No abstract

Instability is an inevitable and common problem in all different kinds of lasers when they are oscillating in both single-and multi-mode states. Here, the stability conditions are investigated for a three-mode class-A laser. A set of linear equations is derived for the stable oscillation of the cavity central mode together with its left and right adjacent longitudinal modes. The coefficient determinant of stability equations is Hermitian and equal to zero for the roots of two diagonal arrays. In other words, the novelty of our work is to expand the stability coefficient determinant in terms of main diagonal arrays rather than for one row or one column. These diagonal roots lead to two lower and upper boundary curves in the form of a bifurcation. The lower boundary curve mimics the single-mode laser and delimits the instability region (with no above-threshold oscillating mode) from the bistability region (with two above-threshold oscillating modes). The upper boundary curve mimics the two-mode laser and delimits the bistability region from the stability region, in which all three-longitudinal modes are simultaneously oscillating in the above-threshold state.

The random oscillations of many longitudinal modes are inevitable in both class -A and -B lasers due to their broadened atomic bandwidths. The destructive superposition of electric field components that are incoherently oscillating at the different longitudinal modes can be converted into a constructive one by using the mode-locking technique. Here, the Maxwell-Bloch equations of motion are solved for a three-mode class-B laser under the mode-locking conditions. The results indicate that the cavity oscillating modes are shifted by changing the laser pumping rate. On the other hand, the frequency components of cavity electric field simultaneously form the various bifurcations. These bifurcations satisfy the well-known mode-locking conditions as well. The atomic population inversion forms only one bifurcation, which is responsible for shaping the cavity electric field bifurcations.