Fig. 1: S-TAI denoising concept.
a Originally observed in the spatial domain, a Talbot Array Illuminator (TAI) focuses a uniform beam into an array of bright spots21. A TAI consists of a discrete phase mask ϕ(x) applied along the transversal spatial direction x, followed by free-space diffraction, which imposes a continuous quadratic phase φ(kx) along the angular frequency variable kx, the Fourier-conjugate of x26. The involved Fourier transformation is indicated by the symbol ℱ. We show here that the well-known TAI mechanism can be applied on a non-uniform spatial wavefront. b Through a mathematical analogy between the space and temporal frequency domains, a similar process is implemented along the frequency spectrum representation of a temporal waveform (i.e., S-TAI). In this case, the observation domain is along the radial frequency variable ω, and its Fourier-dual domain is along the time variable, t. Thus, the S-TAI can be constructed by a suitable discrete spectral phase filter ϕ(ω), followed by a continuous quadratic temporal phase modulation φ(t), as detailed in the main text. This creates a sampled version of the input spectrum, with peaks of width νs separated by νq = qνs, outlining a copy of the waveform of interest amplified by a factor q. On the other hand, the incoherent noise content (here illustrated as a gray background) is left untouched, thus enabling the recovery of a waveform initially buried under noise. c Experimental realization of the concept for optical waveforms, with the acronyms defined in the text. For convenience, the S-TAI mechanism can be implemented through a continuous quadratic spectral phase transformation (dispersive phase filtering), followed by a periodic discrete temporal phase modulation process (see “Methods”).
