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\begin{document} 

\title{\vspace*{-3cm} \bf Model Equations Scatterer Laser 
}
\author{Ede Wünsche}
\date{\today}

\maketitle

 \sf  


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\includegraphics*[width=\textwidth]{equations} \\[-9ex]
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\section*{explanations:}
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$ E_r(z,t) $ is the full optical amplitude along ray $ r $. Each ray has its own local space coordinate $ z \in [0,l] $. 

$ \omega_0 $: reference frequency (freely choosable in the emission band of the laser, typical value $2\pi\cdot 10^{15} $ s$ ^{-1} $ ), \\
$ k=k_0 - i \bar g/2 $ with $ k_0=\omega_0/c $, $ c $: velocity of light in the semiconductor, 
\\ $ \bar g $: reference gain (amplification per length). 

It holds $ \bar g \ll \omega_0/c $ (typically $ \bar g = 10^4 $ m$ ^{-1} $ and $ k_0 = 10^7 $ m$ ^{-1} $). 

$ g(z,t) = g_d(t) $ if $ z\in d $. $ \alpha_0 $: back ground absorption. (Both of same order of magnitude as $ \bar g $).\\
$ \alpha_H $: so called ''alpha-factor'' modeling amplitude-phase coupling (typical $ \alpha_H=0$ to $ 5 $). 

$\ce(z,t) $ varies slow (time scale: ps; space scale 0.1 mm) compared to $G(k,z)e^{-i \omega_0 t	} $ (fs; 0.1 $ \mu $m). \\ $ g_d(t) $ varies even slower (time scale: $ \tau_n = $ some $ 10^2 $ ps). 

$ A_{rr'} $: scattering strength from ray $ r' $ into ray $ r $. $ r_\text{in}(r) $: set of rays arriving at $ r $. $ l $: ray length.

$ \beta_\text{spont} $: small stochastic source modeling spontaneous emission.

$ S_d(t) = \int_{\r \in d} |E(\r,t)|^2 d^2 r$: mean intensity in domain $ d $. It is approximated in the program  by the mean intensity impinging to the scatters in the domain. 

$ g_0 $: unsaturated gain (solution  of (2) for $S_d=0$).

\section*{instantaneous modes:}
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In general, optical modes are solutions of the homogeneous optical equations with time-independent coefficients, i.e. (1) with $g$ independent of $ t $ and  with $ \beta_\text{spont} = 0  $ in (1').

Since $ g_d(t) $ is slow, it is ''nearly'' constant, and Equ. (1)  can be approximately solved by the exponential ansatz $ \ce_r(z,t)=e^{-i\int \Omega(t) dt}\psi_d(z,t) $. Neglecting the temporal derivative of $ \psi_d $	yields 
\begin{align} \label{instmode}
\partial_z\psi_r(z,t)-\left[ \frac{1-i \alpha_H}{2} (g(z,t) - \bar g) - \frac{ \alpha_0 }{2}\right]\psi_r(z,t) = i\frac{\Omega}{c} \psi_r(z,t)
\end{align}
with boundary conditions (1') with $ \beta_\text{spont}=0 $. This is the eigenvalue problem that defines what we call instantaneous light modes. In general, the eigenvalues $ \Omega $ are complex. Re($\omega_0+\Omega  $) defines the optical frequencies (wavelengths) where in expermiment we see peaks in the optical spectra, and -Im($ \Omega $ is the damping (decay constant) of the corresponding mode. 


Now I turn over to the exponential random matrix that we discussed. For simplicity, I assume a constant $ g $ (as it is at threshold) and $ \alpha_0 = \alpha_H = 0 $. Then, Eq.(\ref{instmode}) has a true exponential solution. Inserting into (1'), using $G(k,r)=e^{\bar g/2}G(k_0,r)$, yields
\begin{align} \label{homeq1}
\psi_r(0) = \sum_{r'}  A_{rr'} G(\tilde k,l_{r'})\psi_{r'}(0)
~~~~ \text{ with } \tilde k = k_0+\frac{ \Omega }{c}-i\frac{g}{2}.
\end{align}
This homogeneous system for the initial amplitudes on the rays has nontrivial solutions only if
\begin{align}\label{det1}
\det{ \left( \delta_{rr'} - A_{rr'}G(\tilde k,l_{r'} \right) } =0.
\end{align}
This is equivalent to the nonlinear eigenvalue equation of a random laser considered in our PRL paper.
Note: $ A_{rr'}\ne 0 $ only if the end point of $ r' $ agrees with the start point of $ r $.

In my numerical calculations I considered point scatterers. In this case all nonvanishing elements of $ A_{rr'} = A $ are identical to each other, which allows a considerable simplification. We write (\ref{homeq1}) in the form
\begin{align}
\psi_{s\bs}(0) = A \sum_{s'} G(\tilde k, l_{\bs s'}) \psi_{\bs s'}(0),
\end{align}
where $ s $ and $ \bs $ are end and start scatterer of ray $ r $, and $\bs$ and $ s' $ accordingly of $ r' $. The sum is the total amplitude $\phi_\bs$ arriving at scatterer $ \bs $. Multiplying the whol equation with $  G(\tilde k, l_{s\bs}) $ and summing over $ \bs $ yields
\begin{align}
\phi_s = A \sum_\bs G(\tilde k, l_{s\bs}) \phi_\bs, 
\end{align}
yielding the nonlinear eigenvalue equation
\begin{align}
\det{ \left( \delta_{s\bs} - A G(\tilde k,l_{s\bs} \right) } =0.
\end{align}
I think, the last case is simpler and interesting enough. 


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