Editing Laser diode rate equations (section)

==The modal gain==
<math>{G_\mu}</math>, the gain of the <math>\mu</math><sup>th</sup> mode, can be modelled by a parabolic dependence of gain
on wavelength as follows:

:<math> G_\mu = \frac{\alpha N [1-(2\frac{\lambda(t)-\lambda_\mu}{\delta\lambda_g})^2] - \alpha N_0}{1 + \epsilon \sum_{\mu=1}^{\mu=M}P_\mu}</math>

where:
<math>{\alpha}</math> is the gain coefficient and ε is the gain compression factor (see below). <math>{\lambda_\mu}</math> is the wavelength of the <math>\mu</math><sup>th</sup> mode, <math>\delta\lambda_g</math> is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by

:<math>\lambda(t)=\lambda_0 + \frac{k(N_{th} - N(t))}{N_{th}}</math>

where <math>\lambda_0</math> is the centre wavelength for <math>{N = N_{th}}</math> and k is the spectral shift constant (see below). <math>N_{th}</math> is the carrier density at threshold and is given by

:<math>N_{th}=N_{tr} + \frac{1}{\alpha\tau_p\Gamma}</math>

where <math>N_{th}</math> is the carrier density at transparency.

<math>\beta_{\mu}</math> is given by
:<math>\beta_\mu=\frac{\beta_0}{1+(2(\lambda_s-\lambda_\mu)/\delta\lambda_s)^2}</math>

where

<math>\beta_{0}</math> is the spontaneous emission factor, <math>\lambda_s</math> is the centre wavelength for spontaneous emission and <math>\delta\lambda_s</math> is the spontaneous emission FWHM. Finally, <math>\lambda_{}</math> is the wavelength of the <math>\mu</math><sup>th</sup> mode and is given by

:<math>\lambda_\mu=\lambda_0 - \mu\delta\lambda + \frac{(n-1)\delta\lambda}{2}</math>

where <math>\delta\lambda</math> is the mode spacing.