Appendix B: The Question of Calibration

See the Open Hardware wiki for a full explanation of White Rabbit calibration: https://ohwr.org/projects/white-rabbit/wiki/Calibration and the associated document for the standard calibration procedure: https://ohwr.org/project/white-rabbit/uploads/76cdbdbadccc9d6c54d5caf246550fbf/WR_Calibration-v1.1-20151109.pdf

The calibration of the White Rabbit timing node (or “slave”) concerns the differential delay in transmission (Master-Slave) and reception (Slave-Master), with several elements being concerned:

  • for the node itself, so calibration should be done if the firmware changes

  • for the SFP (optical link) used on the node

  • for the fibre, especially since different wavelengths are used in each direction, so the variation in index with wavelength can affect the differential transit time (so that the round-trip time needs some correction to properly derive the length).

For the first two points, these can be calibrated using a comparison to a standard, with a short optical fibre length. For the third point, this requires a measurement on the fibre itself.

The question is how accurate does this calibration need to be, and what happens if it isn’t done?

The measurement on the fibre is characterized by \(\alpha\) where

\[\alpha = \frac{ \delta_\mathrm{MS} - \delta_\mathrm{SM} } {\delta_\mathrm{SM}}\]

… where \(\delta_\mathrm{MS}\) is the Master-to-Slave fibre latency, and similarly \(\delta_\mathrm{SM}\) the Slave-to-Master fibre latency.

The GRAPPA/DESY group has measured several batches of fibres which may be used in CTA (see slide 8): https://indico.cta-observatory.org/event/2249/contributions/21602/attachments/16317/21153/HP_d20191022_ArrayClock_General.pdf, and found values of \(\alpha=0.000230\rightarrow 0.000265\).

Rearranging the equation gives:

\[\delta_\mathrm{MS} = \delta_\mathrm{SM} (1+\alpha )\]

We can take a typical value of the speed of light in a fibre: \(c_\mathrm{fibre} = 204.1 \pm 0.2\mathrm{m/\mu s}\) to be that for \(\delta_\mathrm{SM}\), then for a fibre of length L, the transit time in the fibre from slave to master is:

\[\Delta_\mathrm{SM}= \frac{L}{c_\mathrm{fibre}}.\]

If we have the minimum value of \(\alpha\) in our fibre, but we apply the maximum value, this gives an estimation of the error. This is equivalent the error in the estimation of the return journey time due to the change in \(c_\mathrm{fibre}\) in the reverse direction.

So, an error in the estimation of \(\alpha\) gives the following error in the estimation of the reverse journey time \(\Delta_\mathrm{MS}\):

\[\mathrm{Time\ Error} = \Delta_\mathrm{SM}(\alpha_\mathrm{high}-\alpha_\mathrm{low}) = \frac{L}{c_\mathrm{fibre}}(\alpha_\mathrm{high}-\alpha_\mathrm{low})\]

For the CTA-N site the radius of the array is 400m, while for CTA-S it is 1200m. Taking 2km as a maximum length of a fibre, then we find that for the maximum range of \(\alpha\) of fibres tested at GRAPPA/DESY, the systematic error induced would be:

\[\mathrm{Time\ Error}_{2\mathrm{km}} = \frac{3.5\times10^{-5}\times2\mathrm{km}}{204.1\mathrm{m/\mu s}} = 3.5\times10^{-5}\times9.8\mathrm{\mu s} = 343\mathrm{ps}\]

This is quite a minor correction, and indeed for the CTA-N site would be quite negligible at the order of \(100\mathrm{ps}\).