Simulations of PRIMA operation with the ATs

In order to investigate potential sources of noise in the fringe tracking process, the simulations described in Section 3 were modified to include a simple fringe tracking algorithm. Optical propagation through the VLTI was not modelled - the spatially filtered output from two telescope simulations was simply combined to form an interferometer. In the fringe tracking algorithm, the phases of the seven spectral channels used in the simulation were adjusted so as to subtract the fringe offset measured in the previous timestep (or according to the piston mode if it was the first timestep). The seven channels were then combined to produce the two group delay tracking channels and the one phase tracking channel analogous to the PRIMA FSU spectral channels, as described in Section 3.2. The group delay tracking channels were then used to produce an estimate of the change in group delay, and this was used to find the nearest zero-point in the fringe phase.

One of the first interesting results to come out of these simulations was just how difficult group delay estimation is using the two widely spaced wavelength channels from the PRIMA FSU design, particularly during periods of less-than-ideal seeing. When the stellar images at the telescopes are distorted by wavefront corrugations across the telescope apertures, the phases of the fringes in the group delay tracking channels are also perturbed. This perturbation is different in the two different spectral channels, which produces a substantial error in the calculated group delay. This problem was partially solved by low-pass filtering the group delay measurements, causing perturbations due to the atmosphere to be averaged out over $5$--$10$ time units (where one time unit corresponds to the time taken for each atmospheric layer to move $r_{0}$, as discussed in Section 3). The fringe tracking algorithm was then found to perform very well, with fringe jumps detected only every few hundred fringe coherence times. Example plots of the fringe tracking performance are shown in Figures 21 and 22. If the group delay measurements are averaged over too long a period, the algorithm is not able to track the motion of the fringes due to the atmospheric piston mode fluctuations.

図 21: The measured optical phase compared with the piston mode in simulations. The curve labelled Phase, $2r_{0}/v$ average shows the optical phase in a simulation where the group delay used in unwrapping the phase is the average group delay over a time period $2$ time units ($2r_{0}/v$). For Phase, $8r_{0}/v$ average the averaging is performed over $8$ time units. Also shown is the piston mode component. All curves are for the same atmosphere. Photon shot noise and detector noise were not included (the optical phase measurements were noiseless).

図 22: The same data as Figure 21 but the piston mode component has been subtracted from the optical phase data for clarity.

These preliminary simulations have also been useful in determining which factors will have the most impact on fringe tracking performance, and hence which should be studied in more detail. Simulations were performed with seeing of $r_{0}=0.45\mbox{ m}$ and seeing of $r_{0}=0.9\mbox{ m}$ at K band (see Section 2.1 for an introduction to atmospheric seeing - these numbers correspond approximately to the following two sets of conditions: observing a target at low elevation under below average seeing; and observing a target at the Zenith under excellent seeing). Both the full aperture ($1.8\mbox{ m}$) and a stopped-down aperture ($1.2\mbox{ m}$) were simulated. A summary of the simulation characteristics is presented in Table 4. Temporal power spectra of the fringe motion are plotted in Figure 23 for these simulations.

It is clear from the power spectra in Figure 23 that the fringe motion at high frequencies is dominated by the wavefront corrugations across the aperture in all the simulations, and the piston mode component (discussed in e.g. [9]) is negligible. Note that the piston mode component has a larger high frequency component with smaller aperture sizes as expected, but this is swamped by the effects of wavefront corrugations across the AT aperture, which are much smaller with the smaller aperture size. It is clear that the fringe tracking performance will be determined mostly by the effects of these wavefront corrugations, and that the slow drifts from the piston mode component will be much less important in optimising the fringe tracking algorithm.

図 23: Temporal power spectra of fringe motion. A and B show the temporal power spectrum of the piston mode and of the fringe phase for simulation 4 (small aperture, good seeing). C and D show the same for simulation 3 (small aperture, poor seeing). E and F show the same for simulation 2 (large aperture, good seeing). G and H show the same for simulation 1 (large aperture, poor seeing). At high frequencies many the datapoints have been binned together and averaged for clarity. The frequency axis is normalised for $v/r_{0}$ with the $r_{0}$ for the poorer seeing conditions (the Taylor screens moved at the same physical velocity with the good seeing conditions).

表 4: Simulations of fringe tracking.

	Simulation 1	Simulation 2	Simulation 3	Simulation 4
K band $r_{0}$	$0.45\mbox{ m}$	$0.9\mbox{ m}$	$0.45\mbox{ m}$	$0.9\mbox{ m}$
Seeing$^{*}$ if $Z=0^{\circ}$	$1.4$''	$0.68$''	$1.4$''	$0.68$''
Seeing$^{*}$ if $Z=60^{\circ}$	$0.89$''	$0.45$''	$0.89$''	$0.45$''
Aperture diameter	$1.8\mbox{ m}$	$1.8\mbox{ m}$	$1.2\mbox{ m}$	$1.2\mbox{ m}$

$^{*}500\mbox{ nm}$ Zenith seeing which would give the appropriate K band $r_{0}$ for observations at the Zenith angle $Z$ listed.

The maximum aperture diameter which can successfully used for fringe tracking will be set by the amount of fringe jitter introduced by the wavefront corrugations across the aperture. It is interesting to compare the fringe jitter under different seeing conditions and with different aperture diameters. In this case I have defined the jitter as the residual fringe phase after the piston mode component has been subtracted. There was no photon shot noise or detector noise in these simulations, so this phase difference corresponded directly to the effect of the wavefront corrugations across the aperture plane. A summary of the simulation characteristics and results is presented in Table 5 (the simulations are the same as those presented in Table 4). The likelihood of fringe jumps should be small as long as the RMS fringe jitter $\ll 1$ radian.

表 5: Phase jitter resulting from wavefront corrugations in the aperture plane.

	Simulation 1	Simulation 2	Simulation 3	Simulation 4
K band $r_{0}$	$0.45\mbox{ m}$	$0.9\mbox{ m}$	$0.45\mbox{ m}$	$0.9\mbox{ m}$
Aperture diameter	$1.8\mbox{ m}$	$1.8\mbox{ m}$	$1.2\mbox{ m}$	$1.2\mbox{ m}$
RMS phase jitter	$0.81\mbox{ radians}$	$0.31\mbox{ radians}$	$0.38\mbox{ radians}$	$0.30\mbox{ radians}$

The phase jitter with the full diameter aperture under the poorer seeing conditions (Simulation 1) is clearly much worse than the other cases. Stopping down the AT aperture provides a very significant reduction in the phase jitter, and appears a promising option at times when fringe tracking becomes very unreliable. Note that the very high phase jitter for Simulation 1 is partly due to a small number of very large phase excursions including fringe jumps, an example of which is shown in Figure 24. Note that the RMS phase jitter is lower for the $1.2\mbox{ m}$ diameter aperture at all times, however. The loss of stellar flux would make stopping down the telescope aperture less appealing for faint stars.

図 24: The amount of phase jitter introduced by wavefront corrugations in the aperture plane of an AT. The jitter with the full $1.8\mbox{ m}$ is shown along with the result when the aperture is stopped-down to $1.2\mbox{ m}$ diameter. Both simulations used the same atmosphere. Photon shot noise and detector noise were not included (the optical phase measurements were noiseless).