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Rate Equations in a Four-Level Laser. Consider a four-level laser with an active volume \(V=1 \mathrm{~cm}^{3}\). The population densities of the upper and lower laser levels are \(N_{2}\) and \(N_{1}\) and \(N=N_{2}-N_{1}\). The pumping rate is such that the steady-state population difference \(N\) in the absence of the stimulated emission and absorption is \(N_{0}\). The photon- number density is \(\mathrm{n}\) and the photon lifetime is \(\tau_{p}\). Write the rate equations for \(N_{2}, N_{1}, N\), and \(\mathrm{n}\) in terms of \(N_{0}\), the transition cross section \(\sigma(\nu)\), and the times \(t_{\text {sp. }} \tau_{1}, \tau_{2}, \tau_{21}\), and \(\tau_{p}\). Determine the steady- state values of \(N\) and \(\mathrm{n}\).

Rate Equations in a Four-Level Laser. Consider a four-level laser with an active volume \(V=1 \mathrm{~cm}^{3}\). The population densities of the upper and lower laser levels are \(N_{2}\) and \(N_{1}\) and \(N=N_{2}-N_{1}\). The pumping rate is such that the steady-state population difference \(N\) in the absence of the stimulated emission and absorption is \(N_{0}\). The photon- number density is \(\mathrm{n}\) and the photon lifetime is \(\tau_{p}\). Write the rate equations for \(N_{2}, N_{1}, N\), and \(\mathrm{n}\) in terms of \(N_{0}\), the transition cross section \(\sigma(\nu)\), and the times \(t_{\text {sp. }} \tau_{1}, \tau_{2}, \tau_{21}\), and \(\tau_{p}\). Determine the steady- state values of \(N\) and \(\mathrm{n}\).