The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictible zone

TL;DR

This paper derives an analytical solution for the Lorenz-Haken equations describing a single-mode homogeneously broadened laser, identifying conditions for regular, periodic, and chaotic pulsing regimes, and exploring the limits of harmonic expansion methods.

Contribution

It extends harmonic expansion methods to higher harmonics in the Lorenz-Haken model and analyzes the transition to chaos in laser dynamics.

Findings

Periodic solutions occur when the population decay rate ratio is below 0.11.

The system exhibits period-doubling leading to chaos as the pumping parameter increases.

Analytical solutions diverge from numerical results beyond fifth harmonic, indicating method limitations.

Abstract

We have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rates is smaller than 0.11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate "2CP" [4], close to the second laser threshold "2C2th" (threshold of instability). When the pumping parameter "2C" increases, the laser undergoes a period-doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and the fifth order harmonics for the laser-field expansion [4, 14]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.