Operation of a two-mode laser in a three-level atomic system with a common upper level

The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables, $n_1$, the photon number of the first mode, and $n_2$, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable ($n_1$ or $n_2$) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter $H$, the equation of motion is simplified and a formal solution can be deduced which is dependent on $H$. It appears that the parameter $H$ cannot be fully determined, but some properties of it can be deduced which prove that the introduction of $H$ is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.