A two dimensional theoretical model for describing gain coefficient in N2-lasers and the model validity for CVL and excimer lasers

Based on our previously reported measurements on the gain-value in a N₂- laser and numerical calculations, we introduce a method to obtain an analytical expression for the small signal gain, g₀, where the dependency of g₀ on the laser geometrical configuration, including electrodes length and gap separation, is demonstrated. For this study one- and two-dimensional approaches for the photon density have been applied independently to determine gain-parameter, where for explaining the observed dependency of the gain-parameter on the laser electrodes separation, d_AMP, which was found experimentally and explained by an empirical expression of the type g₀ = r + q/d_AMP, with r and q some constants, realization of introducing an extra dimension along the gap separation was unavoidable. For the electrodes length, l_AMP, we have already shown that an empirical equation of the type g₀ = m + n/l_AMP, with m and n some constants, is consistent with the measurements corresponding to N₂-lasers. With this realization, it is proved that the gain-parameter in N₂-lasers can be written as g₀^{above threshold} = m″ + g_0z(γ_L^z, z) + g_0y(γ_L^y, y), where it consists of two independent gain-values along the electrodes length (z) and gap separation (y) with the corresponding power losses given by γ_L^z and γ_L^y. m″ is a very small quantity showing that laser is operating slightly above the threshold. The results of this calculation are consistent with our recent measurements and also other reported N₂-laser gain values measured under moderate current density conditions. To check the validity of the model for other types of lasers, the reported gain-values for copper vapor lasers of different laser tube radii, R_AMP, and tube lengths, l_AMP, have been examined using the one-dimensional model of either g₀(R_AMP) or g₀(l_AMP) and the consistency with the observed measurements was found to be quite satisfactory. The model was also found to be valid for the excimer lasers of different types, different gas mixtures and pressures at a constant input operational voltage. Due to limited numbers of the reported experimental measurements, for the graphs preparation of g₀(l_AMP) in excimer lasers we used the observed data at V₀ = 30 kV and also some variations of the input voltages in the range of ΔV ≅ 20 kV have been adopted. The results for both cases were found to be consistent with the proposed one-dimensional model.