A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation

The transport of intensity equation (TIE) provides a very straight forward way to computationally reconstruct wavefronts from measurements of the intensity and the derivative of this intensity along the optical axis of the system. However, solving the TIE requires knowledge of boundary conditions which cannot easily be obtained experimentally. The solution one obtains is therefore not guaranteed to be accurate. In addition, noise and systematic measurement errors can very easily lead to low-frequency artefacts. In this paper we solve the TIE by the finite element method (FEM). The flexibility of this approach allows us to define additional boundary conditions (e.g. a flat phase in areas where there is no object) that lead to a correct solution of the TIE, even in the presence of noise.