## Theory
### Electron Streaking
From Equation (4), the corresponding spectrum is derived, and after rearranging the different terms of the integral in the exponential of Equation (4), we obtain  
\[S(\left|\vec{p}\,\right|,\tau) \propto \left|\int\limits_{-\infty}^{\infty}\mathrm{d}t\,\vec{E}_{X}(t- \tau)\,\vec{d}\left[\vec{p}+e\vec{A}_{S}(t)\right]P(t)\right. \tag{5}\] \[\left.\exp\left[\frac{\mathrm{i}}{\hbar}\left(\frac{\left|\vec{p }\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^{2}.\]  
Since the number of electrons is conserved, the gate, \(P(t)\), is a pure phase gate  
\[P(t)=\exp\left[-\frac{\mathrm{i}}{\hbar}\int\limits_{t}^{\infty}\mathrm{d}t^{ \prime}\,\left(\frac{e\vec{p}\cdot\vec{A}_{S}(t^{\prime})}{m}+\frac{e^{2}|\vec{ A}_{S}(t^{\prime})|^{2}}{2m}\right)\right]. \tag{6}\]The measured electron spectra, \(S(|\vec{p}\,|,\tau)\), can be readily converted to \(S(\omega,\tau)\). When a constant transition matrix element, \(\vec{d}\), is assumed, which is approximately true for streak fields that are not too high, and a linearly polarized ionization pulse, Equation (5) is the spectrogram of the product field, \(E_{X}(t-\tau)P(t)\), and suitable iterative algorithms may be used to reconstruct the slowly varying envelope of the ionization pulse. In the following text, we derive the gate function for an elliptically polarized laser streak field, which includes a linear or a circular polarized streak field as special cases. The electric field of an elliptically polarized streak pulse can be written as  
\[\vec{E}_{S}(t)=E_{0}(t)\frac{1}{\sqrt{1+\varepsilon^{2}}}\left[\hat{x}\cos \omega_{0}t+\hat{y}\varepsilon\sin\omega_{0}t\right] \tag{7}\]  
with the unit vectors, \(\hat{x}\) and \(\hat{y}\), and the ellipticity parameter, \(\varepsilon\). The two special cases are a linear polarized, i.e., \(\varepsilon=0\), and a circular polarized streak pulse, i.e., \(\varepsilon=1\). The corresponding vector potential in the Coulomb gauge reads  
\[\vec{A}_{S}(t)=\frac{E_{0}(t)}{\omega_{0}\sqrt{1+\varepsilon^{2}}}\left[-\hat {x}\sin\omega_{0}t+\hat{y}\varepsilon\cos\omega_{0}t\right]. \tag{8}\]  
Substituting Equation (8) into Equation (6) results in  
\[P(t) = \exp\Bigg{\{}-\frac{\mathrm{i}}{\hbar}\Bigg{[}\frac{\sqrt{8 WU_{p} (t)}}{\omega_{0}\sqrt{1+\varepsilon^{2}}}\hat{p}\cdot\left(\hat{x}\cos\omega_{ 0}t+\hat{y}\varepsilon\sin\omega_{0}t\right) \tag{9}\] \[S(\left|\vec{p}\,\right|)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^ {2}. \tag{3}\]0  
with the momentum unit vector, \(\hat{p}\), the energy of the final continuum state, \(W=|\vec{p}\,|^{2}/(2m)\) and the ponderomotive potential,  
\[S(\left|\vec{p}\,\right|)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^ {2}. \tag{3}\]1  
In angular electron streaking, the streak field has circular polarization, i.e., \(\varepsilon=1\), which simplifies the gate function to  
\[S(\left|\vec{p}\,\right|)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^ {2}. \tag{3}\]2 \[S(\left|\vec{p}\,\right|)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^ {2}. \tag{3}\]3  
Moreover, the time delay, \(\tau\), is mapped onto the azimuthal angle, \(\theta\), through an appropriate arrangement of electron spectrometers. Since the continuously rotating streak field vector completes one full revolution after one oscillation period, the accessible time delay range is \(\tau\in[-\pi/\omega_{0},\dots,\pi/\omega_{0}]\).