Two popular methods to analyse the operation of CW CO$_2$ lasers use the temperature model and the rate equation model. Among the two, the latter model directly calculates the population densities in the various vibrational levels connected with the lasing action, and provides a clearer illustration of the processes involved. Rate equation models used earlier grouped a number of vibration levels together, on the basis of normal modes of vibrations of CO$_2$. However, such grouping has an inherent disadvantage as it requires that theselevels be in thermal equilibrium. Here we report a new approach for modelling CW CO$_2$ lasers wherein the relevant vibration levels are identified and independently treated. They are connected with each other through theprocesses of excitation, relaxation and radiative transitions. We use the universally accepted rate coefficients to describe these processes. The other distinguishing feature of our model is the methodology adopted for carryingout the calculations. For instance, the CW case being a steady state, all the rate equations are thus equated to zero. In the prior works, researchers derived analytical expressions for the vibration level population densities, thatbecomes quite a tedious task with increasing number of levels. Grouping of the vibration levels helped in restricting the number of equations and this facilitated the derivation of these analytical expressions. We show that insteady state, these rate equations form a set of linear algebric equations. Instead of deriving analytical expressions, these can be elegantly solved using the matrix method. The population inversion calculated in this manner alongwith the relaxation rate of the upper laser level determines the output power of the laser. We have applied the model to an experimental CW laser reported in literature. Our results match the experimentally reported power.
Volume 96, 2022
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