A new theoretical basis of higher-derivative optical differentiators

This paper presents a new theoretical basis of higher-derivative finite-impulse response (FIR) optical differentiators for high-speed optical signal processing. A numerical differentiation algorithm, namely, the backward Taylor series expansion and the digital signal processing technique are employed, for the first time, to develop a generalized theory of a pth-derivative FIR optical differentiator. The proposed differentiators are synthesized using a simple integrated-optic transversal filter structure based on the silica-based planar lightwave circuit (PLC) technology, which is compact and compatible with fiber optics. The proposed pth-derivative differentiator only requires p+1 waveguide arms of the transversal filter. By means of computer simulations, the differentiators are shown to be capable of processing Gaussian pulses with high accuracy. We also show the application of a first-derivative differentiator for optical dark-soliton detection and for the generation of an ultrashort optical pulse train. Although the analysis is directed at integrated optical differentiators, the methodology and the results are applicable to other physical systems such as free-space and guided-wave optical signal processors.