Now we will go back to the concept of the interaction point, since we care about the star values \(\beta^{*} \text{ and } s^{*}\). To maximize luminosity, we would like both of these values to match the design optics.
Star Values
The Interaction Region, the region containing the BPMs around the interaction point, is a drift space. The equation can be derived from accelerator concepts, but is stated here:
\[\beta(s) = \beta^{*}(1 + (\frac{s - s^{*}}{\beta^{*}})^2)\]Using the two bpms (1, 2) that surround the IP, one could solve two simultaneous equations using \(\beta_1, s_1, \beta_2, s_2\). One could use a curve fitting method to solve for the star values. Another method is to use \(\alpha\) to solve for \(s^{*} \text{ then } \beta^{*}\), although that increases the error on the measurement of all three variables.
Statistical Analysis of Star Values
The star values could actually be solved for with a bit of algebra:
\[\begin{aligned} \beta^* = \frac{(s_1 - s_2)^2 \left(\beta_1 + \beta_2 - 2Q\right)}{D} = \frac{N}{D} \\ s^* = \frac{s_1 + s_2}{2} - \frac{(\beta_1 - \beta_2)\beta^*}{2(s_1 - s_2)} \end{aligned}\]Where: \(N = (s_1 - s_2)^2 (\beta_1 + \beta_2 - 2Q), D = (\beta_1 - \beta_2)^2 + 4(s_1 - s_2)^2, \text{ and } Q = \sqrt{\beta_1\beta_2 - (s_1 - s_2)^2}\).
From these, the following errors can be calculated:
\[\begin{aligned} \frac{\partial \beta^*}{\partial \beta_i} = \frac{1}{D^2}(\frac{\partial N}{\partial \beta_i}D \mp 2N(\beta_1 - \beta_2)) \\ \frac{\partial s^*}{\partial \beta_i} = \mp \frac{1}{2(s_1 - s_2)}(\beta^* \pm (\beta_1 - \beta_2)\frac{\partial \beta^*}{\partial \beta_i}) \end{aligned}\]Where \(i = 1, 2\) for the upstream and downstream BPMs. For symmetric beta at IP8, \(\Delta \beta \rightarrow 0, \beta_1 \approx \beta_2 \approx \beta_0, \text{ and } s_1 + s_2 = 0\). \(\sigma_{\beta_1} \approx \sigma_{\beta_2} \approx \sigma_{\beta_0}\) for most measurement methods. \(\sigma_{\beta^*} \text{ and } \sigma_{s^*}\) can then be approximated as:
\[\begin{align} \sigma_{\beta^*} \approx (1 + \frac{\beta_0}{Q_0})\sigma_{\beta_0}\sqrt{\frac{1 + \rho}{8}} \\ \sigma_{s^*} \approx \frac{\beta^*}{2\Delta s}\sigma_{\beta_0}\sqrt{2(1 - \rho)} \end{align}\]Where \(Q_0 = \sqrt{\beta_0^2 - (\Delta s)^2}\) and \(\rho\) is the correlation coefficient between \(\beta_1 \text{ and } \beta_2\). These approximations demonstrate that \(\sigma_{\beta}\) from both BPMs scale linearly with both \(\sigma_{\beta^*} \text{ and } \sigma_{s^*}\). If \(\beta_1 \text{ and } \beta_2\) are correlated, the covariance term will drive the error of the star values in opposing directions, as indicated by the equations.
Beta Function at the Interaction Region
This is an interactive plot of the beta function at an interaction region in relationship to the longitudinal coordinate (\(s\)) of the beam. The user can change the star values along with the error of \(\beta\) at the BPMs. The user can also change the covariance coefficient of \(\beta_1 \text{ and } \beta_2\) (\(\rho\)). This varies between optics measurement methods: CF and HA are around 0, OLS around 0.5, and TLS/GTLS around 0.9.