Global wavefront sensing for interferometers and mosaic telescopes: the dispersed speckles principle

The piston-sensing algorithm proposed by Labeyrie (Lardière et al 2002 IAU Technical Workshop vol 266, p 608) for diluted interferometric arrays can be extended to redundant segmented apertures such as those of extremely large telescopes and periodic hypertelescopes (Labeyrie 1996 Astron. Astrophys. 118 (Suppl.) 517). We first show that his ‘dispersed image’ input data cube is the tri-dimensional Fourier transform, in squared modulus form, of a ‘wavefront bumpiness’ function, a binary function of both aperture coordinates and the local optical path (or piston) error. It follows that Labeyrie’s output cube is the tri-dimensional autocorrelation of the wavefront bumpiness. Using a tri-dimensional extension of Fienup’s (1978 Opt. Lett. 3 1) phase retrieval algorithm, we show through numerical simulations that the multiple peaks appearing in columns of the output cube can be discriminated to retrieve the piston errors. About ten photons are needed per exposure, per sub-aperture and per spectral channel.