Reconstruction of the magnetization in a thin film by a combination of Lorentz microscopy and external field measurements

A combination of two measurements is shown to be sufficient to determine the magnetization in a thin film except for a constant out-of-plane component, under the condition that the magnetization is uniform through the thickness. The measurements consist of DPC (differential phase contrast) Lorentz microscopy, which measures the path integral of the in-plane B field, and some measurements of the out-of-plane component of H above and below the film, for example by Hall probe. Field measurements on one side only are sufficient if the magnetization is in-plane. The in-plane magnetization is decomposed into a divergence-free part and a part with zero component of curl perpendicular to the plane of the film. The DPC measurement directly yields the divergence-free component, and the curl-free as well as the out-of-plane parts are found from a deconvolution of the perpendicular field. Several examples are given, and the interpretation of the two magnetization components in terms of B and H provides a qualitative description of the physics