  The control of structured waves has recently opened innovative scenarios in
the perspective of radiation propagation and light-matter interaction. In
particular, the transmission of customized electromagnetic fields is
investigated for telecommunications, with the aim of exploring new modulation
formats besides the traditional, almost saturated, division multiplexing
techniques. Beams carrying twisted wavefronts have long been recognized as the
promising candidates, however their phase singularities and efficient
multiplexing still raise open issues. In a more general insight into
structured-phase beams, we introduce and develop here a new and unique paradigm
based on the transmission of beams with harmonic phases having a multipole
structure. The outlined framework encompasses multiplexing, transmission, and
demultiplexing as a whole for the first time, describing wavefields evolution
in terms of conformal mappings, and solving straightforwardly the critical
issues of previous solutions. Because of its potentialities, versatility, and
ease of implementation, we expect this completely new paradigm to find
widespread applications for space division multiplexing especially in free
space, from the optical to the microwave and radio regimes.
