Figure 9: Unidirectional reflectionlessness of a non-*PT*-symmetric system consisting of a MDM plasmonic waveguide side coupled to two MDM stub resonators.

(A) Schematic of an MDM plasmonic waveguide side coupled to two MDM stub resonators. (B) Reflection spectra for the structure of (A) calculated for both forward and backward directions using the finite-difference frequency-domain (FDFD) method (solid lines) and the scattering matrix theory (circles). Results are shown for *w*=50 nm, *w*_{1}=20 nm, *w*_{2}=100 nm, *L*_{1}=175 nm, *L*_{2}=365 nm, and *L*=561 nm. Also shown are the reflection spectra calculated using FDFD for lossless metal (blue solid line). (C and D) Real and imaginary parts of the eigenvalues of the scattering matrix *S* as a function of the distance *L* between the two MDM stub resonators. The black and red lines correspond to eigenvalues ${\lambda}_{\text{s}}^{+}=t+\sqrt{{r}_{f}{r}_{b}},$ and ${\lambda}_{\text{s}}^{-}=t-\sqrt{{r}_{f}{r}_{b}},$ respectively. All other parameters are as in (B). (E) Spectra of the generalized power $T+\sqrt{{R}_{f}{R}_{b}}$ (black) and of the differential generalized power (red), defined as the derivative of the generalized power with respect to frequency $d\left[T+\sqrt{{R}_{f}{R}_{b}}\right]/df,$ calculated using FDFD. All parameters are as in (B). (F) Phase spectra of the reflection coefficients in the forward (*r*_{f}, black) and backward (*r*_{b}, red) directions. All parameters are as in (B) [107].