state then an amplification of the input intensity results. The population inversion, in units of atoms per cubic meter, is where g1 and g2 are the degeneracies of energy levels 1 and 2, respectively. Small signal gain equation The intensity (in watts per square meter) of the stimulated emission is governed by the following differential equation: as long as the intensity I(z) is small enough so that it does not have a significant effect on the magnitude of the population inversion. Grouping the first two factors together, this equation simplifies as where is the small-signal gain coefficient (in units of radians per meter). We can solve the differential equation using separation of variables: Integrating, we find: or where is the optical intensity of the input signal (in watts per square meter). Saturation intensity The saturation intensity IS is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain. We can compute the saturation intensity as where is Planck's constant, and is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification. is the frequency in Hz The minimum value of occurs on resonance, where the cross section is the largest. This minimum value is: For a simple two-level atom with a natural linewidth , the saturation time constant . General gain equation The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity I as a function of position z in the gain medium: where is saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensity I and position z: Integrating both sides, we obtain or The gain G of the amplifier is defined as the optical intensity I at position z divided by the input intensity: Substituting this definition into the prior equation, we find the general gain equation': Small signal approximation In the special case where the input signal is small compared to the saturation intensity, in other words, then the general gain equation gives the small signal gain as or which is identical to the small signal gain equation (see above). Large signal asymptotic behavior For large input signals, where the gain approaches unity and the general gain equation approaches a linear asymptote: See also