Fig. 1

Exceptional points enabled by a pair of Mie scatterers embedded in channel waveguides and resonators. a Scanning electron microscope (SEM) image of a channel waveguide (WG) with a lithographically defined rectangular-shaped symmetric Mie-scatterer on SOI substrate. b SEM image of a typical asymmetric Mie scatterer with right triangle shape. c A microring resonator (MRR) based add-drop filter with Mie scatterers. The Mie scatterers (a, b) defined on the ring perimeter control the non-Hermiticity and spectral degeneracy of the MRR state. Depths (D) and widths (W) are first chosen such that the two Mie scatterers have the same reflectance (Additional file 1: Fig. S1). Then the optical path difference ΔФ between the Mie scatterer pair is tuned to bring the system to an EP or DP. (d, e) A system with a pair of identical symmetric or asymmetric Mie scatterers leads only to a diabolic point (DP) spectral degeneracy (see Additional file 1: Section 2 for detailed derivations). f, g A combination of one asymmetric and one symmetric Mie scatterer or two non-identical asymmetric Mie scatterers may be tuned to create Eps (see Additional file 1: Section 2 for detailed derivations). The real and imaginary parts of the eigenmodes are plotted in black and blue, respectively. h Measured asymmetric reflection spectra (port 1 to 4 and port 2 to 3) and (i) transmission spectrum (port 1 to 3) of the add-drop filter with optimized geometric parameters creating an EP. The asymmetry in the reflection spectra for clockwise (CW) and counterclockwise (CCW) inputs in h suggests that the system is at an EP