Observation of superconducting fluxons by transmission electron microscopy: A Fourier space approach to calculate the electron optical phase shifts and images

An approach is presented for the calculation of the electron optical phase shift experienced by high-energy electrons in a transmission electron microscope, when they interact with the magnetic field associated with superconducting fluxons in a thin specimen tilted with respect to the beam. It is shown 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 this method can be used to recover the result, previously obtained by the real-space approach, relative to the case of a straight flux tube perpendicular to the specimen surfaces. Then the method is applied to the case of a London fluxon in a thin film, where the bending and the broadening of the magnetic-field lines due to the finite specimen thickness are now correctly taken 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 the fluxon can be analyzed within this framework and the main features of electron transmission images predicted.