High-order autocorrelation by a cascade time-lens

High-order temporal autocorrelation functions provide critical insight into the structure and phase of ultrafast signals and are essential in fields ranging from optics to signal processing. In this work, we present a single-shot method for measuring high-order temporal autocorrelation using a cascade time-lens system based on four-wave mixing. We derive the imaging conditions for each cascade order, demonstrating that the system functions as distinct temporal imaging systems, simultaneously, each deriving a different correlation order. Numerical simulations confirm that the cascade outputs represent time-reversed images and high-order temporal autocorrelations of the input signal. Experimental results using double-pulsed input validate our model, showing reconstruction of the high-order temporal autocorrelation. This system enables high-resolution temporal characterization with a single nonlinear component and has applications in ultrafast analysis and optical communication.