# Optically Injected Solid-State Laser – Simulation

This simulation was performed at the University of Helsinki, Finland by Simo Valling, Thomas Fordell and Asa Lindberg [1,2].

A full description of the equations and parameters used is given in [1]. This is reproduced here:

The experimental results are compared to a phenomenological single mode rate equation model of a class B laser [3,4].

$\frac{{da}}{{dt}} = \left[ {\frac{1}{2}\left( {1 – i\alpha } \right)\frac{{{\gamma _c}{\gamma _n}}}{{{\gamma _s}\tilde J}}\left( {n – 1} \right) – \frac{1}{2}{\gamma _p}\left( {{a^2} – 1} \right) + i\Omega } \right]a + \kappa + {F_a}\\$     (1)
$\frac{{dn}}{{dt}} = {\gamma _s}\left( {1 – n} \right) + {\gamma _s}\tilde J\left( {1 – {a^2}} \right) + {\gamma _n}{a^2}\left( {1 – n} \right) + \frac{{{\gamma _p}{\gamma _s}\tilde J}}{{{\gamma _c}}}{a^2}\left( {{a^2} – 1} \right) + {F_n}$     (2)

where $a$ and $n$ are the amplitudes of the slowly varying field envelope and the population inversion density, both normalized to their steady state values, $\tilde J = \left( {J – {J_{th}}} \right)/{J_{th}}$ is the pump power $J$ normalized to the threshold value $J_{th}$, $\alpha$ is the linewidth enhancement factor, $\gamma _c$ and $\gamma _s$ are the decay rates for the cavity and for the upper laser level, $\gamma _n$ and $\gamma _p$ are the relaxation rates for the differential and nonlinear gain. The experimentally controllable injection parameters $\kappa$ and $\Omega = 2\pi \left( {{\upsilon _{ML}} – {\upsilon _{SL}}} \right)$ stand for the coupling of the injected field and the angular frequency detuning between the master and the slave lasers, respectively. Noise is introduced into the system through the Gaussian white noise source terms $F_a$ and $F_n$.

The parameter values are those given in [5] and shown in the table below.

 Parameter Symbol Value Relaxation oscillation frequency $f_R$ 4.3 MHz Normalized pump power $\tilde J$ 3.3 Cavity decay rate $\gamma _c$ 2.0 x 1010 s-1 Atomic decay rate $\gamma _s$ 1.11 x 104 s-1 Differential gain relaxation rate $\gamma _n$ 3.66 x 104 s-1 Nonlinear gain relaxation rate $\gamma _p$ 0.56 x 106 s-1 Linewidth enhancement factor $\alpha$ 0.2 Noise terms ${F_n},{F_a}$ 0