Imaging with hypertelescopes: a simple modal approach

Aims. We give a simple analysis of imaging with hypertelescopes, a technique proposed by Labeyrie to produce snapshot images using arrays of telescopes. The approach is modal: we describe the transformations induced by the densification onto a sinusoidal decomposition of the focal image instead of the usual point spread function approach. Methods. We first express the image formed at the focus of a diluted array of apertures as the product R-0(alpha) X-F(alpha) of the diffraction pattern of the elementary apertures R-0(alpha) by the object-dependent interference term X-F(alpha) between all apertures. The interference term, which can be written in the form of a Fourier Series for an extremely diluted array, produces replications of the object, which makes observing the image difficult. We express the focal image after the densification using the approach of Tallon and Tallon-Bosc. Results. The result is very simple for an extremely diluted array. We show that the focal image in a periscopic densification of the array can be written as R-0(alpha)X-F(alpha/gamma), where gamma is the factor of densification. There is a dilatation of the interference term while the diffraction term is unchanged. After de-zooming, the image can be written as gamma X-2(F)(alpha)R-0(gamma alpha), an expression which clearly indicates that the final image corresponds to the center of the Fizeau image intensified by gamma(2). The imaging limitations of hypertelescopes are therefore those of the original configuration. The effect of the suppression of image replications is illustrated in a numerical simulation for a fully redundant configuration and a non-redundant one.