Separation of electrostatic and magnetic phase shifts using a modified transport-of-intensity equation

• We provide a new way to separate electrostatic and magnetic phase shifts in Lorentz microscopy.
• We derive two new transport-of-intensity style equations, one for electrostatic phase shifts and the other for magnetic phase shifts.
• We provide a new way to determine the longitudinal intensity derivative that automatically includes time reversal symmetry.
• This approach allows for the Tikhonov regularization parameter to be selected for each phase shift separately.
• We provide two example application on Permalloy and CoFeB patterned islands.

We introduce a new approach for the separation of the electrostatic and magnetic components of the electron wave phase shift, based on the transport-of-intensity equation (TIE) formalism. We derive two separate TIE-like equations, one for each of the phase shift components. We use experimental results on FeCoB and Permalloy patterned islands to illustrate how the magnetic and electrostatic longitudinal derivatives can be computed. The main advantage of this new approach is the fact that the differences in the power spectra of the two phase components (electrostatic phase shifts often have significant power in the higher frequencies) can be accommodated by the selection of two different Tikhonov regularization parameters for the two phase reconstructions. The extra computational demands of the method are more than compensated by the improved phase reconstruction results.