Measurement of multipole phases in optics and electron microscopy by means of conformal transformations

Conformal mappings have been recently rediscovered as a practical solution to measure physical quantities in an efficient manner by implementing a unitary transformation on structured wavefields. For instance, one of the most known is the log-pol mapping, enabling the measurement of orbital angular momentum of optical and electron vortices. We report our latest research on a new family of conformal mappings, the circular-sector transformations, applied to wave-functions endowed with multipole phases, showing disruptive applications in optics and matter-waves physics, in particular electron microscopy. The results suggest an innovative and promising method to measure astigmatisms and electric/magnetic dipoles in a fast and direct way.