# Extended SSH Model in Non-Hermitian Waveguides with Alternating Real and Imaginary Couplings
## 4 Topological Edge Modes  
We then analyzed the optical modes in the waveguides with open boundary conditions. The most prominent feature was that the edge modes in the array had no vanished topological invariants, which is known as bulk-edge correspondence. In Figure 4, we plot the propagation constants of all supermodes for topological trivial and nontrivial structures. Figure 4a,b exhibits the real and imaginary parts of propagation constants as \(c_{1}=8\times 10^{-3}\)\(\upmu\)m\({}^{-1}\) and \(c_{2}=4\times 10^{-3}\)\(\upmu\)m\({}^{-1}\), respectively. In this case, the structure is topological nontrivial and can support topological edge modes. One can see the real part of propagation constants has an energy gap, and two zero modes are in the gap. The propagation constants of these two modes are equal to that in a single waveguide and can be regarded as zero modes. Figure 4c,d shows the real and imaginary parts of propagation constants as \(c_{1}=1\times 10^{-3}\)\(\upmu\)m\({}^{-1}\) and \(c_{2}=4\times 10^{-3}\)\(\upmu\)m\({}^{-1}\)respectively. The Berry phase vanishes, and the structure will not allow for topological edge modes. We cannot see zero mode in the gap of spectra of propagation constants. Figure 5 illustrates the field distributions of two topological edge modes. In Figure 5a,b, we plot the distributions of electric field norm \(|\mathbf{E}|\) and component \(E_{x}\) for one topological edge mode. The values of norm of the electric field are positive, while the component of the electric field can be positive and negative. Therefore, we used different colormaps. In Figure 5a, the dark color represents zero valued field, and red color stands for the strong field. The energy is confined at two terminations of the structure and exponentially decreases away from the terminations. The energy at two ends are almost equal. In Figure 5b, the blue color denotes the negative values, and the red stands for positive values. One can see this mode is anti-symmetric with respect to the center axis of the structure. Figure 5c,d shows the distributions of electric field norm \(|\mathbf{E}|\) and component \(E_{x}\) for the other edge mode. The norm of the field is also distributed at two terminations of the structure. One can see this mode is symmetric from the distributions of \(E_{x}\). The above discussed the extended SSH model has chiral symmetry, which can be broken by adding onsite energy. In waveguides, the onsite energy is propagation constants, which can be altered by real and imaginary parts of permittivity. Next, we considered the case by tuning the imaginary part of propagation constants. In each unit cell, the two bare waveguides have alternating detuned propagation constants -\(i\delta\)/2 and \(i\delta\)/2. Other parameters remain unchanged from that used in Figure 4a. Figure 6a plots the propagation constants in complex plane as \(\delta=0\). Blue dots denote the bulk modes, and the red dots represent the edge mode. The two edge modes are of zero energy. Figure 6b shows the propagation constants of all supermodes for \(\delta=0.015\). The imaginary part of propagation constants of edge modes now has the largest absolute value, while the real part of propagation constants remains zero. The mode with positive imaginary propagation constants is decayed, while the mode with negative imaginary is amplified. Therefore, one of the edge modes will dominate after suitable evolution, which is important in many applications. In Figure 6c, we plot Im(\(\beta\)) of the edge mode and the largest value of imaginary propagation constant of bulk mode as a function of detuning \(\delta\). The imaginary part of the edge mode linearly increases with the increase of detuning \(\delta\)  
Figure 4: The propagation constants in finite waveguide arrays for two different coupling strengths. (**a**) and (**b**) are the real and imaginary parts of propagation constants as \(c_{1}>c_{2}\). (**c**) and (**d**) are the real and imaginary parts of propagation constants as \(c_{1}<c_{2}\).