## 4 Outlook
### Tandem Phase-Shifted Geometry for Infrared Streaking and Attosecond Resolution  
As discussed, to further improve the resolution of the linear THz streaking measurements to the sub-femtosecond level, streaking pulses with a shorter rise time in the IR spectral region must be used. These pulses, however, consist of multiple electric field cycles and are typically not carrier-envelope phase stabilized. As a result, it not possible to control the streaking field amplitude and phase due to the temporal jitter at XFELs. Crucially, without defined streaking parameters it would not be possible to perform a calibrated transformation from energy to time. This problem can be addressed by recording two streaked photoelectron spectra for each X-ray pulse, using a phase-shifted streaking pulse in a tandem streaking geometry as illustrated in Figure 28. As usual, the instantaneous vector potential \(A(t)\) is determined by analysing the shift in the centroid of the dressed photoemission. Qualitatively, due to the fact that the measurements are made with phase-shifted streaking fields, the value of the instantaneous vector potential of one measurement gives the gradient of the field in the other and vice versa, thus providing all of the information required to yield a calibrated measurement of the temporal profile of the photoemission and corresponding X-ray pulse. The instantaneous amplitude and phase of the streaking field can also be combined to determine the cycle-averaged amplitude of the IR streaking laser pulse, which gives the precise overlap between the X-ray pulse and envelope of the streaking laser pulse for use in pump/probe experiments. In practice, the Gouy phase shift can be used to introduce a controlled and fixed phase shift between the two different stages of the streaking measurement required for self-calibration. To achieve a \(\pi/2\) phase shift, streaking measurements would be performed at one Rayleigh length upstream and downstream of the laser focus. A precise \(\pi/2\) phase shift between the measurements allows for full  
Figure 28: Principle of self-calibrated streaking measurements made with multi-cycle infrared fields. In the first stage the X-ray pulse overlaps with a maximum of the vector potential, resulting in a maximal energy shift of the streaked photoelectron spectrum by an amount proportional to the instantaneous vector potential. In the second stage the photoemission occurs at a zero-crossing of the vector potential, resulting in a maximally broadened photoelectron spectrum. The magnitude of the shift in kinetic energy, observed in the first stage measurement, gives the gradient of the vector potential in the second stage measurement. Combined, this information enables a calibrated transformation of the second stage measurement from energy to time. temporal characterization of the X-ray pulse, but the technique is applicable for any phase shift as long as it is constant. For a \(\pi/2\) phase shift the instantaneous streaking fields in each stage of the streaking measurement for an arbitrary overlap \(t_{0}\) between streaking pulse (of arbitrary carrier-envelope offset \(\varphi_{0}\)) and the X-ray pulse are:  
\[A_{1}(t_{0})=a(t_{0})\cos(\omega_{L}t_{0}+\phi_{0})\]  
and  
\[A_{2}(t_{0})=a(t_{0})\cos(\omega_{L}t_{0}+\phi_{0}+\pi/2)=a(t_{0})\sin(\omega_ {L}t_{0}+\phi_{0})\]  
As usual, \(A_{1}(t_{0})\) and \(A_{2}(t_{0})\) would be found by inverting the expression for the energy shift in the streaked photoemission, which is to first order: \(\Delta W(t_{0})\approx-p_{c}A(t_{0})\), compare also Equation (1). Here, \(p_{c}\) is the initial photoelectron momentum. Conceptually, to transform broadened or streaked spectra from kinetic energy to time, the streaking strength or derivative of the instantaneous streaking field must also be known.