### Effect of Saturated Amplification in KrF, in the Presence of Nonsaturable Absorption  
It is known from former studies [15; 46] that, from the point of view of energy extraction, _KrF_ is a superior amplifying medium for short-pulse amplification compared to other excimers. Even this medium equal amplification for a short signal pulse and the (long) ASE in _KrF_ cannot be realized for two reasons. One reason is associated with the fact that _KrF_ is an amplifying medium, exhibiting saturable amplification with considerable nonsaturable absorption [1; 47; 48]. Amplification in such a medium can be described by the  
\[\frac{\partial n_{2}}{\partial t}=\frac{I}{h\
u}\sigma_{e}n_{2} \tag{1}\]  
\[\frac{\partial I}{\partial z}+\frac{1}{\
u}\frac{\partial I}{\partial t}=I \sigma_{e}n_{2}-I\alpha \tag{2}\]  
coupled differential equations [47; 48], where \(I\) is intensity of the short-pulse, \(\sigma_{e}\) is stimulated emission cross-section, \(n_{2}\) is the population density of the upper lasing level (accessibleby a short pulse), \(\alpha\) is the absorption coefficient, and v is the velocity of light in the active medium. Without the \(l\alpha\) quantity, these equations are the well-known Frantz-Nodvik equations [49], which describe the amplifications of short pulses in an ideal four-level system, whose analytical solution is  
\[\varepsilon_{out}^{*}=\ln\Bigl{[}1+e^{\delta_{0}L}\Bigl{(}e^{\varepsilon_{in}^{* }}-1\Bigr{)}\Bigr{]}. \tag{3}\]  
Here, \(g_{0}L=\int_{0}^{L}\sigma_{e}n_{2}(z)dz\), where \(g_{0}\) is the small-signal gain coefficient for a (weak) short pulse, \(L\) is the length of the medium, \(\varepsilon_{out}^{*}\) and \(\varepsilon_{in}^{*}\) are the normalized output and input energy densities, defined by the  
\[\varepsilon_{out}^{*}=\frac{\varepsilon_{out}}{\varepsilon_{sat}}=\frac{\int I _{out}(t)dt}{\varepsilon_{sat}}\text{ and }\varepsilon_{in}^{*}=\frac{\varepsilon_{in}}{ \varepsilon_{sat}}=\frac{\int I_{in}(t)dt}{\varepsilon_{sat}} \tag{4}\]  
equations, where \(\varepsilon_{out}\) and \(\varepsilon_{in}\) are the energy densities of the output and incoming pulses, respectively, and \(\varepsilon_{sat}=h\
u/\sigma_{e}\) is the saturation energy density (see later). Solving (1) and (2), the amplification properties of \(KrF\) can be derived. For such a medium, both the extraction efficiency (\(\eta\)) and the contrast coefficient (\(c\))--the ratio of the effective gain coefficients for an \"energetic\" and a weak signal--deteriorate rapidly with increasing energy density when approaching saturation. This feature can be followed in Figure 4, taken from [1, 48], where the local efficiency [48]  
\[\eta(\varepsilon)=\frac{1}{g_{0}}\frac{d\varepsilon}{dz}, \tag{5}\]  
and the \(c\) contrast coefficient [48]  
\[c(\varepsilon)=\frac{g_{eff}}{g_{0}-\alpha} \tag{6}\]  
are shown as functions of the normalized energy density (\(\varepsilon^{*}\)). Figure 4: Local extraction efficiency \(\eta\) (dashed line) and contrast coefficient \(c\) (solid line) as functions of the normalized energy density \(\varepsilon^{*}\) in a \(KrF\) amplifier. The optimum operations, with regard to both efficiency and contrast for a preamplifier and a power amplifier, are marked by the arrows \(X\) and \(Y\), respectively (based on [48]). The \"effective gain coefficient\" [48] is defined as  
\[g_{eff}=\underset{\Delta L\to 0}{\lim}\ln(\varepsilon_{out}/\varepsilon_{in})/ \Delta L. \tag{7}\]  
The optimum operation with regard to both efficiency and contrast for a preamplifier is marked by the arrow \(X\). Here, the contrast coefficient is below 0.6. On the other hand, a power amplifier primarily must be optimized for efficiency. This condition is marked by the arrow \(Y\) in the Figure 4, where the contrast coefficient falls to below 0.4.