Optical Time-Mapped Spectrograms (II): Fractional Talbot Designs

Real-time implementations of joint time-frequency analysis over instantaneous bandwidths above the GHz range remain challenging. In a companion paper, we have proposed an analog photonic processing scheme that enables computing a short-time Fourier transform (STFT), or spectrogram (SP), of an incoming arbitrary broadband signal, over tens-of-GHz analysis bandwidths, in a continuous, gapless and real-time manner. The proposed method involves a temporal sampling of the signal under test (SUT) with a periodic train of interfering, linearly chirped optical pulses followed by group-velocity dispersion to map the spectra of consecutive and overlapping truncated sections of the SUT along the time domain. This scheme offers a notable design versatility to customize the performance specifications of the computed SP, but it generally involves a sub-optimal non-uniform sampling of the SUT and it requires the use of a bulky and expensive pulsed optical source. In this communication, we show that this previous general scheme can be easily configured to ensure an optimal, uniform sampling of the SUT by simply setting the involved dispersive lines to satisfy a fractional self-imaging condition, while keeping all the advantages (e.g., design versatility) of the original scheme. Moreover, the resulting design is further adapted to entirely avoid the need for a pulsed source, using instead a more efficient and simpler phase-only temporal sampling of the SUT, e.g., implemented through electro-optic phase modulation. We derive the design conditions and performance trade-offs of the proposed time-mapped STFT schemes based on dispersion-induced fractional Talbot self-imaging. Through numerical simulations and experimental demonstration, we confirm the potential of this simple and efficient approach for real-time SP analysis of arbitrary signals over instantaneous bandwidths above a few tens of GHz, with MHz frequency resolutions and ultrahigh processing speeds, approaching billions of FTs per second.