## 2 Amplification Properties of Excimers for Long and Intense Short Pulses  
### Contrast Issues for Short-Pulse Amplification in Excimers  
Before going into detail about the correct evaluation of the gain dynamics measurements, which also needs to consider the eventual population of the X state by the saturating pulse, we do believe that the best way is to determine the \(K\) value in the small-signal regime. The \(K\) value is only dependent on the ratio of the accessible molecules in the B state by a weak short pulse; therefore, one has to compare the small-signal gain of the excimers for short and long pulses. From the input-output curves measured for both excimers in [15; 46], small-signal gains of 6500, 450 and 3300 were obtained for short-pulse amplification in _KrF_, _XeCl_ and _XeF_, respectively. These correspond to gain-length products of 8.7, 6.1, and 8.1 in the _KrF_, _XeCl_ and _XeF_ case. If it is compared to the usual gain-length product of ASE-depleted excimer gain modules, whose value is typically around 11 for all cases, as seen in Figure 1 of [52], different \(K\) values are obtained for the _KrF_, _XeCl_ and _XeF_ with \(K_{Krf}=0.76\), \(K_{X\_{\_{\rm{e}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\! Regarding _XeCl_ and _XeF_, the same way of calculations leads to a contradiction; it results in a left-in energy, which is significantly different from that obtained from the small-signal gain measurements. In those cases--connected to the slightly bound feature of the X state--only partial extraction of the \"accessible\" population of the B state can be assumed due to the significant population of the X state, whose balance value--in case of strong saturation--is determined by the ratio of the stimulated transition rates of the B and X states, by the emission (\(\sigma_{e}\)) and absorption cross-sections (\(\sigma_{a}\)). In this way, the remaining population (\(n^{\prime}\)) is the sum of the inaccessible initial population and a \((1-\beta)\) fraction of the \"accessible\" initial population, as  
\[(1-K)n_{2}+(1-\beta)\;Kn_{2}=n^{\prime}{}_{2}, \tag{13}\]  
where  
\[\beta=\frac{\sigma_{e}}{\sigma_{e}+\sigma_{a}}. \tag{14}\]  
In the _XeCl_ case, substituting \(K_{XeCl}=0.53\) (obtained from the gain measurements) to this equation, \(\beta=0.66\) is obtained. This gives  
\[\sigma_{a}/\sigma_{e}=0.5. \tag{15}\]  
From the measurement data obtained by [PERSON] et al. [15] for _XeF_, a similar calculation results in \(K_{XeF}\approx 0.74\), a value that is comparably high to that in _KrF_; however, the energy extraction is limited by the (rapidly populated) bound X state, which only takes a \(\beta_{XeF}=0.28\) fraction, and leaves as high as a \((1-\beta)=0.72\) fraction of the population in the B state, resulting in a ratio of  
\[\sigma_{a}/\sigma_{e}=2.6. \tag{16}\]  
From the gain dynamics of the three excimers, a conclusion, to some extent similar to that of [46], can be drawn: it is only _KrF_ that behaves as a real, four-level system, offering good energy extraction; therefore, it is preferable for the generation of high contrast, short pulses. In _KrF_, beyond the saturation of amplification, the only limitation for energy extraction and temporal contrast is imposed by the partial access to the excited molecules in the B state. However, regarding _XeCl_ and _XeF_, not only the partial access in the B state, but also absorption in the transiently populated X state limits the efficiency and contrast. Their relative contributions are different; in _XeCl_ the former effect, in _XeF_ the latter one are found dominant, as will be shown later. Then, the final ratio of gain coefficients for intense, short pulses and weak, long pulses, called as overall contrast coefficient (\(c^{*}\)), is the product of \(K\) and \(c^{*}\), as defined by Equation (9).