## 2 Amplification Properties of Excimers for Long and Intense Short Pulses
### Practical Consequences of the Gain Dynamics for Short-Pulse KrF Amplifiers  
Going back to the case of the most practical _KrF_, the relatively low value of \(c^{*}\) suggests a considerably higher gain for the weak ASE than for the intense subpicosecond signal pulse. It means that the temporal filter of the laser system must be used as close to the end of the amplifier chain as possible, or preferably after the amplifier chain. In solid-state systems, even a minor amplification following the temporal filter necessitates the use of CPA (i.e., a subsequent use of a stretcher, an amplifier and a compressor). That is why in solid-state systems the only possible position of an effective temporal filter (usually plasma mirror) is after the CPA amplifier chain. In such cases, a high-throughput operation is absolutely required since no additional amplification can compensate for the energy loss. In contrast to solid-state systems, short-pulse UV laser systems use direct amplification; therefore, the use of a power amplifier following the pulse cleaning element seems possible, since deterioration of the temporal contrast in a power amplifier is moderate compared to that of the CPA scheme. For these reasons, amplification in _KrF_ of a limited \(g_{0}L\) product  
Figure 7: Overall gain contrast coefficient \(c^{*}\) and overall extraction efficiency (\(\eta^{*}\) ) as a function of the normalized energy density (\(c^{*}\) ) in a _XeCl_ amplifier (**a**) and in a _XeF_ amplifier (**b**). following the temporal filter is feasible, without significant deterioration of the contrast. For the calculation of the maximum value of short-pulse amplification, we have to refer to Equation (18) as \(c_{power}^{*}=0.3\) and \(c_{pre}^{*}=0.5\). Assuming a _KrF_ amplifier, which typically allows the development of ASE in a solid angle \(\Omega=10^{4}\) as seen in Figure 1 of [52], one can set a limit of \(g_{0}L\approx 4\), where ASE remains in the low intensity, slowly rising region. This, however, also limits the effective gain length product (for saturated short-pulse amplification) to \(g_{power}L\sim 1.2\), allowing a power amplification of somewhat more than \(G_{power}\sim 3\) for short pulses. This can compensate for the typical energy loss of both plasma mirrors [41] and NFF [34] at this wavelength. The most important features of a temporal filter are the throughput and the contrast enhancement. In a similar manner, a _KrF_ amplifier can best be characterized by its short-pulse amplification (\(G\)) and by the associated deterioration of the contrast (\(D\)). From the former considerations, even \"local\" values (normalized to the unit length) can be defined. For the most important power amplifier case:  
\[G_{power}=e^{g_{power}L}=e^{0.3}s_{0}^{*}L \tag{26}\]  
\[D=\frac{e^{g_{0}^{*}L}}{e^{g_{power}L}}=e^{(g_{0}^{*}-g_{power})L}=e^{0.7}s_{0 }^{*}L. \tag{27}\]  
It is seen from the equations that a _KrF_ power amplifier of a given gain coefficient deteriorates the contrast with a twice larger \"coefficient\". A similar calculation for a _KrF_ preamplifier, using Equation (17), as \(c_{pre}^{*}=0.5\), yields \(G_{pre}=D_{pre}=e^{0.5}s_{0}^{*}L\), i.e., a _KrF_ preamplifier deteriorates the contrast with the same coefficient as it amplifies the short-pulse. For the above detailed reasons, in excimer amplifier systems, additional amplification following the temporal filter is possible, but its gain must be minimized. A good compromise is to set the gain to be somewhat more than necessary to compensate for the loss of the filter. In this case, the deterioration of the contrast caused by the ASE generated by this \"following\" amplifier has minor contribution to the final temporal contrast. Considering the high contrast improvement and high throughput of NFF, such a combination of the filter and the final amplifier offers a much higher final temporal contrast (in excess of \(10^{12}\)) than reported before.