the solid-state lasers. Have be concentration of active centers in the ground state. Have be concentration of excited centers. Have . The relative concentrations can be defined as and . The rate of transitions of an active center from ground state to the excited state can be expressed with and The rate of transitions back to the ground state can be expressed with , where and are effective cross-sections of absorption at the frequencies of the signal and the pump. and are the same for stimulated emission; is rate of the spontaneous decay of the upper level. Then, the kinetic equation for relative populations can be written as follows: , However, these equations keep . The absorption at the pump frequency and the gain at the signal frequency can be written as follows: , . Steady-state solution In many cases the gain medium works in a continuous-wave or quasi-continuous regime, causing the time derivatives of populations to be negligible. The steady-state solution can be written: , The dynamic saturation intensities can be defined: , . The absorption at strong signal: . The gain at strong pump: , where is determinant of cross-section. Gain never exceeds value , and absorption never exceeds value . At given intensities , of pump and signal, the gain and absorption can be expressed as follows: , , where , , , . Identities The following identities take place: , The state of gain medium can be characterized with a single parameter, such as population of the upper level, gain or absorption. Efficiency of the gain medium The efficiency of a gain medium can be defined as . Within the same model, the efficiency can be expressed as follows: . For the efficient operation both intensities, pump and signal should exceed their saturation intensities; , and . The estimates above are valid for a medium uniformly filled with pump and signal light. Spatial hole burning may slightly reduce the efficiency because some regions are pumped well, but the pump is not efficiently withdrawn by the signal in the nodes of the interference of counter-propagating waves. See also Population inversion Laser construction Laser science List of laser articles List of laser types References and notes External links Gain media Encyclopedia of Laser Physics and Technology Laser science