Design of Warped Stretch Transform

Abstract

Time stretch dispersive Fourier transform enables real-time spectroscopy at the repetition rate of million scans per second. High-speed real-time instruments ranging from analog-to-digital converters to cameras and single-shot rare-phenomena capture equipment with record performance have been empowered by it. Its warped stretch variant, realized with nonlinear group delay dispersion, offers variable-rate spectral domain sampling, as well as the ability to engineer the time-bandwidth product of the signal's envelope to match that of the data acquisition systems. To be able to reconstruct the signal with low loss, the spectrotemporal distribution of the signal spectrum needs to be sparse. Here, for the first time, we show how to design the kernel of the transform and specifically, the nonlinear group delay profile dictated by the signal sparsity. Such a kernel leads to smart stretching with nonuniform spectral resolution, having direct utility in improvement of data acquisition rate, real-time data compression, and enhancement of ultrafast data capture accuracy. We also discuss the application of warped stretch transform in spectrotemporal analysis of continuous-time signals.Time stretch dispersive Fourier transform 1-3 addresses the analog-to-digital converter (ADC) bottleneck in real-time acquisition of ultrafast signals. It leads to fast real-time spectral measurements of wideband signals by mapping the signal into a waveform that is slow enough to be digitized in real-time. Combined with temporal or spatial encoding, time stretch dispersive Fourier transform has been used to create instruments that capture extremely fast optical phenomena at high throughput. By doing so, it has led to the discovery of optical rogue waves 4 , the creation of a new imaging modality known as the time stretch camera 5 , which has enabled detection of cancer cells in blood with record sensitivity [6][7][8] , a portfolio of other fast real-time instruments such as an ultrafast vibrometer 9,10 , and world record performance in analog-to-digital conversion 11,12 . The key feature that enables fast real-time measurements is not the Fourier transform, but rather the time stretch. For example, direct frequency-to-time mapping can be replaced by phase retrieval 13 or coherent detection after the dispersion 14 followed by back propagation. Using warped group delay dispersion as a photonic hardware accelerator 15 , an optical signal's intensity envelope can be engineered to match the specifications of the data acquisition back-end [16][17][18] . One can slow down an ultra-fast burst of data, and at the same time, achieve data compression by exploiting sparsity in the original data 19 . Also called anamorphic stretch transform 16,17 , the warped stretch transform performs a nonuniform frequency-to-time mapping followed by a uniform sampler. The combined effect of the transform is that the signal's Fourier spectrum is sampled at a nonuniform rate and resolution. By designing the group delay profile according to the sparsity in the spectrum of...

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