# Two-Color Operation of a Soft X-ray FEL with Alternation of Undulator Tunes
## 4 Simultaneous Measurements of Two Colors  
### Intensity Measurement  
As it was mentioned, the two colors are not generated independently, so that one can not measure their intensities by turning off one color and measuring the other one (and vice versa). Moreover, in case of user experiments, one should have online nondisruptive measurements. Thus, one of the goals of our studies was to develop methods for such measurements. It is obvious that we can determine pulse energies of each of the X-ray beams using two linear detectors having a different response for the different wavelengths. In this case, we have a system of two linear equations with two unknowns and can easily retrieve pulse energies at \(\lambda_{1}\) and \(\lambda_{2}\). We have four of such detectors available in the FLASH2 tunnel and experimental hall: two gas monitor detectors (GMDs) for measurements of absolute pulse energy [24], the on-line photoionization spectrometer (OPIS) for non-invasive wavelength measurements [25], and a micro-channel-plate (MCP) detector for pulse energy measurements with a large dynamic range [26]. Note that, in the following, we refer to ensemble-averaged pulse energies throughout the text. As an example, let us consider the measurement we did with the tunnel GMD and OPIS. Note that both devices are placed in the tunnel next to each other, and there are no propagation effects that can be different for the two X-ray beams. In the time-of-flight spectra of the OPIS, relative intensities of the two X-ray colors can be determined since the signals of the respective photoelectrons are separated in arrival time due to different kinetic energies. From these signal intensities, a ratio of number of photons of the two wavelengths can be evaluated. Thus, we have the first linear equation:  
\[N_{2}=pN_{1}\, \tag{1}\]  
where \(N_{1}\) and \(N_{2}\) are unknown average photon numbers, and \(p\) is the measured coefficient. If the tunnel GMD is set to the measurements of pulse energy at \(\lambda_{1}\), it will show spurious pulse energy \(\mathcal{E}_{gmd}\), when the second color is present:  
\[\mathcal{E}_{gmd}=\hbar\omega_{1}\bigg{(}N_{1}+N_{2}\frac{\sigma_{2}\gamma_{2 }}{\sigma_{1}\gamma_{1}}\bigg{)}. \tag{2}\]  
Here, \(\sigma_{1,2}\) are the photoionization cross sections and \(\gamma_{1,2}\) are the mean charges for a given gas and the two photon energies (see, e.g., [27]), \(\omega_{1}=2\pi c/\lambda_{1}\). Solving Equations (1) and (2) for \(N_{1}\) and \(N_{2}\), we obtain the final result for the actual pulse energies \(\mathcal{E}_{1}=\hbar\omega_{1}N_{1}\) and \(\mathcal{E}_{2}=\hbar\omega_{2}N_{2}\):  
\[\mathcal{E}_{1}=\frac{\mathcal{E}_{gmd}}{1+p\frac{\sigma_{2}\gamma_{2}}{ \sigma_{1}\gamma_{1}}}\quad,\quad\quad\mathcal{E}_{2}=p\mathcal{E}_{1}\frac{ \omega_{2}}{\omega_{1}}\.\]  
In a similar way, one can obtain simple expressions for the combination of the MCP-based detector and the tunnel GMD. The MCP detector can be cross-calibrated with the GMD at a certain wavelength, say \(\lambda_{1}\). When two X-ray beams are present, it shows a linear combination of two actual pulse energies: \(\mathcal{E}_{mcp}=\mathcal{E}_{1}+s\mathcal{E}_{2}\). It is interesting to note that, in the wavelength range of our experiments, we had \(s\simeq 1\). Combining this equation with Equation (2), we obtain the actual pulse energies. We also did the measurements with two GMDs, one in the tunnel and one in the experimental hall. They have to be filled with different gases; then, Equation (2) can be also used for the second GMD but with the different constants \(\sigma\) and \(\gamma\). Moreover, we should correct the photon numbers \(N_{1}\) and \(N_{2}\) in that equation for the beamline transmission that has to be measured once for each wavelength individually. Then, we have again the system of two equations from which we obtain actual pulse energies.