## 2 Amplification Properties of Excimers for Long and Intense Short Pulses
### Contrast Issues for Short-Pulse Amplification in Excimers
It means that, in both the _XeCl_ and _XeF_ cases, instead of solving Equations (1) and (2), one must solve the somewhat different  
\[\frac{\partial n_{2}}{\partial t}=-\frac{I}{h\
u}(\sigma_{e}n_{2}-\sigma_{a}n_{ 1}) \tag{21}\]  
\[\beta=\frac{\sigma_{e}}{\sigma_{e}+\sigma_{a}}. \tag{14}\]0  
coupled differential equations (see for example [53]), which account for transient ground-state absorption, but assume no initial nonsaturable absorption. In these equations, \(n_{2}\) and \(n_{1}\) are the population densities in the upper and lower lasing states, \(\sigma_{e}\) and \(\sigma_{a}\) are the corresponding cross-sections. With regard to the short pulse duration and the slightly bound ground states of _XeCl_ and _XeF_, within the duration of the pulse, no relaxation from the ground state is assumed:  
\[\beta=\frac{\sigma_{e}}{\sigma_{e}+\sigma_{a}}. \tag{14}\]1  
Introducing the \(x=\frac{\sigma_{a}}{\sigma_{e}}\) and the \(N=n_{2}-xn_{1}\) quantities, the differential equations can be simplified to  
\[\beta=\frac{\sigma_{e}}{\sigma_{e}+\sigma_{a}}. \tag{14}\]2  
and  
\[\beta=\frac{\sigma_{e}}{\sigma_{e}+\sigma_{a}}. \tag{14}\]3  
(similarly to [53]). Introducing \(\varepsilon_{sat}=\frac{h\
u}{\sigma_{e}(1+x)}=\frac{h\
u}{\sigma_{e}+ \sigma_{a}}\), one arrives to the well-known Frantz-Nodvik equations [49], whose solution is the same as Equation (3), but instead of \(n_{2}\), one has to consider \(N=n_{2}-xn_{1}\) as the \"weighted\" differential population, moreover the saturation energy density is lowered from \(\varepsilon_{sat}=\frac{h\
u}{\sigma_{e}}\) to \(\varepsilon_{sat}=\frac{h\
u}{\sigma_{e}+\sigma_{a}}\). It means that from the value of \(\varepsilon_{sat}\), obtained by fitting the Frantz-Nodvik curves to the measured input-output characteristics [15; 46], one can define not only \(\sigma_{e}\) (such as in the _KrF_ case), but both \(\sigma_{e}\) and \(\sigma_{a}\), by knowing their ratios from the gain dynamics measurements, as \(\frac{\sigma_{a}}{\sigma_{e}}=0.5\) for _XeCl_ and \(\frac{\sigma_{a}}{\sigma_{e}}=2.6\) for _XeF_ (see Equations (15) and (16)). From this approach, the values listed in Table 1 are obtained. Comparing our value of \(\sigma_{e}=4\times 10^{-16}\)cm\({}^{2}\) for _KrF_ and \(\sigma_{e}=5\times 10^{-16}\)cm\({}^{2}\) for _XeCl_ with that found in the literature (\(\sigma_{e}=2\times 10^{-16}\)cm\({}^{2}\) for _KrF_ in [19], and \(\sigma_{e}=1.25\times 10^{-16}\)cm\({}^{2}\) for _XeCl_ in [54]), the deviations can be explained by the completely different pulse durations used for the measurements in the present and the cited two former publications. We expect the present data to be more accurate, since our values for the emission cross-section are directly derived from the experimentally obtained saturation energy density measured/defined by pulses of relevant (short) pulse duration. Note, that one can hardly obtain data for \(\sigma_{a}\) (X \(\rightarrow\) B) with other methods, since the X state can only be populated for a short time through the B state. In such a case, however, transient absorption can contribute to saturation of amplification; from the former considerations, the overall local efficiency (\(\eta^{*}\)) and the overall local contrast coefficient (\(c^{*}\)) can be calculated as a function of the normalized local energy density (for a given small \begin{table}
\begin{tabular}{c c c c} \hline  & \(\varepsilon_{sat}\) & \(\sigma_{e}\) & \(\sigma_{a}\) \\  & **(mJ/cm\({}^{2}\))** & **(cm\({}^{2}\))** & **(cm\({}^{2}\))** \\ \hline _KrF_ & 2.0 & \(4\times 10^{-16}\) & \(*\) \\ \hline _XeCl_ & 0.85 & \(5\times 10^{-16}\) & \(2.5\times 10^{-16}\) \\ \hline _XeF_ & 0.2 & \(7.9\times 10^{-16}\) & \(2.0\times 10^{-15}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Emission and absorption cross-sections for _KrF_, _XeCl_, and _XeF_ excimers associated with the B \(\rightarrow\) X laser transitions (* has no contribution; therefore, cannot be measured on this time scale). volume of the excimer amplifier) in a similar way as in the _KrF_ case (using Equations (5)-(7)).