Alias-Free Basis for Modal Sensorless Adaptive Optics Using the Second Moment of Intensity

TL;DR

This paper investigates the problem of aliasing errors in wavefront measurement techniques and proposes that alias-free bases should be composed of Laplace eigenfunctions, with implications for optical aberration correction.

Contribution

It demonstrates that alias-free bases for second-moment wavefront sensing are formed by Laplace eigenfunctions orthogonal in two dot-products, improving measurement accuracy.

Findings

Alias-free basis functions are eigenfunctions of the Laplace operator.

Numerical simulations show improved aberration measurement accuracy.

Traditional bases like Zernike or Lukosz-Braat may still introduce aliasing errors.

Abstract

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. The article 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 analysed by means of numerical simulations on typical aberrations occurring in microscopy and astronomy.