# Attosecond Streaking Time Delays: Finite-Range Interpretation and Applications
## 3 Applications
### Role of Attochirp  
The analytical formula has also been applied to study the effects of properties of the ionizing XUV pulse on the streaking time delay. For example, it has recently been suggested [27] that attosecond time measurements may depend on the chirp of the XUV pulse. This conjecture was related to the presumption that the asymptotic kinetic energy of the photoelectron would depend on the time of transition into the continuum during the interaction with a chirped XUV pulse. To study the effect of the attochirp on the streaking time delay, the expectation value in the chirped laser pulse was determined as [28]:  
\[\langle\Delta t_{s}\rangle(\omega_{0},b)=\int d\omega\int dt_{i}P(\omega, \omega_{0},t_{i},b)\Delta t_{s}(\omega,t_{i})\simeq\int d\omega P_{avg}(\omega,\omega_{0},b)\Delta t_{s}(\omega), \tag{18}\]  
where \(P(\omega,\omega_{0},t_{i},b)\) is the probability distribution for a transition at a certain frequency \(\omega\) at a given time \(t_{i}\); \(\omega_{0}\) is the central frequency of the pulse, and \(b\) is related to the chirp in the spectral distribution of a linearly chirped Gaussian pulse,  
Figure 5: (**a**) Analytical predictions for the streaking time delay (in attoseconds) of hydrogen atom as a function of time \(t^{\prime}-t_{i}\) (in atomic units) for electron kinetic energies corresponding to XUV central frequencies of \(\omega=44.22\) eV (blue), \(\omega=55.27\) eV (black), \(\omega=68.03\) eV (red), \(\omega=82.48\) eV (green), \(\omega=98.64\) eV (yellow), and \(\omega=116.49\) eV (pink) and a three-cycle streaking pulse. The times \(t_{90\%}\) and \(t_{90\%}\) are shown as stars and circles, respectively, and \(T/4\) and \(T/2\) are denoted with dashed vertical lines; (**b**) time \(t_{90\%}\) (in atomic units) is shown as a function of XUV central frequency for different streaking pulse durations; \(N_{s}=3\) cycles (circles), \(N_{s}=4\) cycles (stars), \(N_{s}=5\) cycles (diamonds), and \(N_{s}=6\) cycles (squares). Further shown is \(T/2\) (dashed light blue) and \(T/4\) (solid green); (**c**) same as panel (**b**) but for helium atom and the separate contributions, short-range (circles), Coulomb-phase (squares), logarithmic (diamonds), and the total (stars) in a three-cycle streaking pulse. For all calculations, a streaking pulse of wavelength \(\lambda_{s}=800\) nm and \(I_{s}=1\times 10^{12}\) W/cm\({}^{2}\) has been considered. \[g(\omega,\omega_{0},t,b)=\exp\left[-\frac{(\omega-\omega_{0}-bt)^{2}}{\gamma^{2}/4 ln2}\right]\, \tag{19}\]  
with \(\gamma^{2}=\Gamma^{2}-\frac{\gamma^{2}b^{2}}{4}\), where \(\Gamma=\frac{1}{\tau}\left[64(\ln 2)^{2}+\tau^{4}b^{2}\right]^{1/2}\) and \(\tau\) is the full-width at half maximum (FWHM) pulse duration. Since it has been shown that the streaking time delay does not vary significantly with the time of ionization, in Equation (18), the first expression has been further approximated using the time averaged probability distribution \(P_{avg}\). Using the analytical estimates, Equation (11), in Equation (18) and the time averaged probability distribution for photoionization in a Gaussian pulse [29], the time delay has been evaluated as a function of the XUV central frequency \(\omega_{0}\) and chirp parameter \(b\). From Equation (18), one expects that the influence of the chirp is large when the time delay and/or the photoionization cross section changes rapidly as a function of the laser frequency. It is well known that the cross section varies strongly near the frequency for a Cooper minimum. Indeed, the predictions for the time delay, presented as a difference between delays obtained in a chirped and an unchirped pulse in Figure 6, show that the attochirp has a rather small effect for (a) Ne(\(2p\)) while it is significant in a certain range of frequencies for (b) Ar(\(3p\)). This is in agreement with the fact that the photoionization cross section for neon atom is rather smooth as a function of laser frequency.