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In this paper, we propose and demonstrate an all-optical second-order differentiator based on a two-stage self-coupled optical waveguide on a silicon-on-insulator platform. The transmission spectrum of the fabricated device has a parabola-like filtering notch with a 3-dB bandwidth of up to ~ 100 GHz and a depth of ~ 12 dB. Experiments are carried out for 10-, 20-, and 40-Gb/s optical time-domain multiplexing (OTDM) picosecond pulse trains, and the second-order differentiations are achieved using the fabricated device.