On-Chip Optical Feedback Systems for Solving Systems of Ordinary Differential Equations

In this paper, we propose a scalable configuration of all-optical solver for systems of ordinary differential equations (SODE) based on the integrated feedback system, which can be easily extended to solve SODEs with various dimensions and forms. We use different optical ordinary differential equation (ODE) solvers to construct different ODEs and then introduce optical feedback waveguides to realize the coupling of them, thus forming the SODE solvers. For a proof of concept, we design and fabricate a two-dimensional (2-D) and a four-dimensional (4-D) SODE solver based on the silicon-on-insulator platform. The corresponding expressions of the SODEs are deduced based on the temporal coupled mode theory. In addition, aiming at the 2-D SODE solver, we also use transfer matrix method to examine the correctness of the SODE and estimate the working bandwidth at the same time. At last, a system experiment is carried out to get the solutions of the SODEs when we use 10-GHz non-return-to-zero (NRZ)-like pulses as input signals, and all the measured output waveforms agree well with the calculated results.