The interferometric method of image formation and associated mathematical problems

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Increase of informativeness of the images obtained by means of astronomical instruments in all wavelength ranges is invariably an actual problem of observational astronomy. The informativeness of an image is formed of the resolution, with which the image is obtained, and the precision, with which the brightness of each element of the image is measured. From this, two traditional problems arise immediately: the increase of the resolution and decrease of measurement errors. The present report is devoted to these questions considered in connection with the interferometric method of image formation, suggested not so long ago [I, 21. As it is known, the principal limit of the resolution of an astronomical instrument is put by the diffraction of the coming wave on the instrument aperture. However, when observing from the Earth surface, this limit is seldom achievable because of phase distortions of wave in the Earth atmosphere. At one time there were hopes of subsequent computer processing such images. However, gradually it became clear, that the loss of information, caused by the atmospheric distortions at conventional ways of observation, is irreplaceable, and therefore the efforts of investigators should be directed to the struggle against the atmospheric distortions immediately at the time of the observation. There exist a set of ideas on how one has to form an astronomical image in order to reduce or exclude the influence of the phase distortions. They differ in wavelength range and class of objects for which they are more aplicable, as well as in the degree in which their authors are ready to break off with traditions and move in unknown direction. One of such methods, apparently, the most radical, is the interferometric method of image formation, suggested in [1,2]. In its initial form this method is intended for image formation in the optical range. It is based on the fact, that by virtue of the van Cittert - Zernike theorem, the brightness distribution over the object surface and the coherence function of the field of the wave coming from it are related by Fourier-transformation (for an infinitely remote object and free space of wave propagation). The coherence function depends on the pair of points in the plane, passing through the observation point perpendicularly to the direction to the object. However, this dependence has the special form: actually the coherence function depends not on both radius-vectors of these points separately, but only on their difference. Therefore, measurment of the coherence function for the different pairs of points with the same value of the base vector has to deliver the same value (that is infringed only by errors of the measurement and phase distortions in the propagation medium.) It means that the information on the brightness distribution over the object surface is coded in the field of the incoherent wave coming from the object with some redundancy which gives rise to certain noise immunity of the coding and opens the possibility to restore the information damaged by the atmospheric distortion. The idea to obtain the object image by direct measurement of the coherence function and subsequent Fourier-transformation of it, is known in radio astronomy long ago and is the only real way of imaging objects in the long wave range. The idea of excluding the phase distortions produced by the atmosphere, on the basis of van Cittert - Zernike theorem, was stated in [3]. However, for the long time it was not clear, how this ideas may be extended to the optical range distinguished in other physical principles of signal detecting and parallel nature of measuring the energy flow from all the elements of the object image. This question was resolved, when in [1,2], and then in [4] the optical scheme of the interferometer (interferometric telescope) was suggested, enabling one to perform the parallel measurement of the coherence function values for all points of the frequency domain accessible for the given aperture, with redundancy, sufficient to exclude unknown phase distortions caused by atmospheric inhomogenities. (In principle, the most correct variant of this scheme is the one suggested in [2].) Though such a statement of the problem is typical to the optical range, actually this problem and its solution have the direct relation to the radio wave range because of advance of radio astronomy to short-wave end of the radio wave range and inevitable hereafter transition to optical methods of image formation. Furthermore, it is important to have in mind that, though this method is found on the way of struggle against the atmospheric distortions, it is also actual for design of orbital telescopes since the phase distortions and problems associated with them will inevitably arise in the space because of deformation of the construction at attempt to build an optical (radiooptical) system of the largest technically accessible size.