Optical Time-Mapped Spectrograms (I): From the Time-Lens Fourier Transformer to the Talbot-Based Design

The short-time Fourier transform (STFT), or spectrogram (SP), is the prime method for joint time-frequency signal analysis and processing. Real-time implementation of this powerful tool over instantaneous bandwidths above the GHz range remains challenging. We propose here a universal analog optical processing approach to obtain the STFT of a high-speed temporal waveform (typically, a microwave signal) in a continuous and real-time manner, and with no gaps in the signal acquisition and analysis. The proposed method is based on a photonics time-mapped Fourier transformer, involving temporal modulation of the signal under test (SUT) with a chirped optical pulse (i.e., a time lens) followed by group-velocity dispersion, in which consecutive, overlapping chirped pulses (or time lenses) are utilized for realization of a continuous and gap-free STFT analysis. We derive the design conditions and performance trade-offs of this general scheme, especially concerning its instantaneous bandwidth, time-frequency resolutions and processing speed (number of FTs per second). Moreover, we also show that a previous, simpler time-mapped STFT design based on Talbot effects, in which the SUT is directly sampled with unchirped pulses before dispersion, can be interpreted and evaluated as a particular case of the time-lens STFT scheme introduced here. This suggests a way of implementing an array of overlapping time lenses by simply sampling the SUT with a suitable periodic train of short pulses. The time-lens STFT scheme offers a remarkable versatility to tailor the specifications of the obtained SP, whereas the Talbot design is particularly interesting for SP analysis with enhanced time resolution and ultrahigh processing speed. Our findings are validated through numerical simulations. The proposed method would enable performing the STFT of signals over instantaneous bandwidths above a few tens of GHz, with MHz frequency resolutions and processing speeds exceeding hundreds of millions of FTs per second using realistic fiber-optics technologies.