Cherenkov angle reconstruction

Individual HPD hits are used to determine Cherenkov angles. The aim is to reconstruct the precision with which $\theta_{\rm c}^{}$ is determined for single photons. The observed resolution in data is then compared with that expected from the simulation described in Section 5. Data are used where four HPDs, (2,3,5,6), have no filter in front of their input windows, but where two HPDs, 4 & 7, have mylar filters. The method used to reconstruct Cherenkov angles in data is illustrated in Figure 11 and requires the following parameters to be determined :

Figure 11: Schematic of the Cherenkov angle reconstruction in the tilted spherical mirror geometry.

1. The detection point, D supplied by the physical hit position on the surface of the HPD, subject to alignment uncertainties.
2. The centre of curvature of the mirror, C, defined by the measured focal length and the rotation angle relative to the nominal beam direction. The focal length has a precision of $\pm$10 mm, whereas the rotation angle is nominally 18 degrees, but is subject to small uncertainties when centering the ring image on the detector plane using micrometer screws.
3. The direction of the particle through the system determined using, either the nominal direction, (0, 0, - 1), or an improved estimator, such as from the silicon telescope or event-by-event reconstruction.
4. The emission point, E, of the photon is assumed to be the centre of the radiator, traversed by the particle. This is the position which minimises systematic uncertainties.

The reflection point, M, is constrained to lie on the same plane as E, D and C. The two-dimensional problem is solved [11] for $\theta$ under the condition that incident and reflected angles ($\alpha$) are equal.