Amplitude, Competition, Self-Locking, Beat Frequency, and Time Development in a Three-Mode Gas Laser

The unabbreviated Lamb semiclassical equations for the case of three interacting modes are numerically solved for a variety of laser parameters. Steady-state solutions are obtained for the amplitudes, beat frequencies, and time development of the modes by using the Kutta-Merson method of integration. It is found that for some solutions the relative phase angle $\psi$ becomes constant, and so it is clear under what conditions self-locking is possible. When the modes are unlocked, the value of $\psi$ varies with time even after the steady-state amplitudes have been achieved. Comparison is made with the locking criteria predicted by approaches to the theory where approximations have been made. A physical interpretation is given for the presence or absence of self-locking in terms of competition and of the ratio $\frac{\Delta }{\Delta {\nu }_D}$. Some general rules are established for the behavior of the rise time and delay of the onset of oscillation of the three modes.