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.
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