.. _pixel-calibration: Pixel Calibration ================= .. _pixel calibration: After the correction of DRS4 systematic effects, the R0 waveform must be converted from units in ADC to units in number of photo-electrons (pe) : .. math:: \mathrm{waveform_{pe}} = (\mathrm{waveform_{ADC}} - pedestal_{ADC}) \times dc\_to\_pe :label: calib This step is generally called \`Cat-A pixel calibration\` or \`DC to pe\`` conversion. In LST the estimation of the calibration coefficients is performed offline with so called \`F-factor\` (or \`photon statistics\`) method :cite:`BENCHEIKH1992349`, which permits to estimate the coefficients *dc_to_pe* in eq. :eq:`calib`. This method is however affected by some systematic deviations that must be estimated and corrected. In the next sections we briefly describe the methods, see :cite:`Kobayashi:2021jc` for more details. F-factor method ............... This method permits to estimate the effective gain of the pixels, defined as *gain = 1/ dc_to_pe*. It makes use of the statistical correlation between the charge dispersion of a signal and the number of photo-electrons (:math:`{N_{pe}}`) originally produced by the light in the photo-cathode. In fact, in the case of an ideal poissonian detector, the integrated charge :math:`{Q}` produced by an incident light pulse on a PMT and its standard deviation, :math:`{\sigma_{Q}}` (when repeated several times) are related to the *gain* (*g*) by the following simple relations: .. math:: \begin{eqnarray} Q & = & g & \; \cdot & N_{pe} \\ \sigma_{Q} & = & g & \; \cdot & \sqrt{N_{pe}} \end{eqnarray} :label: idealdet which permits to estimate the gain as .. math:: \begin{eqnarray} g & = & \frac{\sigma_{Q}^{2}}{Q} \end{eqnarray} :label: idealgain For a real detector the relation is more complicated due to added noise components. In particular, for LST, where :math:`{Q}` is produced by flat-field events, the variance :math:`{\sigma_{Q}^2}` is given by: .. math:: \begin{eqnarray} \sigma_{Q}^2 = \sigma_{Q_{ped}}^2 + F^{2} g (Q-Q_{ped}) + B^2 (Q-Q_{ped})^2 \end{eqnarray} :label: lst_variance where :math:`Q_{ped}` is the pedestal charge, :math:`\sigma_{Q_{ped}}^2` is its variance, :math:`F = \sqrt{1 + \sigma_{spe}}` is called \`excess noise factor\` and depends on :math:`\sigma_{spe}`, which is the single photon electron width in pe, *B* is a quadratic noise term due to the DRS4 time sampling irregularity and the intrinsic pulse light dispersion of the laser (see :cite:`Kobayashi:2021jc` for more details). Hence, the formula used to estimate the gain in LST is : .. math:: \begin{eqnarray} g & = & \frac{\sigma_{Q}^{2}-\sigma_{ped}^{2}}{Q-Q_{ped}} \frac{1}{F^2} - \frac{B^2}{F^2} (Q-Q_{ped}) \end{eqnarray} :label: lst_gain In short: * Input data : flat-field and NSB pedestal events * Correction applied by : EventBuilder * Coefficient production : see :ref:`How to ` F-factor systematics correction ............................... The systematic term B in eq. :eq:`lst_variance` is estimated, per pixel, by the fit of the charge variance of an intensity scan obtained by changing the Calibox filters in front of the laser. .. figure:: ../figures/FFactor_corrections.png :scale: 60 % :alt: FFactor systematics Example of intensity scan fit, based on eq. :eq:`lst_variance`, for both channels of one pixel. In short: * Input data : flat-field and NSB pedestal events from an intensity scan * Correction applied by : lscam_calib * Coefficient production : see :ref:`How to ` Cat-B pixel calibration ....................... This calibration is performed in order to improve the Cat-A calibration (based on fixed coefficients for the full night) with a continuous estimation of the camera gain during the night based on interleaved calibration events. In this case, the F-factor method is applied to calibrated waveforms, the estimated Cat-B gain is then relative to the Cat-A gain. Therefore, if no changes are present during the night, the Cat-B gain is expected to be one. In general, smooth changes of less then 2% are observed. .. figure:: ../figures/Cat_B_gain.png :scale: 60 % :alt: Cat-B gain Example of Cat-B gain estimation for some runs of a night. In short: * Input data : interleaved (calibrated) flat-field and NSB pedestals events * Correction applied by : lscam_calib * Coefficient production : see :ref:`How to `