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 \title{Mode Competition in a Fabry P\'{e}rot Semiconductor laser: \\Travelling 
Wave Model with asymmetric dynamical gain}
 \author{M. Homar, S. Balle and M. San Miguel} 
\address{ Departament de F\'\i sica, Universitat de les Illes Balears and\\
Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA (CSIC-UIB)\\ 
 E--07071 Palma de Mallorca, Spain\\ }
 \date{}
 \maketitle

\begin{abstract}
We report and discuss our main results for the 
multimode dynamics of a Fabry-P\'{e}rot laser in 
different situations above threshold: mode beating and mode-hopping.  
A Travelling Wave approximation has been used together with a model with 
an asymmetric dynamical gain for the semiconductor laser which captures 
important features needed for the understanding of multimode dynamics. 
\end{abstract}
 
\newpage

\section{Introduction} 

Fabry-Perot semiconductor lasers usually exhibit a multi-longitudinal mode behavior
which is revealed in their power spectra. The relative power of the longitudinal
modes varies with pump current, while the competition among them is influenced by
several physical mechanisms such as spontaneous emission noise, spatial hole burning
(SHB), spectral hole burning, and carrier diffusion. This competition gives rise to 
a variety of dynamical phenomena which
most notably include mode-beating and mode-hopping. Mode-beating modulates in fast
time scales the total output power, while mode-hopping involves jumps of
dominant mode \cite{ohtsu,gray}. 
Multimode semiconductor dynamics has been usually described within a rate equation 
framework \cite{lee,TSM}. In this framework a fixed number of modes is considered.
The modes have different and given gain coefficients which introduce in the model
the frequency dependence of the gain curve. This approach neglects the changes in the 
spectral dependence of the gain and refraction index with carrier number. It also neglects
SHB and coherent couplings associated with phase dynamics and
phase-amplitude coupling. Some of these shortcomings can be partially overcome in
more ellaborated descriptions \cite{mandel}, which are still based on a mode
decomposition of the field and neglect the fast polarization dynamics. A rather
different approach involves the direct solution of the traveling wave (TW) equation
for the field within the laser cavity \cite{fleck,marga} which avoids a
priori assumptions on the spectral structure of the field. In the TW description, the field
equation is coupled to equations for the material variables describing the active
medium, so that the frequency dependent gain is dynamically determined and
coherent couplings are fully taken into account.

An analysis of multimode dynamics based on a TW model has been recently reported
\cite{marga}. However, the active medium was described by Bloch equations for a two
level system. This is a conceptually simple situation which captures important
mechanisms needed for the understanding of multimode dynamics, but it does not
account for specific properties of the semiconductor medium. In particular, a 
two level model gives
a lorentzian lineshape with maximum gain (for zero detuning) at the frequency where
the carrier-induced refraction index vanishes. A semiconductor medium is known to
have an asymmetric gain curve with maximum gain at non-zero refraction index. This
gives rise to the linewidth-enhancement factor $\alpha$ \cite{henry}. In order to
match the typical measured values of $\alpha$ with the detuning in a two level
system, lasing away from resonance must be artificially enforced, which is not
possible within a TW model \cite{marga}. It is then natural that some of the rich
phenomenolgy of multimode semiconductor dynamics is not described when using a two
level system. In particular, in the study of \cite{marga} mode hopping was never
observed. A more refined description of the semiconductor medium can follow two
different avenues. One is a fully microscopic approach \cite{koch}, with its obvious
advantages and the disadvantages of computational cost, complication and associated
low level of physical intuition. A different approach is to use simpler
models \cite{agrawal,salvador}, which within the simplicity of a two-level model, 
incorporate in a phenomenological way the main results of microscopic analyses.

In this paper we use one of those models, which has been recently introduced 
for the semiconductor medium dynamics \cite{salvador}, together with a TW approach
for the field within the laser cavity. Ref. \cite{salvador} discusses the physical
basis of the model and describes its consequences for the steady state complex
susceptibility and subthreshold behavior (amplified spontaneous emission spectrum).
Here we report results for the multimode dynamics above threshold. Thus, our
full modeling incorporates a wave description of the field, a selfconsistent
frequency dependent gain and refraction index (with asymmetric shapes), full
coherent couplings, carrier diffusion and the effects of spontaneous emission noise.
In addition to the main qualitative results of \cite{marga} on the factors
influencing multimode behavior, we find that complicated mode-hopping dynamics can
occur, mostly as a consequence of the changes induced in the modal gain by spatial 
hole burning. The number of active modes increases 
with pumping level, and it is strongly influenced by carrier diffusion. Strong diffusion 
washes out mode competition leading to essentially
single mode behavior. A low diffusion level allows for several competing modes, 
leading to mode-beating and mode-hopping.

The paper is organized as follows. Sect.II describes the main features of our model.
In Sect. III we summarize the subthreshold behavior of 
our model and we determine the threshold characteristics.
Sect. IV contains the discusssion of our main results for the multimode dynamics in
different situations above threshold.

 
\section{Model}

For the semiconductor laser we consider a planar structure which, for angular
frequencies around a reference frequency $\omega_c$ supports only 
the fundamental TE mode \cite{agrawal2}, whose propagation constant is close to 
$k_c \equiv \omega_c n/c$, $n$ being the effective background refraction index 
for the TE mode. The slowly-time-varying amplitude of the
TE-mode, ${\rm E}(z,t)$, along the Fabry Perot cavity ($z$ direction) is governed by
Maxwell's equations, which contain a source term associated with the nonlinear
electrical polarization of the semiconductor medium. We assume perfect confinement of
the carrier density to the active zone, which we consider thin enough for carrier
diffusion to homogenize the carrier density in the transverse direction, so that
within the active zone the carrier density varies only longitudinally. With these
considerations, the relevant variables for the modeling of the semiconductor medium
are the longitudinal carrier density, ${\rm N}(z,t)$, and ${\rm P}(z,t)$, the
slowly-time-varying distribution of the projection of the nonlinear polarization of
the medium onto the fundamental TE mode.

Our model is based on coupled equations for $E(z,t)$, $N(z,t)$, 
and $P(z,t)$. It distinctive feature is an evolution equation for the nonlinear 
polarization based on an approximate calculation of the nonlinear electric 
susceptibility of a semiconductor medium \cite{salvador},  
\begin{equation} 
\partial_t {\rm P} = \left[ -\gamma_p + i (\omega_c - \omega_g) \right] {\rm P} - 
   i a ({\rm N} - {\rm N_t} + i \alpha_0 {\rm N_t}) {\rm E} \; . 
\end{equation} 
This equation is similar to that of a two-level system except for the emergence of a new  parameter
$\alpha_0$. This parameter yields an asymmetric gain spectrum, shifting the
frequency of the gain peak away from the zero dispersion frequency and originating a
carrier-induced shift of the frequency of the gain peak.  The parameter $a$ 
determines the differential gain and it is given by an effective electric 
dipole element which incorporates a geometrical factor, $\gamma_p$ is the 
decay rate for the nonlinear polarization, and $\omega_g$ locates the gain peak 
with respect to $\omega_c$. In addition, the carrier density dependence of $\gamma$ 
(arising from the enhancement of the carrier-carrier scattering rate for increasing 
carrier density) and $\omega_g$ (associated with carrier-induced band gap 
renormalization) are considered in a linear approximation of the form
\begin{eqnarray*} 
\gamma_p &=& \gamma_0 ( 1 + \rho D ) \\ 
\omega_g &=& \omega^0_g +  \gamma_0 \sigma D 
\end{eqnarray*}
where $D\equiv ({\rm N-N_t})/\rm N_t $ and $\rm N_t $ is the material 
transparency carrier density at frequency 
$\omega^0_g$. The polarization decay rate (which sets the bandwidth
of the gain spectrum) when $\rm N = N_t$ is $\gamma_0$. The parameter $\rho$ 
modifies the width and
height of the gain curve, thus reducing the differential gain, while $\sigma$ gives a
carrier induced shift of the peak of the gain curve. We note that 
the lorentzian shape of the gain curve in the case $\alpha_0=0$ is not changed by 
$\rho$ and $\sigma$. 

Using the slowly time varying approximation in the wave equation for the TE field and 
scaling space ($s=z/L$, $L$ being the length of the FP cavity) and time 
($\tau = c t / L n_g$, where $n_g = [dn/dw]_{\omega=\omega_c}$ is the group index), 
the coupled equation for $E$, $P$ and $D$ become, 
\begin{eqnarray}
\partial_{\tau} E &=& {i\over 2 q_c} \left(\partial_s^2 E + q_c^2 E \right) + 
  (P - \Lambda_0 E) - \kappa E \; , \label{eqnarray:e} 
\\
\partial_{\tau} D &=& - \gamma_e (D - D_0 + E ( P - \Lambda_0 E )^* + E^* ( P - \Lambda_0 E ) 
- \Delta \partial_s^2 D) \; , \label{eqnarray:d} 
\\
\partial_{\tau} P &=& 
   -\gamma \left\{ \left[ 1 + i \delta + (\rho + i \sigma ) D \right] P -
    a'(D + i \alpha_0) E - \xi (s, \tau) \right\}\; . \label{eqnarray:p}
\end{eqnarray}
In these equations $\gamma_e$ ($\gamma$) are the scaled decay rates for the carrier density  
(polarization), $D_0$ corresponds to the injection current and $\Delta$ is the scaled diffusion 
constant for the carrier density. In the following we neglect internal losses ($\kappa =0 $) and 
we set the normalized detuning $\delta=0 $ ($\omega_g^0=\omega_g$), and choose 
$N_t$ such that $a'=1$. 

Finally we mention that $\Lambda_0 E$ gives a linear polarization which has to be 
substracted from $P$ so that there is no not gain when there are no injected carriers 
($D=-1$). The constant parameter $\Lambda_0 $ is given by 
\begin{equation}
\Lambda_0 = {1 \over 2} {\alpha_0 \over 1 - \rho}
\left( {\alpha_0 \over 1 + \sqrt{1 + \alpha_0 ^2}} + i \right) 
\end{equation}
The 
spatially distributed noise term $\xi (s, \tau)$ acts as a 
polarization source and it models
independent spontaneous emission process in different points of the cavity. It
is taken to be a Gaussian white noise in space and time with zero mean and
correlations $\langle \xi(s, \tau) \xi^{*}(s', \tau') \rangle = \beta
\delta(\tau - \tau') \delta(s - s')$. The spontaneous emission noise intensity
is measured by $\beta$, which we approximate by a constant parameter
\cite{lax}. 

The Travelling-Wave description is obtained by decomposing the electric field
amplitude into two counter-propagating waves,
\begin{equation}
E(s, \tau) = E^{+}(s, \tau)e^{i q_c s} + E^{-}(s, \tau)e^{-i q_c s} \; , \nonumber 
\end{equation}
where $E^{+}(s, \tau)$ is the slowly varying envelope of the electric field 
propagating forward and $E^{-}(s, \tau)$ is the slowly varying envelope of the 
electric field propagating backwards. The set of coupled equations for 
$E^{+}(s, \tau)$, $E^{+}(s, \tau)$, $P(s, \tau)$ and $D$ are obtained as in 
\cite{fleck,marga}:
\begin{eqnarray}
\partial_{\tau} E^{+} + \partial_s E^{+} &=& 
\langle P e^{-i q_c s} \rangle - \Lambda_0  E^{+} \; , \label{eqnarray:e2+} \\
\partial_{\tau} E^{-} - \partial_s E^{-} &=& 
\langle P e^{i q_c s} \rangle - \Lambda_0  E^{-} \; , \label{eqnarray:e2-} \\
\partial_{\tau} P &=& -\gamma \left\{ 
\left[ 1 + \left(\rho + i \sigma \right) D \right] P - 
(D +i \alpha_0)\left( E^{+}e^{i q_c s} + E^{-}e^{-i q_c s} \right) 
- \xi(s,\tau) \right\} \; ,
\label{eqnarray:pol2} \\
\partial_{\tau} D &=& - \gamma_e \left[ D - D_0  +  
\left( E^{+} P ^{*} + E^{-*} P \right)e^{i q_c s} + 
\left( E^{-} P ^{*} + E^{+*} P \right) e^{-i q_c s}\right. \nonumber \\
&-& \left. 2 Re(\Lambda_0) \left(\vert E^{+}\vert^{2} +  \vert E^{-}\vert^{2} +
E^{+}E^{-*}e^{2 i q_c s} + E^{-}E^{+*}e^{-2 i q_c s} \right) + 
\Delta \partial_s^2  D \right] \; .
\label{eqnarray:port2}
\end{eqnarray}

The above set of equations include TW equations for the field which must be solved 
in conjunction with boundary conditions imposed by the cavity mirrors. This will
result in a structure of the field power spectrum without a priori assumptions on 
the laser longitudinal mode structure. We denote by  $r_1$ and $r_2$ the field
amplitude reflectivities of the cavity facets placed at $s=0$ and  $s=1$,
respectively. Thus, our boundary conditions for the field are,
\begin{eqnarray}
E^{+}(s=0,\tau)&=& r_1 E^{-}(s=0,\tau) \; , \\
E^{-}(s=1,\tau)&=& r_2 E^{+}(s=1,\tau) \; ,
\end{eqnarray}
together with the boundary condition for the longitudinal carrier density
\begin{equation}
\left[ \partial_s D \right]_{s=0, 1} = 0 \; .
\end{equation}
% which amounts to require no carrier flow through the laser facets.

The spatial variations in polarization and carrier number over wavelength distances 
can be treated by means of a Fourier series expansion,  
\begin{eqnarray}
P(s,\tau)= e^{i q_c s} \sum_{p=0}^{\infty}P^{+}_{(p)}e^{2i p q_c s} + 
e^{-i q_c s} \sum_{p=0}^{\infty}P^{-}_{(p)}e^{-2i p q_c s} \; , \label{eqnarray:f1}
\\
D(s,\tau)=D_{(0)}(s,\tau) + \sum_{p=1}^{\infty} \left[ 
  D_{(p)}(s,\tau)e^{2i p q_c s} + D_{(p)}(s,\tau)^* e^{-2i p q_c s}  
                                                \right] \; . \label{eqnarray:f2}
\end{eqnarray}
Following the procedure in \cite{marga} we truncate the  expansion after the first
harmonic, keeping only the "grating terms" $p=1$. This truncation gives rise to a
set of coupled equations for $E^{\pm}$, $P^{\pm}_{(0)}$, $P^{\pm}_{(1)}$, $D_{(0)}$ 
and $D_{(1)}$. The set of equations is the same than the one discussed in
\cite{marga}, except by the occurrence of terms proportional to $\alpha_0$, $\rho$
and $\sigma$ in the equations for $P^{\pm}_0$ and $P^{\pm}_1$. These new terms
incorporate in the TW model the specific properties of gain dynamics in a
semiconductor medium. We use the same semi-implicit finite difference integration
scheme and integration procedures described in \cite{marga}. Noise sources are only
included for $P^{\pm}_0$. 

The parameter values that we will keep fixed throughtout this paper are given 
in table I. In Sect. IV we will consider different injection current and carrier 
diffusion constants. 

\section{Subthreshold behavior and threshold determination}

A basic undestanding of our dynamical model for the semiconductor medium and its 
implications for the lasing properties and mode competition can be obtained 
from the analysis of the gain and the refraction index curves obatined in the 
subthreshold regime \cite{salvador}. In this regime, the only deterministic 
stable state is the ``off'' 
solution, characterized by zero values of both $E$ and $P$ and a constant and
uniform value of $D = D_0$. Hence, the scaled electrical susceptibility obtained 
from (\ref{eqnarray:e})-(\ref{eqnarray:p}) is 
\begin{equation} \label{susc}
\Lambda (\omega , D_0) = {D_0 + i \alpha_0 \over 1+\rho D_0 - 
  i (\omega / \gamma - \sigma D_0)} \; ,
\end{equation}
which determines the spectral shape of the material gain, $g (\omega, D) = Re[\Lambda
(\omega , D) - \Lambda_0]$, and the spectral dependence of the carrier-induced
refraction index change, $\delta n (\omega, D) = Im[\Lambda (\omega , D) -
\Lambda_0]$. Therefore, the frequency and carrier density dependent linewidth
enhancement factor, $\alpha$, can be obtained as
\begin{equation} \label{alfa}
\alpha \equiv {\partial \delta n / \partial D \over \partial g / \partial D} \; .
\end{equation}

In Fig.~1a we show the spectral curve of the material gain for different values of
$D_0$ displaying a clear nonlorentzian shape. As can be observed in Fig.~1b, 
the frequency of the maximum in the gain curve
shifts from $\omega_p = -1.4 \gamma$ for $D_0 = -1$ to $\omega_p = 0.2 \gamma$ for 
$D_0 = 3$, which represents a blueshift of $\sim 7 nm$ (emission wavelength $\sim
0.88 \mu m$). In addition, the maximum value of the gain increases quasi linearly with
carrier injection for $D_0 > 0$, reflecting an almost constant differential gain. In
Fig.~1d we plot the value of $\alpha$ at the frequency of maximum gain as a function
of $D_0$.  

The threshold for the device can be readily obtained from the deterministic version
(i. e. $\xi(s, \tau) = 0$) of equations (\ref{eqnarray:e})-(\ref{eqnarray:p}). By 
substituting $D = D_0$ in these equations and Fourier transforming in time and space
one has the (complex) dispersion relation 
\begin{equation} \label{qw}
\omega - q - i 
   \left[ \Lambda(\omega, D_0) 
% {D_0 + i \alpha_0 \over 1 + \rho D_0 - i (\omega / \gamma - \sigma D_0)}
           - \Lambda_0 \right] = 0 \; ,
\end{equation}
where $q \simeq q_c$ has been considered. Also, from the boundary  conditions for
the electric field, \cite{agrawal2}, one finds $r_1 r_2 e^{2 i q} = 1$, which determines the
longitudinal mode complex wavevectors as
\begin{equation} \label{modalq}
q_m = m \pi - {i \over 2} ln \left( {1 \over r_1 r_2} \right) \; , 
\; m = 0, \pm 1, \pm 2, ...
\end{equation}
Eqs. (\ref{qw}) and (\ref{modalq}) allow to determine the modal frequencies
$\omega_m$. When mode $m$ reaches threshold, which happens for the carrier density
value $D_0 = D_m$, its frequency $\omega_m = \Omega_m$ becomes purely real; then, 
equating to zero the real and imaginary parts of (\ref{qw}) one obtains
\begin{eqnarray*}
\Omega_m - m \pi &=& Im\left[\Lambda_0 - \Lambda(\Omega_m, D_m) \right]\; , \\
%  {\alpha_0 (1 + \rho D_m) + D_m (\Omega_m / \gamma - \sigma D_m)
%  \over
%  (1 + \rho D_m)^2 + (\Omega_m / \gamma - \sigma D_m)^2} \; , \\
Re\left[ \Lambda(\Omega_m, D_m) - \Lambda_0 \right]&=& 
 {1 \over 2} ln \left( {1 \over r_1 r_2} \right) \; .
%  &=&
%  {D_m (1 + \rho D_m) - \alpha_0 (\Omega_m / \gamma - \sigma D_m)
%  \over
%  (1 + \rho D_m)^2 + (\Omega_m / \gamma - \sigma D_m)^2} \; . 
\end{eqnarray*}

This set of equations can be solved numerically for diferent values of $m$, the
minimum value of $D_m$ corresponding to the laser threshold, which determines the
lasing mode. For our parameter values, we obtain the threshold carrier density 
$D_{th} \equiv min(D_m) = 0.751$, where mode
$m = -7$ (frequency $\nu_{-7} \simeq -1161 GHz$) becomes unstable. Then, from
Fig.~1d we expect $\alpha \sim 3.5$ when the laser is on, which should result in 
important frequency chirp during optical pulse emission.

\section{Mode competition} 

In order to discuss the phenomena of mode competition we have integrated the TW
equations starting from random initial conditions for the electric field and for
various values of the injection current and carrier diffusion constant. We monitor
the time evolution of the electric field amplitude at each point during a time
interval of 30ns. The whole set of data points for the output complex electric field
amplitude at the right facet, $E^+ (s=1,t)$, is divided in 150 time windows,   
each of these corresponding to a real time window of $0.18$  
$ns$. An FFT in each time window is performed to obtain a time resolved Field Power
Spectrum (FPS) which describes the competition among modes. 

We first recall that
for a two level model ($\alpha_0=\sigma=\rho=0$), SHB results in multimode operation
that can be washed out for large enough carrier diffusion. For large diffusion
single mode operation is obtained not too close to threshold after 4ns. For smaller
diffusion the number of active modes increases with injection current and a steady
state is reached in which the time resolved FPS is essentially constant. 
A dominant mode remains fixed and no mode-hopping is observed \cite{marga}.

A sample of the main different situations that can be found 
with the model of Sect. II is summarized in Fig.2
where the the time evolution of the total output power $I(t) = \vert E^+(s=1, t)
\vert^2$ and dominant frequency (inset) are shown for different values 
of the injection current and of the carrier
diffusion coefficient. The crucial role of carrier diffusion is again evidentiated. For
a diffusion $\Delta = 1.6\ 10^{-5}$ the laser reaches single mode
emission after a short transient (Fig. 2a). Residual chirping associated with the
carrier density modulation due to relaxation oscillations can be observed in the
dominant frequency (see inset). The same qualitative behavior is observed for two
levels of injection current, $D=1.1 D_{th}$ or $D=1.8 D_{th}$. For lower carrier
diffusion ($\Delta = 0.4\ 10^{-5}$) laser emission becomes multimode, with a number
of active modes that depends on the carrier injection level (Figs. 2b and 2c). Close
to threshold mode beating between two main modes is observed in the total output
power (Fig. 2b) with a  fast modulation of $I(t)$ at an oscillation frequency of
$171~GHz$ corresponding to the longitudinal mode spacing. The dominant mode remains
fixed for long time intervals with occasional exchages with the adjacent mode. 
The two active modes compete but coexist at all times. Higher above
threshold (Fig. 2c) $I(t)$ displays a complicated sustained dynamics which is
associated with a recurrent hopping among four (and even five) active modes, as can
be observed in the inset. 

A more direct description of the mode competition in the situation of Fig. 2.e 
is given by looking at the output power associated with each of the active
modes. This is obtained by convolution of the output field with a lorentzian filter
($50~GHz$ HWHM) centered at the different modal frequencies. 
There are time windows where the laser emits almost in a single mode
(e. g. between 12 and 13~ns), but  in other time windows as shown in Fig. 3 
there are two or more modes
actively contributing to the total power. In general, the dominant mode hops in a
recurrent way, and it is worth noting that differently to the case in Fig.~2b,
some of the active modes switch-on and off without coexisting at all times.

The mode hopping dynamics in Fig. 2c can be further evidentiated by looking at the
time-resolved FPS, as shown in Fig.4. Mode competition and hopping are clearly 
displayed, showing time windows of quasi single-mode operation followed by mode
hops and time windows of mode coexistence. One can also see the effects 
of frequency chirp associated with transient
relaxation oscillations as a drift in the positions of the modal peaks; this effect
is especially clear in the initial regime (before 9~ns, say), and it also induces the
characteristic chirp-induced broadening of the modal peaks in the time resolved FPS.
We have also plotted the time-resolved FPS corresponding to some of the different
time windows considered, which may help the reader to visualize the above dynamics. 
The same description in terms of FPS (Figs.~5 and 6) for the  other cases considered
identifies single-mode operation after a short multimode initial transient (Fig.~5,
corresponding to Fig.~2a) and the coexistence of two dominant modes (Fig.~6,
corresponding to Fig.~2b). In Fig. 6 the Side-Mode Suppresion Ratio (SMSR) is of 
the order of $10-20 dB$ at most, which means that the two active modes coexist. 
In the last stages of Figs. 5 and 6, the carrier density has almost relaxed
to its steady state, and the FPS clearly displays an asymetric (not lorentzian) shape
induced by the parameter $\alpha_0$. 

The usual discussion of mode competition is based on characteristics of the gain curve. 
From this point of view one focuses on the changes in the spectral gain curve 
as the carrier density varies. In our model such variations 
induce a shift in the location of the maximum of the gain curve
(through the combined effect of $\alpha_0$ and $\sigma$) and also modify its width
and height (due to $\rho$). In this way, the number of modes which are close to
threshold (and which of these modes is the closest to the gain peak) 
change as the carrier density is modified. The shift in the frequency of the gain 
peak naturally results in a change of dominant mode (mode-hopping). In our results, 
this shift is mainly due to the effect of $\sigma$, which we recall that, together 
with $\alpha_0$, is associated with band-gap renormalization. We have checked 
that by artificially setting $\alpha_0=0$,
mode hopping is still possible within our model when $\sigma \not= 0$. However in this case 
the resulting linewidth enhancement falctor is very small. In fact, $\alpha_0 \not= 0$ 
is a necessary  ingredient to observe chirping and associated
broadened peaks in transient spectra by phase amplitude coupling. We note however that 
the above explanation for mode hopping relies implicitly on a constant carrier 
density, as assumed in the calculation of a subthreshold gain curve. This picture 
might break down for enough above threshold, where in spite of the carrier density 
being the slow variable of the system, spatial hole burning results in complicated 
spatiotemporal dynamical behavior. Such complicated dynamics is fully taken into 
account in our numerical calculations but might not admit a simple intuitive 
explanation. 

In summary, we have used a travelling wave model for the optic field inside the 
cavity together with a modified two-level model for the material variables that 
yields an asymmmetric gain spectrum. After determining the threshold characteristics 
for different modes, we have discussed our main results for the multimode dynamics in 
different situations above threshold: mode beating and mode-hopping. 

\acknowledgements
We have greatly benefited from discussions from N. B. Abraham.
This work has been partially supported by  Comision Interministerial de
Ciencia y Tecnologia (CICYT, Spain) project TIC95-0563-CO5 
and Direccion General de Investigacion Cientifica y Tecnica (DGICYT, Spain) project 
PB94-1167. 
We also acknowledge financial support from the Human Capital and Mobility Program
 of the  European Union,
Contract CHRX-CT94-0594 
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\end{thebibliography}

\newpage

\begin{table}
\caption{Meaning and values of the different parameters appearing in the 
model}
 \begin{tabular}{|c|c|c|}
Parameter & Meaning & Value  \\
 \hline \hline
 $\alpha_0$ &	& 3    \\
 $\rho$ &	& 0.22 \\
 $\sigma$ &	& 0.3  \\
 $r_1$ & Field reflectivity of left side mirror & 99.5$\%$ \\
 $r_2$ & Field reflectivity of right side mirror & 56.6$\%$ \\
 L & Length of the diode cavity ($\mu m$) & 250   \\
 $\gamma_{e}$ & Scaled carrier decay rate & $5.83$ $\times 10^{-4}$  \\
 $\gamma$ & Scaled polarization decay rate & $29.1$  \\
 $\beta $ & Spontaneous emission rate & 6.32$\times 10^{-8}$  \\ 
 $n, n_g$ & Index of refraction  & 3.5 \\
 \end{tabular}
 \end{table}
\newpage

\begin{figure}
\caption{a) Spectral gain curve different values of $D_0$. 
b) Frequency of the maximum in the gain curve as a fuction of $D_0$ (solid line). 
In addition, we plot the maximum value in the gain, (dashed line). 
c) Spectral shape of the carrier-induced  refraction index change. 
d)$\alpha$-value at the frequency of maximum gain as a function of $D_0$. }
\end{figure}
\begin{figure}
\caption{Time evolution of the total output power, $I(t)$, for different levels of injection current 
and diffusion coefficients: 
a)$ D=1.8 D_{th}$ and $D_c= 2 cm^2s^{-1}$ ($\Delta=1.6\ 10^{-5} $); 
b)$ D=1.1 D_{th}$ and $D_c= 0.5 cm^2s^{-1}$ ($\Delta=0.4\ 10^{-5} $); 
c)$ D=1.8 D_{th}$ and $D_c= 0.5 cm^2s^{-1}$. In the insets we plot the 
frequency carrying the highest power in the time-resolved FPS for each case.}
\end{figure}
\begin{figure}
\caption{Output power associated with different active modes as a fuction of time for 
$ D=1.8 D_{th}$ and $\Delta=0.4\ 10^{-5} $. (The zero mode is considered the 
one closest to zero frequency).}
\end{figure}
\begin{figure}
\caption{Left pannel: Time-resolved FPS for the case $D=1.8 D_{th}$ and $\Delta=0.4\ 10^{-5} $. 
The power level is displayed in a time and frequency plot on a 16-level gray scale, 
with black corresponding to the lowest value and white to the highest. Time runs from 
$t=0$ to $t=28$ $ns$ on the vertical axis while frequency runs from $-2777.78$ $GHz$ 
to $0$ $GHz$ on the horitzontal axis. Right pannel: selected samples of the time 
resolved FPS taken over diferent time windows. The starting time for each window is 
displayed, and their positions in the left pannel indicated by arrows. }
\end{figure}
\begin{figure}
\caption{Same as Fig.4 but for $ D=1.1 D_{th}$ and  $\Delta=0.4\ 10^{-5} $.}  
\end{figure}
\begin{figure}
\caption{Same as Fig.4 but for $ D=1.8 D_{th}$ and $\Delta=1.6\ 10^{-5} $. } 
\end{figure}

\end{document}