Tunable spatial fractional derivatives with graphene-based transmit arrays.

The optical implementation of mathematical spatial operators is a critical step toward achieving practical high-speed, low-energy analog optical processors. In recent years, it has been shown that using fractional derivatives in many engineering and science applications leads to more accurate results. In the case of optical spatial mathematical operators, the derivatives of the first and second orders have been investigated. But no research has been performed on fractional derivatives. On the other hand, in previous studies, each structure is dedicated to a single integer order derivative. This paper proposes a tunable structure made of graphene arrays on silica to implement fractional derivative orders smaller than two, as well as first and second orders. The approach used for derivatives implementation is based on the Fourier transform with two graded index lenses positioned at the structure's sides and three stacked periodic graphene-based transmit arrays in middle. The distance between the graded index lenses and the nearest graphene array is different for the derivatives of order smaller than one and between one and two. In fact, to implement all derivatives, we need two devices with the same structure having a slight difference in parameters. Simulation results based on the finite element method closely match the desired values. Given the tunability of the transmission coefficient of the proposed structure in the approximate amplitude range of [0,1] and phase range of [-180, 180], on top of the acceptable implementation of the derivative operator, this structure allows obtaining other spatial multi-purpose operators, which are a prelude to achieving analog optical processors and even improving the optical studies performed in image processing.