Study on longitudinal modes oscillation of laser diodes based on separate and multi-mode rate differential equations

Rate equations serve as a bridge between microscopic and macroscopic quantities, but it is difficult to establish a direct correlation between them. The steady-state and transient responses of multi-mode laser diodes are modeled based on multi-mode rate equations. Using the Newton iteration method, the intensities of modes of arbitrary order can be simulated, and the carrier and photon densities can be calculated. The spectrum is fitted using a symmetric distribution of different-order modes. In addition, the phenomenon of relaxation oscillation is analyzed theoretically using a numerical approach derived under the large-signal approximation. Under the small-signal approximation, the nonlinear differential equations can be simplified to linear ordinary differential equations, yielding analytical solutions from which the oscillation and damping frequencies of arbitrary orders can be determined. We demonstrate that the system is a Hamiltonian system and present its phase diagram. We reveal a time-independent relationship between the carrier and photon densities. We find that the microscopic \delta factor can be characterized by the spectral width of laser diodes (LDs). The model and calculations are verified experimentally, indicating that the \delta factor is estimated to be approximately 3\times 10^2 under free-running conditions and approximately 5\times 10^4 when the spectrum is locked.