1

In the absence of instrumental perturbation, the visibility is a measure of the degree of mutual coherence. In practice, the experimental visibility (as measured) is the product of three terms:

• the intrinsic visibility of the object

• a term linked to the instrument function of the interferometer

• a term linked to atmospheric perturbations (for observation from the ground) or to the environment.

The last two terms are estimated by observing a 'calibration' object, which is a star whose intrinsic visibility is known, and generally a star that is not resolved with the interferometric baseline, and which is thus completely coherent (having an intrinsic visibility of 1).

optical path difference (microns)

Fig. 2.23 An example of an interferogram obtained by varying the path difference by use of a delay line. For a path difference greater than ±5|m, there is no temporal coherence. The spatial coherence is not perfect (the visibility is about 0.7)

optical path difference (microns)

Fig. 2.23 An example of an interferogram obtained by varying the path difference by use of a delay line. For a path difference greater than ±5|m, there is no temporal coherence. The spatial coherence is not perfect (the visibility is about 0.7)

Various studies may be found in the literature of the factors that degrade visibility, in particular, atmospheric turbulence. We shall not, therefore, discuss this aspect in detail here.

Visibility of the Source's Structure

The relationship between what may be observed (the experimental visibility) and the astrophysical source is obtained by employing the Zernicke-Van Cittert theorem, which may be expressed as follows:

If both the linear dimensions of the source of quasi-monochromatic radiation and the distance between the two points on the screen (here, the distance between the two telescopes, i.e., the distance between P1 and P2) are small relative to the distance between the source and the screen (here, between the source and the Earth), then the modulus of the complex degree of coherence (the experimental visibility) is equal to the modulus of the spatial Fourier transform of the intensity of the source, normalized to the total intensity of the source.

In other words, there is a simple Fourier relationship between the spatial structure of the source and the visibility function, measured at several different baselines (several spatial frequencies) as shown in Fig. 2.24.

In practice, the opposite problem has to be tackled. The visibility curve is obtained, and then, generally by adapting the model, one can deduce the structure and parameters of the source. For example, in the case of a stellar source, the simplest model is one consisting of an evenly illuminated disk. In this case, interpolation from the measured visibility points enables one to determine the diameter of the source (by determining the point at which the visibility function first becomes zero). Obviously, this requires the source to be resolved by the interferometer; that is that the object's angular resolution should be greater than the angular resolution determined by the interferometer's baseline (the value X/B, where B is the distance between the telescopes and X is the observational wavelength).

It should, however, be noted that the object space (the plane of the sky, where the coordinates are denoted by x and y), as well as the associated Fourier space (i.e., the one dealing with spatial frequencies and where the conjugate coordinates in x and y are denoted by u and v, described as the (u, v) plane), are two-dimensional

' Intensity

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