7.3 Phase calibration

We now turn to the more difficult problem of correcting for the time dependence of the complex gains, contained in the functions $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}} }_{ij}(t)$. Variations occur in both amplitude and phase. Let us summarize the various effects to be calibrated:

• Interferometer geometry:
  Small errors in the baseline determination will cause slow phase drifts (period 24 hours); unfortunately these errors are dependent on the source direction so they cannot be properly calibrated out by phase referencing on a calibrator, only reduced by a factor of the order of the source to calibrator distance expressed in radians.

• Atmosphere
  The atmosphere introduces phase fluctuations on time scales 1s to a few hours, depending on baseline length and atmospheric conditions (see previous lecture). The effect of fluctuations on short time scales is to cause loss of amplitude by decorrelation, while on the long term the phase fluctuations can be mistaken for structure in the source itself. The critical time there is the time it takes for the projected baseline vector to move by half the diameter of one antenna:
  

  $$\Delta t_C = \frac{D}{2} \frac{86400}{2 \pi b} = 224 \frac{D}{15} \frac{450}{b}$$ (7.21)

  
  
  This time ranges from 4 minutes to 1 hour for Plateau de Bure, depending on the baseline length. The phase fluctuations on short time scales may be corrected by applying the radiometric phase correction method, if it is available.

• Antennas
  The antennas may cause variations in amplitude gains due to degradation in the pointing and in the focusing due to thermal deformations in the antennas. These can be corrected to first order by amplitude calibration but it is much better to keep the errors low by proper frequent monitoring of the pointing and focus, since these errors affect differently sources at the center and at the edges of the field. Focus errors also cause strong phase errors due to the additional path length which is twice the sub reflector motion in the axial direction.

• Electronics
  Phase and amplitude drifts in the electronics are kept low by efficient design (see the lectures on receivers and signal and LO transport). The electronics phase drifts are generally slow and of low amplitude, expect for hardware problems.

A detailed analysis can be found in [Lay 1997].