Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings

In exact analogy with their electronic counterparts, photonic temporal integrators are fundamental building blocks for constructing all-optical circuits for ultrafast information processing and computing. In this work, we introduce a simple and general approach for realizing all-optical arbitrary-order temporal integrators. We demonstrate that the N(th) cumulative time integral of the complex field envelope of an input optical waveform can be obtained by simply propagating this waveform through a single uniform fiber/waveguide Bragg grating (BG) incorporating N pi-phase shifts along its axial profile. We derive here the design specifications of photonic integrators based on multiple-phase-shifted BGs. We show that the phase shifts in the BG structure can be arbitrarily located along the grating length provided that each uniform grating section (sections separated by the phase shifts) is sufficiently long so that its associated peak reflectivity reaches nearly 100%. The resulting designs are demonstrated by numerical simulations assuming all-fiber implementations. Our simulations show that the proposed approach can provide optical operation bandwidths in the tens-of-GHz regime using readily feasible photo-induced fiber BG structures.