Tutorial on Electromagnetic Nonreciprocity and its Origins

Loss-induced nonreciprocity

“…When only one loss channel exists ( N = 1), the effective coupling amplitudes for the forward and backward couplings are still the same, i.e., , meaning that a nonreciprocal energy flow does not exist. This is consistent with the conclusion in the previous literature 2 , 3 , in which lossy systems are regarded as reciprocal in terms of restricted time-reversal symmetry. However, when more than one channel exists, the interference between different channels ultimately leads to unequal effective coupling amplitudes for the forward and backward couplings, i.e., for N ≥ 2.…”

“…Although loss breaks the time-reversal symmetry according to the traditional definition of time reversal, it is commonly believed that in classical electrodynamics, only restricted time reversal is valid. In this framework, lossy materials remain lossy under time-reversal transformation and restricted time-reversal symmetry still holds, which implies reciprocity 2 , 3 . Here we demonstrate that nonreciprocity can be achieved by making use of loss combined with multichannel interference.…”

“…Optical nonreciprocity, which prohibits a light field from returning along its original path after passing through an optical system in one direction, implying the breaking of the Lorentz reciprocity theorem, is crucially important for both fundamental studies and applied sciences 1 – 3 . For example, nonreciprocal devices, such as optical isolators 1 , optical circulators 4 , and directional amplifiers 5 , play important roles in optical communication and optical information processing.…”

“…Despite these achievements, the basic principles for realizing optical nonreciprocity remain limited as a result of the time-reversal symmetry and linear nature of Maxwell’s equations. The existing approaches can be grouped into three categories with the following requirements 1 – 3 : (i) magnetic-field-induced breaking of time-reversal symmetry 26 – 29 , (ii) spatiotemporal modulation of system permittivity 30 – 36 , and (iii) nonlinearity 37 , 38 . However, these principles either encounter difficulties in integration 39 , require stringent experimental conditions 40 , or have limited performance 38 .…”

“…Here we devise a new principle for realizing optical nonreciprocity by making use of loss. Although it is obvious that loss breaks time-reversal symmetry, it is generally believed that loss cannot lead to optical nonreciprocity as a result of restricted time-reversal symmetry 2 , 3 , in which the field amplitudes are reduced while the field ratios are conserved. However, we show that loss under multiple channels with interference gives rise to optical nonreciprocity.…”