Root-locus analysis of exceptional points in coupled-resonator networks

Recently, the engineering of exceptional points (EP) in optical and electrical systems becomes a multi-disciplinary field in physical sciences, because phase transitions often occur near such points. In this paper, we indicate that the engineering of EPs can be treated as a root-locus problem that is well-know in control theory. We formulate two root-locus problems that arise from the tuning of coupled-resonator networks, which are interesting in that the characteristic polynomial may have complex-number coefficients or is quadratic in the root-locus parameter. Observing that EPs in the system correspond to the break-in and break-out points of the root locus, we analyze the geometry of the root locus and, for three-resonator networks, we derive all possible networks that possess at least one EP. These results provide a systematic way of designing coupled-resonator networks with desired EPs.