Programmable Photonic Processor for Discrete 2 Fractional Fourier Transform with 𝜋/8-order 3 Resolution

The fractional Fourier transform (FrFT) plays a crucial role in multidimensional signal processing for applications ranging from synthetic aperture radar to optical encryption. However, conventional implementations-such as those based on bulky 4f lens systems or kilometer-scale fiber arrays-suffer from limitations in reconfigurability and integrability. In this work, we present a programmable discrete FrFT (DFrFT) processor based on a fixed array of basic transformation units (BTUs) interconnected via a dynamically reconfigurable architecture. Leveraging the order additivity of DFrFT, this processor synthesizes arbitrary DFrFT matrices with orders spanning the full 0 to 2<i>π</i> range at <i>π</i>/8-order resolution. Using the inverse design method, four BTUs (<i>π</i>/8, <i>π</i>/4, <i>π</i>/2, <i>π</i>) were designed with fidelity values exceeding 0.995 in simulation. Numerical analysis of assembled high-order DFrFT matrices demonstrates fidelities>0.989 across all 16 transformation orders. Experimental characterization on a silicon photonic platform verified both individual BTU performance (fidelity >0.85) and assembled DFrFT operations (fidelity>0.8), confirming scalable performance for integrated photonic signal processing. Order mapping strategy and fabrication tolerance are further evaluated to confirm the system's modularity, reconfigurability, and robustness. This processor establishes a scalable and programmable platform for integrated optical signal processing, particularly suited to time-frequency transformations and non-stationary signal analysis in future optical computing systems.