The Response Matrix

The basic idea behind a response matrix measurement is to introduce a small perturbation to the storage ring (i.e. exciting a corrector magnet) and then measure the position of the new closed orbit. In presence of such a kick [4] $x=0$ or $y=0$ is now longer a solution. If we look at this kick as a field error we can calculate a new closed orbit:

$$\ x(s)=\frac{\theta\sqrt{\beta (s)\beta_\theta}}{2\sin (\pi Q)}\cos [\phi (s)-\pi Q]$$ (6)

where $\theta$ is the kick in radians, $\beta_\theta$ is the value of the $\beta$-function at the position of the corrector magnet where the kick is applied, $Q$ is the betatron tune, and $\phi(s)$ is the phase advance. Particles will now perform betatron oscillations around this new closed orbit. So basically a response Matrix will result if (after giving a certain kick to each corrector magnet) the orbit displacements are read at various spots along the storage ring, e.g. if we have m corrector magnets and n BPMs then the value $r_{ij}(\theta )$ will represent the displacement of the beam at BPM $j$ induced by corrector $i$ when applying a kick strength $\theta$. So the result of such a response measurement will be the Response Matrix

$$R(\theta)=[r(\theta)_{ij}]_{i=1...m, j=1...n}$$

The measurements conducted here are only two column vectors out of such a matrix,

$$\vec{r}_{horizontal}=[r(\theta)_{ik}]_{i=1...m, 1\le{k}\le{n}}$$

$$\vec{r}_{vertikal}=[r(\theta)_{il}]_{i=1...m, 1\le{l}\le{n}}$$

i.e. one corrector (horizontal and vertical) kicks the beam with a certain amplitude and then the response of the beam is measured on all of the 72 BPMs. The experiment consisted of taking data from all of the 72 BPMs after having given once a horizontal kick of +1A on the first horizontal corrector magnet and once a vertical kick of +2A on a vertical corrector magnet. A kick of +1A on a horizontal corrector corresponds to $\theta =-0.1mrad$ whereas a kick of +2A on a vertical corrector corresponds to $\theta =-0.25mrad$. The result of these kicks can be seen in figures 13 and 14.

Figure 13: Response Matrix measurement for a kick on a horizontal corrector.

Figure 14: Response Matrix measurement for a kick on a vertical corrector.

To compare the theoretical model with measured data, it is a good idea to divide the position of the particles by the square root of the $\beta$-function. Recall equation 6 which will then be transformed to:

$$\frac{x(s)}{\sqrt{\beta(s)}}=\frac{\theta\sqrt{\beta_\theta}}{2\sin (\pi Q)}\cos [\phi (s)-\pi Q]$$ (7)

All factors outside the cosine are constant and so using equation 7 we can study the form of the cosine with respect to the phase advance $\phi(s)$. A comparison of these values (see figures 15 and 16) shows that the phase is well predicted by theory (the tunes differ by less than a half-integer).

Figure 15: Comparison of results of equation 7 for a horizontal kick between measured data and theoretical prediction.

Figure 16: Comparison of results of equation 7 for a vertical kick between measured data and theoretical prediction.

If we want to compare the amplitudes of these values it is better to look at their difference as in figures 17 and 18. However it is important to keep in mind that there is also a resolution problem: There are only 72 BPMs even though the beam position is oscillating very quickly, so that readings may look offset even though they are in good agreement with theory; this can be seen in figures 15 and 16.

Figure 17: Comparison of the amplitudes of measured data and theoretical prediction for a horizontal kick.

Figure 18: Comparison of the amplitudes of measured data and theoretical prediction for a vertical kick.

Figures 17 and 18 show a modulation of the beam position amplitudes over the period of the storage ring. This comes from the so-called beta-beat which can be derived from these plots.