Optics Measurement Methods from BPMs
Optics Measurement Methods are used to measure the optics of an accelerator at different locations. \(\beta\) is the quantity we will focus on measuring at the interaction regions. There are many methods to measure this quantity in a ring using BPMs.
Beta from Amplitude
Beta from amplitude methods usually follow a certain pattern. First, a form of the amplitude function \(A\) is measured, which depends on the method. The amplitudes are related to the beta function at each BPM multiplied by a factor, usually known as the action. The resulting beta function is calculated at RHIC in the following way:
\[\beta = A^2 \frac{\sum_{arc} \beta_{model}}{\sum_{arc} A}\]The arc values refer to the bpm values that are located in the arc sections of the ring (i.e. not near the IR regions). These regions are designed to have a lower beta function and thus have lower room for error. They are also usually unmodified when changes to the lattice are made.
1. Harmonic Analysis (HA)
The fourier transform is applied to all working BPM data, and the $\boldsymbol{A}$ is calculated according to (cite paper):
The conversion from amplitude to beta function is then applied.
2. Curve Fitting (CF)
A least-squares fit to the closed orbit equation (\(x_{co}(N)\)) is applied to all working BPMs. The amplitudes of each BPM are then collected into a vector $\boldsymbol{A}$ and the beta function depends on the square of the amplitudes. The conversion from amplitude to beta function is then applied.
3. Model Independent Analysis
3.1 Principal Component Analysis
3.2 Independent Component Analysis
3.3 Dynamic Mode Decomposition
Beta from Phase
4. N-BPM Methods
Beta from phase instead of amplitude
- Independent of BPM calibration and tilt errors and of being insensitive to betatron coupling to first order
- assumes perfect timing synchronization and no optics errors