The purpose of this test is to determine the uncertainty with which the TDC's internally are measuring time intervals. For inputs to the TDC's we took the internal test which was provided by the TDC's themselves. The test was programmed in such a way that 8 equally spaced pulses were sent simultaneously to each of the 96 channels of the TDC. The TDC recorded the leading and trailing edges of the pulses.
We calculated the differences between the first leading edge (see diagram) and all the other edges. The results were 15 time differences (Time n) for each channel. In order to see the channel to channel variation of these time differences, we made 15 histograms of Time n (n=2,16). Please see the diagram for an illustration of Time 2, Time 3, etc. This figure is an example of a histogram of Time 2 for one TDC.
We tested all 20 TDC's which will be used in the Superkamiokande project. In order to look at the performance of all TDC's at a glance we summarized the 300 histograms in the following way. We took the mean and the root-mean-square (RMS) of each Time n from each TDC module. We distinguished between even time numbers and odd time numbers. As we see in the diagram even (odd) time numbers represent the intervals between the first leading edge (T1) and the trailing (leading) edges.
We made 1-dimensional histograms containing the leading edge: mean and the RMS; plus a set of histograms for the trailing edge: mean and RMS. One can notice that there is a difference whether the TDC measures the interval of the leading edges or the trailing edges. The mean of the leading edges is very sharp. The spread of the means goes up to not more than 3 bins (1.5 ns). The values of the RMS lie between 0 and 0.7 counts. The means of the trailing edges on the other hand show a different behavior. They are spread over maximum 5 bins (2.5 ns). The RMS of the trailing edges is between 0.3 and 1.
The result of this time resolution test is that the performance of the TDC's without further corrections is much better than we'll need for the Superkamiokande experiment. The required time resolution will be about 1/4 of the expected 11 ns time resolution of the PMTs. Of course, if the higher time resolution is necessary, we will likely be able to achieve this through using off/line calibration.
Prior to the calibration we proved the linearity of the TDC's. Therefore, we plotted the measured TDC counts versus the pulse edges. As before, we distinguished between the leading and trailing edges, respectively (see plots). We fitted the data with a least square fit. You can find the parameters of the fit in plot 1 and 2. The value of sigma describes the uncertainty with which the TDC's convert time into counts.
This table summarizes the results of this calculation. Since each count is proportional to 0.5 ns we obtain an error of the measurement of sigma = 0.04 ns and sigma = 0.05 ns, respectively.