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Photonic Hilbert transformers are fundamental components for optical computing and signal processing. Here, for the first time we propose all-optical arbitrary-order Hilbert transformers using phase-modulated fiber Bragg gratings (PM-FBGs) in transmission to our best knowledge. The PM-FBG is a kind of fiber grating, whose coupling strength remains almost uniform and period varies along the fiber length. For demonstration, we have designed and numerically simulated 0.5th-order, first-order, and 1.5th-order photonic Hilbert transformers, respectively. The profiles of those PM-FBGs are obtained employing quasi-Newton optimization algorithm. Simulation results show that the designed three Hilbert transformers are all in good agreement with the ideal results in bandwidths up to 500 GHz and can tolerate a large range of input pulse width.