Editing Laser diode rate equations (section)

==Multimode rate equations==
In the multimode formulation, the rate equations<ref>G. P. Agrawal, "Fiber-Optic Communication Systems", Wiley Interscience, Chap. 3</ref> model a laser with multiple optical [[normal mode|modes]]. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the [[optical cavity]] modes:

:<math>\frac{dN}{dt} = \frac{I}{eV} - \frac{N}{\tau_n} - \sum_{\mu=1}^{\mu=M}\Gamma_\mu G_\mu P_\mu</math>
:<math>\frac{dP_\mu}{dt} = (\Gamma_\mu G_\mu - \frac{1}{\tau_p})P_\mu + \beta_\mu \frac{N}{\tau_r}</math>

where: 
<math>{N}</math> is the carrier density, <math>{P}</math> is the photon density, <math>{I}</math> is the applied current, <math>{e}</math> is the [[elementary charge]], <math>{V}</math> is the volume of the [[Active laser medium|active]] region, <math>{\tau_n}</math> is the carrier lifetime, <math>{G}</math> is the gain coefficient (s<sup>−1</sup>), <math>{\Gamma}</math> is the confinement factor, <math>{\tau_p}</math> is the photon lifetime, <math>{\beta}</math> is the spontaneous emission factor, <math>{\tau_r}</math> is the radiative recombination time constant, <math>{M}</math> is the number of modes modelled, μ is the mode number, and 
subscript <math>{\mu}</math> has been added to <math>{G}</math>, <math>{\Gamma}</math>, and <math>{\beta}</math> to indicate these properties may vary for the different modes.

The first term on the right side of the carrier rate equation is the injected electrons rate (<math>{I/eV}</math>), the second term is the carrier depletion rate due to all recombination processes (described by the decay time <math>{\tau_n}</math>) and the third term is the carrier depletion due to [[Stimulated emission|stimulated recombination]], which is proportional to the photon density and medium gain.

In the photon density rate equation, the first term <math>{\Gamma_\mu G_\mu P_\mu}</math> is the rate at which photon density increases due to stimulated emission (the same term in carrier rate equation, with positive sign and multiplied for the confinement factor <math>{\Gamma}</math>), the second term is the rate at which photons leave the cavity, for internal absorption or exiting the mirrors, expressed via the decay time constant <math>{\tau_p}</math> and the third term is the contribution of spontaneous emission from the carrier radiative recombination into the laser mode.