On the Design of Efficient and Accurate Arbitrary-Order Temporal Optical Integrators Using Fiber Bragg Gratings

In this paper, we propose and numerically investigate a simple and practical all-fiber design for implementing first-order and higher order all-optical passive temporal integrators with optimized energetic efficiencies. The proposed solution is based on a high-reflectivity fiber Bragg grating (FBG) providing a reflection spectral response that approaches the frequency transfer function of a time-limited Nth-order optical integrator (N = 1, 2, 3 ...). A closed-form analytical expression has been derived for the frequency response to be targeted for implementing an optical integrator of any given integration order operating over a prescribed limited time window. The required grating profile can then be designed using a layer-peeling FBG synthesis algorithm. Our simulations show that for a sufficiently long FBG, a relatively smooth amplitude-only apodization profile is required for any desired integration order even when an FBG peak reflectivity > 99% is targeted. The resulting FBG integrators can provide at least a sixfold increase in energetic efficiency as compared with previously proposed FBG designs while offering a similar or superior performance in terms of processing accuracy. We estimate that ultrafast highly efficient arbitrary-order all-optical temporal integrators capable of accurate operation over nanosecond time windows could be implemented using readily feasible, centimeters-long FBGs.