Fundamentals and theories for the design of an arbitrary fractional-order all-optical differentiator based on a microring resonator.

In this paper, we present a complete theory for designing an all-optical time differentiator that enables the realization of arbitrary differentiation orders using the microring resonator (MRR). Based on this theory, a detailed quantitative analysis of the correspondence between the differentiator orders and the microring parameters is provided. All combinations of the amplitude transmission coefficient and single-pass amplitude transmission can be determined for desired orders. Furthermore, the theory is extended to cascaded rings for high-order differentiators, facilitating the design of arbitrary order differentiators guided by this theory. As an example, a 0.54th-order differentiator is demonstrated, with two parameter sets derived from the theory validated through a FDTD simulation, yielding errors of less than 10%. Additionally, the theory is applied to the design of higher-order differentiators, with the 1.44th order and the 2.1th order used as case studies. The errors are 4.05% and 1.54%, respectively. The proposed theory provides good guidance for designing an arbitrary order differentiator, enabling significant new applications.