From Multimode Near-Field Coupling to Friis

Near-field propagation between finite apertures can support multiple spatial channels, while far-field transmission is effectively single mode and follows the Friis transmission formula. This letter establishes a direct connection between these two regimes through the mutual shadow area between the transmitting and receiving apertures. The mutual shadow determines the asymptotic number of spatial degrees of freedom and, in the paraxial regime, leads to a simple beamforming distance that provides a characteristic scale for the transition from multimode to effectively single-mode propagation. The channel singular-value sum rule further shows that, below this distance, increasing separation primarily reduces the number of supported spatial channels while their average normalized strength remains approximately constant. Beyond the transition, the remaining channel strength decreases according to the conventional free-space quadratic power law, recovering the Friis transmission formula. Numerical results for circular apertures demonstrate the transition, the scaling with aperture size, and the dependence on the transmitter and receiver apertures. The mutual-shadow formulation also extends the description beyond the paraxial regime.

Source: arXiv:2609.18837v1 - http://arxiv.org/abs/2609.18837v1 PDF: https://arxiv.org/pdf/2609.18837v1 Original Link: http://arxiv.org/abs/2609.18837v1