam is increased by the value of the inner kick, which needs to be included in calculations such as evaluating background due to low energy pairs created in collision.

III. SYNCHROTRON RADIATION EFFECTS DUE TO THE VERTICAL DEFLECTION
Synchrotron radiation is emitted whenever there is a change in the beam trajectory producing both a decrease in the average energy and an increase in the rms energy spread.At the IP, the energy change causes an orbit offset that can be easily corrected, but the energy spread will increase the beam size.The vertical orbit shift and synchrotron radiation emitted in the detector solenoid field have already been discussed in Ref. [2].In this paper, the derivations presented in [2] are generalized to a more realistic situation where the solenoid field overlaps with the final doublet (FD) magnets, thus the effects of the FD focusing must be included.
The horizontal field which drives the vertical orbit change is given by where the first two terms correspond to the field of the solenoid, and the last term is due to any correction field.This field, together with the field created by the orbit offset in the final doublet quadrupoles with gradient Gz, defines the vertical orbit, and determines its curvature z.The SR-induced increase in energy spread for a slice dz is then given by 55r e e 5 dz 24 3 p jzj 3 ; (3) where r e is the classical electron radius, e r e =, and is the fine structure constant.The corresponding addition to the vertical beam size is where R 36 is the transport matrix element from the particular slice to the IP.The latter is given by which includes the effect of FD focusing.
Adding in quadrature the contributions from each slice, we obtain the IP beam size increase due to synchrotron radiation: where C E 55r e e =24 3 p and the integral is over the full extent of the solenoid and correcting fields.The quantity SR y is to be added to the nominal IP size in quadrature.This quantity is proportional to B 0 L c 5=2 and does not depend on the beam energy.However, the nominal beam size typically decreases with energy, and therefore this SR beam size growth must be kept small enough not to limit performance at higher energies.

IV. VERTICAL ORBIT COMPENSATION IN A REALISTIC DETECTOR
As a realistic example, we consider the NLC interaction region optics and parameters, and the field configuration of the proposed silicon detector (SiD).Figure 2 shows the ANSYS model of the SiD detector and the field lines, and Fig. 3 shows the fields on the beam trajectory and locations of the final doublet focusing elements of the NLC beam delivery system (BDS).In addition to the fields of the bare SiD solenoid, we will also consider two other cases of detector field combined with the field of a weak antisolenoid.
As discussed in Ref. [1], most of the beam size distortion due to the solenoid is generated because the solenoid field overlaps with the final quadrupole and breaks the natural cancellation of coupling and other beam correlations.Correction of the distortions requires using the linear combinations of skew quadrupoles and displacements of sextupoles (so-called linear knobs).When the beam energy is changed, these corrections need to be reoptimized.A short weak antisolenoid, coaxial with the detector and overlapping with QD0, can be matched to cancel the integral effect of the solenoid-FD overlap, and restore cancellation of the COMPENSATION OF THE EFFECTS OF A DETECTOR . . .Phys.Rev. ST Accel.Beams 8, 041001 (2005) distortions.This cancellation then works for any beam energy.In the SiD case and NLC beam parameters, the vertical beam size increases 30 times in the bare SiD and only by 30% if a weak antisolenoid is used (thus this antisolenoid compensates 99% of the beam size distortions).In this paper we use the antisolenoid with the parameters suggested in [1] (the red line in Fig. 3) and also an alternative configuration, optimized to reduce the SR effects (dash-dotted green line in Fig. 3).The beam trajectory in the bare SiD is shown in Fig. 4. If there were no focusing elements in the detector field, the vertical trajectory through the detector solenoid would primarily be determined by the horizontal field B x B r ÿ c B z (shown on the top plot).The corresponding vertical trajectory, obtained by simple integration of the SiD horizontal field, is shown by the dashed line on the bottom plot.Just as for the sharp edge solenoid field, the IP offset is exactly canceled but the IP angle is nonzero.This cancellation of the IP offset is extremely important because the coupling and other beam distortions introduced by the solenoid are also canceled.The presence of focusing elements that overlap with the solenoid field destroys this perfect cancellation of the orbit and beam distortions.Reference [1] contains more discussion of the solenoid effects on the beam size and compensation methods.For realistic simulations of the orbits and beam sizes, tracking with DIMAD [4] was used, where the FD region was modeled by a sequence of short, typically 1 cm long, slices containing all the solenoid, dipole, quadrupole, sextupole, and octupole fields, using the realistic solenoid field map FIG. 3. (Color) Longitudinal and radial magnetic field in SiD calculated by ANSYS, without and with the weak antisolenoid which cancels the beam distortions produced by the detector solenoid.The red line shows the field with the antisolenoid parameters suggested in [1], and the green dot-dashed line shows the field with another configuration of the antisolenoids, optimized to reduce SR effects (see text).The radial field is at the nominal beam trajectory with half crossing angle c 10 mrad.Locations of the final doublet elements (quadrupoles QD0 and QF1, sextupole SD0, octupole OC0, and an optional dipole corrector BXMID) are also shown.The IP is at z 0 m. and the design BDS optics.For the SiD example, the beam orbit obtained by tracking with DIMAD is shown in Fig. 4 (bottom, solid line).The vertical angle at the IP is about 100 rad, and the vertical IP position is not zero (approximately ÿ20 m), due to the overlap of the solenoid field with the final quadrupole QD0.The beam size growth due to synchrotron radiation, calculated analytically using formulas from the previous section is SR y 0:31 nm.The horizontal orbit deviation is much smaller than the vertical one, and is neglected in this paper.
As seen in Fig. 4, the SiD detector solenoid field produces a vertical trajectory which has its maximum curvature about 2 m from the IP.Thus, the most effective compensation will be local.For example, if one attempts to compensate the IP angle by offsets of the FD quads QD0 and QF1, as illustrated in Fig. 5, the resulting orbit deviation would be too large and synchrotron radiation would increase the beam size by an unacceptable amount.In this example, due to the nonlocal nature of the compensation, the vertical orbit deviation reaches 0.4 mm, and the beam size increases by SR y 5:2 nm, an amount that would significantly reduce the luminosity.
Local compensation of the IP angles can be done by a pair of dipoles embedded in the detector coil at large radius (a few meters).The fields should be antisymmetrical around the IP.Such a field can be created by a pair of dipole windings integrated with the detector solenoid in its cryostat, referred to as a detector integrated dipole corrector.(This has also been called a serpentine corrector due to a particular winding technique, which inspired the idea of local compensation of the IP angles).Design considerations for the DID corrector are given in Sec.V.The DID field shape shown in Fig. 6 was calculated with the OPERA3D code and the coil geometry is determined by the SiD dimensions.This field represents only one of the possible solutions for local correction, and the particular field shape is not important as long as it is local to the detector.
For the DID corrector field to compensate the IP angle, this field has to be combined with external kicks of opposite sign, which together produce an angle at the IP, with no offset.The FD quadrupoles can be offset to produce the external kicks.Figure 7 illustrates compensation of the IP angles in SiD using the DID corrector together with offsets of FD quadrupoles QD0 and QF1.While, in principle, only the QD0 needs to be offset to correct the angle and offset at the IP, a small QF1 offset cancels the vertical second order dispersion produced by the total correcting field.In this example, the optimal displacement of QF1 is about 2% of that of QD0.The combined effect of DID and quads resembles the effect of the solenoid itself.The IP angle is compensated to less than 1 rad.The local character of the correction limits the orbit deviation near the IP, which also limits the beam size growth (in this case SR y 0:26 nm).
Another obvious application of the DID corrector is to minimize the beam size growth caused by synchrotron radiation.This may be especially important for the second IR or for a larger size or stronger field detector, since the beam size growth depends sharply on the angle and detector length: SR y / B 0 c L 5=2 .For the SiD example and with c 10 mrad, the SR beam size growth is SR y 0:31 nm without and 0.26 nm with the IP angle compensation.The DID strength optimized to flatten the vertical orbit and thus to minimize the SR beam size growth corresponds to 60% of the field which minimizes the IP angle; see Fig. 6.The corresponding orbits are shown in Fig. 8 (blue curve).The SR beam size growth is reduced by about an order of magnitude, to SR y 0:034 nm.The vertical IP angle is also reduced by approximately a factor of 3 with this solution.Figure 9, which shows the dependence of the IP vertical angle and of SR y on the DID strength, demonstrates that with the SiD detector and DID geometry considered here, there is a wide region of parameters where both the IP angle and SR effects can be made small simultaneously.In practice the balance between the two corrections will be dictated by the specific needs of the experiment.
Let us now discuss the vertical orbit compensation in the case when weak antisolenoids are used to cancel the beam (ii) SiD with antisolenoid (parameters from [1]) (red line), SR y 0:83 nm; (iii) SiD with antisolenoid optimized to minimize SR effects (green dash-dotted line), SR y 0:33 nm.In the last two cases the IP angle is compensated by the DID, FD offsets, and BXMID without introducing any linear or second order dispersion.distortions produced by detector solenoid.In this case, if the antisolenoid solution is to remain distortion free when combined with the DID corrector, the latter should also be made distortion free.The DID arrangement considered earlier does not generate any second order vertical dispersion; however, it produces linear dispersion.To make the DID arrangement completely distortion free, one more dipole corrector must be added in the middle of the FD and then the three parameters (offsets of QD0, QF1, and middle dipole field) matched simultaneously to cancel the first and second order dispersion and the IP offset produced by the DID corrector.
Figure 8 (red curves) shows the horizontal fields on the beam axis and the orbit calculated by tracking in SiD with antisolenoid (with parameters from [1]: z 0 4:14 m, z w 0:5 m, B z0 ÿ1:56 T and with the field profile given by B asol z B z0 =f1 z ÿ z 0 =z w 4 g).One can see that the IP angle compensation is as effective as before.However, in this case the SR-induced beam size growth is SR y 0:83 nm (Fig. 9).This still gives a small effect with c 10 mrad.But when scaled to the maximum considered crossing angle of 35 mrad, this SR contribution becomes SR y 3:4 nm, which is already a significant increase of the beam size.One can also see that the DID strength cannot be optimized to significantly decrease the SR beam size growth in this case.
The field of the weak antisolenoid can be reoptimized to reduce the SR effects.One of the possible solutions consists of four antisolenoids with the following parameters: z 0 3:64; 3:89; 4:14; 4:64 m, B z0 0:30; ÿ0:19; ÿ0:43; ÿ0:60 T, and z w 0:5 m for all four of them.This solution simultaneously minimizes the IP vertical offset, the SR beam size growth, and two major beam distortions (hyx 0 i and hyEi).The beam size before applying any additional linear correction is only 52% larger than the nominal, thus this antisolenoid compensates more than 98% of the beam size distortions.The SiD field with these antisolenoids is shown in Fig. 3, and the beam orbit in the case when DID is used to zero the IP angle is shown in Fig. 8 (green dash-dotted line).The SR beam size growth in this case is SR y 0:33 nm, which is about 3 times smaller that with the previous version of the antisolenoid.Figure 9 also shows that the curve of SR y versus DID strength has a wide minimum; i.e., there is a region of parameters where both the IP angle and SR beam size growth are small.

V. DETECTOR INTEGRATED CORRECTOR, DESIGN CONSIDERATIONS
There are several advantages in integrating the dipole correction coils with the cold mass inside the detector solenoid cryostat.First and foremost, a small diameter magnet placed close to the IP would introduce extra radiation lengths of uninstrumented material and reduce the detector acceptance.The large diameter dipole corrector coils proposed are quite thin and present only a negligible addition to the already considerable thickness of the solenoid itself.
Secondly, the interaction of the solenoidal field with the coil ends causes net torques in the horizontal plane that have to be resisted in addition to supporting the weight of the dipole itself.Co-winding the corrector coils with the solenoid in the same cold mass ensures that no new torque is transferred to the outside world and the corrector weight is again a small perturbation for the solenoid coil supports.
Finally, the large dipole coil radius ensures that even for a relatively crude coil configuration the field seen by the colliding beams is very uniform.For the coil configurations investigated so far, the field nonuniformity is less than a few parts in 10 000 at the coil longitudinal midpoints and half the coil radius (1.4 m).Near the beam pipe, the nonuniformity approaches a few parts per 10 6 .However, since the length of each dipole coil is almost equal to its radius and since there is strong cancellation of the field from the dipole coils at the IP symmetry point, the dipole field profile exhibits a marked longitudinal dependence that is nearly independent of the other details and dimensions of the coil structure.
The assumed DID corrector coil pattern is shown at the top of Fig. 10.The number of dipole turns is chosen in such a way that the width of the area unfilled with conductor is about a meter, as shown.Adding additional turns to the winding pattern quickly becomes counterproductive as then the dipole ends become too long and the increase of transfer function is balanced by the reduction of the straight section length.
Initially, the 3D field profiles were calculated based upon the positions of each conductor segment in space (i.e., in effect an air coil).In order to evaluate the effect of the solenoid yoke on the field distribution, a simplified 3D conductor model was generated by averaging the conductor locations to a smaller number of coil packs and this coil was then inserted inside a simplified 3D model of the SiD yoke as shown at the bottom of Fig. 10.
There was a concern that the yoke end cap, that goes down to small radius, might rob too much flux from the body of the DID corrector thereby reducing its efficiency in creating a dipole field.In fact, it happens that the increase of efficiency that comes from the yoke for the body of the magnet more than makes up for any loss near the end cap.In comparison with the field used for tracking, there is a small deviation in the assumed field shape that occurs near the inner edge of the end cap, but overall the results from the 3D field calculations match the heuristically motivated field shape very well.
Finally, it is useful to mention that the DID corrector can be used to optimize the IR trajectories not only in the linear collider, but in other machines as well.For example, it has been suggested to use a DID corrector to optimize the IR region of eRHIC [5] where the trajectories of electrons need to be carefully optimized to allow clean beam sepa-BRETT PARKER AND ANDREI SERYI Phys.Rev. ST Accel.Beams 8, 041001 (2005) 041001-8 ration while reducing SR power.It has also been suggested to use the DID corrector to upgrade the SLAC B Factory [6].In this case the symmetric version of the DID field would provide separation of the beams at the parasitic collisions, and thus the first dipole from the IP, which provides beam separation in the present design, can be replaced by a quadrupole that would give a possibility of stronger focusing and thus higher luminosity.

VI. CONCLUSION
A special dipole can be added in the linear collider IR to correct the effect of the vertical deflection caused by the beam passing through the detector solenoid field with a horizontal crossing angle.To be most effective, the correction (detector integrated dipole) needs to be local, and thus is incorporated into the detector solenoid winding.The DID corrector can be used to compensate for rotation of the beam polarization or to minimize the beam size growth due to synchrotron radiation.The solution presented uses the DID corrector to provide local compensation of the orbit and works both for e e ÿ and e ÿ e ÿ cases.This method is compatible with beam size compensation using weak antisolenoids.The DID corrector can also be used for upgrades of other colliders, such as B factories.

FIG. 1 .
FIG. 1. (Color) Illustration of e e ÿ and e ÿ e ÿ collisions in a detector solenoid field with sharp edges (schematically shown in the top plot) without (middle) and with (bottom) compensation of the IP angle by the detector integrated dipole and two external correctors.The model parameters are L 3 m, c 10 mrad, B 0 5 T, beam energy 250 GeV.The uncompensated vertical angle at the IP is approximately 45 rad.The compensation kicks are shown on the top plot by the blue arrows; they are located at z 2 m and z 5 m and their magnitudes are 75 and 30 rad.The IP is at z 0 m.Outgoing beams are shown by thick dashed curves.

FIG. 2 .
FIG. 2. (Color)Schematic of the silicon detector (SiD) with magnetic field lines calculated byANSYS.The IP is at z 0 m.

FIG. 4 .FIG. 5 .
FIG. 4. (Color) Horizontal field strength on the beam axis of the SiD detector (top), and vertical beam orbits (bottom), in the absence of vertical IP angle compensation.The orbit calculated in the absence of any focusing elements is shown by a dashed line, and the orbit determined by tracking with DIMAD is shown by a solid line on the bottom plot.For the tracked beam, the IP beam coordinates are x 0:65 m, y ÿ18:5 m, x 0 ÿ0:21 rad, and y 0 ÿ104 rad.The beam size growth due to synchrotron radiation is SR y 0:31 nm.

FIG. 6 .FIG
FIG. 6. (Color) Horizontal field of the detector integrated dipole (DID) for the SiD detector.The DID strength is optimized to zero the IP vertical angle (blue line) or to minimize the SR vertical beam size growth (red line) in the case of bare SiD without antisolenoids.