## 3 Microwave Photonics Applications of DFT
### DFT for Real-Time Spectroscopy
To overcome the fundamental tradeoff between the interrogation speed and resolution, a new technique to achieve simultaneously ultrafast and ultrahigh-resolution interrogation of FBG sensors based on interferometric temporal spectroscopy has been proposed and experimentally demonstrated [48]. In the system, a  
Figure 3: Schematic diagram to show the real-time spectroscopy based on DFT. PD: photodetector. reference FBG is introduced to produce a temporal interference pattern between two stretched waveforms reflected by the two FBGs. The temporal shift of the optical pulse can then be evaluated from the carrier frequency of the interference pattern, which is usually within microwave frequency range. Therefore, the interrogation resolution is greatly improved due to the facts that an interferometer has an inherently high sensitivity and that the equivalent resolution of an electrical spectrum analyzer is much higher than that of an oscilloscope. Moreover, since the generated interference pattern does not rely on the amount of dispersion, a high measurement speed is maintained, hence eliminating the above mentioned tradeoff. DFT spectroscopy can also be employed in real-time characterization of optical pulses [65]. Ultrashort optical pulses with durations in the picosecond, femtosecond, and even attosecond regimes are powerful tools for studying fast dynamic phenomena in science and engineering. In such applications, fast, and complete (_i.e._, amplitude and phase) characterization of ultrashort pulses is essential for evaluation of optical systems that employ them. However, precise pulse measurement is challenging due to lack of fast enough photodetector to directly capture the ultrafast optical pulses. One widely used method is spectral phase interferometry for direct electric-field reconstruction (SPIDER) [90]. This technique relies on spectral interferometry measurement and its update rate is limited by the refresh rate of the optical spectrometer (typically up to \(\sim\) 10 kHz), hence not capable of real-time single-shot characterization of ultrashort optical pulses. A promising solution is real-time spectral interferometry [54, 91], which combines SPIDER and DFT techniques, hence enabling pulse-by-pulse characterization of the complex field (amplitude and phase) with a significantly higher update rate of up to several GHz. Figure 4: An alternative real-time spectral interferometry system for complete pulse characterization based on unbalanced temporal pulse shaping (UB-TPS). PUT: pulse under test, DE: dispersive element, PD: photodetector. [65]. However, an optical interferometer is required to generate two time-delayed replicas of the original pulse for equivalent spectral shearing. This makes the system not stable due to the inherent high sensitivity of the interferometer to environmental perturbations. To remove this difficulty, in [54,91], an alternative real-time spectral interferometry method without using optical interferometers has been reported [65]. Figure 4 illustrates the principle of the method. An unbalanced temporal pulse shaping (UB-TPS) system [92] is used to simultaneously generate two time-delayed replicas of the input pulse under test and to implement DFT. The system is unbalanced as two dispersive elements (DEs) with opposite dispersion but nonidentical in magnitude are employed. As shown in Figure 4, the UB-TPS system is equivalent to a normal TPS system to perform a real-time Fourier transformation of the microwave driving signal for the generation of two time-delayed replica pulses, followed by a residual DE to stretch the pulses for the implementation of DFT. Therefore, the spectral interferometry can be implemented in the time domain based on DFT technique, enabling continuous single-shot characterization of ultrashort pulses at a high scan rate. More importantly, the system stability is greatly improved as no optical interferometers are needed anymore.