Fig. 1. Summary of our proposed approach. A discrete SUSY transformation is applied to a set of N coupled quantum oscillators (for demonstration, we take N=3). The resultant partner network made of N−1 elements exhibits a subset of the spectrum of the original system. By populating both quantum networks with multiple bosons (2 bosons in the example shown here), we can construct classical arrays that exhibit partial spectral overlap.

Fig. 2. Optical implementation of the SUSY arrays example shown in Fig. 1 using a waveguide platform. The left panel shows the original array while right panel shows the BD-SUSY partner obtained as described in the text. The waveguides are all identical, having an elliptic geometry with main/minor diameters of 12 and 6 μm, respectively. The core and cladding refractive indices are taken to be ncore=1.461 and nclad=1.46, respectively [24,25]. Each waveguide supports only one optical mode for each polarization direction. Finally, the distances shown in panel (a) are: d1=23.765  μm, d2=18.275  μm, and d3=16.190  μm. These design parameters result in the following coupling coefficients: κ12=14.143  m−1, κ23=13.141  m−1, and κ24=10.000  m−1. The second order nearest next neighbor coupling is found to be below 10% of the above values. Similarly, in panel (b) we have: d4=22.425  μm and κ12=20.012  m−1. Note that we list the above values with high precision as per our numerical simulations; however, in practice the weakly guiding nature of the structure provides reasonable robustness against fabrication tolerance.

Fig. 3. Eigenmode structure of the waveguide arrays shown in Figs. 2(a) and 2(b) are depicted in the left and right panels, respectively (obtained by full-wave finite element simulations). The figures also indicate the values of the associated propagation constants as measured from the isolated waveguide value, i.e., Δβ=βm−βo (in units of m−1), where βm,o are the propagation constants of array mode m and the isolated waveguide mode, correspondingly. As anticipated from the coupled mode analysis, modes ② and ⑤ in the main array have no partner modes in the BD-SUSY partner array. Moreover, mode ② of the partner array corresponds to two degenerate states in the main array. These results confirm the feasibility of our approach for building quasi-2D supersymmetric optical systems.

Fig. 4. Light propagation dynamics in a waveguide array formed by introducing a weak coupling between the main structure and its partner, as shown in (a)–(e). When mode ① of the main array is excited, we observe an efficient optical power transfer to the partner array after a propagation distance corresponding to z=4π (in units of meter). On the other hand, if mode ② of the main array is excited, no appreciable power transfer to the partner array is observed (not shown here). The power transfer efficiency between the modes is illustrated in (f) where near perfect transfer is observed. The blue and red lines are the total power in the main and pattern arrays, respectively.