Fig. 2
Optical impedance matching towards EP. a Complex reflectivity of a Mie scatterers embedded channel waveguide with forward (rf) and backward (rb) excitation. The symmetric rectangular shaped Mie scatterers with dimensions D = 200 nm, W = 20 ~ 400 nm have the same rf (light blue solid curve) and rb (blue dashed curve). An asymmetric (right triangle) Mie scatterer with dimension D = 200 nm and W = 140 nm has different rf (black curve) and rb (gray). Distance L between the Mie scatterers is varied in the range L = 0 ~ λ. b Asymmetric reflectivity of a channel waveguide with two asymmetric (i.e., right triangle) Mie scatterers. Blue curves: DAsym’ = 160 nm, WAsym’ = 20 to 400 nm. Black and grey curves: DAsym = 200 nm, WAsym = 140 nm. The optical path length between the input point and the scatter varies from 0 to λ. (c,d), The real (black dashed curve) and imaginary (blue dots) parts of the eigenvalues versus optical path length ΔФ (c) between a symmetric and an asymmetric Mie scatterers, and (d) between two nonidentical asymmetric Mie scatterers. W of the symmetric Mie scatterer in c is taken as 80 nm, and that of the asymmetric Mie scatterer in d is taken as 240 nm. The dimensions of the Mie scatterers used in c and d are chosen from the intersection points of reflectivity plots in a and b. EP: exceptional point; AC: avoided crossing. e Complex reflectivity of a channel waveguide with an embedded symmetric Mie scatterer (red: forward scattering; blue: backward scattering) and that of a channel waveguide with surface roughness (black: forward scattering; gray: backward scattering). The Mie scatterer depth D is fixed at 80 nm, and its width W is varied from 20 to 400 nm. The reflectivity trajectory of the Mie-scatterer overlaps with the reflectivity of the channel waveguide with sidewall surface roughness (\({L}_{c}\) = 100 nm, \(\sigma =10 nm\)) at W = 180 nm. Inset: Magnitude of the complex frequency splitting \(\pm \left|\omega \right|\) versus D of the Mie-scatterer with W = 180 nm defined on the perimeter of the MRR. EP emerges at D value where \(\pm \left|\omega \right|=0\). (f) \(\pm \left|\omega \right|\) versus W of the Mie-scatterer with D equals to 30 nm (black), 80 nm (red), 120 nm (blue) and 160 nm (light blue)