Acceptance

The second additional factor that allows us to reduce the undulator length is the larger acceptance in CLIC, compared to ILC, for positrons produced from the source. The acceptance defines those particles that will survive from the positron source (exit of the matching device) to the exit of the damping ring. The acceptance is determined by physical and dynamic aperture limitations in the transport lines and damping rings. The reasons that CLIC has a larger acceptance than ILC are as follows.

In the case of ILC, the superconducting RF linac works most efficiently for long beam pulses. Although the beam is compressed to reduce the size of the damping ring, large (6.4 km) rings are still needed to accommodate each machine pulse. This makes it impractical to use a pre-damping ring, which could be optimised for a large

acceptance, without needing to achieve the very low emittances needed in the beams at the interaction point.

However, for CLIC, the machine pulses are much shorter, and the damping rings are correspondingly smaller. This makes it possible to use a pre-damping ring for the positron beam. The pre-damping ring is designed to have a large acceptance, but does not achieve a very small final emittance. But the final emittance is small enough that the beam can be injected efficiently into the main damping ring, which must have a very small final emittance, and will therefore have a smaller acceptance than the pre-damping ring.

We should emphasise that the acceptance of the systems downstream of the positron source is a complicated issue, depending on many different factors, including features of the design of the matching device, acceleration section, transfer line and damping ring. At the moment, the designs of all these systems are incomplete. Therefore, it is impossible to state with certainty what the acceptance will be. To some extent, design choices will depend on how the acceptance affects the positron source. For example, a larger aperture in the transfer line could improve the acceptance (if this were the limiting factor) and allow a reduction in the length of the undulator. However, increasing the aperture of the transfer line would increase the costs of the vacuum and magnet systems in the transfer line. Without a great deal of study and design work it is not possible to say what the optimum designs for all the various systems will be. Therefore, we consider the impact on the positron source of nominal values of the acceptance for CLIC, shown in Table 4.1. The results should provide some guidance for the acceptance actually required for the transport lines and damping rings in CLIC. The acceptance of a damping ring is specified in terms of the maximum betatron amplitude and maximum energy deviation for a particle that will be captured by the damping ring. The horizontal betatron amplitudeAx is defined by:

Ax=γ γxx2+ 2αxxpx+βxp2x

, (4.4)

where γ is the relativistic factor, x and px the horizontal coordinate and normalised

momentum of the particle, andαx,βx and γx are the Twiss parameters. The vertical

betatron amplitude is defined in a similar way. Note that the betatron amplitudes are constant for a particle moving through a simple (linear) magnetic lattice, or being accelerated in a linac. The specified acceptance values for the ILC damping rings and

Table 4.1: Nominal acceptance specifications for ILC damping rings and CLIC pre- damping ring.

ILC CLIC Maximum total betatron amplitude,Ax+Ay 0.09 m 0.57 m

Maximum energy deviation _±0.5% _±1.6%

CLIC pre-damping rings [72] are given in Table 4.1. The values given here are the nominal specifications: in practice, the real acceptance will have a complicated shape in phase space, but particles with the betatron amplitudes and energy deviations shown in Table4.1should always be captured by the damping ring.

Note that the longitudinal acceptance for the damping ring is specified as a maxi- mum energy deviation. This actually places a limit on the length of the bunch from the positron source: if the length of the bunch is significant compared to the wavelength of the RF in the accelerating section, then as the bunch is accelerated, particles at the head and tail of the bunch (which will be slightly off-crest) will get less energy than particles at the centre of the bunch (which will be on-crest). For the ILC and CLIC parameters, this “RF curvature” effect dominates the energy spread of the bunch by the time it reaches the damping ring. The longer the bunch, the larger the energy difference between the head or tail, and the centre of the bunch. For a maximum en- ergy deviation of _±0.5%, particles for the positron source in ILC (with RF frequency 1.3 GHz) should be no more than _±5 mm from the centre of the bunch. For a given bunch length, the energy spread from the RF curvature increases with RF frequency. For CLIC, the bunch spacing is 0.5 ns, so the minimum RF frequency is 2 GHz: this means that the energy spread on a bunch with particles within_±5 mm of the centre of the bunch will be_±1.4%. However, this is still within the expected acceptance of the predamping ring. In the case that a smaller energy spread is required, then a section of linac operating at a higher harmonic frequency can be used to “flatten” the RF curvature. For example, with a 4 GHz section in CLIC, operating at 25% of the voltage of the 2 GHz RF (and phased to decelerate the beam), the energy spread on a bunch (with particles within_±5 mm of the centre) can be reduced to less than_±0.006%. The drawback is the additional cost, and possible aperture and wake field issues with a linac operating at 4 GHz. However, in principle, higher harmonic RF could be used to mitigate the effect of energy acceptance limitations in the predamping ring.

0.4 0.45 3.5 4 4.5 5 m P 0.3 0.35 1.5 2 2.5 3 Y ie ld /1 0 0 m Yield/100mundulator Polarisation P o la ris a tio n 0.2 0.25 0 0.5 1 0 50 100 150 200 250 300 l ( )

ElectronEnergy(GeV)

Figure 4.3: Positron yield and polarisation from 100 m of undulator (deflection parameter

0.92, and period 11.5 mm) as a function of electron beam energy. ILC damping ring acceptance is applied. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 9 0 50 100 150 200 250 300 Y ie ld /1 0 0 m

Electron Energy (GeV)

Yield/100m undulator Polarisation P o lar is ati o n

Figure 4.4: Positron yield and polarisation from 100 m of undulator (deflection parameter

0.92, and period 11.5 mm) as a function of electron beam energy. CLIC predamping ring acceptance is applied.

The larger acceptance in CLIC compared to ILC increases the yield by roughly a factor of 2, for the same photon parameters: compare the results shown in Fig. 4.3

(yield and polarisation as functions of energy for ILC) with those shown in Fig. 4.4

(yield and polarisation using the same parameters as Fig. 4.4, except that the CLIC predamping ring acceptance is applied). Therefore, we need only half the number of photons, and this means that we can reduce the length of the undulator by two. In Stage 1, the total length of the undulator would be 27 m; the period stays fixed at 11.5 mm. In Stage 2, the total length of the undulator would be 460 m, and the period would remain fixed at 288 mm. Now the undulator parameters in Stage 1 seem very realistic. For Stage 2, the undulator is still rather long. Note that in both cases, the radiation power on the target is about half what it will be for ILC, because of the large acceptance of the pre-damping ring.