## 3 Sidebands and Their Correlation Analysis
### Evolution of the Correlation Plots  
The amplitudes and phases of the harmonics H\({}_{q-1}\), H\({}_{q}\), and H\({}_{q+1}\) are needed to reconstruct the APTs. The correlation plot of the oscillating components (\(P_{q-1,q},P_{q,q+1}\)) is constructed using the single-shot intensities to extract the phase differences. The amplitudes are extracted from the photoelectron spectra acquired in the absence of the NIR dressing field. The correlation between two random variables refers to how the two signals behave with respect to each other. Using Equation (2), it can be easily seen that the analytical form of the correlation plot of the oscillating components \(P_{q-1,q}\) and \(P_{q,q+1}\) describes an ellipse aligned at an angle to the \(P_{q-1,q}\) axis, with the length of the major and minor axes depending on the phase difference \(\Delta\varphi_{q-1,q,q+1}\), defined as:  
\[\Delta\varphi_{q-1,q,q+1}=\varphi_{q,q+1}-\varphi_{q-1,q}=(\varphi_{q+1}- \varphi_{q})-(\varphi_{q}-\varphi_{q-1}), \tag{3}\]  
where \(\varphi_{q,q+1}\) is the relative phase between the harmonics H\({}_{q+1}\) and H\({}_{q}\). The overall shape of the ellipse varies as the phase difference \(\Delta\varphi_{q-1,q,q+1}\) is changed. When \(\Delta\varphi_{q-1,q,q+1}\) approaches zero, the ellipse collapses to a straight line aligned at an angle \(\tan(\beta)=\frac{P_{q,q+1}}{P_{q-1,q}}\), and at \(\Delta\varphi_{q-1,q,q+1}\approx\pi\), the ellipse collapses to a straight line aligned at an angle \(\tan(\beta)=-\frac{P_{q,q+1}}{P_{q-1,q}}\). The correlation distribution is periodic as a function of \(\Delta\varphi_{q-1,q,q+1}\) with periodicity \(2\pi\), implying the generation of the same correlation distribution by increasing or decreasing the phase by \(2\pi\). Therefore, from the shape of the correlation distribution of the oscillating components, the phase difference \(\Delta\varphi_{q-1,q,q+1}\) can be extracted. By definition, \(\Delta\varphi_{q-1,q,q+1}\) is proportional to the group-delay dispersion (GDD) of the harmonic spectrum. The shape of the correlation distribution is a good indicator of a phase difference, but only qualitatively. To quantitatively understand the variation of the phase difference, the so-called correlation coefficient is introduced. The method of associating the correlation coefficient with the shape of the correlation plots and phase difference extraction was presented in [23; 26]. In Figure 1, the undulator configuration to generate a harmonic comb with three harmonics is presented. In the CoBRA scheme, the phase difference \(\Delta\varphi_{7,8,9}\) is extracted using the evolution of the correlation distributions of the oscillating components. The evolution is achieved by varying the phase difference between the harmonics using either the phase shifter PS\({}_{2}\) or PS\({}_{4}\) (shown in red). The variation in the phase difference \(\Delta\varphi_{7,8,9}\) by the delay introduced by the two phase shifters is given by:  
\[\Delta\varphi_{7,8,9}=\varphi_{9}^{(0)}+\varphi_{7}^{(0)}-2\varphi_{8}^{(0)}+9 \omega_{\text{UV}}\tau_{22}-7\omega_{\text{UV}}\tau_{54} \tag{4}\]  
where \(\varphi_{q}^{(0)}\) is the initial phase of the harmonic H\({}_{q}\) and \(\tau_{si}\) is the delay introduced by the phase shifter PS\({}_{i}\). The evolution of the shape of the correlation distribution of the oscillating components \(P_{7,8}\) and \(P_{8,9}\) as a function of the delay introduced by phase shifter PS\({}_{4}\) for a fixed phase shifter PS\({}_{2}\) (\(\tau_{22}^{*}\)) is shown in Figure 2. A noticeable change in shape can be observed in the different panels. A delay of 13.76 as was added in each step. Assuming a phase difference \(\Delta\varphi_{7,8,9}^{(a)}\) attributed to Panel a, the addition of nine delay steps results in the shape of the correlation distribution in Panel j. The total delay accounts for 123.84 as, corresponding to a phase change of 6.21 rad \(\approx 2\pi\), making the phase difference \(\Delta\varphi_{7,8,9}=\Delta\varphi_{7,8,9}^{(a)}\)-6.21 rad (from Equation (4)).