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.

An energy-stable pseudospectral scheme for Swift-Hohenberg equation with its Lyapunov functional

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. The?oretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l2-norm for the proposed scheme is given.

Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

We propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.