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1\section{Calibration \label{sec:calibration}}
2
3
4In this section, we describe the tests performed using light pulses of different colour,
5pulse shapes and intensities with the MAGIC LED Calibration Pulser Box \cite{hardware-manual}.
6\par
7The LED pulser system is able to provide fast light pulses of 2--4\,ns FWHM
8with intensities ranging from 3--4 to more than 600 photo-electrons in one inner photo-multiplier of the
9camera. These pulses can be produced in three colors {\textit {\bf green, blue}} and
10{\textit{\bf UV}}.
11
12\begin{table}[htp]
13\centering
14\begin{tabular}{|c|c|c|c|c|c|c|}
15\hline
16\hline
17\multicolumn{7}{|c|}{The possible pulsed light colors} \\
18\hline
19\hline
20Colour & Wavelength & Spectral Width & Min. Nr. & Max. Nr. & Secondary & FWHM \\
21 & [nm] & [nm] & Phe's & Phe's & Pulses & Pulse [ns]\\
22\hline
23Green & 520 & 40 & 6 & 120 & yes & 3--4 \\
24\hline
25Blue & 460 & 30 & 6 & 600 & yes & 3--4 \\
26\hline
27UV & 375 & 12 & 3 & 50 & no & 2--3 \\
28\hline
29\hline
30\end{tabular}
31\caption{The pulser colors available from the calibration system}
32\label{tab:pulsercolours}
33\end{table}
34
35Table~\ref{tab:pulsercolours} lists the available colors and intensities and
36figures~\ref{fig:pulseexample1leduv} and~\ref{fig:pulseexample23ledblue} show typical pulses
37as registered by the FADCs.
38Whereas the UV-pulse is rather stable, the green and blue pulses can show smaller secondary
39pulses after about 10--40\,ns from the main pulse.
40One can see that the stable UV-pulses are unfortunately only available in such intensities as to
41not saturate the high-gain readout channel. However, the brightest combination of light pulses easily
42saturates all channels in the camera, but does not reach a saturation of the low-gain readout.
43\par
44Our tests can be classified into three subsections:
45
46\begin{enumerate}
47\item Un-calibrated pixels and events: These tests measure the percentage of failures of the extractor
48resulting either in a pixel declared as un-calibrated or in an event which produces a signal outside
49of the expected Gaussian distribution.
50\item Number of photo-electrons: These tests measure the reconstructed numbers of photo-electrons, their
51spread over the camera and the ratio of the obtained mean values for outer and inner pixels, respectively.
52\item Linearity tests: These tests measure the linearity of the extractor with respect to pulses of
53different intensity and colour.
54\item Time resolution: These tests show the time resolution and stability obtained with different
55intensities and colors.
56\end{enumerate}
57
58\begin{figure}[htp]
59\centering
60\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Inner.eps}
61\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Outer.eps}
62\caption{Example of a calibration pulse from the lowest available intensity (1\,Led UV).
63The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
64Note that the pulse height fluctuates much more than suggested from these pictures. Especially, a
65zero-pulse is also possible.}
66\label{fig:pulseexample1leduv}
67\end{figure}
68
69\begin{figure}[htp]
70\centering
71\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Inner.eps}
72\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Outer.eps}
73\caption{Example of a calibration pulse from the highest available mono-chromatic intensity (23\,Leds Blue).
74The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
75One the left side of both plots, the (saturated) high-gain channel is visible,
76on the right side from FADC slice 18 on,
77the delayed low-gain
78pulse appears. Note that in the left plot, there is a secondary pulses visible in the tail of the
79high-gain pulse. }
80\label{fig:pulseexample23ledblue}
81\end{figure}
82
83We used data taken on the 7$^{\mathrm{th}}$ of June, 2004 with different pulser LED combinations, each taken with
8416384 events. 19 different calibration configurations have been tested.
85The corresponding MAGIC data run numbers range from nr. 31741 to 31772. These data have been taken
86before the latest camera repair access which resulted in a replacement of about 2\% of the pixels known to be
87mal-functioning at that time.
88There is thus a lower limit to the number of un-calibrated pixels of about 1.5--2\% of known
89mal-functioning photo-multipliers.
90\par
91Although we had looked at and tested all colour and extractor combinations resulting from these data,
92we restrict ourselves to show here only typical behaviour and results of extractors.
93All plots, including those which are not displayed in this TDAS, can be retrieved from the following
94locations:
95
96\begin{verbatim}
97http://www.magic.ifae.es/~markus/pheplots/
98http://www.magic.ifae.es/~markus/timeplots/
99\end{verbatim}
100
101%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
102
103\subsection{Un-Calibrated Pixels and Events \label{sec:uncalibrated}}
104
105The MAGIC calibration software incorporates a series of checks to sort out mal-functioning pixels.
106Except for the software bug searching criteria, the following exclusion criteria can apply:
107
108\begin{enumerate}
109\item The reconstructed mean signal $\widehat{Q}$ is less than 2.5 times the extractor resolution $R$: $\widehat{Q}<2.5\cdot R$.
110(2.5 Pedestal RMS in the case of the simple fixed window extractors, see section~\ref{sec:pedestals}).
111This criterium essentially cuts out
112dead pixels.
113\item The error of the mean reconstructed signal $\delta \widehat{Q}$ is larger than the mean reconstructed signal $\widehat{Q}$:
114 $\delta \widehat{Q} > \widehat{Q}$. This criterion cuts out
115signal distributions which fluctuate so much that their RMS is bigger than its mean value. This
116criterium cuts out ``ringing'' pixels or mal-functioning extractors.
117\item The reconstructed mean number of photo-electrons lies 4.5 sigma outside
118the distribution of photo-electrons obtained with the inner or outer pixels in the camera, respectively.
119This criterium cuts out channels with apparently deviating (hardware) behaviour compared to
120the rest of the camera readout\footnote{This criteria is not applied any more in the standard analysis,
121although we kept using it here}.
122\item All pixels with reconstructed negative mean signal or with a
123mean numbers of photo-electrons smaller than one. Pixels with a negative pedestal RMS subtracted
124sigma occur, especially when stars are focused onto that pixel during the pedestal run (resulting
125in a large pedestal RMS), but have moved to another pixel during the calibration run. In this case, the
126number of photo-electrons would result artificially negative. If these
127channels do not show any other deviating behaviour, their number of photo-electrons gets replaced by the
128mean number of photo-electrons in the camera, and the channel is further calibrated as normal.
129\end{enumerate}
130
131Moreover, the number of events are counted which have been reconstructed outside a 5$\sigma$ region
132from the mean signal $<\widehat{Q}>$. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical
133outlier obtained with the digital filter applied on a low-gain signal, and figure~\ref{fig:unsuited:all}
134shows the average number of all excluded pixels and outliers obtained from all 19 calibration configurations.
135One can already see that the largest window sizes yield a high number of un-calibrated pixels, mostly
136due to the missing ability to recognize the low-intensity pulses (see later). One can also see that
137the amplitude extracting spline yields a higher number of outliers than the rest of the extractors.
138\par
139The global champion in lowest number of un-calibrated pixels results to be
140{\textit{\bf MExtractTimeAndChargeSpline}} extracting the integral over two FADC slices (extractor \#25).
141The one with the lowest number of outliers is
142{\textit{\bf MExtractFixedWindowPeakSearch}} with an extraction range of 2 slices (extractor \#11).
143
144\begin{figure}[htp]
145\centering
146\includegraphics[width=0.95\linewidth]{Outlier.eps}
147\caption{Example of an event classified as ``outlier''. The histogram has been obtained
148using the digital filter (extractor \#32) applied to a high-intensity blue pulse (run 31772).
149The event marked as ``outlier'' clearly has been mis-reconstructed. It lies outside the 5$\sigma$--region from the fitted mean.}
150\label{fig:outlier}
151\end{figure}
152
153\begin{figure}[htp]
154\centering
155\includegraphics[height=0.75\textheight]{UnsuitVsExtractor-all.eps}
156\caption{Un-calibrated pixels and outlier events averaged over all available
157calibration runs.}
158\label{fig:unsuited:all}
159\end{figure}
160
161The following figures~\ref{fig:unsuited:5ledsuv},~\ref{fig:unsuited:1leduv},~\ref{fig:unsuited:2ledsgreen}
162and~\ref{fig:unsuited:23ledsblue} show the resulting numbers of un-calibrated pixels and events for
163different colors and intensities. Because there is a strong anti-correlation between the number of
164excluded pixels and the number of outliers per event, we have chosen to show these numbers together.
165
166\par
167
168\begin{figure}[htp]
169\centering
170\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-5LedsUV-Colour-12.eps}
171\caption{Un-calibrated pixels and outlier events for a typical calibration
172pulse of UV-light which does not saturate the high-gain readout.}
173\label{fig:unsuited:5ledsuv}
174\end{figure}
175
176\begin{figure}[htp]
177\centering
178\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-1LedUV-Colour-04.eps}
179\caption{Un-calibrated pixels and outlier events for a very low
180intensity pulse.}
181\label{fig:unsuited:1leduv}
182\end{figure}
183
184\begin{figure}[htp]
185\centering
186\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-2LedsGreen-Colour-02.eps}
187\caption{Un-calibrated pixels and outlier events for a typical green pulse.}
188\label{fig:unsuited:2ledsgreen}
189\end{figure}
190
191\begin{figure}[htp]
192\centering
193\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-23LedsBlue-Colour-00.eps}
194\caption{Un-calibrated pixels and outlier events for a high-intensity blue pulse.}
195\label{fig:unsuited:23ledsblue}
196\end{figure}
197
198One can see that in general, big extraction windows raise the
199number of un-calibrated pixels and are thus less stable. Especially for the very low-intensity
200\textit{\bf 1\,Led\,UV}-pulse, the big extraction windows -- summing 8 or more slices -- cannot calibrate more
201than 50\% of the inner pixels (fig.~\ref{fig:unsuited:1leduv}).
202This is an expected behavior since big windows
203sum up more noise which in turn makes the search for the small signal more difficult.
204\par
205In general, one can also find that all ``sliding window''-algorithms (extractors \#17-32) discard
206less pixels than the corresponding ``fixed window''-ones (extractors \#1--16).
207
208The spline (extractors \#23--27) and the digital filter with the correct weights (extractors \#30-31) discard
209the least number of pixels and are also robust against slight modifications of the pulse form
210(of the weights for the digital filter).
211\par
212Concerning the numbers of outliers, one can conclude that in general, the numbers are very low never exceeding
2130.1\% except for the amplitude-extracting spline which seems to mis-reconstruct a certain type of events.
214\par
215In conclusion, already this first test excludes all extractors with too large window sizes because
216they are not able to extract cleanly small signals produced by about 4 photo-electrons. Moreover,
217the amplitude extracting spline produces a significantly higher number of outlier events.
218
219\clearpage
220
221%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
222
223\subsection{Number of Photo-Electrons \label{sec:photo-electrons}}
224
225Assuming that the readout chain adds only negligible noise to the one
226introduced by the photo-multiplier itself, one can make the assumption that the variance of the
227true signal, $S$, is the amplified Poisson variance of the number of photo-electrons,
228multiplied with the excess noise of the photo-multiplier which itself is
229characterized by the excess-noise factor $F$:
230
231\begin{equation}
232Var[S] = F^2 \cdot Var[N_{phe}] \cdot \frac{<S>^2}{<N_{phe}>^2}
233\label{eq:excessnoise}
234\end{equation}
235
236After introducing the effect of the night-sky background (eq.~\ref{eq:rmssubtraction})
237and assuming that the variance of the number of photo-electrons is equal
238to the mean number of photo-electrons (because of the Poisson distribution),
239one obtains an expression to retrieve the mean number of photo-electrons released at the photo-multiplier cathode from the
240mean extracted signal, $\widehat{S}$, and the RMS of the extracted signal obtained from
241pure pedestal runs $R$ (see section~\ref{sec:ffactor}):
242
243\begin{equation}
244<N_{phe}> \approx F^2 \cdot \frac{<\widehat{S}>^2}{Var[\widehat{S}] - R^2}
245\label{eq:pheffactor}
246\end{equation}
247
248In theory, eq.~\ref{eq:pheffactor} must not depend on the extractor! Effectively, we will use it to test the
249quality of our extractors by requiring that a valid extractor yields the same number of photo-electrons
250for all pixels individually and does not deviate from the number obtained with other extractors.
251As the camera is flat-fielded, but the number of photo-electrons impinging on an inner and an outer pixel is
252different, we also use the ratio of the mean numbers of photo-electrons from the outer pixels to the one
253obtained from the inner pixels as a test variable. In the ideal case, it should always yield its central
254value of about 2.6$\pm$0.1~\cite{michele-diploma}.
255\par
256In our case, there is an additional complication due to the fact that the green and blue colored light pulses
257show secondary pulses which destroy the Poisson behaviour of the number of photo-electrons. We will
258have to split our sample of extractors into those being affected by the secondary pulses and those
259being immune to this effect.
260\par
261Figures~\ref{fig:phe:5ledsuv},~\ref{fig:phe:1leduv},~\ref{fig:phe:2ledsgreen}~and~\ref{fig:phe:23ledsblue} show
262some of the obtained results. One can see a rather good stability for the standard
263{\textit{\bf 5\,Leds\,UV}}\ pulse, except for the extractors {\textit{\bf MExtractFixedWindowPeakSearch}}, initialized
264with an extraction window of 2 slices.
265\par
266There is a considerable difference for all shown non-standard pulses. Especially the pulses from green
267and blue LEDs
268show a clear dependence of the number of photo-electrons on the extraction window. Only the largest
269extraction windows seem to catch the entire range of (jittering) secondary pulses and get the ratio
270of outer vs. inner pixels right. However, they (obviously) over-estimate the number of photo-electrons
271in the primary pulse.
272\par
273The strongest discrepancy is observed in the low-gain extraction (fig.~\ref{fig:phe:23ledsblue}) where all
274fixed window extractors with extraction windows smaller than 8 FADC slices fail to reconstruct the correct numbers.
275This has to do with the fact that
276the fixed window extractors fail to catch a significant part of the (larger) pulse because of the
2771~FADC slice event-to-event jitter and the larger pulse width covering about 6 FADC slices.
278Also the sliding windows smaller than 6 FADC slices and the spline smaller than
2792 FADC slices reproduce too small numbers of photo-electrons. Moreover, the digital filter shows a small dependency
280of the number of photo-electrons w.r.t. the extraction window.
281\par
282
283
284\begin{figure}[htp]
285\centering
286\includegraphics[height=0.92\textheight]{PheVsExtractor-5LedsUV-Colour-12.eps}
287\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour UV,
288reconstructed with each of the tested signal extractors.
289The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
290for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
291outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
292denote the mean of all not-excluded pixels, the error bars their RMS.}
293\label{fig:phe:5ledsuv}
294\end{figure}
295
296\begin{figure}[htp]
297\centering
298\includegraphics[height=0.92\textheight]{PheVsExtractor-1LedUV-Colour-04.eps}
299\caption{Number of photo-electrons from a typical, very low-intensity calibration pulse of colour UV,
300reconstructed with each of the tested signal extractors.
301The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
302for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
303outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
304denote the mean of all not-excluded pixels, the error bars their RMS.}
305\label{fig:phe:1leduv}
306\end{figure}
307
308\begin{figure}[htp]
309\centering
310\includegraphics[height=0.92\textheight]{PheVsExtractor-2LedsGreen-Colour-02.eps}
311\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour green,
312reconstructed with each of the tested signal extractors.
313The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
314for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
315outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
316denote the mean of all not-excluded pixels, the error bars their RMS.}
317\label{fig:phe:2ledsgreen}
318\end{figure}
319
320
321\begin{figure}[htp]
322\centering
323\includegraphics[height=0.92\textheight]{PheVsExtractor-23LedsBlue-Colour-00.eps}
324\caption{Number of photo-electrons from a typical, high-gain saturating calibration pulse of colour blue,
325reconstructed with each of the tested signal extractors.
326The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
327for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
328outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
329denote the mean of all not-excluded pixels, the error bars their RMS.}
330\label{fig:phe:23ledsblue}
331\end{figure}
332
333One can see that all extractors using a large window belong to the class of extractors being affected
334by the secondary pulses, except for the digital filter.
335\par
336The extractor {\textit{\bf MExtractTimeAndChargeDigitalFilter}} seems to be sufficiently stable against modifications of the
337exact form of the weights in the high-gain readout channel since all applied weights yield about
338the same number of photo-electrons and the same ratio of outer vs. inner pixels.
339\par
340All sliding window and spline algorithms yield a stable ratio of outer vs. inner pixels in the high and the low-gain.
341\par
342Concluding, there is no fixed window extractor yielding always the correct number of photo-electrons,
343except for the extraction window of 8 FADC slices.
344Either the number of photo-electrons itself is wrong or the ratio of outer vs. inner pixels is
345not correct. All sliding window algorithms seem to reproduce the correct numbers if one takes into
346account the after-pulse behaviour of the light pulser itself. The digital filter seems to be
347stable against modifications of the intrinsic pulse width from 1~to~4\,ns. This is the expected range within which the pulses from
348realistic cosmics signals may vary.
349
350\clearpage
351
352%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
353
354\subsection{Linearity \label{sec:calibration:linearity}}
355
356\begin{figure}[htp]
357\centering
358\includegraphics[width=0.99\linewidth]{PheVsCharge-4.eps}
359\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
360and three typical outer ones (lower plots) obtained with the extractor
361{\textit{MExtractFixedWindow}} on a window size of 8 high-gain and 8 low-gain slices
362(extractor \#4). }
363\label{fig:linear:phevscharge4}
364\end{figure}
365
366In this section, we test the linearity of the conversion factors FADC counts to photo-electrons:
367
368\begin{equation}
369c_{phe} =\ <N_{phe}> / <\widehat{S}>
370\end{equation}
371
372As the photo-multiplier and the subsequent
373optical transmission devices~\cite{david} is a relatively linear device over a
374wide dynamic range, the number of photo-electrons per charge has to remain constant over the tested
375linearity region.
376\par
377A first test concerns the stability of the conversion factor: mean number of averaged photo-electrons
378per FADC counts over the tested intensity region. This test includes all systematic uncertainties
379in the calculation of the number of photo-electrons and the computation of the mean signal.
380A more detailed investigation of the linearity will be shown in a
381separate TDAS~\cite{tdas-calibration}, although there, the number of photo-electrons will be calculated
382in a more independent way.
383
384\par
385Figure~\ref{fig:linear:phevscharge4} shows the conversion factor $c_{phe}$ obtained for different light intensities
386and colors for three typical inner and three typical outer pixels using a fixed window on
3878 FADC slices. The conversion factor seems to be linear to a good approximation, with the following restrictions:
388\begin{itemize}
389\item The green pulses yield systematically low conversion factors
390\item Some of the pixels show a difference
391between the high-gain ($<$100\ phes for the inner, $<$300\ phes for the outer pixels) and the low-gain
392($>$100\ phes for the inner, $>$300\ phes for the outer pixels) region and
393a rather good stability of $c_{phe}$ for each region separately.
394\end{itemize}
395
396We conclude that, with the above restrictions,
397the fixed window extractor \#4 is a linear extractor for both high-gain
398and low-gain regions, separately.
399\par
400
401Figures~\ref{fig:linear:phevscharge9} and~\ref{fig:linear:phevscharge15} show the conversion factors
402using an integrated spline and a fixed window with global peak search, respectively, over
403an extraction window of 8 FADC slices. The same behaviour is obtained as before. These extractors are
404linear to a good approximation, except for the two cases mentioned above.
405\par
406
407\begin{figure}[h!]
408\centering
409\includegraphics[width=0.99\linewidth]{PheVsCharge-9.eps}
410\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
411and three typical outer ones (lower plots) obtained with the extractor
412{\textit{MExtractFixedWindowSpline}}
413on a window size of 8 high-gain and 8 low-gain slices (extractor \#9). }
414\label{fig:linear:phevscharge9}
415\end{figure}
416
417\begin{figure}[h!]
418\centering
419\includegraphics[width=0.99\linewidth]{PheVsCharge-15.eps}
420\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
421and three typical outer ones (lower plots) obtained with the extractor
422{\textit{MExtractFixedWindowPeakSearch}} on a window size of 8 high-gain and 8 low-gain slices
423(extractor \#15). }
424\label{fig:linear:phevscharge15}
425\end{figure}
426
427\begin{figure}[h!]
428\centering
429\includegraphics[width=0.99\linewidth]{PheVsCharge-14.eps}
430\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three
431typical inner pixels (upper plots) and three typical outer ones (lower plots) obtained with the extractor
432{\textit{MExtractFixedWindowPeakSearch}}
433on a window size of 6 high-gain and 6 low-gain slices (extractor \#11). }
434\label{fig:linear:phevscharge11}
435\end{figure}
436
437Figure~\ref{fig:linear:phevscharge11} shows the conversion factors using a fixed window with global peak search
438integrating a window of 6 FADC slices. One can see that the linearity is completely lost above 300 photo-electrons in the
439outer pixels. Especially in the low-gain,
440the reconstructed mean charge is too low and the conversion factors bend down. We show this extractor especially because it has
441been used in the analysis and to derive a Crab spectrum with the consequence that the spectrum bends down at high energies. We
442suppose that the loss of linearity due to usage of this extractor is responsible for the encountered problems.
443A similar behaviour can be found for all extractors with window sizes smaller than 6 FADC slices, especially in the low-gain region.
444This is understandable since the low-gain pulse covers at least 6 FADC slices.
445(This behaviour
446was already visible in the investigations on the number of photo-electrons in the previous section~\ref{sec:photo-electrons}).
447\par
448Figure~\ref{fig:linear:phevscharge20} shows the conversion factors using a sliding window of 6 FADC slices.
449The linearity is maintained like in the previous examples, except that for the smallest signals the effect
450of the bias is already visible.
451\par
452
453\begin{figure}[h!]
454\centering
455\includegraphics[width=0.99\linewidth]{PheVsCharge-20.eps}
456\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three
457typical inner pixels (upper plots) and three typical outer ones (lower plots) obtained with the extractor
458{\textit{MExtractTimeAndChargeSlidingWindow}}
459on a window size of 6 high-gain and 6 low-gain slices (extractor \#20). }
460\label{fig:linear:phevscharge20}
461\end{figure}
462
463Figure~\ref{fig:linear:phevscharge23} shows the conversion factors using the amplitude-extracting spline
464(extractor \#23).
465Here, the linearity is worse than in the previous examples. A very clear difference between high-gain and
466low-gain regions can be seen as well as a bigger general spread in conversion factors. In order to investigate
467if there is a common, systematic effect of the extractor, we show the averaged conversion factors over all
468inner and outer pixels in figure~\ref{fig:linear:phevschargearea23}. Both characteristics are maintained
469there. Although the differences between high-gain and low-gain could be easily corrected for, we conclude
470that extractor \#23 is still unstable against the linearity tests.
471\par
472
473\begin{figure}[h!]
474\centering
475\includegraphics[width=0.99\linewidth]{PheVsCharge-23.eps}
476\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
477and three typical outer ones (lower plots) obtained with the extractor
478{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
479\label{fig:linear:phevscharge23}
480\vspace{\floatsep}
481\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-23.eps}
482\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
483obtained with the extractor
484{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
485\label{fig:linear:phevschargearea23}
486\end{figure}
487
488Figure~\ref{fig:linear:phevscharge24} shows the conversion factors using a spline integrating over
489one effective FADC slice in the high-gain and 1.5 effective FADC slices in the low-gain region (extractor \#24).
490The same problems are found as with extractor \#23, however to a much lower extent.
491The difference between high-gain and low-gain regions is less pronounced and the spread
492in conversion factors is smaller.
493Figure~\ref{fig:linear:phevschargearea24} shows already rather good stability except for the two
494lowest intensity pulses in green and blue. We conclude that extractor \#24 is still un-stable, but
495preferable to the amplitude extractor.
496\par
497
498\begin{figure}[h!]
499\centering
500\includegraphics[width=0.99\linewidth]{PheVsCharge-24.eps}
501\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
502and three typical outer ones (lower plots) obtained with the extractor
503{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices
504(extractor \#24). }
505\label{fig:linear:phevscharge24}
506\vspace{\floatsep}
507\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-24.eps}
508\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
509obtained with the extractor
510{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices
511(extractor \#24). }
512\label{fig:linear:phevschargearea24}
513\end{figure}
514
515Looking at figure~\ref{fig:linear:phevscharge25}, one can see that raising the integration window
516by two effective FADC slices in the high-gain and three effective FADC slices in the low-gain
517(extractor \#25), the stability is completely resumed, except for
518a systematic increase of the conversion factor above 200 photo-electrons.
519We conclude that extractor \#25 is almost as stable as the fixed window extractors.
520\par
521
522\begin{figure}[htp]
523\centering
524\includegraphics[width=0.99\linewidth]{PheVsCharge-25.eps}
525\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
526and three typical outer ones (lower plots) obtained with the extractor
527{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices
528(extractor \#25). }
529\label{fig:linear:phevscharge25}
530\vspace{\floatsep}
531\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-25.eps}
532\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
533obtained with the extractor
534{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices
535(extractor \#25). }
536\label{fig:linear:phevschargearea25}
537\end{figure}
538
539Figure~\ref{fig:linear:phevscharge30} and~\ref{fig:linear:phevscharge31} show the conversion factors using a digital filter,
540applied on 6 FADC slices and respectively 4 FADC slices with weights calculated from the UV-calibration pulse in the
541high-gain region and from the blue calibration pulse in the low-gain region.
542One can see that one or two blue calibration pulses at low and intermediate intensity fall
543out of the linear region, moreover there is a small systematic offset between the high-gain and low-gain region.
544It seems that the digital filter does not pass this test if the pulse form changes for more than 2\,ns from the
545expected one. The effect is not as problematic as it may appear here, because the actual calibration
546will not calculate the number of photo-electrons (with the F-Factor method) for every signal intensity.
547Thus, one possible reason for the instability is not relevant in the cosmics analysis. However, the limits
548of this extraction are visible here and should be monitored further.
549
550\par
551
552\begin{figure}[htp]
553\centering
554\includegraphics[width=0.99\linewidth]{PheVsCharge-30.eps}
555\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
556and three typical outer ones (lower plots) obtained with the extractor
557{\textit{MExtractTimeAndChargeDigitalFilter}}
558using a window size of 6 high-gain and 6 low-gain slices with UV-weights (extractor \#30). }
559\label{fig:linear:phevscharge30}
560\vspace{\floatsep}
561\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-30.eps}
562\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
563obtained with the extractor
564{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and UV-weight
565(extractor \#30). }
566\label{fig:linear:phevschargearea30}
567\end{figure}
568
569
570\begin{figure}[htp]
571\centering
572\includegraphics[width=0.99\linewidth]{PheVsCharge-31.eps}
573\caption{Conversion factor $c_{phe}$ for three typical inner pixels (upper plots)
574and three typical outer ones (lower plots) obtained with the extractor
575{\textit{MExtractTimeAndChargeDigitalFilter}} using a window size of
5764 high-gain and 4 low-gain slices (extractor \#31). }
577\label{fig:linear:phevscharge31}
578\vspace{\floatsep}
579\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-31.eps}
580\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
581obtained with the extractor
582{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and blue weights
583(extractor \#31). }
584\label{fig:linear:phevschargearea3}
585\end{figure}
586
587\clearpage
588
589\subsection{High-Gain vs. Low-Gain Calibration \label{sec:cal:hivslo}}
590
591The High-gain vs. Low-gain calibration is performed with events which on the one side do not yet
592saturate the high-gain channel, and on the other side are intense enough to trigger the low-gain switch
593in the electronics. Assuming that the signal reconstruction bias is negligible in any low-gain event
594(see also chapter~\ref{sec:mc}), one can then build the ratio of the reconstructed signal from the high-gain
595channel vs. the one reconstructed from the low-gain channel.
596\par
597For the following tests, we applied the following criteria:
598
599\begin{itemize}
600\item The content of the FADC slice with the largest signal has to be greater than 200 FADC counts
601\item The content of the FADC slice with the largest signal has to be smaller than 245 FADC counts
602\end{itemize}
603
604One of the used calibration runs (run \# 31762, {\textit{\bf 1\,Led\,Blue}})
605was especially apt to test the high-gain vs. low-gain
606inter-calibration of the reconstructed signals since there, the two requirements were fulfilled by
607more than 100~pixels in a reasonable number of events such that enough statistics could be accumulated.
608\par
609
610Figure~\ref{fig:ratio:sliding} shows some of the obtained results for all pixels with enough statistics:
611The results obtained with two spline algorithms and with two digital filter initializations are plotted
612against those obtained with a sliding window over 8 FADC slices in high-gain and low-gain. One can see that
613there is a rather good correlation for:
614
615\begin{figure}[htp]
616\centering
617\includegraphics[width=0.45\linewidth]{Ratio-21vs25.eps}
618\includegraphics[width=0.45\linewidth]{Ratio-21vs27.eps}
619\includegraphics[width=0.45\linewidth]{Ratio-21vs28.eps}
620\includegraphics[width=0.45\linewidth]{Ratio-21vs32.eps}
621\caption{Distributions of the calibrated high-gain vs. low-gain signal ratio, calculated with one test extractor
622vs. a reference extractor (sliding window over 8 high-gain and 8 low-gain FADC slices, extractor \#21).
623The tested extractors are: top left: integrating spline over 0.5 FADC slices left from maximum and 1.5
624FADC slice right from maximum (extractor \#25), top right: integrating spline over 1.5 FADC slices left
625from maximum and 4.5 FADC slices right from maximum (extractor \#27), bottom left: digital filter fitting
626cosmics pulses over 6 FADC slices, bottom left: digital filter fitting a blue calibration pulse over
6276 FADC slices.}
628\label{fig:ratio:sliding}
629\end{figure}
630
631\begin{figure}[htp]
632\centering
633\includegraphics[width=0.45\linewidth]{Ratio-28vs29.eps}
634\includegraphics[width=0.45\linewidth]{Ratio-32vs33.eps}
635\caption{Distributions of the calibrated high-gain vs. low-gain signal ratio, calculated with the
636digital filter. For the values on x-axis the integration over 6 FADC slices has been applied, for those
637one the y-axis, the integration over 4 FADC slices. Left: Digital filter fitting
638cosmics pulses, right: Digital filter fitting a blue calibration pulse.}
639\label{fig:ratio:df}
640\end{figure}
641
642\begin{figure}[htp]
643\centering
644\includegraphics[width=0.45\linewidth]{Ratio-SW88.eps}
645\includegraphics[width=0.45\linewidth]{Ratio-DF66.eps}
646\caption{Distributions of the calibrated high-gain vs. low-gain signal ratio for cosmics, calculated with a
647sliding window (left) and the digital filter (right). }
648\label{fig:ratio:cosmics}
649\end{figure}
650
651
652
653
654\clearpage
655
656\subsection{Relative Arrival Time Calibration}
657
658The calibration LEDs
659deliver fast-rising pulses, uniform over the camera in signal size and time.
660We estimate the time-uniformity to as good as about~30\,ps, a limit due to the different travel times of the light
661from the light source to the inner and outer parts of the camera. For cosmics data, however, the staggering of the
662mirrors limits the time uniformity to about 600\,ps.
663\par
664The extractors \#17--33 are able to compute the arrival time of each pulse.
665Since the calibration does not permit a precise measurement of the absolute arrival time, we measure
666the relative arrival time for every channel with respect to a reference channel (usually pixel no.\,1):
667
668\begin{equation}
669\delta t_i = t_i - t_1
670\end{equation}
671
672where $t_i$ denotes the reconstructed arrival time of pixel number $i$ and $t_1$ the reconstructed
673arrival time of the reference pixel no. 1 (software numbering). In one calibration run, one can then fill
674histograms of $\delta t_i$ and fit them to the expected Gaussian distribution. The fits
675yield a mean $\mu(\delta t_i)$, comparable to
676systematic delays in the signal travel time, and a sigma $\sigma(\delta t_i)$, a measure of the
677combined time resolutions of pixel $i$ and pixel 1. Assuming that the PMTs and readout channels are
678of the same kind, we obtain an approximate time resolution of pixel $i$:
679
680\begin{equation}
681t^{res}_i \approx \sigma(\delta t_i)/\sqrt{2}
682\end{equation}
683
684Figures~\ref{fig:reltimesinnerleduv} show distributions of $\delta t_i$
685for a typical inner pixel and a non-saturating calibration pulse of UV-light,
686obtained with six different extractors.
687One can see that all of them yield acceptable Gaussian distributions,
688except for the sliding window extracting 2 slices which shows a three-peak structure and cannot be fitted.
689We discarded that particular extractor from the further studies of this section.
690
691\begin{figure}[htp]
692\centering
693\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor17.eps}
694\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18.eps}
695\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23.eps}
696\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24.eps}
697\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30.eps}
698\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31.eps}
699\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
700Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 2 slices (\#17) and 4 slices (\#18) \protect\\
701Center: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\
702Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to a UV-calibration pulse over 6 slices (\#30) and 4 slices (\#31) \protect\\
703A medium sized UV-pulse (5\,Leds UV) has been used which does not saturate the high-gain readout channel.}
704\label{fig:reltimesinnerleduv}
705\end{figure}
706
707Figures~\ref{fig:reltimesinnerledblue1} and~\ref{fig:reltimesinnerledblue2} show
708the distributions of $\delta t_i$ for a typical inner pixel and an intense, high-gain-saturating calibration
709pulse of blue light, obtained from the low-gain readout channel.
710One can see that the sliding window extractors yield double Gaussian structures, except for the
711largest window sizes of 8 and 10 FADC slices. Even then, the distributions are not exactly Gaussian.
712The maximum position extracting spline also yields distributions which are not exactly Gaussian and seems
713to miss the exact arrival time in some events. Only the position of the half-maximum gives the
714expected result of a single Gaussian distribution.
715A similar problem occurs in the case of the digital filter: If one takes the correct weights
716(fig.~\ref{fig:reltimesinnerledblue2} bottom), the distribution is perfectly Gaussian and the resolution good,
717however a rather slight change from the blue calibration pulse weights to cosmics pulses weights (top)
718adds a secondary peak of events with mis-reconstructed arrival times. In principle, the $\chi^2$ of the digital filter
719fit gives an information about whether the correct shape has been used.
720
721\begin{figure}[htp]
722\centering
723\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18_logain.eps}
724\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor19_logain.eps}
725\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor21_logain.eps}
726\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor22_logain.eps}
727\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23_logain.eps}
728\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24_logain.eps}
729\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
730Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 4 slices (\#18) and 6 slices (\#19) \protect\\
731Center: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 8 slices (\#20) and 10 slices (\#21)\protect\\
732Bottom: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\
733A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
734\label{fig:reltimesinnerledblue1}
735\end{figure}
736
737\begin{figure}[htp]
738\centering
739\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30_logain.eps}
740\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31_logain.eps}
741\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor32_logain.eps}
742\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor33_logain.eps}
743\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
744Top: {\textit{\bf MExtractTimeAndChargeDigitalFilter}}
745fitted to cosmics pulses over 6 slices (\#30) and 4 slices (\#31) \protect\\
746Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to the correct blue calibration pulse over 6 slices (\#30) and 4 slices (\#31)
747A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
748\label{fig:reltimesinnerledblue2}
749\end{figure}
750
751%\begin{figure}[htp]
752%\centering
753%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor32.eps}
754%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor23.eps}
755%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor17.eps}
756%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to
757%the arrival time of the reference pixel no. 1. The left plot shows the result using the digital filter
758% (extractor \#32), the central plot shows the result obtained with the half-maximum of the spline and the
759%right plot the result of the sliding window with a window size of 2 slices (extractor \#17). A
760%medium sized UV-pulse (10Leds UV) has been used which does not saturate the high-gain readout channel.}
761%\label{fig:reltimesouter10leduv}
762%\end{figure}
763
764%\begin{figure}[htp]
765%\centering
766%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor23.eps}
767%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor32.eps}
768%\caption{Example of a two distributions of relative arrival times of an inner pixel with respect to
769%the arrival time of the reference pixel no. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
770%(extractor \#32). A
771%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
772%\label{fig:reltimesinner10ledsblue}
773%\end{figure}
774
775
776
777%\begin{figure}[htp]
778%\centering
779%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor23.eps}
780%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor32.eps}
781%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to
782%the arrival time of the reference pixel no. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
783%(extractor \#32). A
784%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
785%\label{fig:reltimesouter10ledsblue}
786%\end{figure}
787
788\clearpage
789
790%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
791
792\subsection{Number of Outliers}
793
794As in section~\ref{sec:uncalibrated}, we tested the number of outliers from the Gaussian distribution
795in order to count how many times the extractor has failed to reconstruct the correct arrival time.
796\par
797Figure~\ref{fig:timeunsuit:5ledsuv} shows the number of outliers for the different time extractors, obtained with
798a UV pulse of about 20 photo-electrons. One can see that all time extractors yield an acceptable mis-reconstruction
799rate of about 0.5\%, except for the maximum searching spline yields three times more mis-reconstructions.
800\par
801If one goes to very low-intensity pulses, as shown in figure~\ref{fig:timeunsuit:1leduv}, obtained with on average 4 photo-electrons,
802the number of mis-reconstructions increases considerably up to 20\% for some extractors. We interpret this high mis-reconstruction
803rate to the increase possibility to mis-reconstruct a pulse from the night sky background noise instead of the signal pulse from the
804calibration LEDs. One can see that the digital filter using weights on 4 FADC slices is clear inferior to the one using 6 FADC slices
805in that respect.
806\par
807The same conclusion seems to hold for the green pulse of about 20 photo-electrons (figure~\ref{fig:timeunsuit:2ledsgreen})
808where the digital filter over 6 FADC slices seems to
809yield more stable results than the one over 4 FADC slices. The half-maximum searching spline seems to be superior to the maximum-searching
810one.
811\par
812In figure~\ref{fig:timeunsuit:23ledsblue}, one can see the number of outliers from an intense calibration pulse of blue light yielding about
813600 photo-electrons per inner pixel. All extractors seem to be stable, except for the digital filter with weights over 4 FADC slices. This
814is expected, since the low-gain pulse is wider than 4 FADC slices.
815\par
816In all previous plots, the sliding window yielded the most stable results, however later we will see that this stability is only due to
817an increased time spread.
818
819\begin{figure}[htp]
820\centering
821\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-5LedsUV-Colour-12.eps}
822\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
823of colour UV, reconstructed with each of the tested arrival time extractors.
824The first plots shows the time resolutions obtained for the inner pixels, the second one
825for the outer pixels. Points
826denote the mean of all not-excluded pixels, the error bars their RMS.}
827\label{fig:timeunsuit:5ledsuv}
828\end{figure}
829
830\begin{figure}[htp]
831\centering
832\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-1LedUV-Colour-04.eps}
833\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse
834of colour UV (carrying a mean number of 4 photo-electrons),
835reconstructed with each of the tested arrival time extractors.
836The first plots shows the time resolutions obtained for the inner pixels, the second one
837for the outer pixels. Points
838denote the mean of all not-excluded pixels, the error bars their RMS.}
839\label{fig:timeunsuit:1leduv}
840\end{figure}
841
842\begin{figure}[htp]
843\centering
844\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-2LedsGreen-Colour-02.eps}
845\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
846of colour Green, reconstructed with each of the tested arrival time extractors.
847The first plots shows the time resolutions obtained for the inner pixels, the second one
848for the outer pixels. Points
849denote the mean of all not-excluded pixels, the error bars their RMS.}
850\label{fig:timeunsuit:2ledsgreen}
851\end{figure}
852
853\begin{figure}[htp]
854\centering
855\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-23LedsBlue-Colour-00.eps}
856\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse
857of colour blue, reconstructed with each of the tested arrival time extractors.
858The first plots shows the time resolutions obtained for the inner pixels, the second one
859for the outer pixels. Points
860denote the mean of all not-excluded pixels, the error bars their RMS.}
861\label{fig:timeunsuit:23ledsblue}
862\end{figure}
863
864\clearpage
865
866%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
867
868\subsection{Time Resolution \label{sec:cal:timeres}}
869
870There are three intrinsic contributions to the timing accuracy of the signal:
871
872\begin{enumerate}
873\item The intrinsic arrival time spread of the photons on the PMT: This time spread
874can be estimated roughly by the intrinsic width $\delta t_{\mathrm{IN}}$ of the
875input light pulse.
876The resulting time
877resolution is given by:
878\begin{equation}
879\Delta t \approx \frac{\delta t_{\mathrm{IN}}}{\sqrt{Q/{\mathrm{phe}}}}
880\end{equation}
881The width $\delta t_{\mathrm{LED}}$ of the calibration pulses of about 2\,ns
882for the faster UV pulses and 3--4\,ns for the green and blue pulses,
883for muons it is a few hundred ps, for gammas about 1\,ns and for hadrons a few ns.
884\item The intrinsic transit time spread $\mathrm{\it TTS}$ of the photo-multiplier:
885It can be of the order of a few hundreds of ps per single photo electron, depending on the
886wavelength of the incident light. As in the case of the photon arrival time spread, the total
887time spread scales with the inverse of the square root of the number of photo-electrons:
888\begin{equation}
889\Delta t \approx \frac{\delta t_{\mathrm{TTS}}}{\sqrt{Q/{\mathrm{phe}}}}
890\end{equation}
891\item The reconstruction error due to the background noise and limited extractor resolution:
892This contribution is inversely proportional to the signal to square root of background light intensities.
893\begin{equation}
894\Delta t \approx \frac{\delta t_{\mathrm{rec}} \cdot R/\mathrm{phe}}{Q/{\mathrm{phe}}}
895\end{equation}
896where $R$ is the resolution defined in equation~\ref{eq:def:r}.
897\item A constant offset due to the residual FADC clock jitter~\cite{florian}
898\begin{equation}
899\Delta t \approx \delta t_0
900\end{equation}
901\end{enumerate}
902
903In the following, we show measurements of the time resolutions at different
904signal intensities in real conditions for the calibration pulses. These set upper limits to the time resolution for cosmics since their
905intrinsic arrival time spread is smaller.
906
907Figures~\ref{fig:time:5ledsuv} through~\ref{fig:time:23ledsblue} show the measured time resolutions for very different calibration
908pulse intensities and colors. One can see that the sliding window resolutions are always worse than the spline and digital filter
909algorithms. Moreover, the half-maximum position search by the spline is always slightly better than the maximum position search. The
910digital filter does not show notable differences with respect to the pulse form or the extraction window size, except for the low-gain
911extraction where the 4 slices seem to yield a better resolution. This is only after excluding about 30\% of the events, as shown in
912figure~\ref{fig:timeunsuit:23ledsblue}.
913
914\begin{figure}[htp]
915\centering
916\includegraphics[height=0.38\textheight]{TimeResExtractor-5LedsUV-Colour-12.eps}
917\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
918of colour UV, reconstructed with each of the tested arrival time extractors.
919The first plots shows the time resolutions obtained for the inner pixels, the second one
920for the outer pixels. Points
921denote the mean of all not-excluded pixels, the error bars their RMS.}
922\label{fig:time:5ledsuv}
923\end{figure}
924
925\begin{figure}[htp]
926\centering
927\includegraphics[height=0.38\textheight]{TimeResExtractor-1LedUV-Colour-04.eps}
928\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse
929of colour UV (carrying a mean number of 4 photo-electrons),
930reconstructed with each of the tested arrival time extractors.
931The first plots shows the time resolutions obtained for the inner pixels, the second one
932for the outer pixels. Points
933denote the mean of all not-excluded pixels, the error bars their RMS.}
934\label{fig:time:1leduv}
935\end{figure}
936
937\begin{figure}[htp]
938\centering
939\includegraphics[height=0.38\textheight]{TimeResExtractor-2LedsGreen-Colour-02.eps}
940\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
941of colour Green, reconstructed with each of the tested arrival time extractors.
942The first plots shows the time resolutions obtained for the inner pixels, the second one
943for the outer pixels. Points
944denote the mean of all not-excluded pixels, the error bars their RMS.}
945\label{fig:time:2ledsgreen}
946\end{figure}
947
948\begin{figure}[htp]
949\centering
950\includegraphics[height=0.38\textheight]{TimeResExtractor-23LedsBlue-Colour-00.eps}
951\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse
952of colour blue, reconstructed with each of the tested arrival time extractors.
953The first plots shows the time resolutions obtained for the inner pixels, the second one
954for the outer pixels. Points
955denote the mean of all not-excluded pixels, the error bars their RMS.}
956\label{fig:time:23ledsblue}
957\end{figure}
958
959\clearpage
960
961The following figure~\ref{fig:time:dep} shows the time resolution for various calibration runs taken with different colors
962and light intensities as a function of the mean number of photo-electrons --
963reconstructed with the F-Factor method -- for four different time extractors. The dependencies have been fit to the following
964empirical relation:
965
966\begin{equation}
967\Delta T = \sqrt{\frac{A^2}{<Q>/{\mathrm{phe}}} + \frac{B^2}{<Q>^2/{\mathrm{phe^2}}} + C^2} .
968\label{eq:time:fit}
969\end{equation}
970
971The fit results are summarized in table~\ref{tab:time:fitresults}.
972
973\begin{table}[htp]
974\scriptsize{%
975\centering
976\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
977\hline
978\hline
979\multicolumn{10}{|c|}{\large Time Fit Results} \rule{0mm}{6mm} \rule[-2mm]{0mm}{6mm} \hspace{-3mm}\\
980\hline
981\hline
982\multicolumn{2}{|c|}{} & \multicolumn{4}{|c|}{\normalsize Inner Pixels} & \multicolumn{4}{|c|}{\normalsize Outer Pixels} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm}\\
983\hline
984{\normalsize Nr.} & {\normalsize Name } & {\normalsize A} & {\normalsize B } & {\normalsize C }& {\normalsize $\chi^2$/NDF }
985& {\normalsize A } &{\normalsize B} & {\normalsize C} &{\normalsize $\chi^2$/NDF} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm} \\
986\hline
98721 & Sliding Window (8,8) & 3.5$\pm$0.4 & 29$\pm$1 & 0.24$\pm$0.05 & 10.2 &6.0$\pm$0.7 & 52$\pm$4 & 0.23$\pm$0.04 & 4.3 \\
98825 & Spline Half Max. & 1.9$\pm$0.2 & 3.8$\pm$1.0 & 0.15$\pm$0.02 & 1.6 &2.6$\pm$0.2 &8.3$\pm$1.9 & 0.15$\pm$0.01 & 2.3 \\
98932 & Digital Filter (6 sl.) & 1.7$\pm$0.2 & 5.7$\pm$0.8 & 0.21$\pm$0.02 & 5.0 &2.3$\pm$0.3 &13 $\pm$2 & 0.20$\pm$0.01 & 4.0 \\
99033 & Digital Filter (4 sl.) & 1.7$\pm$0.1 & 4.6$\pm$0.7 & 0.21$\pm$0.02 & 6.2 &2.3$\pm$0.2 &11 $\pm$2 & 0.20$\pm$0.01 & 5.3 \\
991\hline
992\hline
993\end{tabular}
994\caption{The fit results obtained from the fit of equation~\ref{eq:time:fit} to the time resolutions obtained for various
995intensities and colors. The fit probabilities are very small mainly because of the different intrinsic arrival time spreads of
996the photon pulses from different colors. }
997\label{tab:time:fitresults}.
998}
999\end{table}
1000
1001The low fit probabilities are partly due to the systematic differences in the pulse forms in intrinsic arrival time spreads between
1002pulses of different LED colors. Nevertheless, we had to include all colors in the fit to cover the full dynamic range. In general,
1003one can see that the time resolutions for the UV pulses are systematically better than for the other colors which we attribute to the fact
1004the these pulses have a smaller intrinsic pulse width -- which is very close to pulses from cosmics. Moreover, there are clear differences
1005visible between different time extractors, especially the sliding window extractor yields poor resolutions. The other three extractors are
1006compatible within the errors, with the half-maximum of the spline being slightly better.
1007
1008\par
1009
1010To summarize, we find that we can obtain a time resolution of better than 1\,ns for all pulses above a threshold of 5\ photo-electrons.
1011This corresponds roughly to the image cleaning threshold in case of using the best signal extractor. At the largest signals, we can
1012reach a time resolution of as good as 200\,ps.
1013\par
1014The expected time resolution for inner pixels and cosmics pulses can thus be conservatively estimated to be:
1015
1016\begin{equation}
1017\Delta T_{\mathrm{cosmics}} \approx \sqrt{\frac{4\,\mathrm{ns}^2}{<Q>/{\mathrm{phe}}}
1018+ \frac{20\,\mathrm{ns}^2}{<Q>^2/{\mathrm{phe^2}}} + 0.04\,\mathrm{ns}^2} .
1019\label{eq:time:fitprediction}
1020\end{equation}
1021
1022\begin{landscape}
1023\begin{figure}[htp]
1024\centering
1025\includegraphics[width=0.24\linewidth]{TimeResFitted-21.eps}
1026\includegraphics[width=0.24\linewidth]{TimeResFitted-25.eps}
1027\includegraphics[width=0.24\linewidth]{TimeResFitted-32.eps}
1028\includegraphics[width=0.24\linewidth]{TimeResFitted-33.eps}
1029\caption{Reconstructed mean arrival time resolutions as a function of the extracted mean number of
1030photo-electrons for the weighted sliding window with a window size of 8 slices (extractor \#21, top left),
1031the half-maximum searching spline (extractor~\#25, top right),
1032the digital filter with correct pulse weights over 6 slices (extractor~\#30 and~\#32, bottom left)
1033and the digital filter with UV calibration-pulse weights over 4 slices (extractor~\#31 and~\#33, bottom right).
1034Error bars denote the spread (RMS) of time resolutions of the investigated channels.
1035The marker colors show the applied
1036pulser colour, except for the last (green) point where all three colors were used.}
1037\label{fig:time:dep}
1038\end{figure}
1039\end{landscape}
1040
1041The above resolution seems to be already limited by the intrinsic resolution of the photo-multipliers and the staggering of the
1042mirrors in case of the MAGIC-I telescope.
1043
1044%\begin{figure}[htp]
1045%\centering
1046%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-24.eps}
1047%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-30.eps}
1048%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-31.eps}
1049%\caption{Reconstructed arrival time resolutions as a function of the square root of the
1050%extimated number of photo-electrons for the half-maximum searching spline (extractor \#24, left) a
1051%and the digital filter with the calibration pulse weigths fitted to UV pulses over 6 FADC slices (extractor \#30, center)
1052%and the digital filter with the calibration pulse weigths fitted to UV pulses over 4 FADC slices (extractor \#31, right).
1053%The time resolutions have been fitted from
1054%The marker colours show the applied
1055%pulser colour, except for the last (green) point where all three colours were used.}
1056%\label{fig:time:fit2430}
1057%\end{figure}
1058
1059
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