Wave-front reconstruction for the non-linear curvature wave-front sensor (2013)

Non-linear curvature wave-front sensing (nlCWFS) delivers outstanding sensitivity and high dynamic range by lifting the linearity constraint of standard curvature wave-front sensing and working in the non-linear Fresnel (near-field) regime [Guyon, 2010]. The goals of this paper are twofold: 1) revisit the phase-diversity PD formalism and attempt to use this framework, originally developed for the Fraunhofer (far-field) regime, with nlCWFS signals and 2) develop formulae making explicit use of the Fresnel regime for later use with gradient-based non-linear minimisation methods. 1 Curvature wave-front sensing: linear and non-linear regimes The highly sensitive non-linear curvature wave-front sensing (nlCWFS) takes full advantage of diffraction, which encodes wave-front aberrations into patterns of diffraction-limited interference speckles [Guyon, 2010]. To do so, a set of images are acquired away from the pupil-plane, beyond the linear regime used in traditional linear curvature WFS, but not reaching the Fraunhofer regime used in phase-diversity (PD). In nlCWFS images are acquired in the intermediate regime at locations that optimise the sensitivity to a large range of spatial frequencies. The trade is set by a non-linear wave-front reconstruction process which involves computationally intensive non-linear algorithms. One such example is the Gerchberg-Saxton iterative algorithm proposed originally in [Guyon, 2010]. The linear, non-linear and Phase Diversity techniques are thus linked by the same principle of acquiring multiple images at different propagation distances and work from there to reconstruct the phase. The differences lie on the locations of the planes where those images are acquired. To enforce linearity linear CWFS uses largely defocused images that correspond to distances closer to the pupil-plane [Roddier and Roddier, 1993],[Guyon et al., 2008]. On the other hand, PD relies on a forward model of image formation using the far-field (Fraunhofer) approximation thus operating near the focal plane where the defocus terms are small. [Fienup et al., 1998] gives a thorough account of the similarities between PD and linCWFS . The massive computation required makes its use more amenable to non-real-time applications, as is the case of non-common path aberration estimation. In this paper we investigate approximations that could allows us to use the framework developed for phase-diversity (PD) to compute the maximum a-posteriori (MAP) wave-front estimate [Correia, 2013]. Since the latter proves insufficient, we use instead a minimisation criterion that explicitly uses the image formation model in the near-field (Fresnel) regime and derive its partial a ccorreia@astro.up.pt, Formerly with HIA derivatives with respect to the parameters of the wave-front, expanded onto a Zernike polynomial basis of functions. At present, only the mono-chromatic case is dealt with. 2 Image formation models The incoherent image formation model is approximated by a cyclic discrete convolution ik(x,α) = ∑ x′∈Ω o(x)sk(x − x,α) + ηk(x) = o(x) ∗ sk(x,α) + ηk(x) (1) where ik(x,α) the kth image, o(x) is the object, s(x,α) is an indicial response function, η(x) is additive noise and ∗ represents 2-D convolution. Variables in the pupil and image planes are indexed by r and x respectively1. The pupil-plane wave-front aberrations are parametrised as