Observation of superconducting fluxons by transmission electron microscopy: A Fourier space approach to calculate the electron optical phase shifts and images - art. no. 054507

An approach is presented for the calculation of the electron optical phase shift experienced by high-energy electrons in a transmission electron micro scope, when they interact with the magnetic field associated with supercond ucting fluxons in a thin specimen tilted with respect to the beam. It is sh own that by decomposing the vector potential in its Fourier components and by calculating the phase shift of each component separately, it is possible to obtain the Fourier transform of the electron optical phase shift, which can be inverted either analytically or numerically. It will be shown how t his method can be used to recover the result, previously obtained by the re al-space approach, relative to the case of a straight flux tube perpendicul ar to the specimen surfaces. Then the method is applied to the case of a Lo ndon fluxon in a thin film, where the bending and the broadening of the mag netic-field lines due to the finite specimen thickness are now correctly ta ken into account and not treated approximately by means of a parabolic fit. Finally, it will be shown how simple models for the pancake structure of t he fluxon can be analyzed within this framework and the main features of el ectron transmission images predicted.