Visibility estimation

The MIDI interferometer is a Michelson interferometer, also called a pupil interferometer. The afokal beams coming from the telescopes are combined at a beam splitter, and then focused onto the array detector. Therefore, to measure the complex degree of coherence between the two beams, also called the visibility (of the fringes), it is necessary to vary the optical path length difference (OPD) between the telescopes and the beam combiner from its nominal value (i.e. the geometric delay) which combines the light within their coherence length. This variation, called fringe scanning, produces a quasi-sinusoidal modulation of the signal S detected by each detector pixel as follows.

Here F is the total flux from the source at wavenumber k=2&pi /λ, and V is the visibility amplitude. D is a residual delay of the fringe packet, and it can be written as the sum of the instrumental delay Di which represents the effect of the fast-scanning piezos, and the more slowly changing atmospheric delay Da which is related to refractive index fluctuations. The object visibility phase as well as other systematic phase offsets (if any) are lumped together in Φ.

To compute the visibility in a band pass, or the broad band "white light" visibility, we integrate over k.

One way to estimate FV, the correlated flux in either "white" light or in a limited band, is the so-called incoherent or total power method. Here, you vary D in a (hopefully controlled) way and then measure the amount S varies by taking its mean square. The GARBAGE contributes directly to this estimate. To remove its influence, two steps are usually necessary:

(1) filter out the frequencies that should not be present. This obviously involves removing the low frequency components we just saw, but in a more sophisticated system we first Fourier Transform the fringe pattern. We then consider which wavelengths are present in the band, and the rate at which the OPD is changing (e.g. in microns/sec) and include only the power in the right frequency interval (with a little leeway for atmospheric OPD velocities).

(2) After due precautions there will still be excess power in the band from any number of sources (sky variations, photon noise, amplifier hum, amplifier gain fluctuations). We estimate these by measuring the fringe power when there should be no fringe. This can be done when there is no star in the beam, but it is better to have the star there, but the OPD so far offset that no fringes should be visible. For narrow wavelength bins, this can be as much as 500-1000 μ.