# Attosecond Streaking Time Delays: Finite-Range Interpretation and Applications
## 2 Finite-Range Interpretation  
### Analysis of Streaking Time Delays Using Classical Electron Dynamics  
It has been shown (e.g., [19; 20]) that predictions for the streaking time delay determined by solving the classical dynamics of the electron ionized by the XUV field and streaked by the IR field are in good agreement with results from ab initio solutions of the time-dependent Schrodinger equation (TDSE) simulating a streaking measurement. For further analysis of the physical meaning of the time delay, one may therefore consider the solution of the Newton equation for the propagation of the photoelectron in the streaking field after its transition in the continuum:  
\[\frac{d\mathbf{k}(\mathbf{t})}{dt}=-\mathbf{E}_{s}(t)-\
abla V(\mathbf{r}). \tag{3}\]  
In Equation (3), \(\mathbf{k}(t)\) represents the momentum of the photoelectron, \(\mathbf{E}_{s}\) is the streaking field, and \(V(\mathbf{r})\) is the Coulomb potential of the residual atom (in the single-active-electron approximation). For a linearly polarized streaking field the interaction with the streaking field takes effect along the direction of the laser polarization, say \(z\)-direction, and the corresponding component of the final momentum \(k_{f,z}\) can be written as:  
\[k_{f,z}(t_{i})=\sqrt{k_{0,z}^{2}-2\int\limits_{t_{i}}^{T_{s}}E_{s}(t)k_{z}(t) dt}=k_{0,z}-A_{s}(t_{i}+\Delta t_{s})\, \tag{4}\]  
where \(k_{0,z}\) is the field-free asymptotic momentum, \(t_{i}\) is the time of transition of the photoelectron in the continuum, \(A_{s}\) and \(T_{s}\) are the vector potential and duration of the streaking field, and \(\Delta t_{s}\) is the streaking time delay, as in Equation (1). Assuming that the streaking field is weak, the vector potential in Equation (4) can be expanded as:  
\[A_{s}(t_{i}+\Delta t_{s})\simeq A_{s}(t_{i})-E_{s}(t_{i})\Delta t_{s}, \tag{5}\]  
and the streaking time delay can be written as (for \(E_{s}(t_{i})\
eq 0\)):  
\[\Delta t_{s}\simeq\frac{\sqrt{k_{0,z}^{2}-2\int\limits_{t_{i}}^{T_{s}}E_{s}(t) k_{z}(t)dt}-k_{0,z}+A_{s}(t_{i})}{E_{s}(t_{i})}. \tag{6}\]This expression can be further simplified by noting that the momentum shift is usually small if the streaking field is weak, and expanding the square root in Equation (6) to first order to obtain (for \(E_{\rm s}(t_{i})\
eq 0\)) [20; 21]:  
\[\Delta t_{\rm s}\simeq\frac{1}{E_{\rm s}(t_{i})}\int\limits_{t_{i}}^{T_{\rm s}} E_{\rm s}(t)\left[1-\frac{k_{z}(t)}{k_{0,z}}\right]dt. \tag{7}\]  
Although Equation (7) already provides the interpretation of the streaking time delay as a finite-range time delay, it has been sometimes found instructive to also write it using a finite-difference approximation (for \(E_{\rm s}(t_{i})\
eq 0\)):  
\[\Delta t_{\rm s}\simeq\sum_{j=1}^{N}\frac{E_{\rm s}(t_{j})}{E_{\rm s}(t_{i})} \left[1-\frac{k_{z}(t)}{k_{0,z}}\right]\delta t\simeq\sum_{j=1}^{N}\frac{E_{ \rm s}(t_{j})}{E_{\rm s}(t_{i})}\Delta t_{\rm ff}^{(j)}\, \tag{8}\]  
in which one assumes that the streaking field and the electron momentum are approximately constant for the short time intervals \([t_{j},t_{j}+\delta t]=[t_{j},t_{j+1}]\) while propagating over the short distance \([r_{j},r_{j}+\delta r]=[r_{j},r_{j+1}]\), i.e., \(E_{\rm s}(t)\simeq E_{\rm s}(t_{j})\) and \(k(t)\simeq k(t_{j})\). In the last part of Equation (8), we identified the term in the square brackets times \(\delta t\) to be approximately equal to the instantaneous field-free time delay \(\Delta t_{ff}^{(j)}\), which the electron accumulates in the potential \(V\) as compared to the propagation of a free particle. The field-free time delays can be also written as  
\[\Delta t_{\rm ff}^{(j)}=\Delta t_{\rm ff}(k_{j+1},r_{j+1})-\Delta t_{\rm ff}(k _{j},r_{j}), \tag{9}\]  
where \(\Delta t_{ff}(k,r)\) is the field-free time delay accumulated during the propagation from the origin of the transition into the continuum to location at a _finite_ distance \(r\) with momentum \(k=k(r)=\sqrt{2(E_{asym}-V(r))}\) and \(E_{asym}\) is the asymptotic kinetic energy of the photoelectron.