Phenomenological Rate Equations for a Semiconductor Laser

K. Myneni

$\framebox{\it Intensity Rate Equation}$

$\begin{displaymath} {d\vert E\vert^2 \over dt} = \underbrace{G(n)\vert E\vert^2}... ...\vert E\vert^2 \over \tau_p}}_{\mbox{\tiny cavity loss rate}} \end{displaymath}$

(1)

$\begin{displaymath} {d\vert E\vert \over dt} = {1 \over 2} \left( G(n)\vert E\vert - {\vert E\vert \over \tau_p} \right) \end{displaymath}$

$\framebox{\it Linear Gain Approximation}$

$\begin{displaymath} G(n) \approx G(n_{th}) + \left.{\partial G \over \partial n}\right\vert _{n=n_{th}} (n - n_{th}) \end{displaymath}$

$\begin{displaymath} G_N \equiv \left.{\partial G \over \partial n} \right\vert _{n=n_{th}} \end{displaymath}$

$\begin{displaymath} {d\vert E\vert \over dt} = {1 \over 2} \left( G(n_{th}) + G_N(n - n_{th}) - {1 \over \tau_p} \right) \vert E\vert \end{displaymath}$

In the steady state,

$\begin{displaymath}{d\vert E\vert \over dt} = 0 \end{displaymath}$

Therefore,

$\begin{displaymath}G(n_{th}) = {1 \over \tau_p} \end{displaymath}$

$\framebox{\it Amplitude Rate Equation}$

$\begin{displaymath} {d\vert E\vert \over dt} = {1 \over 2} G_N(n-n_{th})\vert E\vert \end{displaymath}$

(2)

$\framebox{\it Fabry-Perot Cavity Mode Frequency}$

$\begin{displaymath} \omega_m = m 2\pi {c \over 2\mu L} \end{displaymath}$

$\begin{displaymath} d\omega_m = m\pi {c \over L} d\left({1 \over \mu}\right) = - m\pi {c \over L\mu^2} d\mu \end{displaymath}$

$\framebox{\it Linear Index Approximation}$

$\begin{displaymath} \mu(n) \approx \mu_f + bn \end{displaymath}$

$\begin{displaymath} b = \left.{\partial \mu \over \partial n}\right\vert _{n=0} \end{displaymath}$

The envelope phase rate equation is given by

$\begin{displaymath} {d \phi \over dt} = \Delta \omega \end{displaymath}$

$\begin{displaymath} \Delta \omega = \omega (n) - \omega (n_{th}) = m \pi {c \over \mu_{th}^2 L} (\mu - \mu_{th}) \end{displaymath}$

Therefore, the mode frequency shift is

$\begin{displaymath} \Delta \omega = r(n-n_{th}) \end{displaymath}$

from which follows:

$\framebox{\it Phase Rate Equation}$

$\begin{displaymath} {d \phi \over dt} = r(n-n_{th}) \end{displaymath}$

(3)

Since, $E = \vert E\vert e^{i\phi}$ ,

$\begin{displaymath} {dE \over dt} = e^{i\phi}\left({d\vert E\vert \over dt} + i\vert E\vert{d\phi \over dt}\right) \end{displaymath}$

Combining the amplitude and phase equations (2) and (3) gives:

$\begin{displaymath} {dE \over dt} = ({1 \over 2}G_N + ir)(n-n_{th})E \end{displaymath}$

$\framebox{\it Complex Field Rate Equation}$

$\begin{displaymath} {dE \over dt} = {1 \over 2}(1 + i\alpha)G_N(n-n_{th})E \end{displaymath}$

(4)

$\begin{displaymath} \alpha = {2r \over G_N} \propto {\left.\left({\partial \mu ... ...t({\partial G \over \partial n}\right)\right\vert _{n=n_{th}}} \end{displaymath}$

$\framebox{\it Carrier Density Rate Equation}$

$\begin{displaymath} {dn \over dt} = \underbrace{{I \over eV_a}}_{\mbox{\tiny inj... ...ce{G(n)\vert E\vert^2}_{\mbox{\tiny stimulated emission rate}} \end{displaymath}$

(5)

$\begin{displaymath} G(n) \approx {1 \over \tau_p} + G_N(n - n_{th}) \end{displaymath}$