21.1 The basic equation of interferometry

In a two element interferometer, the signal coming on the detectors from telescopes 1 and 2 is the sum of contributions from the background $B_{1,2}$ (either the atmosphere or the instrumentation itself) and from the astronomical source $I_{1,2}$.

21.1.1 Additive interferometry

For direct detection (or additive) interferometry, as in the optical domain, an interferometer measures on-source on the baseline ${\cal B}_{12}$:

$$I_{12} = I_1 + I_2 + 2 \sqrt{I_1 I_2}V_o \vert{\cal V}_{12}\vert \cos{\Phi_{12} } + B_1 + B_2$$ (21.1)

The term $I_1 + I_2 + B_1 + B_2$ is the continuum term while $2 \sqrt{I_1 I_2}V_o \vert{\cal V}_{12}\vert \cos{\Phi_{12}}$ is the interferometric term.

After doing an on-off (also called the ``sky calibration''), Eq.21.1 becomes:

$$I_{12} = I_1 + I_2 + 2 \sqrt{I_1 I_2}V_o \vert{\cal V}_{12}\vert \cos{\Phi_{12}}$$ (21.2)

Where ${\cal V}_{12}$ is the visibility of the astronomical source measured on baseline ${\cal B}_{12}$ of amplitude $\vert{\cal V}_{12}\vert$ and phase $\Phi_{12}$.

The visibility to calibrate can be expressed by:

$$V_{corr} = \frac{2 \sqrt{I_1 I_2}}{I_1 + I_2} V_o \vert{\cal V}_{12}\vert \cos{\Phi_{12}}$$ (21.3)

$V_o$ is the contrast which takes into account the calibration of all the system (instrumentation + atmosphere). The photometric term is given by $\frac{2 \sqrt{I_1 I_2}}{I_1 + I_2}$ (note that $I_1$ and $I_2$ are relatively easily measured).

The visibility ${\cal V}_{12} = \vert{\cal V}_{12}\vert\cos{\Phi_{12}}$ appears as a fringe contrast (which is flux calibrated), therefore it is normalized to unity. Note finally that in the optical case $B_{1,2} \ll I_{1,2}$.

21.1.2 Multiplicative interferometry

For heterodyne or multiplicative detection, the output of the interferometer (correlator) gives a correlation rate $r_{12}$ which is a dimension less number (this uses a simple correlation between two antennas, not a ``bi-spectrum'').

The correlation corresponding to $\sqrt{I_1I_2} V_o {\cal V}_{12}$ is the term of astronomical interest, and is related to $r_{12}$ by:

$$\sqrt{I_1 I_2}V_o \vert{\cal V}_{12}\vert e^{i\Phi_{12}} = r_{12} \sqrt{(B_1 + I_1)(B_2 + I_2 )}$$ (21.4)

Where $(B_1 + I_1)$ and $(B_2 + I_2 )$ are the autocorrelations mesured on telescopes 1 and 2, respectively.

At mm waves, $B_{1,2} \gg I_{1,2}$ because the atmospheric thermal emission strongly dominates with typically $I_{1,2}/B_{1,2} \sim 10^{-3}- 10^{-4}$ (except for the Sun and bright planets). Therefore, Eq.21.3 simplifies as:

$$\sqrt{I_1 I_2}V_o \vert{\cal V}_{12}\vert e^{i\Phi_{12}} = r_{12} \sqrt{B_1 B_2}$$ (21.5)

The heterodyne technique does not allow to measure the continuum term but preserves the phase (thanks to the use of a complex correlator, see Chapter 2). $V_o$ can be seen as the correlation efficiency of the interferometer (instrumental + atmospheric). The calibrated visibilities (as defined in previous chapters) $V_{12} = \sqrt{I_1 I_2}{\cal V}_{12}$ are expressed in unit of flux density (Jy) while ${\sqrt{B_1 B_2}}$ can by considered as the photometric term (including the photometric calibration of the atmosphere).