Lang-Kobayashi モデル
Lang-Kobayashi 方程式
\begin{align}
\frac{dE}{dt} &= \frac{1+i\alpha}{2} \left(G-\frac{\kappa}{\tau_p}\right)E
+ \frac{\kappa}{\tau_{in}} E(t-\tau_D) e^{-i\theta}
+ \sqrt{\frac{C_sN}{\tau_s}} \xi \\
\frac{dN}{dt} &= J - \frac{1}{\tau_s} N - G |E|^2 \\
& G \equiv \frac{G_0(N-N_0)}{1+\epsilon|E|^2}
\end{align}
- $\xi$ : white Gaussina noise with zero mean and unitary variance;
the effect of quantum noise of sponteneous emission.
- $\alpha$ : the linewidth enhancement factor
- $G_0$ : the differential gain
- $\epsilon$ : the gain saturation coefficient
- $\tau_{in}$ : the propagation time in the DFB laser
- $\tau_D$ : the delay time
- $\theta$ : the delay phase shift
- $\tau_s$ : the carrie life time
- $\kappa$ : the feedback strength
- $N_0$ : the transparent carrier density
- $C_s$ : the spontaneous emission factor
- $\tau_p$ : the photon life time
Lang-Kobayashi 方程式の導出
Nonlinear Dynamics of Semiconductor Lasers Subject to
Optical Feedback, Fabien ROGISTER (学位論文)を参考にした。
レート方程式
共鳴条件
The forward and backward propagation power
\begin{align}
& {\cal P}_f(z) = {\cal P}_{f_0}\exp[gz - \alpha_sz] \\
& {\cal P}_b(z) = {\cal P}_{b_0}\exp[g(L-z) - \alpha_s(L-z)]
\end{align}
$g$ : gain due to stimulated emission
$\alpha_s$ : optical loss \\
The amplitudes of the forward and backward traveling complex electric fields
\begin{align}
& {\cal E}_f(z) = {\cal E}_{f_0}\exp\left[-i\frac{n\omega}{c}z + \frac{1}{2}(gz - \alpha_sz)\right] \\
& {\cal E}_b(z) = {\cal E}_{b_0}\exp\left[-i\frac{n\omega}{c}z + \frac{1}{2}(g(L-z) - \alpha_s(L-z))\right]
\end{align}
$c$ : the velocity of the light
$n$ : real part of the reflective index
$\nu$ : optical frequency
$\omega=a\pi\nu$ : optical angular velocity \\
The imaginary part of the index of refraction
\begin{align}
n' = \frac{c}{2\omega} (g-\alpha_s)
\end{align}
\begin{align}
& {\cal E}_{f_0} = r_1 {\cal E}_{b}(0) \;\;\;\;\;
\mbox{and} \;\;\;\;\;
{\cal E}_{b_0} = r_2 {\cal E}_{f}(L) \\
& r_1 r_2 \exp\left[-2i\frac{n\omega}{c}L + (g-\alpha_s)L\right] = 1
\end{align}
$r_1$ and $r_2$ are the reflection coefficients at the laser facets \\
the round trip gain
\begin{align}
{\cal G} = r_1 r_2 \exp\left[-2i\frac{n\omega}{c}L + (g-\alpha_s)L\right]
\end{align}
\mbox{}\\
\underline{the gain $g_{th}$ at laser threshold $|{\cal G}|=1$}
\begin{align}
g_{th} = \alpha_s + \frac{1}{2L}\ln\left(\frac{1}{R_1R_2}\right) \label{eqn:gth}
\end{align}
$r_1$ and $r_2$ are real and $R_1=r_1^2$ and $R_2=r_2^2$
\mbox{}\\
\underline{The resonance condition for the phase (the Fabry Perot cavity)}
\begin{align}
\frac{2\pi\omega}{c}L = 2\pi m
\end{align}
\begin{align}
d\left(\frac{nL\omega}{m\pi c}\right)
=
\frac{L}{\pi c}\left( \frac{n}{m}d\omega + \frac{\omega}{m}dn - \frac{n\omega}{m^2}dm \right) = 0
\;\;\;\;\;
dm = 1
\end{align}
the group refractive index
\begin{align}
n_g = \frac{c}{v_g} = c \df{k}{\omega} = c \df{}{\omega}\left(\frac{n\omega}{c}\right)
= n + w\df{n}{\omega} = \frac{n\omega}{m}\frac{1}{d\omega}
\end{align}
the mode spacing frequency
\begin{align}
\delta \nu = \frac{\delta \omega}{2\pi} = \frac{n\omega}{2\pi n_gm}
= \frac{c}{2n_gL}
\end{align}
The roundtrip time of a mode with frequency $\nu=\omega_m/2\pi$
\begin{align}
\tau_{in} = \frac{2n_gL}{c} = \frac{1}{\delta\nu} \label{eqn:tauin}
\end{align}
\underline{the round trip gain operator ${\cal G}$ for the time dependent electric field}
${\cal G}$ acts differently on each frequency
the dependence of the refractive index $n$
on both the carrier number $N$ and the optical frequency $\omega$
\begin{align}
\frac{n\omega}{c}
&\simeq
\left.\frac{n\omega}{c}\right|_{th}
+ \frac{1}{c} \left.\pd{n\omega}{\omega}\right|_{th} (\omega-\omega_{th})
+ \frac{1}{c} \left.\pd{n\omega}{N}\right|_{th} (N-N_{th}) \\
&=
\frac{n_{th}\omega_{th}}{c}
+ \frac{n_{g th}}{c} (\omega-\omega_{th})
+ \frac{\omega_{th}}{c} \left.\pd{n}{N}\right|_{th} (N-N_{th})
\end{align}
\begin{align}
{\cal G} &\equiv {\cal G}_1{\cal G}_\omega \\
& {\cal G}_1
=
\exp\left[
(g-\alpha_s)L
+ \frac{1}{2} \ln(R_1R_2)
- 2i\frac{\omega_{th}L}{c} \left.\pd{n}{N}\right|_{th} (N-N_{th})
\right] \\
& {\cal G}_\omega
=
\exp\left[
-2i \frac{n_{th}\omega_{th}L}{c}
- 2i \frac{n_gL}{c} (\omega-\omega_{th})
\right]
=
\exp[-i(\omega-\omega_{th})\tau_{in}] \\
&=
\exp(i\omega_{th}\tau_{in})\exp\left(-\tau_{in}\df{}{dt}\right)
\end{align}
$2n_{th}\omega_{th}L/c=2\pi m$
and
$2n_{g}L/c=\tau_{in}$
and
$i\omega=d/dt$ for monochromatic fields with frequency $\omega$
\begin{align}
{\cal G} &\equiv {\cal G}_1 \exp(i\omega_{th}\tau_{in})\exp\left(-\tau_{in}\df{}{dt}\right) \\
\end{align}
The optical gain increases lenearly with the total number of hole-electron pairs $N$.
\begin{align}
g(N) &= \left.\pd{g}{N}\right|_{N_0} (N - N_0) \\
g(N) &= g_{th} + \left.\pd{g}{N}\right|_{N_{th}} (N - N_{th}) \label{eqn:gNth}
\end{align}
$N_0$ : the carier number at transparency
The gain per second
\begin{align}
G(N)
&= \nu_g g(N)
= \nu_g g_{th} + \nu_g \left.\pd{g}{N}\right|_{N_{th}} (N-N_{th})
= \frac{1}{\tau_p} + G_N(N-N_{th}) \\
& G_N
= \nu_g \left.\pd{g}{N}\right|_{N_{0}}
\simeq \nu_g \left.\pd{g}{N}\right|_{N_{th}}
\end{align}
the phton lifetime $\tau_p$ as the inverse of the total loss rate of phtons
Application ofthe round trip gain
to the time dependent electric fields ${\cal E}_f$ of th eforward traveling wave
at $z=0$
\begin{align}
{\cal E}_f(t)
= {\cal G}{\cal E}_f(t)
= {\cal G}_1 \exp(i\omega_{th}\tau_{in}) \exp\left(-\tau_{in}\df{}{t}\right) {\cal E}_f(t)
= {\cal G}_1 \exp(i\omega_{th}\tau_{in}) {\cal E}_f(t-\tau_{in})
\end{align}
\underline{slow-varying complex amplitude $E(t)$}
\begin{align}
{\cal E}_f(t) = E(t) \exp(i\omega_{th}t)
\end{align}
\begin{align}
&E(t)
= {\cal G}_1 E(t-\tau_{in})
= {\cal G}_1 \left(E(t)-\tau_{in}\df{E(t)}{t}\right) \\
&\df{E(t)}{t}
= \frac{1}{\tau_{in}} \left(1-\frac{1}{{\cal G}_1}\right) E(t)
\end{align}
\begin{align}
\frac{1}{{\cal G}_1}
&=
\exp\left[
-(g-\alpha_s)L
- \frac{1}{2} \ln(R_1R_2)
+ 2i\frac{\omega_{th}L}{c} \left.\pd{n}{N}\right|_{th} (N-N_{th})
\right] \\
&=
1
-(g-\alpha_s)L
- \frac{1}{2} \ln(R_1R_2)
+ 2i\frac{\omega_{th}L}{c} \left.\pd{n}{N}\right|_{th} (N-N_{th})
\end{align}
式(\ref{eqn:gth}),(\ref{eqn:tauin}),(\ref{eqn:gNth})より
\begin{align}
\frac{1}{\tau_{in}} \left(1-\frac{1}{{\cal G}_1}\right)
=
\frac{c}{2n_g} \left.\pd{g}{N}\right|_{th} (N-N_{th})
- i\frac{\omega_{th}}{n_g} \left.\pd{g}{N}\right|_{th} (N-N_{th})
\end{align}
\begin{align}
\df{E(t)}{t}
=
\left[
\frac{c}{2n_g} \left.\pd{g}{N}\right|_{th} (N-N_{th})
- i\frac{\omega_{th}}{n_g} \left.\pd{g}{N}\right|_{th} (N-N_{th})
\right]
E(t)
\end{align}
Kramers-Kronig relations
\begin{align}
\alpha = \frac{\delta n}{\delta n'}
= -2 \frac{\omega}{c} \frac{\partial n /\partial N}{\partial g / \partial N}
> 1
\end{align}
\begin{align}
\df{E(t)}{t}
=
\frac{1+i\alpha}{2}
\left[
G(N) - \frac{1}{\tau_p}
\right]
E(t)
\end{align}
\underline{The dynamical behaviour of the electron-hole pair number $N$}
\begin{align}
\pd{N}{t} = D \nabla^2 N + \frac{I}{e} - R_{st}(N,|E|^2)
\end{align}
The first term : carrier diffusion, $D$ diffusion coefficient (negligible in the active region)
The second ter : electron-hole injected into the active region
by means of the electrical current $I$,
$e$ electron charge
The thrid term : losses of carriers by spontaneous emission and (non-radiative transitions)
\begin{align}
R_{st}(N,|E|^2)=G(N)|E|^2
\end{align}
\begin{align}
\pd{N}{t} = D \nabla^2 N + \frac{I}{e} - G(N)|E|^2
\end{align}
The gain increases linearly with the population inversion
but saturates at high intensities
\begin{align}
G(N,|E|^2) = G_N(N-N_0) (1-\epsilon|E|^2)
\end{align}
last equation
Rate equation without noise
\begin{align}
\df{E}{t} &= \frac{1+i\alpha}{2} \left(G(N,|E|^2)-\frac{1}{\tau_p}\right)E \\
\df{N}{t} &= \frac{I}{e} - \frac{1}{\tau_s} N - G(N,|E|^2) |E|^2
\end{align}
Rate equation with laser noise
\begin{align}
\df{E}{t} &= \frac{1+i\alpha}{2} \left(G(N,|E|^2)-\frac{1}{\tau_p}\right)E + F_{E}(t) \\
\df{N}{t} &= \frac{I}{e} - \frac{1}{\tau_s} N - G(N,|E|^2) |E|^2 + F_{N}(t)
\end{align}