## How SUSI works

*This description is brief, but may seem a bit complex to the uninitiated, so proceed at your own risk!*

*A knowledge of the complex degree of coherence allows spatial information about a source to be determined since the van Cittert-Zernike theorem states that the complex degree of coherence is the spatial Fourier transform of the object being observed. * In principle the complex degree of coherence can be determined from the interference fringe pattern formed when the beams of starlight from the two apertures of a stellar interferometer are combined. The

**modulus of the complex degree of coherence**is equal to the true

**fringe visibility**and its

**phase**is the phase of the fringe pattern.

Here we use the terms visibility and phase to refer to the raw or measured quantities. Turbulence in the Earth's atmosphere corrupts the phase of the fringe pattern and, for a two-aperture interferometer like SUSI, it conveys no useful information about the source being observed. The
observed fringe visibility V will be less than expected due to instrumental losses and the effects of atmospheric turbulence. In SUSI we measure V^2 rather than the visibility itself, and we term V^2 the
**correlation** C.

### Measurement of the correlation

In SUSI the beams from the two arms of the interferometer are directed from opposite sides towards a beamsplitting plate where the beams are combined. Two beams emerge from the beamsplitter and each is a superposition of the two input beams. The incident beams are aligned so that they emerge accurately superimposed and parallel. Ideally, the emerging beams will be uniformly illuminated having irradiances, I1 and I2, proportional to [1 + V * cos(phase)] and [1 - V * cos(phase)] where V is a seeing-affected measure of the fringe visibility. The sign difference between the two terms is a consequence of the pi phase shift of one of the reflected beams at the beamsplitting plate.

At optical wavelengths the fringe phase is corrupted since it is the sum of the phase of the complex degree of coherence *and* phase fluctuations introduced by path length variations in the atmosphere. Even for baselines of a few metres these phase fluctuations amount to many radians. If the average of [I1 - I2]^2 is taken over a sufficiently long period, the cos^2 term it contains will average to 1/2, and consequently we can measure the mean correlation and fringe visibility from the expression

{C} = {V^2} = 2 * { [ I1 - I2]^2 } / [{I1} + {I2}]^2

(note that { } is used here to denote ensemble averages to avoid the use of < > which is employed by HTML)

Because of seeing effects, the **sample time** used to measure the correlation must be very short - of the order of 1-10 milli-seconds. As a result, individual measurements are dominated by photon noise and, in addition to the need for a large number of samples to adequately average the cos^2 term, it is essential to integrate over a much longer period to improve the signal to noise ratio. Typically we use an **integration time** of the order of 60-100 sec.

In practice a number of small correction factors must be included to account for the fact that the transmissivities of the various parts of the interferometer are not equal, the quantum efficiencies of the two detectors are different, etc.

### Seeing effects

The distortion of the wavefronts of starlight as they propagate through the earth's turbulent atmosphere
leads to the random phase and amplitude variations that are responsible for **seeing** in conventional telescopes. Seeing can be characterised by two parameters, **ro** and **to** ("r nought" and "t nought"). ro is a measure of the spatial scale of the phase variations across the wavefront. For optical wavelengths ro is ~5 to10cm at typical astronomical sites. A large value of ro implies good seeing conditions. The median seeing for the SUSI site ~1.3 arc seconds with subarcsecond seeing occurring
for ~13% of the time. These seeing figures correspond to ro values of 6.7cm and 9.0cm respectively at the effective measurement wavelength of 440nm. to is a measure of the speed of phase fluctuations. The range of values of to observed is ~0.9 to ~8ms with a median value of 2.7ms.

The effect of seeing is always to *reduce* the observed fringe visibility (i.e. correlation). This
loss can be reduced by using tip-tilt adaptive optics as is done in SUSI. Because the magnitude of the tip-tilt component of the seeing depends on ro, we obtain estimates of ro with SUSI by monitoring the tip-tilt fluctuations. In principle, values of ro can be used to determine correction factors for the observed correlation. Because the basic sampling time is non-zero, the measured correlation will also be reduced because the phase of the fringes will change during the sampling interval.

It should be emphasised that a two-parameter model for the effects of atmospheric turbulence is in most cases an oversimplification. The SUSI results show that, even with short sample times and small apertures, the ro and to corrections only partially account for the overall reduction in correlation due to seeing. Calibration procedures are necessary to obtain unbiased estimates for the correlation.

### Bandwidth effects and optical path stability

The discussion so far has ignored the effect of a finite spectral bandwidth. The correlation will decrease symmetrically about the matched optical path position at a rate which is a function of the spectral bandwidth. Even for bandwidths of ~1nm the optical paths must be matched to within a few tens of micrometres to detect correlation, and to within a very few micrometres if the correlation loss due to the path difference is to be less than 1%.

### Observing procedure

In order to determine the angular diameter of a star, measurements of correlation at more than one baseline are required. If a star is known to be single, the correlation at a single baseline is, in principle, sufficient. However, whether a star is single or not is generally unknown and observations at more than one baseline, and covering a range of orientations of the projected baseline, are desirable to identify previously unknown multiple stars.

Correction of the observed correlation for instrumental and residual seeing effects is achieved with observations of unresolved stars or stars of known angular diameters. Ideally, the calibration stars will be as close as possible in the sky to the programme stars, the stars whose angular diameters are to be determined. Observations of the calibration and programme stars are interleaved to minimise any potential systematic effects.

### A sample result - the angular diameter of delta CMa

As an example of the determination of the angular diameter of a star we describe the observations for the determination of the angular diameter of delta CMa. This star was the faintest star measured with the Narrabri Stellar Intensity Interferometer (NSII). Based on the angular diameters determined with the NSII, epsilon and eta CMa were chosen as the calibration stars for this series of observations.

In principle, one calibrated value of correlation at a baseline will allow an angular diameter to be determined. In practice, the observations of delta CMa were interleaved with observations of epsilon CMa at baselines of 5, 10, 15, 20 and 30m and with observations of eta CMa at 5, 10, 15 and 20m.

Some measured correlation values and the fitted curves are shown in diagram.

Get fig.4(a)- Normalised mean values of observed correlation versus projected baseline for delta Cma, using epsilon CMa as the calibration star.

There is excellent agreement between the final SUSI value for the angular diameter of delta CMa of 3.505 +/- 0.064 milli-arc seconds (mas) and the NSII result of 3.29 +/- 0.46 milli-arc seconds. The maximum correlation values for this star are less than 1, the value expected for a single star. The implication of this result is that delta CMa is a binary system with a significantly fainter secondary component. The SUSI results are consistent with a magnitude difference between two components of 3.25 +/- 0.24 at 442nm.

SUSI Baselines |