Point Spread Function

Eduardo do Couto e Silva - Last revision February 04, 2003

Our requirements indicate that the ratio of PSF 95%/PSF 68% should be smaller than 3 and this was confirmed by the simulation used in the GLAST proposal. However, this requirement was not met for the SLAC 99/00 beam test data or simulation and not for all energies for the SLAC 97 beam test. I personally believe that the beam test PSF measurements should be revisited together with the new simulation program to guide us through new beam tests.

Table below shows the ratio of PSF 95%/PSF 68% for the previous test beam simulation (SLAC 99/00) and test beam data (SLAC 97). Data from SLAC Beam test 99/00 was not shown since ratios was even worse.

	Ratio of PSF95%/PSF68%
	SLAC 99/00 Beam Test (simulation TBSIM)		SLAC 97 Beam Test (data)
comments	32cm x 32 cm detector area More sophisticated Kalman filter In principle same simulation as used for the GLAST proposal		6cm x 6cm detector area (tails are reduced) pancake configuration silicon planes 3 cm apart
Energy	Front (3.6% Pb)	Back (28.1% Pb)	Energy	(0% Pb)	(4% Pb)
150 MeV	3.46	3.39	16 MeV	2.28	2.34
360 MeV	3.18	2.99	37 MeV	2.79	2.69
580 MeV	2.89	3.92	81 MeV	2.59	2.81
900 MeV	2.92	4.61	156 MeV	2.75	2.97
1.20 GeV	3.15	4.38	355 MeV	2.99	3.10
1.56 GeV	2.94	4.66	760 MeV	3.17	3.05
1.97 GeV	2.91	4.22	1.46 GeV	3.30	3.42
2.20 GeV	2.66	4.72	3.34 GeV	3.86	3.37
3.11 GeV	3.00	4.38	7.47 GeV	4.15	4.11
3.89 GeV	4.00 *	4.74	15.77 GeV	4.45	4.87
5.59 GeV	3.04	5.73

* low statistics = 338 photons. All other bins have > 1000 photons

Table below shows the ratio of PSF 68% for the previous test beam data (SLAC 99/00 and SLAC 97). It clearly indicates the need for measuring separately the PSF for the Front and Back sections of the tracker. The latter is 2 times larger than the former.

PSF 68%
Energy	SLAC 99/00 Beam test data (Front 3.6%Pb)	Energy	SLAC 97 Beam test data pancake (4% Pb)
100 MeV	2.98 +- 0.45	79 MeV	2.6
1.09 GeV	0.41 +- 0.06	1.45 GeV	0.3
4.88 GeV	0.18 +- 0.03	3.2 GeV	0.2

One of the main difficulties in comparing the results is that there are cuts used in the reconstruction to generate photon candidates and cuts used in the analysis to select the final photons. Both chain of cuts needs to be described.

To calculate the PSF for the SLAC beam test 1999/2000 we worried about corrections due to

the presence of multiple photons in the beam,
the uncertainty in the reconstructed energy
the bin width of the final histogram
the limited statistics per energy bin
the uncertainty in the Monte Carlo model
the misalignment of silicon planes
the beam divergence

Multiple Photons

Photons were generated after the electron beam impinged onto a Copper targets of different thickness, namely 1%,3% and 9% radiation lengths. Because of that one could get events with 2 photons with different energies but with only one of them pair producing in the tracker. The reconstruction program could measure the energy in the calorimeter for a 2 GeV photon that did not convert in the tracker with the direction of a second photon of 100 MeV that converted in a tracker. This would lead to a broader PSF. Therefore the "true" PSF should correspond to a target of 0% radiation lengths. Ideally one would measure the PSF at different energies and angles for all three possible radiator configurations (1%, 3% and 9%) and extrapolate down to 0% to obtain the "true" PSF value. Employing this correction method, hereafter radiator method, we estimated the "true" PSF to be about 30% less than the measured value for a radiator with a 3% radiation length.

Energy reconstruction

For high energies (> 10 GeV) we had a good calibration of the calorimeter and we could measure the reconstruction energy either by profile fitting or by the correlation method (see SLAC-PUB-8682). However we did not have enough statistics to do a PSF measurement at these energies. For lower energies(< 10 GeV), due to technical reasons, we did not have a good calibration of the calorimeter and the measurement of the energy was done using the sum of the energy in the crystals.

Energy bin width

To evaluate the uncertainty on the energy and therefore to decide in whcih bin one photon gets assigned to, we halved the measured energy of a given simulation run and study its effect on the PSF. An uncertainty of about 50% in the energy translated into a 10% effect on the PSF value.

Limited Statistics

During the test beam planning we estimated about 1-2% efficiencies for tagging photons. However after reconstruction the number of photons dropped to about 0.5% of the original number of level 1 triggers. To increase statistics by a factor of 2, we use a combination of photon tagger and calorimeter to select events. Obviously some of these events had multiple photons as explained above. To get a decent measurement of the PSF we needed at least 500 events per energy bin.

Monte Carlo modeling

For our publication we did not have time to include the beam line elements (e.g copper radiator) in the simulation. We studied the effect of the assignment of a photon to a given energy bin, by generating events with a single monochromatic beam and a brehmstrahlung beam.

Misalignement of silicon planes

Silicon planes were aligned using high energy protons. 25 runs of about 30000 events each were used to map the front face of the BTEM tower. The alignment was performed to the level of 30 mm. Time constraints did not allow us to incorporate this alignment in the measurement of the PSF.

Beam Divergence

The beam was about 10 mm wide and the last quad somewhere about 100 m away so the divergence was about 0.006 degrees. This is a factor of 10 smaller than the estimated error on the PSF for 1 GeV.