Beam focusing limitation from synchrotron radiation in two dimensions

The Oide effect considers the synchrotron radiation in the final focusing quadrupole, and it sets a lower limit on the vertical beam size at the interaction point, particularly relevant for high-energy linear colliders. The theory of the Oide effect was derived considering only the second moment of the radiation in the focusing plane of the magnet. This article addresses the theoretical calculation of the radiation effect on the beam size considering the first and second moments of the radiation and both focusing and defocusing planes of the quadrupole. The effect for a Gaussian beam is referred to as 2D-Oide; however, an alternative beam size figure is given that could represent better the effect on the minimum achievable ${\beta }_y^{*}$. The CLIC 3 TeV final quadrupole (QD0) and beam parameters are used to compare the theoretical results from the Oide effect and the 2D-Oide effect with particle tracking in placet. The 2D-Oide effect is demonstrated to be important, as it increases by 17% the contribution to the beam size. Further insight into the aberrations induced by the synchrotron radiation opens the possibility to partially correct the 2D-Oide effect with octupole magnets. A beam size reduction of 4% is achieved in the simplest configuration, using a single octupole.

Figure 1

The solid and dashed lines represent the particle trajectory without and with radiation, respectively.

Figure 2

Oide effect beam size contribution for CLIC 3 TeV design parameters. (a) Total beam size normalized to the designed linear beam size as a function of the quad length for the minimum focusing $k$ and the current QD0. (b) $k$ in the two previous cases.

Figure 3

The solid and dashed lines represent the particle trajectory without and with radiation, respectively.

Figure 4

Beam size as a function of ${\beta }_y^{*}$. The current design value for CLIC 3 TeV is ${\beta }_y^{*}=0.07\mathrm{mm}$.

Figure 5

CLIC 3 TeV beam at the IP after tracking through QD0 with and without radiation.

Figure 6

Correlation between the phase space coordinates $\mathrm{\Delta }y,y^{\prime }$ for CLIC 3 TeV from particle tracking with two different horizontal emittances.

Figure 7

The solid and dashed lines represent the particle trajectory without and with radiation, respectively. The dotted line represents the correction of the particle trajectory given by C0.

Figure 8

$\beta$ functions and ${\beta }_y/{\beta }_x$ ratio for CLIC 3 TeV FD, QF1, and QD0 in red on top. The dark area is occupied by QD0, and the IP is at $s=-3.5\mathrm{m}$.