## 1 Introduction  
Amplified ultrashort laser pulses are in widespread use for the study of numerous phenomena and have provided important insights across many disciplines. Whether a study involves single-shot or multi-shot excitation of an effect, pulse-train stability is essential for optimal performance in any application. In other words, such measurements require consistency not only in the pulse energy but also in the _pulse shape_, i.e., its intensity and phase evolution during the pulse. For instance, high harmonic generation, which servesas the foundation for many important disciplines, including attosecond light sources [1], ultrafast optoelectronics [2], and attosecond spectroscopy [3; 4; 5], demands high stability in all pulse characteristics. Moreover, spectroscopic methods [6; 7; 8; 9; 10; 11; 12; 13] use ultrashort pulses to enable the exploration of fundamental properties of matter, such as transient photo-induced phenomena, electronic structure, the dynamics of bound and free electrons, quantum coherence, and quantum spin, all of which require pulse-train stability to prevent fluctuating excitations of the sample with each pulse, especially when nonlinear dependencies or small temporal or spectral variations in the ultrashort pulse are being investigated [14]. The generation of ultrashort pulses at exotic wavelengths requires pump lasers, i.e., optical parametric oscillators and amplifiers, whose stability is especially consequential as the output pulses are nonlinearly dependent on the properties of the input light. Stable pump lasers are necessary to achieve consistent supercontinuum using optical fibers, in particular, hollow-core fibers [15; 16], and for applications in high-intensity (>TW) ultrashort pulses, especially as the repetition rates and average power of these systems continue to increase [17]. Indeed, the medical [18; 19; 20; 21; 22] and industrial [23; 24] domains now rely on amplified ultrashort laser pulse trains for surgical procedures, imaging, and micro-fabrication, including ultrashort pulse ablation manufacturing [25; 26; 27], selective laser-induced etching [28], and powder bed alloy fusion [29; 30]. Furthermore, existing short-pulse petawatt-class lasers have now demonstrated new high-intensity laser-matter interactions that lead to secondary sources of high-energy photons [31], neutrons [32], and charged particles [33; 34; 35], with applications to medical imaging [36], cancer radiotherapy [37], multi-modal radiography [38] and tomography [39], and even fast ignition for inertial fusion energy [40; 41]. Looking into the future, the realization of the necessary secondary source flux for such applications will require petawatt lasers with high average power (100 s kW) and increasing repetition rates (>kHz [42; 43]). Over the last few years, high intensity (>10\({}^{18}\) W/cm\({}^{2}\)), higher repetition rate (10-50 Hz) laser systems have been commissioned [44; 45] and are now online [17; 46], with 10 kHz high-average-power petawatt systems now designed [47]. A central need for these high average power systems as their repetition rates approach the MHz level is rapid characterization and feedback [48] to ensure the highest quality pulse trains and the most stable performance. In all the above applications, laser pulse-shape stability is crucial to ensure reliable secondary source generation. Unfortunately, amplified ultrashort laser pulses are often plagued by instability, which can be due to a number of factors, including thermal fluctuations, unstable pump sources, inconsistent mode locking, vibrational pointing jitter, and turbulence in the surrounding air, to name only a few such culprits. The resulting instability tends to be more prevalent in laser systems with higher energies and shorter pulses, and it is especially pronounced in cutting-edge laser systems that advance ultrashort-pulse technology, such as few-cycle-pulse systems and lasers at atypical wavelengths.