We study an all-optical Fredkin gate consisting of a Mach-Zehnder interferometer and a phase modulator based on silicon nitride ring resonators. Our study proposes an accurate, deterministic, and energy-efficient all-optical Fredkin gate utilizing the Kerr effect for optical computing. We firstly derive the mathematical model of the proposed controlled-swap gate from the Lugiato-Lefever equation taking into account the third-order optical nonlinearity. In addition, the mechanism of thermo-optical effect in the silicon nitride ring cavity is also included. The phase modulator built by the ring resonator with third-order nonlinearity can introduce a π phase shift to the signal when the Fredkin Gate is in the swap mode. However, it does not affect the overall phase shift of the parallel operation. Besides, the pump power introduces a phase shift to the signal by altering the intensity-dependent refractive index inside the micro-ring cavity. It dominates the switching of the Mach-Zehnder interferometer via cross-phase modulation. Moreover, we investigate the conditions of optical bistability for the silicon nitride cavities that can lead to the malfunctioning of the performance of the Fredkin gate. Lastly, we optimize the Fredkin gate performance by the pump power, pump detuning, and signal-pump wavelength difference sweeps.
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