## 3 Linear Streaking Measurements at Free-Electron Lasers
### Measurement resolution  
The temporal resolution in THz streaking of X-ray pulses depends primarily on the energy resolution of the photoelectron spectrometer and the THz streaking strength. In addition, the initial FEL pulse bandwidth and the evolution and Gouy phase advance of the THz pulse in the interaction region can also limit the temporal resolution. Moreover, shot-to-shot intensity fluctuation of the streaking pulse, as well as energy jitter in the FEL pulse can degrade the temporal resolution of the measurement. Typically, the streaked photoelectron spectrum is recorded with a time-of-flight (TOF) spectrometer. The energy of the photoelectron is determined by measuring the time t required  
Figure 15: Mapping THz streaked spectrum to temporal profile. When the streaked photoelectron spectrum (blue line) is significantly broader than the initial spectral bandwidth of the FEL photon pulse (blue filled area), the temporal profile (violet filled area) of the ionizing FEL pulse can be easily retrieved. The red curve indicates the streaking vector potential used as a transformation map, to obtain temporal information from a measurement in the spectral domain. (Figure from [95]). by the electron to travel from the interaction region to the detector placed at a known distance \(L\). The kinetic energy of the electron is given by:  
\[E_{\mathrm{kin}}=\frac{1}{2}m_{e}\frac{L^{2}}{t^{2}} \tag{5}\]  
From this simple kinematic equation, a measure of the energy resolution \(\Delta E_{\mathrm{kin}}\) can be written:  
\[\Delta E_{\mathrm{kin}}=\frac{2\sqrt{2}}{\sqrt{m_{e}}}\frac{\Delta t}{L}E_{ \mathrm{kin}}^{\frac{3}{2}}, \tag{6}\]  
where \(\Delta t\) is the smallest interval that can be resolved by the electronic detection system. The acceptance angle of the TOF spectrometer is defined by its geometry (length of the drift tube and detector diameter). In order to enhance the sensitivity, the TOF is equipped with an electrostatic lens at the entrance of the drift tube, which increases the effective acceptance angle for electrons within a specific kinetic energy range. Additionally, a post acceleration stage in the TOF ensures a flat response of the electron detector (multi-channel plate, MCP) regardless of the initial electrons' kinetic energy. Wide acceptance angle, and flat detector response is especially important for measurements at FELs, as the full spectrum of photoelectrons must be collected in a single shot. High efficiency through a larger acceptance angle is usually accompanied by a decrease of the energy resolution due to variations in the electron flight path for electrons of same energy, but with different initial momentum. Moreover, the need to collect single-shot spectra, means that many photoelectrons must be generated during each exposure which can lead to space charge and additional degradation of the energy resolution. Beyond these fundamental factors, the relationship written above for \(\Delta E_{\mathrm{kin}}\) indicates that the energy resolution of the TOF is limited by the length of the drift tube and the temporal resolution of the detection electronics. The latter one includes the response time of the MCP electron detector and the resolution of the analog-to-digital (ADC) converter [106]. To improve the energy resolution, retardation potentials can be applied to increase the flight time of the electrons, before they enter the drift tube. At FELs, the relative energy resolution of electron TOF spectrometers is usually not much better than \(\sim\)1%. The dominant issue lies in the temporal resolution of the detection electronics for multiple photoelectron acquisition. The issue is illustrated in Figure 16. In the first panel, the signal resulting from a single electron response is illustrated. The single electron response is amplified and a pulse with a characteristic duration determined by the capacitive properties of the MCP amplifier is produced.