First order optical differentiator based on an FBG in transmission

Optical differentiators constitute a basic device for analog all-optical signal processing [1]. Fiber grating approaches, both fiber Bragg grating (FBG) and long period grating (LPG), constitute an attractive solution because of their low cost, low insertion losses, and full compatibility with fiber optic systems. A first order differentiator LPG approach was proposed and demonstrated in [2], but FBGs may be preferred in applications with a bandwidth up to few nm because of the extreme sensitivity of LPGs to environmental fluctuations [3]. Several FBG approaches have been proposed in [3-6], requiring one or more additional optical elements to create a first-order differentiator. A very simple, single optical element FBG approach was proposed in [7] for first order differentiation, applying the well-known logarithmic Hilbert transform relation of the amplitude and phase of an FBG in transmission [8]. Using this relationship in the design process, it was theoretically and numerically demonstrated that a single FBG in transmission can be designed to simultaneously approach the amplitude and phase of a first-order differentiator spectral response, without need of any additional elements