Semiclassical study of the laser transition

A semiclassical, nonlinear theory of a single-mode laser is developed so that the transition around threshold can be studied. The field is expressed in the frequency domain which allows us to emphasize the role of the Fabry-P\'erot cavity. A generalized Airy function is obtained for the laser line shape: it contains the source line shape and the empty cavity line shape. It describes the laser both above and below threshold, including the transition region where there is an abrupt increase of intensity and decrease of the laser linewidth. The general arguments are illustrated by detailed numerical calculations for the 3.39-{μ}m line of the He-Ne laser. The intensity of the spontaneous emission source which plays a central role is computed from first principles.