Alias-free basis for modal sensorless adaptive optics using the second moment of intensity

In theory of optical aberrations, an aberrated wavefront is represented by its coefficients in some orthogonal basis, for instance by Zernike polynomials. However, many wavefront measurement techniques implicitly approximate the gradient of the wavefront by the gradients of the basis functions. For a finite number of approximation terms, the transition from a basis to its gradient might introduce an aliasing error. To simplify the measurements, another set of functions, an "optimal basis"with orthogonal gradients, is often introduced, for instance Lukosz-Braat polynomials. This paper first shows that such bases do not necessarily eliminate the aliasing error and secondly considers the problem of finding an alias-free basis on example of second-moment-based indirect wavefront sensing methods. It demonstrates that for these methods any alias-free basis should be formed by functions simultaneously orthogonal in two dot-products and be composed of the eigenfunctions of the Laplace operator. The fitness of such alias-free basis for optical applications is analyzed by means of numerical simulations on typical aberrations occurring in microscopy and astronomy.