Arbitrary Conformal Transformations of Wave Functions

In this paper we prove that any conformal transformation of a wave can be produced via a suitably arranged cascade of two, or at most four, discrete phase elements satisfying Laplace's equation. Although this result is of general applicability, in the case of charged-matter waves it implies that such transformations can be exactly obtained by employing only electrostatic or magnetostatic phase elements. Furthermore, we illustrate how a basis for such generating phase elements is given by integer and fractional charge multipoles, proving that these transformations can be used to perform the efficient sorting of multipole-induced quantum states. This provides a fast, compact, and direct method to measure the strength and orientation of dipole systems and of astigmatism. It thus adds a further observable to the four whose spectrum can already be directly measured via spatial separation on the detector, i.e., position, momentum, energy, and orbital angular momentum. The results hold true in optics and for all kinds of charged-particle beams of sufficient coherence.

Automated summary

This paper proves that any conformal transformation of a wave can be produced through a cascade of discrete phase elements satisfying Laplace's equation, with the specific implication for charged-matter waves being the use of electrostatic or magnetostatic phase elements. Additionally, the basis for generating such phase elements is shown to be integer and fractional charge multipoles, allowing for efficient sorting of multipole-induced quantum states. The results hold true in optics and for charged-particle beams of sufficient coherence, providing a fast and direct method for measuring dipole systems and astigmatism.