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.