# Extended SSH Model in Non-Hermitian Waveguides with Alternating Real and Imaginary Couplings
## 3 Bloch Mode  
We then used the coupled dielectric waveguide to carry out the above theorical proposal. The main purpose was to work out the parameters desired in adiabatic elimination. As discussed above, the loss added on the mediate waveguide should be extremely large. In practice, the effective system behaved well as \(J>8c_{2}\), and the loss should be \(\delta_{2}>64c_{2}\). We discuss the planar waveguide arrays as illustrated in Figure 1c. The waveguide is periodic in the \(x\) direction, and the light propagated in the \(z\) direction. Every unit comprised three waveguides. Red denotes amplified bare waveguides, and blue denotes the mediate waveguides with loss. The real part of dielectric permittivity of each waveguide is fixed at \(\varepsilon_{0}=12.25\), and the background dielectric is air with \(\varepsilon_{\text{d}}=1\). The wavelength of light is set at \(\lambda=1.55~{}\upmu\)m. The width of all waveguides is constant and fixed at \(w=0.24~{}\upmu\)m. Transverse magnetic polarization is accounted for in this study, and each individual waveguide is determined to sustain a single mode with propagation constant \(\beta_{0}=7.7617~{}\upmu\)m\({}^{-1}\). This can be regarded as the background onsite energy in the Hamiltonian H and will not violate the conclusions. The spatial spacing between bare and assistant waveguides is fixed at \(d_{2}=0.5~{}\upmu\)m. The nearest coupling \(J\) between different waveguides can be extracted from a double-waveguide coupler. Then, we can use Equation (1) to find out the detuned propagation constants for certain imaginary coupling \(c_{2}\). The detuned dielectric permittivity for bare and assistant waveguides can be determined by simulating the propagation constants in a single waveguide. For instance, if the effective imaginary coupling was chosen as \(c_{2}=4\times 10^{-3}~{}\upmu\)m\({}^{-1}\), the imaginary parts of dielectric permittivity in waveguide A(B) and C should be \(-0.0056i\) and \(0.7769i\), respectively. The spacing \(d_{1}\) will be adjusted to satisfy different real coupling strengths. More detail about how to determine the related parameters can be found in our recent studies [25; 37]. We use full wave simulation to calculate the band structures. Figure 3 shows the band structures in the first Brillion zone for periodic boundary condition for several coupling strengths. In all cases, the effective imaginary coupling is fixed at \(c_{2}=4\times 10^{-3}~{}\upmu\)m\({}^{-1}\). The interlayer coupling \(c_{1}\) is modulated by modifying the special spacing \(d_{1}\). There are two energy bands (depicted by red and blue colors) in the structure, corresponding to two different Bloch modes. We define two different modes according to the continuity of complex eigenvalues. For example, in Figure 3e, the real parts of two eigenvalues are always separated and can represent two modes. Then, we plot the corresponding imaginary parts in Figure 3f. The imaginary part may cross each other, but they can be distinguished by their real part and the colours can be determined. The two bands come in pairs with \(\pm\) (\(\beta-\beta_{0}\)) as a result of  
Figure 2: The Berry phase as a function of the ratio of relative coupling strength. The Berry phase of two band structures is equal, which is 0 as \(c_{2}>c_{1}\) and \(\pi\) as \(c_{2}<c_{1}\). chiral symmetry. Both energy bands are complex for all different couplings. This is different from a PT symmetric system where the energy bands can be purely real for special conditions. In Figure 3a,b, the relative coupling strength is \(c_{1}<c_{2}\). In this case, the real part of two energy bands is crossed and the imaginary part is separated. The cross points always arise in the condition of \(\varphi=\pm\,\pi/2\). In Figure 3c,d, the absolute values of intra- and inter-layer couplings are identical. This is the phase transition point. In this case, both the real and imaginary parts of two energy bands coalesce simultaneously at \(\varphi=\pm\,\pi/2\) with \(\beta-\beta_{0}=0\).