# 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}