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 \title{Travelling Wave Model of a Multimode Fabry-P\'{e}rot Laser\\
 in Free Running and External Cavity Configurations}
 \author{M. Homar$^{1,2}$, J. V. Moloney$^{1}$ and M. San Miguel$^{1,2}$ \\
 $^{1}$ Arizona Center for Mathematical Sciences, University of Arizona \\
 $^{2}$ Departament de F\'{\i}sica, Universitat de les Illes Balears\\
 E--07071 Palma de Mallorca, Spain\\ }
 \date{~}
 \maketitle
 
\begin{abstract} 
We report the results of a numerical study of multimode behavior of a Fabry-P\'{e}rot laser. 
The model is based on travelling-wave equations for the slowly varying amplitudes
of the counterpropagating waves in the cavity, coupled to equations for spatially 
dependent population inversion and polarization of a two-level active medium.
Variations in the material variables on the scale of a wavelength are 
taken into account by means of an expansion in a Fourier series. 
Results are given for typical semiconductor laser parameters. Spatially distributed 
spontaneous emission noise  and carrier diffusion are taken into account.
The competing roles of Spatial Hole Burning (SHB), spontaneous emission noise and carrier
diffusion in determining multimode behavior is elucidated: With no carrier diffusion, 
spontaneous emission noise excites a large number of modes close to threshold, while 
SHB leads to a fixed number of significant lasing modes well above threshold. Carrier diffusion 
washes out the gratings in the material variables, and the resulting strengthening of 
the inter-mode coupling (cross-saturation) restores dominant single-mode 
emission well above threshold. We have also studied the effects
of optical feedback and opportunities for mode selection with short external cavities: For an 
external cavity much shorter than the laser cavity length and a small field amplitude reflectivity 
coefficient a single mode can be selected. For a large reflectivity 
coefficient two groups of intracavity modes separated
by the external cavity mode interspacing are selected. 
For an external cavity with a round trip time half that of the laser cavity, the 
laser can be forced with modest feedback to operate on two modes which are both quasi-resonant 
with the external cavity. Mode selection is not found, even for weak feedback, 
when the external mode spacing is about $90 \%$ of the laser mode spacing. 
 
\end{abstract}

\newpage

\section{Introduction} 
The optical spectrum of laser light usually includes several frequencies which are
associated
with longitudinal cavity modes. Multimode emission degrades the 
spectral purity required in some 
laser applications, but allows a higher output power required in  other applications.
Longitudinal multimode laser behavior is well known in solid state 
\cite{baer,TSM,khanin,roy,glorieux}, 
dye\cite{raymer1,raymer2}, semiconductor \cite{agrawal,lee,mandel,yamamoto}  
and gas lasers\cite{refneal}. In principle, the number of possible lasing modes
can be estimated by counting the cavity modes that 
lie in the spectral region where the unsaturated gain exceeds the loss. 
However, multimode laser emission involves strong 
nonlinear mode
interaction and competition for the gain. This may lead to mode locking or to a random 
partition of the total power among modes which oscillate simultaneously 
(anticorrelated fluctuations), with the extreme behavior being mode
hopping. The number
of lasing modes with significant power depends on the operating conditions of the laser, and
it is determined by several physical mechanisms such as spontaneous emission noise, Spatial
Hole Burning (SHB), phase sensitive interactions such as Four Wave Mixing (FWM) and diffusion
processes. A conventional theoretical description of multimode laser behavior is based on an 
expansion of the laser field in a set of 
normal standing-wave modes appropriate to the laser cavity \cite{sigman}.
In this paper we follow a different approach analyzing a Traveling Wave (TW) model for
the laser field in
a two level active medium filling a Fabry-P\'{e}rot (FP) cavity. The partially reflecting cavity 
boundary conditions are incorporated and the mode decomposition is avoided. Within this 
description we discuss the role of different mechanisms in multimode behavior at 
different pumping levels.
  
A popular description of multimode behavior in semiconductor and 
solid state lasers features rate equations \cite{agrawal}
for the photon numbers  
$I_i$ associated with the different lasing modes $i$. The variables $I_i$ are nonlinearly
coupled among themselves and with a global  population inversion variable $N$. 
In this description one introduces as many gain parameters, for the different modes, as
a fixed number of modes are selected {\it a priori} to be relevant. Multimode rate
equations, often used for semiconductor lasers \cite{agrawal,lee,yamamoto,gray,ohtsu},
do not incorporate some important physical phenomena such as phase dynamics,
phase-amplitude coupling and SHB. The effect of carrier diffusion and SHB in semiconductor lasers
was considered in the context of rate equations by phenomenologicaly introducing effective
gain parameters for the modes \cite{lee}. 

A more elaborated approach than standard rate
equations has been used 
to study homogeneously broadened two-level laser models: Studies of
multimode solid-state lasers \cite{baer,TSM,glorieux} are based on a set of equations
for the $I_i$ which incorporates, as additional dynamical variables, a gain variable associated 
with each
mode $i$. These additional gain variables are used to describe the gratings in the inversion 
from the longitudinally inhomogeneous gain saturation caused by the optical standing waves associated
with each lasing mode in the cavity.  A related  modeling has been used for dye lasers 
\cite {raymer1,raymer2},
and justified from Maxwell-Bloch equations by a mode expansion for the lasing
field and an adiabatic elimination of the polarization variables. Coupled equations
for the complex field mode amplitudes and two spatially independent Fourier amplitudes
of the population inversion are obtained in a thin medium approximation which includes
effects such as SHB. The relevance of phase
sensitive mode-mode interaction for semiconductor lasers has been emphasized by Mandel
et al. \cite{mandel} in an analysis similar to the one in \cite {raymer1,raymer2}. They also
take as starting point Maxwell-Bloch equations for a two-level medium interacting with 
a number of cavity modes of a laser field. Carrier diffusion is neglected. Through an
adiabatic elimination of the polarization variables they obtain coupled equations for the 
complex amplitudes of the laser modes and population variables which include phase dependent
couplings. Phase sensitive interactions have  also been discussed in the context
of cubic equations for coupled mode amplitudes for gas lasers \cite{lugiato}. 

All the multimode lasing descriptions mentioned above have in common that they are
based on a decomposition of the laser field into cavity modes and in neglecting the fast polarization
dynamics. In addition, if the spatial dependence of population inversion is considered, 
it is assumed to have the same Fourier modes as that of the laser field. 
A different description of the laser field, which we pursue here, is based upon a 
direct solution of the wave equation for the field with reflecting boundary conditions 
appropriate to the laser 
cavity \cite{fleck,n-m,wfjm}. Such a treatment avoids explicit assumptions regarding the 
longitudinal structure of the laser. The traveling wave equation for the field is coupled
to Bloch equations for population inversion and polarization variables which depend
on the spatial coordinate along the cavity. 
We note that a simple adiabatic elimination of the fast polarization dynamics 
is not possible within
the TW model, since it leads to ill-behaved equations. 
This fact is similar to the unphysical growth of large wavenumber perturbations 
found when studying transverse effects by generalized rate equations
obtained by adiabatic elimination of the polarization
\cite{jakobsen}. In our TW description the frequency dependent gain is then selfconsistently
determined from the polarization dynamics. In addition, all the physical mechanisms influencing
multimode behavior mentioned above (phase sensitive interactions, SHB, etc.) are, in principle,
included in the model. Spontaneous emission noise is also included by means of random polarization
sources. We solve this model directly in the space-time domain and for large-signal situations.
Another analysis which also considers a propagation equation for a semiconductor 
laser field
is given in \cite{tromborg}: There the equation for the field is coupled to a rate equation for the global carrier 
number density  and modulation response and stability of a given mode solution  are discussed
through small-signal 
differential equations which are solved numerically by transforming them into a 
matrix equation. 

An important feature of our TW model is the ease with which optical feedback from
external mirrors or coupled cavities can be taken into account. A large number of studies
exist on optical feedback effects on single-mode semiconductor lasers 
\cite{petterman,mork,elsasser,macinerney,besnard,feedbackregimes}, 
but it is only recently that some desciriptions
of the interplay of multimode behavior and feedback have been given \cite{wu,ryan}.
Strong feedback and multimode behavior have also been considered in dye lasers \cite{munroe}.
The analyses in \cite{wu,ryan} are again based on multimode rate equations in which
optical feedback from mode $i$ is only directly coupled to the same mode $i$. In our TW
model the whole laser field propagates back and forth from the laser cavity to the 
external mirror. We will focus here on properties of mode selection of a multimode 
solitary laser by optical feedback.  We will consider short external cavities 
\cite{munroe,tager}
for which such phenomena such as coherence collapse for moderate feedback 
might be more easily avoided. 
We explore mode selection by changing the length of the  external cavity: 
The field fed back into the laser cavity  interacts 
with different intracavity modes depending on the length of the external cavity. 
We also explore how the strength of the feedback leads to selection of one 
mode and to suppression of the other modes. 

The paper is organized as follows: In section II the TW model is introduced within a 
slowly varying amplitude approximation for the counterpropagating waves in the FP cavity. 
The relevance of the different ingredients
of the model for solid-state and semiconductor lasers is discussed. In particular,
carrier diffusion is taken into account for semiconductor lasers.  Results of the 
numerical analysis are given in Sects. III and IV for typical parameters of a semiconductor
laser. In section III, we discuss the
factors influencing multimode behavior of a solitary Fabry-P\'{e}rot laser: SHB, spontaneous 
emission and, for semicondcutor lasers, carrier diffusion. The effect of optical feedback
and opportunities for mode selection are discussed in Section IV. 

 
\section{ Travelling Wave Model}
We consider a Fabry-P\'{e}rot cavity filled with a two-level atomic medium. 
The Maxwell equation for a linearly polarized electric field in the cavity is 
\begin{eqnarray}
\nabla^{2} E(\vec r, t) -\frac{ \eta ^{2}}{c^{2} }\frac{ \partial ^{2} E(\vec r, t)}
{\partial t^{2} }= \frac{ 1}{\epsilon_0 c^{2}}\frac{ \partial ^{2} \Pi (\vec r, t)}
{\partial t^{2} }
\label{eqnarray:camp} 
\end{eqnarray}
where $E$ is the electrical field, $ \Pi$ is the polarization of the atomic 
medium,   
$\eta$ the refraction index of the host medium, $\epsilon_0$ 
the electrical permittivity and $c$ the speed of light. 

The optical field inside the cavity consists in two identically polarized counter-propagating 
waves of the same frequency, \cite{n-m}
\begin{eqnarray*}
 E(\vec r, t) =  \left[ E^{+}(z, t)\exp{\left(\imath (k_0z-\omega_c t)\right)} + 
E^{-}(z, t)\exp{\left(-\imath (k_0z+\omega_c t)\right)}  + c.c.\right.] 
\nonumber 
\end{eqnarray*}
$E^{+}(z, t)$ is the slowly varying envelope of the electric field propagating forward 
in the Fabry-P\'{e}rot cavity and 
$E^{-}(z, t)$ is the slowly varying envelope of the electric field propagating backwards. The 
envelopes $E^{+}$ and $E^{-}$ are assumed to depend only on the longitudinal coordinate $z$
and we neglect any dependence on the transverse coordinates $x$, $y$.
The carrier frequency  $\omega_c$ is the frequency of the cavity mode closest to the 
maximum of the gain curve. We take $\omega_c$ as the reference frame for the 
optical frequency of the stationary solution, and 
$k_0$ is the wave number associated with 
$\omega_c$, 
$k_0=\eta\omega_c/c$. In the following we derive
equations for $E^{+}(z, t)$ and $E^{-}(z, t)$ coupled to polarization and population inversion
following the general scheme in \cite{fleck,n-m,wfjm} 

The polarization can be written as,  
\begin{eqnarray*}
\Pi(\vec r, t) = P (z, t)\exp{\left(-\imath \omega_c t\right)}  
\nonumber 
\end{eqnarray*}
where $P (z, t)$ is a slowly varying amplitude in $t$, which still keeps
the fast $z$ dependence.		
Performing the slowly varying envelope approximation, we obtain   
\begin{eqnarray}
\frac{ \eta }{c}\frac{\partial E^{+}}{\partial t} + \frac{\partial E^{+}}{\partial z} 
&=& \frac{ \imath \omega_c }{2\epsilon_0 c \eta} 
\langle P \exp{\left(-\imath k_0z\right)} \rangle \label{eqnarray:e+} \\
\frac{\eta}{c}\frac{\partial E^{-}}{\partial t} - \frac{\partial E^{-}}{\partial z} 
&=& \frac{ \imath \omega_c }{2\epsilon_0 c \eta} 
\langle P \exp{\left(\imath k_0z\right)} \rangle 
\label{eqnarray:e-} 
\end{eqnarray}
where $\langle \rangle$ means an average over many wavelenghts. 

Equations (~\ref{eqnarray:e+}) and (~\ref{eqnarray:e-}) must be supplemented 
with boundary conditions for the electric field amplitudes at the facets of the FP cavity 
since we do not make any assumption on the laser longitudinal mode structure. 
We denote by  $r_1$ and $r_2$  the field amplitude 
reflectivities of the cavity mirrors placed at $z=0$ and 
$z=L$, respectively: 
\begin{eqnarray}
E^{+}(z=0,t)&=& r_1E^{-}(z=0,t) \nonumber \\
E^{-}(z=L,t)&=& r_2E^{+}(z=L,t) \label{eqnarray:bc}
\end{eqnarray}
An external cavity can also be taken into account modifying the second boundary 
condition to include the effect of an external 
mirror of reflectivity $r_3$ placed on the right of the laser cavity. Thus, 
our second boundary condition becomes,
\begin{eqnarray*}
E^{-}(z=L,t)&=& r_2E^{+}(z=L,t)-r_3  \sqrt{1-r_2^{2}} E^{+}(z=L,t-\tau)
\end{eqnarray*}
where $\tau$ is the round trip time of the field in the external cavity of length $L_{ext}$. 
Note that these boundary conditions take into account multiple reflections 
on the external mirror.  

Boundary conditions determine the complex phase constant, $k=(k_r +\imath k_i)$, of a 
plane wave solution of the form $E^{\pm}(z,t)=E_0^{\pm}(t) \exp{({\pm}\imath k z)}$. The 
real part of the wave vector, $k_r =\frac{\pi m}{L}$, yields the cavity modes 
of the laser resonator. The  
imaginary part of the wave vector is given by 
$k_i = |ln(R_1R_2)|/2L$, that accounts for the loss due to the cavity mirrors. 
The reference wavenumber $k_0$ associated with the carrier frequency $\omega_{c}$ 
corresponds to a particular cavity mode. 

The interaction of the electromagnetic wave with the medium is governed by the 
Bloch equations for two-level atoms. The optical Bloch equations are 
derived using the expressions for the total field $E(\vec r, t)$ and 
the polarization $P$ given above. The equations for $P$ and the population inversion 
$N$ are, 
\begin{eqnarray}
\frac{ \partial  P}{\partial t } &=& -\left( \gamma_{\bot} +\imath 
(\omega_{12} - \omega_{c}) \right) P + \frac{\Theta_{12}\Theta_{21} }
{\imath \hbar} N \left(  E^{+}e^{\imath k_0z} + 
E^{-}e^{-\imath k_0z}  \right)
\label{eqnarray:pol} \\
\frac{ \partial N}{\partial t } &=& - \gamma_{\|}\left( N -N_0 \right) -  
\frac{2 \imath}{\hbar} 
\left[\left( E^{+} P ^{*} - E^{-*} P \right)e^{\imath k_0z} + 
\left( E^{-} P ^{*} - E^{+*} P \right) e^{-\imath k_0z}
\right] 
\label{eqnarray:port}
\end{eqnarray}
where $\Theta_{12}$ and $\Theta_{21}$ are components of the dipole moment and 
$\omega_{12}$ is the transition frequency between the two levels.
The difference between the transition and carrier frequencies gives the detuning
 $\delta=\omega_{12} - \omega_{c}$. The decay rate for the polarization and
population inversion are $\gamma_{\bot}$ and $\gamma_{\|}$, respectively, and $N_0$
gives the pumping level.

\noindent It is convenient to scale the variables in equations 
(~\ref{eqnarray:e+})--(~\ref{eqnarray:port}) in the following way,
\begin{eqnarray*}
E^{\pm} \longrightarrow \frac{ \hbar}{\sqrt{2\Theta_{12}\Theta_{21}}} 
\frac{1}{\sqrt{\gamma_{\bot}\gamma_{\|} }} E^{\pm} \nonumber \\ 
P \longrightarrow - \frac{2\imath\epsilon_0}{\omega_{c}}
\frac{ \hbar}{\sqrt{2\Theta_{12}\Theta_{21}}}\sqrt{ \frac{\gamma_{\bot}}
{\gamma_{\|} }}P \nonumber \\
N \longrightarrow\frac{2\hbar\epsilon_0}{\omega_{c}}
\frac{1}{2\Theta_{12}\Theta_{21}}N \nonumber
\end{eqnarray*} 
\noindent The longitudinal coordinate is scaled to the length of 
the laser medium, $L$, 
and the time to the group velocity over $L$, $c/(\eta L)$, 
\begin{eqnarray*}
z \longrightarrow z/L \nonumber \\ 
t \longrightarrow \frac{c}{\eta L} t\nonumber
\end{eqnarray*}
With these scalings, equations (~\ref{eqnarray:e+})--(~\ref{eqnarray:port}) 
become, 
\begin{eqnarray}
\frac{\partial E^{+}}{\partial t} + \frac{\partial E^{+}}{\partial z} &=& 
-\langle P \exp{\left(-\imath k_0z\right)} \rangle \label{eqnarray:e2+} \\
\frac{\partial E^{-}}{\partial t} - \frac{\partial E^{-}}{\partial z} &=& 
-\langle P \exp{\left(\imath k_0z\right)} \rangle 
\label{eqnarray:e2-} \\
\frac{ \partial  P}{\partial t } &=& -\gamma_{\bot}\left[\left( 1 +\imath 
\frac{\delta}{\gamma_{\bot}} \right) P - N \left(  E^{+}e^{\imath k_0z} + 
E^{-}e^{-\imath k_0z}  \right)\right]
\label{eqnarray:pol2} \\
\frac{ \partial N}{\partial t } &=& - \gamma_{\|} \left[\left( N -N_0 \right) +  
\left[\left( E^{+} P ^{*} - E^{-*} P \right)e^{\imath k_0z} + 
\left( E^{-} P ^{*} - E^{+*} P \right) e^{-\imath k_0z}
\right] \right]
\label{eqnarray:port2}
\end{eqnarray}
The spatial variations in polarization and population over wavelength distances are 
treated by means of a standard Fourier series expansion \cite{lamb}, 
\begin{eqnarray}
P(z,t)= e^{\imath k_0z} \sum_{p=0}^{\infty}P^{+}_{(p)}e^{2\imath p k_0z} + 
e^{-\imath k_0z} \sum_{p=0}^{\infty}P^{-}_{(p)}e^{-2\imath p k_0z} \label{eqnarray:f1}\\
N(z,t)=N_{(0)} + \sum_{p=1}^{\infty}\left[  N_{(p)}e^{2\imath p k_0z} + (*) \right] 
\label{eqnarray:f2}
\end{eqnarray}
Substituting (~\ref{eqnarray:f1}) and (~\ref{eqnarray:f2}) in 
(~\ref{eqnarray:e2+})--(~\ref{eqnarray:port2}), the following coupled equations are obtained
\begin{eqnarray}
&&\frac{\partial E^{+}}{\partial t} + \frac{\partial E^{+}}{\partial z} 
= - P^{+}_{(0)}  \label{eqnarray:ep+} \\
&&\frac{\partial E^{-}}{\partial t} - \frac{\partial E^{-}}{\partial z} 
= - P^{-}_{(0)} \label{eqnarray:ep-} \\
&&\frac{ \partial  P^{+}_{(p)}}{\partial t } = 
-\gamma_{\bot}\left[\left( 1 +\imath 
\frac{\delta}{\gamma_{\bot}} \right) P^{+}_{(p)} 
+\left( N_{(p)} E^{+} + N_{(p+1)}E^{-}  \right)\right] + \xi_{(p)}^{+}(z,t)
\label{eqnarray:polp+} \\
&&\frac{ \partial  P^{-}_{(p)}}{\partial t } = 
-\gamma_{\bot}\left[\left( 1 +\imath 
\frac{\delta}{\gamma_{\bot}} \right) P^{-}_{(p)} 
+\left( N_{(p+1)}^{*} E^{+} + N_{(p)}^{*}E^{-}  \right)\right]+ \xi_{(p)}^{-}(z,t)
\label{eqnarray:polp-} \\
&&\frac{ \partial N_{(p)}}{\partial t } = - \gamma_{\|} \left[ N_{(p)}  +  
 \left( E^{-} P _{(p)}^{-*} + E^{+} P_{(p-1)}^{-*} -
E^{+*} P _{(p)}^{+} - E^{-*} P_{(p-1)}^{+}\right)  \right]
\label{eqnarray:portp}\\
&&\frac{ \partial N_{(0)}}{\partial t } = - \gamma_{\|} \left[
\left( N_{(0)} -N_0 \right) +  
\left( E^{+} P ^{+*}_{(0)} + E^{-} P^{-*}_{(0)}  + (*)
\right) \right]
\label{eqnarray:port0}
\end{eqnarray}

Equations (~\ref{eqnarray:ep+})-(~\ref{eqnarray:port0}) are numerically solved 
together with the boundary conditions (~\ref{eqnarray:bc}) for the field amplitudes 
$E^{\pm}$.   
In these equations we have also included Langevin noise terms 
$\xi_{(p)} (z,t)$ as polarization sources. Such spatially distributed noise terms
model independent spontaneous emission process in different points of the cavity.
They are taken to be Gaussian white noise in space and time with zero mean and correlations
$\langle \xi_{0} (z, t) \xi_{0}^{*}(z', t') \rangle = 
\beta  \delta(t - t') \delta(z - z')$.  The spontaneous emission noise intensity is measured 
by $\beta$, which depends, in
principle, on the instantaneous value of the population levels \cite{fleck,haken,lax}. In practice we 
approximate the spontaneous emission intensity by a constant parameter \cite{lax} and, in addition,
we will only include 
noise sources for $p=0$.

If we want to take into account spatial 
variations of the material variables, terms beyond the $p=0$ term must be retained in
(~\ref{eqnarray:ep+})-(~\ref{eqnarray:port0}).
Such spatial variations come from the optical wave inside the cavity and from the finite
reflectivity of the mirrors. 
In practice, we will truncate our expansions after the first harmonic $p=1$. We 
refer to the terms of order $p=1$ as "grating terms", because their effect is comparable 
with the introduction of a spatial grating in the cavity at a spatial 
frequency that corresponds to half the natural wavelengths of the fields inside the cavity. 
Adiabatic elimination of the polarization variables within this TW model is not 
possible since it leads to a  
set of equations which are intrinsically
unstable as can be easily seen by a straightforward linear stability analysis of 
the reduced equations. 

The above set of Equations(~\ref{eqnarray:ep+})-(~\ref{eqnarray:port0}) 
contains a complete description (within the
slowly varying approximation) of an homogeneously broadened two-level laser. It should
provide a good description of multimode behavior of homogeneously broadened 
dye or solid state lasers.
When this model is used to learn about multimode behavior of semiconductor lasers, $N$ has
to be identified with carrier number, $N_0$ with the carrier number at transparency 
and $\gamma_{\|}  N_0$ with the injection current $J$.
For semiconductor lasers
some other aspects have to be considered, mainly carrier diffusion and inhomogeneous
broadening from the band structure. 

Carrier diffusion can be taken into account by introducing a diffusion term 
$D \nabla^{2} N$ in (~\ref{eqnarray:port}), where $D$ is the diffusion coefficient. 
With  this term, the equations for the two first orders of 
the Fourier expansion for the carrier number, are modified as follows: 
\begin{eqnarray*}
&&\frac{ \partial N_{(0)}}{\partial t } = D\nabla^{2} N_{(0)}  + J - \gamma_{\|} \left[
 N_{(0)}  -  
\left( E^{+} P ^{+*}_{(0)} + E^{-} P^{-*}_{(0)}  + (*)
\right) \right]
\nonumber\\
&&\frac{ \partial N_{(1)}}{\partial t } = - \gamma_{\|} \left[ 
(1 + \frac{4Dk_0^{2}}{\gamma_{\|}}) N_{(1)}    -  
 \left( E^{-*} P _{(0)}^{+*} + E^{+} P_{(0)}^{+*} +
E^{+*} P _{(1)}^{+} + E^{-*} P_{(1)}^{-*}\right)  \right]
\nonumber 
\end{eqnarray*}
 
For typical parameter values 
(see table $I$), it is easy to see that the term $D\nabla^{2} N_{(0)}$ can be neglected 
in comparison 
with  $\gamma_{\|}N_{(0)}$.  The equation for $N_{(0)}$ remains then essentially unchanged 
and the main effect of carrier diffusion appears in the equation for $N_{(1)}$ through a 
modification, $4Dk_0^{2}$,  of the damping coefficient, where 
 $k_0$ the wavenumber of the mode around which we have made 
the slowly varying amplitude approximation. 

Our TW homogeneously broadened two-level model includes
 the coherent coupling between the optical field and the two-level medium 
by means of the optical Bloch equations, allowing for phase sensitive multimode operation 
and four-wave mixing. 
An important characteristic of semiconductor lasers is the strong phase-amplitude dynamical
coupling often described by the linewidth enhancement factor $\alpha$-factor \cite{henry}. 
Such phase sensitivity is
also produced by cavity detuning in two-level models. In fact, semiconductor laser behavior
has been modeled by identifying a constant $\alpha$-factor with a large detuning (the same
for all modes) in a 
two-level description \cite{mandel}. However, in  such a two-level  model the maximum 
gain occurs at zero detuning,  
where the 
carrier-induced refractive index change vanishes, while for a semiconductor laser there
is  asymmetric gain with maximum gain away from resonance. The origin of the 
$\alpha$-factor in semiconductors
has been explained by approximating the electron-hole recombinations at different frequencies
by an ensemble of  two-level atoms with different transition
energies \cite{vahala}. In order to match the typical
measured values of $\alpha \sim 1-10$ with a detuning in two-level models, lasing away from 
resonance must be artificially enforced. This is not possible in our multimode travelling 
wave model as modes will appear where the different cavity modes find strong gain independent 
of the detuning of a particular reference mode. 

Hence the effects of the $\alpha$-factor are not taken into
account in (~\ref{eqnarray:ep+})-(~\ref{eqnarray:port0}), but, 
on the other hand we note that there are highly doped 
semiconductor lasers for which, at low temperatures, a nearly homogeneously broadened 
gain is expected \cite{yamamoto}.
In summary, (~\ref{eqnarray:ep+})-(~\ref{eqnarray:port0}) supplemented with carrier diffusion
incorporate a number of important mechanisms needed to understand multimode behavior in 
semiconductor lasers, but one should be aware that additional steps might be needed for a full
understanding of multimode behavior of inhomogenously broadened semiconductor lasers.
In our numerical analysis reported in sections 3 and 4 we  use parameter values (see table $I$) 
which  correspond to the characteristic time scales of a semiconductor laser.

Equations 
(~\ref{eqnarray:ep+})--(~\ref{eqnarray:port0}), with the given boundary conditions (~\ref{eqnarray:bc}), 
are integrated numerically with a semi-implicit finite difference integration scheme 
\cite{fleck}. We use an integration step  $\Delta z = \Delta t = 1/300$, which corresponds to a
spatial step of $0.75 \mu m$ and a temporal step of $9$ $fsec$ for $\eta = 3.5$ and $L=250$ $\mu m$.
As initial conditions we use a random distribution of the 
electric field along the cavity. We take a real electric field distributed between $0$ and $1$ with an 
amplitude of $10^{-4}$. For the polarization and the population 
variables we take initial values $P^{\pm}=0$ and $N= 0.95 N_0$. We have checked that steady
state properties do not change when using
a non-random initial field profile of the same amplitude and which satisfies boundary
conditions. The stochastic part of the equation for the polarization is integrated using
an Euler algorithm of order $(\Delta t)^{1/2}$ \cite{maxi82,greiner}. Independent 
Gaussian random numbers
are generated at each point in the spatio-temporal grid using an optimized version of
the Burlish-Stoer algorithm  \cite{raul}.

\section{Factors influencing multimode behavior of a Fabry P\'{e}rot laser}
The TW model described above includes three main factors which influence mutimode behavior:
a) Spatial hole burning (SHB), b) Spontaneous emission noise  and c) Carrier diffusion.
These three effects compete with different relative importance depending on the 
operation point of
the laser above threshold. In this section we consider these factors for a solitary laser with no 
external mirror ($r_3=0$). Spatial Hole Burning appears via the "grating terms" associated with
the Fourier modes in (~\ref{eqnarray:f1})--(~\ref{eqnarray:f2}) for $p>0$. 
The short range spatial variation of the atomic
variables described by these Fourier modes
tends to be washed out by carrier diffusion. On the other hand, 
close to threshold spontaneous emission noise is expected to be more
important in exciting many modes, while it should become less relevant for high
power emission well above threshold. For solid state lasers there is no diffusion of the atoms
of the active medium and spatial hole burning is the dominant effect, while for semiconductor
lasers carrier diffusion is very important and the noise strength is high.

In the following we discuss these factors separately to elucidate their effects and relative
importance. As a reference situation we show in Fig. 1. the Field Power Spectrum (FPS) obtained
when spontaneous emission noise and carrier diffusion are neglected and 
the Fourier expansion in (~\ref{eqnarray:f1})--(~\ref{eqnarray:f2}) is truncated 
at $p=0$ (no SHB). The FPS is obtained by a standard FFT routine with
a time window of $2$ $ns$, ($2^{16}$ points) when the laser has reached a steady state 
$20 ns$ after switch-on. 
We obtain clear single mode emission for the injection current $J=1.5 J_{th}$ shown. We find
such single mode behavior  for different current values, closer 
to threshold and also well above threshold.
\subsection{Multimode emission with no carrier diffusion}

{\bf a) Spatial Hole Burning:}

Fig.~2 shows the FPS obtained by neglecting spontaneous emission noise and carrier diffusion
and taking into account SHB by truncating the Fourier expansion in 
(~\ref{eqnarray:f1})--(~\ref{eqnarray:f2}) at $p=1$.
It is clear that the "grating terms" induce the growth
of additional modes above threshold. Only very close to threshold (Fig.~2.a), (up to 
$J=1.03 J_{th}$), the behavior remains single-mode. Further above threshold SHB causes the
excitation of side modes located at the frequencies given by the cavity length and the
group velocity inside the 
cavity, giving rise to multimode laser emission.
Fig.~2 shows the results obtained for zero detuning. For a  detuning of $\delta=70 GHz$ 
we obtain the same qualitative 
behavior, but the power 
spectrum is no longer symmetric around the cavity frequency in the case of multimode behavior 
since the gain peak is then $\delta=70 GHz$ away from the reference cavity mode (see
also Fig.~5a).   

We point out ( Fig.~3), that for very  high levels of pumping ($J=15 J_{th}$ and above), 
we observe a splitting of the FPS into two groups of modes as reported for solid state lasers in 
\cite{khanin}.

{\bf b) Spontaneous Emission Noise:}

Fig.~4 shows the FPS (also for zero detuning) and the same pumping levels as Fig.~2, but with 
spontaneous emission noise added. Noise excites side modes at pump levels 
for 
which SHB alone does not destroy single mode emission. Noise has no important effects
for large pump values. For $J>1.9 J_{th}$ the Side Mode Suppresion Ratio (SMSR) of the  modes
excited by spontaneous emission is above 20 dB. In general  both SHB and 
spontaneos emission noise contribute to multimode emission. We note that a finite time 
interval of averaging in the presence of noise causes small asymmetries in the FPS while 
did not occur for $\beta=0$ (Figs.2 and 3).

Figs.~5.a-d show the FPS for a detuning $\delta=70 GHz$ taking into account SHB 
and spontaneous emission noise. The nonzero detuning leads to a nonsymmetric FPS. 
As the pumping level increases, the number of lasing modes increases as well as 
the power of the side-modes with respect to the main one. However, there is a level of 
pumping ($J> 2.5 J_{th}$) beyond which additional side-modes do not grow and 
the number of relevant  longitudinal modes 
remains constant.  In addition, for  $J> 4 J_{th}$, the power ratio between the 
different modes remains constant. Except for the different time scales involved, the general 
qualitative behavior as the pumping is increased (as shown in Fig. 5a-d) it is the one  
typically observed in solid state lasers \cite{khanin}. 
 
\subsection{The Effect of Carrier Diffusion}
As discussed in section II, the main effect of diffusion in our mathematical modelling is to
give rise to an effective damping rate $ \gamma_{\|} + 4Dk_0^{2}$ for the first Fourier mode
of the carrier number $N_1$. This higher damping rate reduces the effect of SHB. 
Fig.~5 shows a comparison of the FPS at different current levels for 
the case of zero diffusion, in the left column, and for  
a given diffusion coefficient, in the right column.  In this case, the central mode 
grows faster than the adjacent modes 
as the current increases and eventually the amplitude of the adjacent modes 
effectively saturates. Thus we observe restoration of single mode
emission for large enough pumping level as a consequence of diffusion.
However, smaller values of the diffusion coefficient do not fully kill multimode emission far 
above threshold. For such multimode behavior the total output power of the
laser exhibits fast oscillations associated with mode beating and slower oscillations 
of the
envelope. The latter oscillations are the ordinary relaxation oscillations 
of the laser (see inset Fig.~6a). For intermediate pump levels and for 
the typical diffusion coefficient of
semiconductors used in Fig.~5 we have observed simultaneous operation of few modes,
but we have not observed mode-hopping of the form of switching of most of the 
power from one mode to another with nearly constant total power when 
the average spectrum showed only a few modes.

An early discussion of multimode behavior in semiconductor lasers was given 
in \cite{lee} in terms of rate equations with a parabolic approximation for the dependence
of the gain parameters of different modes as a function of their wavelength. 
Emphasis in that work was on the effect of spontaneous emission noise. They
showed that, as the current increases, the width of the emitted mode spectrum envelope 
narrowed. 
In their model the main mode 
grew faster than the adjacent modes as the current was increased near threshold, and eventually, 
the amplitude of the side modes saturated with increasing the current. Hence, over a certain  
power level, additional power increases with increasing current were due to the 
growth of a single mode. This general behavior is the one we find when we neglect SHB and 
carrier diffusion. The latter two effects are discussed phenomenologically in \cite{lee} 
by decreasing the value of the gain coefficients of the modes as a consequence of diffusion.
The onset of SHB is identified with the output power level for which such a decrease is equal
to the difference in gain between the lasing mode and its nearest neighbor. The analysis in
\cite{lee}
describes different regions of multimode behavior depending on pump level and cavity length.
The general picture was a transition from noise dominated multimode emission close to 
threshold to single mode emission for higher injection. A second
transition to multimode emission dominated by SHB was predicted further above threshold. In our
TW model the different gain at different cavity frequencies and the gain modifications
due to SHB and diffusion appear as a consequence of the consistent dynamical equations.
In our numerical analysis we obtain 
a transiton from multimode to single mode behavior asthe injection current is increased 
keeping all other parameters fixed (Figs.5e-h). For a fixed injection current, single 
or multimode behavior depends on the value of the diffusion coefficient, so that, the 
injection current for which single mode operation is achieved could be changed by the 
possible dependence of the diffusion coefficient on the injection current which we 
have neglected. We finally note that, for the range of parameters
we have explored, we do not observe the second transition from single to multimode behavior 
obtained in \cite{lee} as the injection current is further increased.


\section{Optical feedback and mode selection} 
Optical feedback from an external mirror combines with the 
intracavity fields and  can enhance the power of a particular  
intracavity mode. As a consequence it can lead to mode selection depending on the 
feedback level
and the external cavity length. We first consider weak feedback levels to avoid instabilities
of the output intensity. We have analyzed short external cavities of different 
lengths up to $L_{ext}=1 cm$. For the shortest cavity considered with $L_{ext}=75$ $\mu m$ 
and weak feedback, single mode selection is achieved, while 
for the longest external cavities considered, many intracavity modes are excited by the 
weak feedback. For strong feedback
and the shortest cavity considered, two groups of intracavity modes can be selected. 
For this analysis of
feedback effects we have chosen
an intermediate value of carrier diffusion for which multimode emission occurs in
a large range of pump values ($D=2\times 10^{-4}$ $m^{2}s^{-1}$) and the same detuning used
for Fig.~5 ($\delta=70 GHz$).

For the shortest external cavity of $L_{ext}=75$ $\mu m$, 
the  external cavity modes 
are separated by  $2000 GHz$, while the 
laser cavity modes, for our paremeters values, have an intermode spacing
of $171 GHz$. The laser cavity mode
closest to the maximum of the gain curve which resonates with one of the external 
modes is excited and can be selected. The intracavity mode spectrum is centered in 
such a way that the mode at zero 
frequency coincides with an external cavity peak. In this case, we  
excite the zero order mode, $m=0$. We can select the mode at a 
frequency of $\approx171 GHz$, $m=1$ by
changing the external cavity by $\frac{171}{2000}\frac{ \lambda}{2}$. The selection of the mode $m=1$ 
is seen in Fig.~6, where single mode emission with a SMSR of 20 dB is obtained 
for an external mirror field reflectivity $r_3$ of just $1\%$. 
We show in Fig.~6 the FPS at the rigth mirror for different values of the external mirror 
reflectivity 
(right column), and  the total output intensity as a function of time (left column), 
in the steady state 
regime of laser emission. In the output intensity variations one can identify
the fundamental modulation fequency as being the beat frequency between adjacent 
intracavity modes. From a) to d), 
we observe how the amplitude 
of these oscillations is reduced being almost zero when nearly single mode emission is
achieved (d).  In the inset of Fig.~6.a) and d), we plot the intensity as a function 
of time for 
a longer time interval of 2 nsec. The envelope of the oscillations of the output power
exhibits slower oscillations associated with the relaxation 
oscillations of the laser. These oscillations are present for all weak feedback levels 
with a frequency which is only slightly modified by weak feedback: For 
the solitary laser the frequency is $\approx 1.21 GHz$ (Fig.6a) and it becomes 
$\approx 1.18 GHz$ for $1\%$ field reflectivity (Fig.6e). 
   
Results on the effect of weak feedback from an external cavity of $L_{ext}=450$ $\mu m$
(giving a mode spacing of 333GHz) are shown in Fig.~7.
For this intermediate length of the external cavitites that we have considered, the external 
mode separation coincides with twice the internal one, so that the laser might be forced to
reduce its multimode emission to lasing in two modes. We consider here a lower 
value of pumping level than for the shortest cavity 
in Fig.~6 to have a larger number of lasing modes. We observe
that two modes can be selected  with a SMSR of $\approx 20 dB$  for $r_3 = 2\%$. Note 
that the peaks in the spectrum alternate in height, as alternate modes of the laser are 
nearly resonant with the external cavity as well. 

The behavior shown in Fig.~8 for an external cavity of $L_{ext}= 1$ $cm$ 
(with mode spacing of $\approx 15.4 GHz$) is typical of what is 
observed for $L_{ext}>0.1 cm$. For 
these external cavity lengths mode selection is not found, and instead, weak feedback
undamps many intracavity modes for a given value of the field reflectivity $r_3$.
The excitation of many modes occurs, for the parameter values in Fig.~8, for $r_3=2\%$, 
but time dependence of the output power becomes chaotic for field reflectivities $r_3>5\%$.

Finally, we have performed a
preliminary study for different feedback levels in the region of 
moderate and strong feedback. While a standard classification of feedback regimes is well known
for single mode lasers \cite{feedbackregimes}, no such classification seems available 
for multimode lasers. The consideration of strong  feedback levels 
is not an extra difficulty in our Travelling Wave model
since it properly takes into account multiple external reflections between the two mirrors that
define the external cavity. However, we have checked that these multiple external reflections
do not introduce significant effects for $r_3 \leq 5\%$. 
Our results for the shortest external cavity are shown in Fig.~9
which is a continuation of the sequence detailed 
in Fig.~6 for increasing field reflectivity $r_3$. 
In the region of weak feedback  ($r_3 < 5\%$) shown in Fig.~6 mode selection is achieved.
For moderate feedback levels ($r_3 \approx 5-15\%$) the laser output becomes unstable exhibiting
a broad Field Power Spectrum (Fig.~9.f). However, for high feedback levels as shown in 
Figs.~9.g to ~9.j, two groups of intracavity modes are selected and a few-mode emission is
recovered. The spacing between these two groups of modes ($\approx 2000 GHz$) is given
by the external roundtrip time or separation of external cavity resonances. 
The internal cavity 
modes are still well defined within the selected groups 
due to the broad external cavity spectrum. 

\acknowledgements
We have greatly benefited from discussions with C. Ning and R. Indik and useful comments
from N. B. Abraham.
This work has been supported by AFOSR contract F49620-94-1-0144DEF.
MH and MSM also
acknowledge financial support from Comision Interministerial de
Ciencia y Tecnologia (CICYT, Spain) project
TIC93-0744. 


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\begin{table}
\caption{Meaning and values of the different parameters appearing in the 
model}
 \begin{tabular}{|c|c|c|c|}
Parameter & Meaning & Value & Units \\
 \hline \hline
 $r_1$ & Field reflectivity of left side mirror & 99.5$\%$ &-\\
 $r_2$ & Field reflectivity of right side mirror & 56.6$\%$ &-\\
 L & Length of the diode cavity & 250 & $\mu m$  \\
 $\gamma_{\|}$ & Inverse carrier lifetime & 2 $\times 10^{8}$ &  $s^{-1}$ \\
 $\gamma_{\bot}$ & Inverse polarization characteristic time & 1 $\times 10^{13}$ &  $s^{-1}$ \\
 $\beta $ & Spontaneous emission rate & 1$\times 10^{4}$ & $s^{-1}$ \\ 
 $\eta$ & Index of refraction  & 3.5 &-\\
 $\delta$ & Detuning & 0 or 70 & GHz \\
 \end{tabular}
 \end{table}
\newpage
\begin{figure}
\caption{:Field power spectrum (log scale) for $J=1.5 J_{th}$ neglecting SHB, 
spontaneous emission noise ($\beta=0$) and diffusion ($D=0$). $\delta=0$.}
\end{figure}
\begin{figure}
\caption{:Field power spectrum (log scale) for: a) $J=1.02 J_{th}$, b) $J=1.7 J_{th}$ 
and c) $J=3.5 J_{th}$, neglecting spontaneous emission noise ($\beta=0$) and 
diffusion ($D=0$). $\delta=0$.}
\end{figure}
\begin{figure}
\caption{:Field power spectrum (log scale) for $J=15 J_{th}$ neglecting 
spontaneous emission noise ($\beta=0$) and 
diffusion ($D=0$). $\delta=0$.}
\end{figure}
\begin{figure}
\caption{:Field power spectrum (log scale) for same pumping levels than in Fig.2. 
Spontaneous emission noise $\beta=1\times 10^{4}$ $s^{-1}$. 
Diffusion is neglected ($D=0$) and $\delta=0$.}
\end{figure}
\begin{figure}
\caption{:Field power spectrum (linear scale) for no diffusion (left 
column) and for $D=4\times 10^{-4}  m^2 s^{-1}$ (right column) and different pumps values: 
a,e) $J=1.05 J_{th}$, b,f) $J=1.5 J_{th}$, c,g) $J=2.5 J_{th}$ and d,h) $J=3.5 J_{th}$. 
Spontaneous emission noise $\beta=1\times 10^{4}$ $s^{-1}$ and $\delta=70$ GHz.}  
\end{figure}
\begin{figure}
\caption{: Total intensity as a function of time (left column) and Field Power Spectrum 
(right column) for $J=1.5 J_{th}$, $L_{ext} = 75.055\mu m$, $D=10^{-4} m^{2}s^{-1}$, 
$\delta=70$ GHz, $\beta=1\times 10^{4}$ $s^{-1}$ 
and different external reflectivities: a,e) $r_3=0\%$, 
b,f) $r_3=0.1\%$, c,g) $r_3=0.5\%$ and d,h) $r_3=1\%$.  
The inserts in a) and d) correspond 
to a plot of the total intensity as a function of time in a large time window. } 
\end{figure}
\begin{figure}
\caption{:Total intensity as a function of time (left column) and Field Power Spectrum 
(right column) for $J=1.3 J_{th}$, $L_{ext} = 450\mu m$, $D$, $\delta$ and $\beta$ 
the same than in Fig.5, 
and different external reflectivities: a,e) $r_3=0\%$, 
b,g) $r_3=0.2\%$, c,h) $r_3=0.5\%$, d,i) $r_3=1\%$ and e,j) $r_3=2\%$.}
\end{figure}
\begin{figure}
\caption{:Total intensity as a function of time (left column) and Field Power Spectrum 
(right column) for $J=1.4 J_{th}$, $L_{ext} = 1 cm$, $D$, $\delta$ and $\beta$ 
the same than in Fig.5,  
and different external reflectivities: a,d) $r_3=0\%$, 
b,e) $r_3=0.75\%$ and c,f) $r_3=5\%$.}
\end{figure}
\begin{figure}
\caption{: Total intensity as a function of time (left column) and Field Power Spectrum 
(right column) for $J=1.5 J_{th}$, $L_{ext} = 75\mu m$, $D$, $\delta$ and $\beta$ 
the same than in Fig.5,  
and different external reflectivities: a,f) $r_3=10\%$, 
b,g) $r_3=15\%$, c,h) $r_3=25\%$, d,i) $r_3=35\%$ and e,j) $r_3=40\%$.} 
\end{figure}
\end{document}