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 π / 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.

Incoherent Visibility Estimation

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 (This is what MIA does). 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 μm.

Coherent visibility estimation

Coherent integration of a complex observable employs a vector average, i.e. the averaging of real and imaginary parts separately, but also implies that by some other technique the vectors have been aligned to within there statistical error so that the result doesn't average to zero. Our observable is the complex visibility, and the alignment process is called rotation by the group delay (The group delay is, by definition, equal to the derivative of the visibility phase with respect to the wavenumber).

We know that our measurement of F(k)V(k) is modulated by a cosine function and we begin by looking at the wavenumber dependence of S. If F(k)V(k) doesn't vary too rapidly with k then most of the variation of S is caused by the cos(kD+Φ) term. (For the moment, D=Da, i.e. we ignore the instrumental delay.) If we take a Fourier Transform of S with respect to k we should get a peak of the square modulus near D (the group delay), actually two peaks near +D and -D. Once we know D we know cos(kD) and we can estimate FV. In practice we want to do this for a variety of D values so that we are not dividing by 0 at some values of k.

While it would be possible to use for D the phase delay, i.e. the visibility phase times the mean wavenumber of its bandpass, the group delay method has the advantage that it uses all the measured values of S(k) at one instant to estimate D, which lowers the noise in this estimate. Another advantage, if our spectral resolution is good, is that we don't have to keep D near the zero OPD point, i.e. the quality of the on-line fringe tracking is not very critical. In practise we still modulate D a lot to both modulate most of the GARBAGE to supressed frequencies, and to distinguish between the correct D and the image delay -D. If we change D in a known way with the MIDI piezos, the correct peak of the delay function moves in the right direction.

Now representing the total OPD D as the sum of a known instrumental delay Di and an unknown, but more slowly varying atmospheric component Da, we write the Fourier transform of our signal S with respect to the wavenumber k as follows.

The square modulus of the function G(D') needs to be maximized, giving the group delay. Here we show that this function has two symmetrical peaks, if we set S(k)=cos(kD), and remember that

giving

In reality, G(D') is of course not a delta-function but is convolved with the Fourier-transform of B(k)V(k).

Since our unknown FK is modulated both at high frequency with the variation of Di as well as at low frequency with the variation of Da, giving S, we multiply, for the purpose of demodulation, S with first exp(-i kDi) which will change G(D) to read

Di varies quickly and will be suppressed if we average. More accurately, what we have done is to multiply our signal with the basis function of the FT, and by summing over all samples of a fringe we have determined the complex FT of the signal at the modulation frequency. Second, we estimate Da by averaging together the FTs of several (almost coherent) scans, and finding the peak in the absolute value of G(D'). This is called group delay determination.

Thus we multiply S by exp(-i kDa) so

If we have done this correctly, the residual value of S remaining is

Note thate we have recovered the source phase Φ(k) except that any component of Φ that is linear in k will have been removed when we fitted the delay D. All valid individual measurements of S' can now be averaged together to estimate F(k)V(k)