II. Detector design 40
5.3. An abstract model for DIRC detectors
OE pos w/2
w/2
optical element radiator
active = readout optics passive = coating particle
intersection radiator-segm
ent photon path direct illumination
optical element
approximation for thin radiators:
radiator plane = plane through center of the plate
photon emission
Cherenkov
cone particle
radiator plane
Figure 5.8.: Illustration of terms defined in the abstract DIRC model.
Due to the initial assumptions,ϕis preserved during transport even if the zig-zag path of the photon is reflected at a radiator segment. It is hence sufficient to describe the photon path in the RPP.
In the following discussion it is assumed that the particle trajectory is known from the tracking detector. It is helpful to begin with the simplest path, a photon which travels directly to an optical element without being reflected at a radiator segment (direct photon). The corresponding path in RPP is a straight line from the emission point, given by the intersection of the particle trajectory with the radiator plane, to the optical element. The angle φrel can be directly obtained from the known location of the optical element. If the optical element is of an imaging type measuringϕ, the Cherenkov angle follows directly from eq. 5.4.
However, the hit detected in the optical element is not necessarily a direct photon.
It could also originate from a photon which has been reflected at a radiator segment (compare Fig. 5.8). This raises two questions. How to determine the possible reflected paths and how to resolve the ambiguities. The first question will be addressed in more detail later in section 5.9.
A common way to remove ambiguities is to exploit the time information. The length s2d of the photon path in RPP is related to the real 3d photon track length
5.3. An abstract model for DIRC detectors 69
s3d by:
s3d= s2d
cosϕ (5.5)
The time of propagationtprop=tdet−t0of a photon emitted at timet0and detected at time tdet can be computed by means of the group velocity vg in the radiator material*:
tprop= s3d
vg = s2d
vg·cosϕ (5.6)
However, this approach requires the knowledge oft0which corresponds roughly to the time when the particle arrives at the radiator plane. Another option to remove ambiguities is by a cut on the expected Cherenkov angle spectrum. The typical single photon resolution of a DIRC counter is better than 15 mrad. Ambiguities often result in computed Cherenkov angles which are far-off the expectation for any particle hypothesis.
So far, this model – including the path reconstruction algorithm discussed in section 5.9 – can be used to describe the photon parameters from emission up to the optical element in any DIRC detector using a thin radiator with polygonal outline.
It can be used directly to compute Cherenkov angles in a proximity focusing setup as shown in (Fig. 5.9). In this case, the RPP is valid inside the expansion volume and the imaging plane can be described as radiator-segment. In such a setup, the Cherenkov angles can be directly computed from path information.
In case of focusing optics however, the RPP approach is invalid either due to the nonlinear refraction at a lens or reflections at a focusing mirror which is not orthogonal to the radiator plane. The different optics cannot be described in an abstract manner. In the following, the model will be extended by a description of cylindrical focusing mirrors. These mirrors map the angleϕ0 to a coordinatez(ϕ0) on the focal plane. The function
z(ϕ0)=z(ϕ,αFEL)=z µ
arctan( tanϕ cosαFEL
)
¶
(5.7) will be referred to as imaging functionof the optical element. The connection betweenϕ0 and the angleαFELof the photon path to the optical element in RPP is
*Note that eq. 5.6 can also be used to compute the angleϕand hence the Cherenkov angle from the photons time of propagation.
projection on imaging plane
projection on radiator plane basic imaging options
focusing
lens focusing
mirror focusing
no imaging pinhole/proximity
radiator focusing element
optics radiator
radiator focusing
element
Figure 5.9.: Illustration of different optic options for the DIRC readout (left) and terms defined by the DIRC model (right).
given by
tanϕ0= tanϕ cosαFEL
(5.8) It is obvious thatαFEL has to be known to computeϕand thus the Cherenkov angleθc from eq. 5.4. One option is to assume that the photon entered the optical element at its center. αFELis then defined by the geometry and tracking information.
The width of the optical element defines the error ofαFEL. Fig. 5.10 shows the dependence of the focus position onαFELfor differentϕangles. Each curve results from the incident angles−45◦≤αFEL≤45◦. In case of small focusing elements with reflecting side surfaces, the pattern is folded onto the smaller focal plane (dashed lines).
IfαFELis determined by the position of the focusing element, the uncertainty depends on the distance of the emission point to the optical element. The opening angle of the direct light cone shown in Fig. 5.8 (top left frame) will become larger when the emission point gets closer to the optical element. Hence, it can be advantageous to reconstruct αFEL from the pattern in a wide focusing element
5.3. An abstract model for DIRC detectors 71
−10 −5 0 5 10
focal plane x 0.0
0.2 0.4 0.6 0.8 1.0
focalplaney
Figure 5.10.:Variation of the focus position in a focusing element with cylindrical mirror for differentϕangles in dependence ofαFEL(−45◦≤αFEL≤45◦). Solid lines show the dependence for a focusing element of a widthw>10and the dashed lines correspond to a focusing element of widthw=3where the light is reflected at the side surfaces. The imaging functionz(ϕ0)is linear, mappingϕ0=21◦...41◦to y=0...1. The parameter<LFEL>equals 10.
where the folding of the pattern can be neglected. This is achieved by the formula tanαFEL= d2
<LFEL>cosϕ0+d1 (5.9)
withd1 andd2 as given in Fig. 5.9 and the average photon path length<LFEL>in the imaging plane projection (IPP). This projection is equal to the view plane in Fig. 5.9, top right. <LFEL>is the average distance which light travels from the entrance of the optics to the focal plane, e.g. fromP2 toP3 in the given figure*. In
*Equation 5.9 can be derived by treating the light propagation in the optical element in the same manner as the photon propagation inside the radiator. The two anglesϕ0andχin IPP determine the positionz(ϕ0)on the focal plane and the displacement∆xalong the axis perpendicular to the IPP-plane (∆x=<LFEL>tanχ). This displacement definesP2 in respect toP3. The condition that the resulting angleαFELhas to be compatible with the anglesϕ0andχresults directly in eq. 5.9.
general,<LFEL>will also be a function ofϕ0.
The presented model has been implemented in C++, resulting in a flexible, geometry-independent DIRC model which serves as basis for the implementation of reconstruction algorithms. Hence, the algorithms can work with any radiator shape and instrumentation, as long as a specific optics model is available.