Tailoring Topological Edge States with Photonic Crystal Nanobeam Cavities

The realization of topological edge states (TESs) in photonic systems has provided unprecedented opportunities for manipulating light in novel manners. The Su-Schrieffer-Heeger (SSH) model has recently gained significant attention and has been exploited in a wide range of photonic platforms to create TESs. We develop a photonic topological insulator strategy based on SSH photonic crystal nanobeam cavities. In contrast to the conventional photonic SSH schemes which are based on alternately tuned coupling strength in one-dimensional lattice, our proposal provides higher flexibility and allows tailoring TESs by manipulating mode coupling in a two-dimensional manner. We reveal that the proposed hole-array based nanobeams in a dielectric membrane can selectively tailor single or double TESs in the telecommunication region by controlling the coupling strength of the adjacent SSH nanobeams in both vertical and horizontal directions. Our finding provides an in-depth understanding of the SSH model, and allows an additional degree of freedom in exploiting the SSH model for integrated topological photonic devices with unique properties and functionalities.

The proposed single nanobeam cavity supports two high Q factor resonance modes in the telecommunication region. The first resonance mode has wavelength of = 1.546 µm with symmetrical field distribution with respect to the = 0 µm plane (Fig. S1a), and the second resonance mode is at wavelength of = 1.624 µm and has antisymmetric field profile along the = 0 µm plane (Fig. S1b). When the vertical spacing of the two coupled nanobeams is as large as 1 = 2 µm, the two resonance modes have localized field at the center of each nanobeam and the modes from the two nanobeams do not couple to each other. When the spacing 1 reducesto 0.6 µm, strong mode coupling occurs, and the two resonance modes are split (Fig. 1e in the main text). The first resonance mode is split into a mode at wavelength of , and a mode at wavelength of , . The mode at , has symmetric (antisymmetric) field profile with respect to the = 0 µm ( = 0 µm ) plane, while the mode at , has symmetric field profile with respect to both the = 0 µm and the = 0 µm planes, as depicted in Fig. S2a-b. The second resonance mode is split into two modes at , and , .   The proposed SSH structure has ten nanobeams with alternatively changed spacing between the successive nanobeams, as indicated by structure outline in Fig. 2f in the main text. With the nearest-neighbor approximation, the Hamiltonian of the finite SSH nanobeams can be given by where is the total number of the nanobeams, and 1 ( 2 ) is the intra-cell (inter-cell) coupling strength. The diagonal terms can be viewed as the detuning with respect to resonance frequency.
By including the obtained coupling strength (see Fig. 1F in the main text) in above equation, we can derive the eigenvalues and the eigenvectors of the proposed SSH nanobeam structures.
The field distribution of the two zero-energy TESs at the first and second resonance mode wavelengths have localized field at edge nanobeams in symmetric or antisymmetric manners, as shown in Fig. S4. Figure 2 in the main text and Fig. S5 demonstrate that our tight binding analysis agrees well with the FDTD results.

C. Topological Robustness of The TESs
An important feature of the topological edge mode is its robustness against disorder. Disorder in the coupling strength and direct modification of the edge site hardly vary the properties of TESs in SSH photonic systesms 1-3 . We consider a local perturbation by introducing the random fluctuation to the intra-cell and inter-cell coupling, as illustrated in Fig. S6a. The Hamiltonian of the perturbed SSH nanobeams is determined by = ( 0 1 (1 + 1 ) 1 (1 + 1 ) 0 2 (1 + 2 ) 2 (1 + 2 ) ⋱ 1 (1 + 9 ) 1 (1 + 9 ) 0 2 (1 + 10 ) 2 (1 + 10 ) 0 ) × . (2) The eigenvalues of the Hamiltonian versus disorder factor are obtained and plotted in Figs. S6b and S6c, which demonstrated that the zero modes are robust against the disorder.

D. Optical Properties of the SSH Nanobeams With the Same Vertical Spacing
Since mode coupling strengthen can be tuned by the horizontal shift of the nanobeams (Fig. 3a), we can tailor TESs with SSH structures having the same vertical spacing but different horizontal shift between the adjacent nanobeams, as demonstrated in Fig. 6 in the main text and Fig. S7.