# Attosecond Streaking Time Delays: Finite-Range Interpretation and Applications
## 3 Applications
### Relevance of the First Half-Cycle of Streaking Field  
In an analysis of a RABBITT experiment, it has been recently suggested [26] that the information extracted from the measurement mainly depends on the dynamical interactions during the first half cycle of the IR field after the transition of the electron into continuum at \(t_{i}\). Since it has been shown [8] that RABBITT and streaking measurements provide equivalent information concerning the time delay, the model implies that the observed streaking (or RABBITT) time delay is essentially determined within one half cycle of the streaking field, i.e., according to the finite-range model between \(t_{i}\) and \(t_{i}+T/2\), where \(T\) is the period of the streaking field. Both the classical estimates, Equations (7) and (8), or the analytical formula, Equation (11), allow for testing this assumption used in the model by [PERSON] et al. For example, the streaking time delay can be evaluated as a function of time after transition in the continuum from the expression in Equation (7) by varying the upper limit of the integral between \(t_{i}\) and \(T_{s}\), i.e.,  
\[\Delta t_{s}(t^{\prime}-t_{i})=\frac{1}{E_{s}(t_{i})}\int\limits_{t_{i}}^{t^{ \prime}}E_{s}(t)\left[1-\frac{k_{z}(t)}{k_{0,z}}\right]dt. \tag{16}\]  
Equivalently, in the finite-difference approximations, Equations (8) and (11), the upper limit of the sum can be varied to obtain \(\Delta t_{s}(t^{\prime}-t_{i})\). In the present study, we have considered streaking of the photoionization of the hydrogen atom, applied the analytical formula, Equation (11), and have chosen the time of ionization \(t_{i}\) to coincide with the peak of the streaking pulse. In Figure 5a, we present the accumulated streaking time delay \(\Delta t_{s}\) as a function of time, \(t^{\prime}-t_{i}\), after the transition in the continuum for a streaking wavelength of 800 nm and seven different XUV central frequencies. To guide the eyes and, for further comparison, the first quarter cycle, \(t^{\prime}-t_{i}=T/4=27.5\) a.u. (vertical blue dashed line), and the first half cycle, \(t^{\prime}-t_{i}=T/2=55\) a.u., are also marked (vertical black dashed line), after liberation of the photoelectron. Figure 4: Absolute difference for the different contributions to the time delay of neon atom with and without the effect of the streaking field as a function of electron energy and streaking field wavelength: (**a**) short-range contribution; (**b**) Coulomb phase contribution; and (**c**) Coulomb logarithmic contribution. Parameters of the streaking field: wavelength \(\lambda_{s}=800\) nm, peak intensity \(I_{s}=1\times 10^{12}\) W/cm\({}^{2}\), and pulse duration of three cycles is used [23]. We first discuss the general trend of the results in Figure 5a. It is seen that in each case the streaking time delay oscillates as a function of time before it converges towards its final value. This can be understood as follows: in the region \(0\leq t^{\prime}-t_{i}\leq T/4\), the influence of the Coulomb potential is strong, and, accordingly, there is a sharp increase in the magnitude of the time delay. Beyond \(t^{\prime}-t_{i}=T/4\), the field changes sign and then oscillates until the end of the pulse. Consequently, the accumulated streaking time delay as a function of time oscillates as well. The magnitude of the oscillation decreases since the amplitude of the streaking electric field decreases (reducing the weight of the field-free time delays in Equation (11)) and the Coulomb potential decreases in strength (reducing the magnitude of the field-free time delays in Equation (11)). In order to quantify how fast the time delay is accumulated towards its final value, we have further determined times beyond which the magnitude of the streaking time delay remains within a certain range of its final value.