# Extended SSH Model in Non-Hermitian Waveguides with Alternating Real and Imaginary Couplings
## 1 Introduction  
The Su-Schrieffer-Heeger (SSH) model is an important and basic model in describing band topology in condensed matter physics. It describes spinless fermions hopping on a one-dimensional lattice with staggered hopping amplitudes and provides intuitive ways to understand topological phenomenon such as topological invariant, phase transition, and bulk-boundary correspondence [1; 2]. Recently, the SSH model has been widely investigated in various photonic systems such as optical waveguides [3; 4; 5; 6] and nano particles [7]. The applications of topology in photonic systems are of great importance and offer new possibilities to design novel optical devices [8; 9; 10; 11; 12]. The photonic systems also provide a flexible platform for the SSH model, as the hopping of adjacent lattices can be controlled as desired by tuning distance via nanofabrication. Numerous extended models have been proposed to reveal further topological properties of the SSH model [13; 14; 15; 16; 17; 18; 19]. Anomalous \(\pi\) modes are observed in coupled microwave waveguides by periodically modulating coupling strength [17]. Multiple topologically protected, bound states are theoretically indicated if the unit cell of the arrays is composed of multiple lattices [16]. The two-dimensional SSH model is able to retain nontrivial topological phase in the absence of Berry curvature and further can support second-order topological bound modes at its corners [18; 19].  
Another important extension is to examine the SSH model in non-Hermitian systems [20; 21; 22; 23]. In contrast to Hermitian systems, the eigen energies may not be real, and the eigenstates become nonorthogonal [24; 25; 26; 27; 28]. Both eigen energies and eigenstates coalesce at the singularity of a non-Hermitian system, known as the exceptional point [24]. Interesting phenomena are found at the exceptional point such as giant Goos-Hanchen shifts [29], topological mode switching [30; 31; 32; 33],and strong absorption [34]. In non-Hermitian systems, the topological properties are proven to be closely related to exceptional points and exhibit exceptional physics that has not manifested in Hermitian systems [20]. Non-Hermiticity can be introduced by adding onsite gain and loss at different lattices or by utilizing non-Hermitian hopping. Many fascinating and unusual topological phenomena are uncovered in the non-Hermitian SSH model. For example, all states, including bulk and edge sates, tend to locate at the boundary of the system, which is known as \"skin effect\" [20]. Topological lasing is experimentally realized in the systems with particle-hole symmetry [35]. The topological modes become extended in the whole system when the imaginary gauge potential is introduced [36]. When hopping is transformed into imaginary values, the SSH model is anti-parity-time-symmetric, and the topological phase transition would associate with the imaginary band structures rather than its real part [37].  
In this work, we studied another extension of the non-Hermitian SSH model with alternating real and imaginary couplings. Imaginary coupling is accomplished by indirectly interacting two waveguides via a damped waveguide, which is known as adiabatic elimination. The technique was recently employed to realize both negative and imaginary couplings [38; 39], and subsequently interesting phenomena are revealed such as synchronized oscillation [35] and extremely localized states in an Aharonov-Bohm cage [40]. Here, we use coupled mode theory to analyze the band structure and calculate its topological invariant in binary arrays. Furthermore, we also present full wave simulation by using planar optical waveguides to carry out the extended SSH model and numerically investigate the topological bound modes. The topological aspects with and without chiral symmetry are discussed in detail.