# Full Characterization of a Molecular Cooper Minimum Using High-Harmonic Spectroscopy
## 2 Materials and Methods
### High-Harmonic Spectral Analysis  
In this section, we outline how to use RABBITT as a spectroscopic probe of the XUV generation gas. At its core, RABBITT is a measurement of the discrete difference in the spectral phase of the photoelectron wavepacket ionized by the XUV comb in the detection gas. The discrete phase difference between the \((q-1)\)th and \((q+1)\)th harmonics appeared as a phase offset in the intensity oscillation of the qth sideband at twice the driving frequency \(\omega_{0}\). \[S_{q}(\tau_{XUV-MIR})=\alpha+\beta\cos\left(-2\omega_{0}\Delta\tau_{XUV-MIR}+ \Delta\varphi_{q}^{XUV}+\Delta\varphi_{q}^{at}\right), \tag{1}\]  
where \(\Delta\tau_{XUV-MIR}\) is the time delay between the XUV and MIR fields, \(\Delta\varphi_{q}^{XUV}=\varphi_{q+1}-\varphi_{q-1}\) is the offset containing the relative phase information between neighboring odd harmonics, \(\Delta\varphi_{q}^{at}\) is the atomic phase contribution of the detection gas, and \(\alpha\) and \(\beta\) are fit constants. The phase of this sideband oscillation was converted to the measured GD using a discrete derivative approximation,  
\[\tau_{q}=\left.\frac{d\varphi(\omega)}{d\omega}\right|_{\omega=q\omega_{0}} \cong\frac{\Delta\varphi(q\omega_{0})}{\Delta(q\omega_{0})}=\frac{\left( \Delta\varphi_{q}^{XUV}+\Delta\varphi_{q}^{at}\right)}{2\omega_{0}}=\tau_{q} ^{XUV}+\tau_{q}^{at}, \tag{2}\]  
due to the \(2\omega_{0}\) spacing inherent to the odd-harmonic comb. The delay contribution from the high-harmonic process alone was then obtained by subtracting the known contributions from the filter, \(\tau_{filt}\), [26] and the detection gas, \(\tau_{q}^{at}=\Delta\varphi_{q}^{at}/2\omega_{0}\)[25; 27]:  
\[\tau_{HHG}=\tau_{q}-\tau_{filt}-\tau_{q}^{at}. \tag{3}\]  
The resulting phases of all sidebands were then concatenated to retrieve the GD of the full harmonic spectrum relative to an offset. In our measurements, we observed that the GDs of molecular targets had a trend similar to that of their atomic counterparts, over which smaller target-specific features were superimposed. This is consistent with quantitative rescattering (QRS) theory [28] and similar factorizations [29], which showed that the complex HHG spectrum signal (i.e., including intensities and phases) can be decomposed into the product of a normalized HHG spectrum, \(HHG_{ref}\), from a reference (atomic) system, an ionization rate, \(\Gamma\), for both the target and reference, and the target's photoelectron scattering cross-section, \(\sigma_{target}\), directly in the spectral domain:  
\[HHG_{target}(q\omega)=HHG_{ref}(q\omega)\times\frac{\Gamma_{target}}{\Gamma _{\text{ref}}}\times\sigma_{target}(q\omega). \tag{4}\]  
Normalizing the spectra removed the contributions of \(\Gamma_{target}\) and \(\Gamma_{ref}\), such that the ratio of the HHG spectral envelopes revealed \(\sigma_{target}\). In Section 3, it was confirmed experimentally that the reference was well approximated by the corresponding semiclassical SFA [28] (e.g., as defined in the Lewenstein model [15]). Since the GD is defined as the energy derivative of the harmonic phases, from the equation above, we deduced that,  
\[\tau_{HHG}=\tau_{ref}+\tau_{target}\approx\tau_{SFA}+\tau_{target}, \tag{5}\]  
where \(\tau_{target}\) is, therefore, the target-specific GD information we wanted to extract. Similar to the filter and detection-gas contributions, knowledge of the generic \(\tau_{SFA}\)--the derivative of which is also known as the \"attochirp\" [20]--would allow one to access the target-specific information. Following the treatment of Lewenstein [15], the attochirp was parameterized using the target IP, laser frequency (\(\omega\)), and laser intensity (\(I_{0}\)). While the first two parameters were known with very good precision, the intensity in HHG focus, as estimated by standard techniques, was too imprecise to be useful in removing the attochirp. Instead, we developed a high-precision algorithm to determine \(I_{0}\), and then proceeded with the attochirp removal.