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\title{Combined effects of semiconductor gain dynamics, spin
dynamics and thermal shift in polarization selection in VCSELs}

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\author{M. San Miguel\supit{$\ast$}\supit{a}, S. Balle\supit{a}, J. Mulet\supit{a},
C.R. Mirasso\supit{b}, E. Tolkachova\supit{c}, and J.R.
Tredicce\supit{c}
\skiplinehalf \noindent
\supit{a}Instituto
Mediterraneo de Estudios Avanzados, IMEDEA, (CSIC-UIB) \\E-07071
Palma de Mallorca, Spain. \\ \noindent \supit{b}Department de
F\'{\i}sica, Universitat de les Illes Balears, E-07071 Palma de
Mallorca, Spain. \\ \noindent \supit{c}Institut Non Lineaire de
Nice, CNRS UMR 6618, 06560 Valbone, France. }

\authorinfo{\supit{$\ast$}Correspondence: Email:
maxi@imedea.uib.es; WWW: http://www.imedea.uib.es/Photonics;
Telephone: 34-971 173229; Fax: 34-971 173426}

\begin{document}
\maketitle

\begin{abstract}
We discuss mechanisms of polarization switching (PS) in Vertical
Cavity Surface Emitting Lasers (VCSELs) within a mesoscopic
approach based on an explicit form of a frequency-dependent
complex susceptibility of the QW semiconductor material.  Cavity
anisotropies, spin carrier dynamics and thermal shift of the gain
curve are also taken into account in this framework. For large
birefringence we find a PS due to thermal shift. For small
birefringence we find a different PS, from the high-gain
to the low-gain polarization state, that occurs at constant
temperature. We characterize polarization partition noise in terms 
of power spectra. Transverse effects for PS in 
gain guided VCSELs are also considered.
\end{abstract}

\keywords{VCSEL, Polarization Switch, Semiconductor
susceptibility, RIN}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1
\section{INTRODUCTION}

Vertical Cavity Surface Emitting Lasers (VCSELs) typically emit
linearly polarized (LP) light and, for bias currents relatively
close to threshold, in the fundamental transverse mode. Often, the
LP state selected at threshold becomes unstable as the injection
current is increased, and a switch to the orthogonal LP state
occurs for some value of the injection current\cite{VCSELs}.
Understanding the physical mechanisms that determine polarization
selection and polarization switching (PS) is a first step towards
the development of applications based on polarization control such
as optical switching, information processing or storage, etc.

Polarization selection has been often attributed to linear cavity
anisotropies, being birefringence one of the dominant\cite{Woerdman}.
A mesoscopic description of polarization, carrier spin and
nonlinear semiconductor gain dynamics has been put forward through
a Spin Flip Model \cite{maxi95}. In this context it has been shown
that nonlinear anisotropies which result from the semiconductor
dynamics are also important mechanisms of polarization selection.
In particular PS has been seen to be caused by phase-amplitude
dynamics that follow from the combined effect of birefringence and
saturable dispersion associated with the linewidth enhancement
factor, $\alpha$ \cite{jmartin97,RegaladoAPL1997}. A limitation of
the original formulation of this explanation is that the model
does not incorporate a frequency dependent gain, therefore it does
not take into account either nonlinear gain differences between
the LP states, or the effects of thermal shift of the cavity
resonances and the gain peak. Gain differences between the two LP
states and thermal shift have been, however, invoked as a dominant
mechanism of PS \cite{Choquette/JSTQE:1995}: Gain differences
appear because of the different frequency of the LP states in a
birefringent optical cavity, and a change of sign in the gain
difference between the two LP states is caused by the thermal shift
of the gain curve when increasing the injection current. A
frequency dependent gain was already introduced within the context
of the SFM model in ref. [7] %\cite{OL97}% 
giving a good description of
general properties of polarization switching and transverse mode
competition. However, the complex susceptibility in ref. [7] %\cite{OL97} 
is based on a simple two-level approach and the choice of the
$\alpha$-factor is justified as an effective description of
thermal lensing effects.

In this paper we elaborate an extension of the SFM which
incorporates a realistic spectral dependence of the gain and index
of refraction of active semiconductor quantum well (QW) structures
\cite{optlett99}. Within this framework there is room for the
different mechanisms of PS described above. We find thermally
induced PS, but for low birefringence and finite values of the
carrier spin flip rate we also find PS from the
high-gain to the low-gain polarization states caused by coupled
phase-amplitude nonlinear dynamics.

The outline of the paper is as follows. In Section 2 we recall the
main features of the complex frequency-dependent quantum well
susceptibility which enters as a main ingredient of our dynamical
model. Section 3 describes our results for the different
mechanisms of PS in the fundamental transverse mode. We also
discuss the changes in the Relative Intensity Noise spectrum
through the PS. Finally Section 4 contains some first results on
transverse effects and transverse beam profile.




%------------------------------------------------------------------------------------------
\section{QUANTUM WELL ELECTRONIC SUSCEPTIBILITY}

We describe the optical field in terms of the slowly varying
amplitudes of its circularly polarized components $E_\pm$. Each
component interacts only with a part, $N_\pm$, of the
total carrier density $N = N_+ + N_-$. The carrier densities
$N_\pm$ are associated with carrier populations of different spin
and tend to be equalized by spin flip processes at a rate
$\gamma_j$ \cite{VCSELs,maxi95}. A scheme of the interactions and
band structure considered is shown in Fig. 1.
\bfig{h}
\bcen \begin{tabular}{c}
\psfig{file=bandes.eps,width=12cm,clip=true}
\end{tabular}\ecen
\caption{An extension of the SFM based on a four level system. We
consider a conduction band (CB) and only one heavy-hole (HH) spin
sub-bands. The spin of the electrons are equalized at rate
$\gamma_j$ and instantaneously for the holes \cite{temps}. The
circularly polarized components of the field $E_\pm$ fields are
coupled by birefringence $\gamma_p$.} \label{fig:bandes} \efig

The interaction of each circularly polarized component of the
field with its corresponding carrier density is described by a
frequency-dependent complex susceptibility $\chi_\pm$ which
incorporates the basic nonlinear properties of the QW gain and
index of refraction \cite{Balle1998}. The form of the
susceptibility described below has been shown to give a good
description, in particular, of semiconductor optical amplifiers
\cite{BallePTL}. The calculation of the susceptibility $\chi_\pm$
involves a summation over all $k-$states in the first Brillouin
zone weighted with the carrier occupancy of each state. It can be
analytically evaluated when the following simplifying assumptions
are made \cite{Balle1998}: i) the dipole matrix element between
the valence and conduction bands is independent of $k$, ii) the
width of each transition is also independent of $k$, iii) all
carriers are in quasi-equilibrium within each band, and iv) low
temperature limit. When only one electron and one hole (parabolic)
bands are considered it is found that,
\begin{equation}
\chi(\omega_{\pm},D_{\pm}) = \chi_0 \left[ \ln \left(1-\frac{2 D_\pm}{u_\pm + i} \right) +
    \ln \left( 1 - \frac{D_+ + D_-} {u_\pm + i}\right)
    -\ln \left( 1-\frac{b}{u_\pm + i} \right) \right] \; ,
\label{eq:QWsusceptibility}
\end{equation}
where $D_\pm \equiv N_\pm/N_t$ is the carrier density per spin orientation
normalized to the (total) transparency carrier density, $N_t$. The dependence on the
frequency of the fields arises through
\begin{equation}
u_\pm = \frac{\omega_\pm}{\gamma_\bot} +
   \Delta + \sigma \left( D_+ + D_- \right)^{1/3} \; ,
\end{equation}
where $\Delta = (\Omega -\omega_t)/\gamma_\bot$ is the detuning
between the cavity resonance $\Omega$ and the nominal transition 
frequency normalized to the linewidth, $\gamma_\bot$, of the optical transitions for
fixed $k$. This linewidth corresponds to the inverse of the
dephasing time of the material polarization. $\Omega + \omega_\pm$ are the optical 
frequencies of each circularly polarized wave. Band-gap
renormalization effects have been included in
(\ref{eq:QWsusceptibility}) through $\sigma$ which describes the
band-gap shrinkage with carrier density. The parameter $b$ is
linked to the total energy span of the conduction and valence
bands and it determines the optical response of the QW material in
the absence of carriers, i.e., it sets the background index of
refraction and absorption of the unpumped active region. Finally,
$\chi_0$ is the device-dependent parameter that determines the
material gain.

From a microscopic point of view one defines lattice and electron
plasma temperatures. Evolution equation for these magnitudes can
indeed be obtained and integrated with the rest of electric and
carrier equations \cite{moloney98}. In the mesoscopic approach
that we follow here only an averaged temperature is taken into
account. In particular, as the main part of generation and
recombination events take place in the active region of the
device, we are interested in the active region temperature. Within
this context, a primary thermal effect, namely, the relative shift
of the cavity frequency and the material gain spectrum, is easily
taken into account in (\ref{eq:QWsusceptibility}) by letting
$\Delta$ to depend on temperature. Since the gain spectrum
redshifts 4-5 times faster than the cavity resonance, increasing
temperature corresponds to increasing $\Delta$. We consider only
temperature changes small enough to neglect thermal changes in
material gain and transparency carrier density.


A subtle technical point is how to make use of the susceptibility
(\ref{eq:QWsusceptibility}) (given in the frequency domain) in
nonlinear dynamical equations written as differential equations in
time. What is needed is an analytical expression for the material
polarization $\tilde{\mathcal P}_\pm (w)$ as function of time. We can
start from the constitutive relation, in frequency domain,
$\tilde{\mathcal P}_\pm (w_\pm )=\varepsilon_{0}\chi(w_\pm )\tilde{\mathcal
E}_\pm (w_\pm )$. Our fields have a narrow interval of frequencies compared
with the gain broad band, so that we can use a Slowly Varying
Approximation (SVA). We take $\Omega$ as the carrier frequency,
$\omega_\pm$ as the SVA frequency, and $w_\pm =\Omega+\omega_\pm$
\begin{eqnarray}
{\mathcal E}_\pm (t)&=&E_\pm (t)e^{-i\Omega t}, \label{eq:svae}\\ 
{\mathcal P}_\pm(t)&=&P_\pm (t)e^{-i\Omega t}. \label{eq:svap}
\end{eqnarray}
By Fourier transform of (\ref{eq:svae}), (\ref{eq:svap}) we obtain
the constitutive relation for the SVA components:
\begin{equation}
\tilde{P}_\pm (\omega_\pm )=\varepsilon_0\chi(\Omega+\omega_\pm )\tilde{E}_\pm (\omega_\pm),
\end{equation}
so that
\begin{equation}
P_\pm(t)=\int_{-\infty}^\infty
\frac{d\omega_\pm}{2\pi}\varepsilon_0\chi(\Omega+\omega_\pm)\tilde{E}_\pm(\omega_\pm)
e^{-i\omega_\pm t}.
\end{equation}
Since $\omega_\pm\ll\Omega$ we can expand $\chi(\Omega+\omega_\pm)$ in a
power series  and integrate the series term by term. The carrier
densities are assumed to be constant in the material polarization
time scales $(\gamma_\bot^{-1})$.
\begin{eqnarray}
P_\pm(t)&=&\varepsilon_0\int_{-\infty}^\infty
\frac{d\omega_\pm}{2\pi}\left[\chi(\Omega)
     +\omega_\pm\chi^\prime(\Omega)+\ldots\right]\tilde{E}_\pm(\omega)e^{-i\omega_\pm t}= \nonumber \\
    &=&\varepsilon_0\left[\chi(\Omega)E_\pm(t)+\chi^\prime(\Omega)i\dot{E}_\pm(t)+\ldots\right]
\approx\varepsilon_0\chi\left(\Omega+i\frac{\dot{E}_\pm(t)}{E_\pm(t)}\right)E_\pm(t),
\label{eq:finalrelation}
\end{eqnarray}
where we have defined
$\chi^\prime(\Omega)\equiv\frac{\partial\chi(\Omega+\omega\pm,D)}{\partial
\omega}|_{\omega_\pm=0}$. Equation (\ref{eq:finalrelation}) allows us
to give a formal way to include the electronic susceptibility in
the laser equations. This essentially amounts to replace
self-consistently in the dynamical equations the frequencies
$\omega_\pm$ by the instantaneous frequencies $\dot{E}_\pm/E_\pm$.\vspace{0.5cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2
\section{POLARIZATION SWITCHING IN THE FUNDAMENTAL TRANSVERSE MODE}

\subsection{Dynamical equations}\label{sec:model1D}
The dynamical evolution equations of the SFM extended to
include the spectral dependence of the material response to the
optical field are \cite{optlett99}: \pagebreak   
\begin{eqnarray}
\dot{E}_{\pm }(t) &=&-\kappa E_\pm -i\frac{a\Gamma}{2}
P_\pm -(\gamma_a+i\gamma_p)E_\mp +\sqrt{\beta D_\pm}
\xi_\pm(t) \label{eq:sfme} \\
%
\dot{D}_\pm(t)&=&\mu/2-AD_{\pm }-BD_{\pm }^{2}-\gamma _{j}(D_{\pm }-D_{\mp
})-\frac{a}{2}(P_{\pm }E_{\pm }^{*}-c.c.)  \label{eq:sfmn} \\
%
P_{\pm }(t) &\simeq &\varepsilon_0\chi(\Omega
+i\frac{\dot{E}_\pm}{E_\pm},D_{\pm }) E_{\pm }.
\label{eq:sfmp}
\end{eqnarray}

The meaning of the different parameters and typical values used in
our numerical simulations are given in Table 1.  We have
normalized the fields such that $\left|E_{\pm}\right|^2$ is
proportional to the photon densities, $n_e$ and $n_g$ are the reference
phase and group refractive indices, $a=\Omega/(\varepsilon_0 n_g
n_e)$ is the gain constant, and $\Gamma$ is the confinement factor to the 
active region.
The injected current divided by the carrier density at
transparency ($N_t$) is $\mu={\mathcal I}/(eVN_t)$ in such a way
that at transparency we have $\mu_t=A+B$. The current normalized
to its transparency value is $J\equiv\mu/\mu_t$. The inverse of
the carriers life-times is $\gamma_e\equiv A+2B D_{\pm}$.
Spontaneous emission processes, $\xi_\pm(t)$, are modeled by two
independent Gaussian white noise sources with properties, $\langle
\xi_\pm(t) \rangle=0$ and $\langle \xi_\pm(t)\xi_\pm^*(t^\prime)
\rangle=2\delta(t-t^\prime)$.

\begin{table}[h]
\begin{center}
\begin{tabular}[b]{|c|c|c|}
\hline
\textbf{Parameter} & \textbf{Meaning} & \textbf{%
Value} \\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\varepsilon_0 a\chi _{0}$ & effective gain constant & $4\cdot 10^{4}$ ns$^{-1}$\\ 
\hline \rule[-1ex]{0pt}{3.5ex}
%
$\Gamma$ & confinement factor & $0.045$ \\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\gamma_\bot$ & material polarization decay & $10$ ps$^{-1}$\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\sigma$ & band-gap shrinkage & $0.2$\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$b$ & unpumped contribution to $\chi$ & $2\times 10^4$\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\kappa $ & mirror loses & $300$ ns$^{-1}$\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\gamma _{a}$ & linear dichroism& $0.5$ ns$^{-1}$
\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\gamma _{p}$ & linear birefringence& $6-30$ ns$^{-1}$
\\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$A$ & non-radiative recombination rate & $1.0$ ns$^{-1}$ \\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$B$ & bimolecular recombination rate & $0.1$ ns$^{-1}$ \\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\gamma _{j}$ & spin flip rate & $50$ ns$^{-1}$ \\ \hline \rule[-1ex]{0pt}{3.5ex}
%
$\beta $ & spontaneous emission rate & $10^{-5}$ ns$^{-1}$  \\ \hline
\end{tabular}
\end{center}
\caption{Parameters involved in Eqs.
(\ref{eq:sfme})-(\ref{eq:sfmp}), their meaning and typical
values.}
\label{tab:table1}

\end{table}


Eqs. (\ref{eq:sfme})-(\ref{eq:sfmp}) together with
(\ref{eq:QWsusceptibility}) have steady state solutions (for
$\beta=0$) which correspond to two orthogonal states of linearly
polarized (LP) emission depending on whether $E_+$ and $E_-$ lock
at a phase difference $\psi = 0$ or $\pi/2$. For definiteness, we
label them as $\hat{x}$ and $\hat{y}$, respectively, and for
$\gamma_p > 0$ they correspond to the shorter and longer
wavelength, respectively. Their frequency splitting is roughly
$\Delta\omega = 2\gamma_p$. These solutions can be written as,
\begin{equation}
E_{\pm }(t)=Qe^{-i\omega_0 t\pm \psi },\;\; D_{\pm }=D_{0},  \label{eq:e_std}
\end{equation}
with $\psi =0$ in $\hat{x}$ -polarized states and  $\psi =\pi /2$ in $%
\hat{y}$-polarized states. Inserting eqs. (\ref{eq:e_std}) into eq. (\ref{eq:sfme})
we obtain with eq. (\ref{eq:sfmp}) a set of two non-linear algebraic equations for 
$\left( \omega_0, D_{0}\right) $:
\begin{eqnarray}
\Gamma \frac{a}{2} Im \chi \left( \Omega +\omega_0 ,D_{0}\right)
&=&\kappa \pm \gamma _{a}  \label{eq:std_eq1}, \\
\Gamma \frac{a}{2} Re \chi \left( \Omega +\omega_0 ,D_{0}\right) &=&\omega_0 -\left( \pm
\gamma _{p}\right).  \label{eq:std_eq2}
\end{eqnarray}
%
From eq. (\ref{eq:sfmn}) we obtain the total optical power proportional to
\begin{equation}
2Q^2=\frac{\Gamma\mu_t}{2}\left[ \frac{%
J-J_{th}}{\kappa \pm \gamma _{a}}\right],   \label{eq:intensity}
\end{equation}
where the sign +(-)  corresponds to $\hat{x}$ -polarized
($\hat{y}$-polarized) solution. The threshold current is given by
$J_{th}=2\left( AD_{0}+BD_{0}^{2}\right)/\mu_t$ which depends on
temperature through the value of $\Delta$.

%------------------------------------------------------------------------------------
\subsection{Non-thermal and thermal switching}

\subsubsection{Stability diagrams}

The linear stability analysis of the linearly polarized solutions
allows to determine the domain in parameter space where each LP
state is stable and thus the possible occurrence of polarization
switching. PS can occur as the system is brought from one
parameter region to another region of different stability
properties.

Typical stability results for a relatively small value of the
birefringence ($\gamma_p/A=6$) are shown in Fig.
\ref{fig:salvador}(a) in the $J-\Delta$ plane. There is a minimum
threshold for $\Delta=\Delta_m$ corresponding to the cavity
resonance being aligned with the gain peak. Generally, for a given
$\Delta$, the LP state selected at threshold is the closest to the
gain peak, i. e.,  $\hat x$ for $\Delta<\Delta_m $ and $\hat y$
for $\Delta>\Delta_m $. This is not true when a positive dichroism
($\gamma_a/A=0.5$) is introduced favoring the $\hat y$ component.
Above the threshold line we find a region ({\bf III}) in which
only $\hat y$ is stable and a region ({\bf I}) where only $\hat x$
is stable. A bistability region ({\bf II}) also appears, extending
from below to above $\Delta_m$, due to
nonlinear gain saturation induced by the lasing mode. PS occurs as
the system is brought from {\bf III} to {\bf I} or viceversa.
Finally, a region {\bf IV} may appear where neither of the
linearly polarized states is stable as was already described in ref. %\cite{jmartin97}.
[4].

Fig. \ref{fig:salvador}(b) displays a typical stability diagram
for a higher birefringence ($\gamma_p/A=30$). An important
difference with Fig. \ref{fig:salvador}(a) is that now, $\hat x$
is stable on almost the whole blue side of the gain peak for
currents near threshold because it is favored by a negative
dichroism. Therefore, no PS at constant temperature appears for
detunings $\Delta<\Delta_m$. Instead, a PS occurs for constant $J$
as $\Delta$ is varied across $\Delta_m$. Within our framework, we
call this PS ``thermally induced'' since it appears by scanning
$\Delta$, either by changing substrate temperature or by device
self-heating as the current is increased (see arrow in Fig.
\ref{fig:salvador}(b)).

Therefore we find {\it two possible, independent ways} to produce
a PS. One that typically occurs for relatively large
birefringence (\ref{fig:salvador}(b)) is ``thermally induced'' and
corresponds to the mechanism discussed by Choquette et al.
\cite{choquette}. A second mechanism can typically occur for small
birefringences keeping
constant $\Delta$ while increasing $J$ (see the arrows in 
Fig. \ref{fig:salvador}(a)). We call this second
mechanism ``non-thermal'' since it occurs at constant $\Delta$ and
while staying on one of the sides of the gain peak. It is
associated with the polarization switching mechanisms discussed in
ref. [4]. It should be noted that in this case, the
switching takes place from the LP state with the longer wavelength
to that with shorter wavelength. For example in the path of arrow
(2) in  \ref{fig:salvador}(a) the mode selected at threshold is
the one with largest unsaturated material gain, but increasing
current it switches to the one of smaller unsaturated gain which
is also unfavorable by the sign of the linear dichroism.
%--------
\bfig{h}
\bcen \begin{tabular}{c}
\epsfig{file=diagrama1.eps,height=8.5cm,clip=true,angle=90}
\epsfig{file=diagrama2.eps,height=8.85cm,clip=true,angle=90}
\end{tabular}\ecen
\caption{Current-detuning linear stability diagram for the
linearly polarized solutions. The lower curves gives the
dependence of threshold current on the detuning $\Delta$.
Parameters used are those in Table \ref{tab:table1} with (a)
$\gamma_p/A=6$, $\gamma_a/A=0.5$ and (b) $\gamma_p/A=30$,
$\gamma_a/A=-0.5$. The stability regions are identified as
follows: {\bf I} only $\hat{x}$ stable, {\bf II} $\hat{x}$ and
$\hat{y}$ stable, {\bf III} only $\hat{y}$ stable, {\bf IV}
$\hat{x}$ and $\hat{y}$ unstable.} \label{fig:salvador} \efig

%--------------------------------------------------------------------------------------------
\subsubsection{Current scanning at constant temperature: Non-thermally induced PS}
Fig. \ref{fig:scantcte} shows examples of dynamical evolution
through a PS obtained from the numerical integration of the
equations (\ref{eq:sfme})-(\ref{eq:sfmp}) in the case of low
birefringece $(\gamma_p/A=6)$. We plot the time evolution of the
intensities $I_x$ and $I_y$ of the linearly polarized field
components as the injected current, $J$, is raised by steps from
0.85$J_{th}$ up to 4$J_{th}$. Results are shown for two values of
the detuning parameter: (a) $\Delta=-0.25$, and (b) $\Delta=2$.
These two values correspond to the two arrows indicated in  Fig. 
\ref{fig:salvador}(a). In both cases a PS ($\hat y
\rightarrow\hat x$)occurs, but the dynamical evolution is
qualitatively different. In (a) the switching is smooth, i.e. the
$\hat y$-linearly polarized solution becomes unstable when the
current is increased, and the switching occurs through
intermediate states with complicated dynamics. 
This complex dynamics can be attributed to the competition between the two
polarization states: in this case dichroism can compensate the difference of 
modal gain for $\Delta<\Delta_m$ yielding two polarization components with nearly
the same effective gain.
On the other hand, in \ref{fig:scantcte}(b) there is a sharp PS between the two
linearly polarized states.
\bfig{h}
\bcen \begin{tabular}{c}
\psfig{file=f1a.eps,width=8.5cm,clip=true}
\psfig{file=f1b.eps,width=8.5cm,clip=true}
\end{tabular}\ecen
\caption{Non-thermal switching obtained increasing the injection
current along the paths indicated in Fig. \ref{fig:salvador}(a).
(a) $\Delta=-0.25$, and (b) $\Delta=2$. ($\alpha$) corresponds to
a $\hat{y}$-linearly polarized state, ($\delta$) to a
$\hat{x}$-linearly polarized, and ($\beta$) is an intermediate
dynamical state.}
\label{fig:scantcte}
\efig

%--------------------------------------------------------------------------------------------
\subsubsection{Current scanning with temperature variations: Thermally induced PS}
When the injection current is slowly increased, the temperature of
the device also increases due to device self-heating. The
consequence is that an effective diagonal path in the $J-\Delta$
stability diagram is followed (see fig. \ref{fig:salvador}(b)).
 We assume a simple dependence of
the detuning on the current given by,
\begin{equation}
    \dot{\Delta}=\frac{(\Delta-\Delta_0)-a(J-J_0)}{\tau_t},
    \label{eq:vardet}
\end{equation}
with $\tau_t$ the thermal time, and where $\Delta_0, J_0$ are
initial conditions. The parameter $a$ is related to the thermal
resistance of the device\cite{progress}. We take $a=2$. A
typical time scale for the thermal response time is $\tau_t=1$
$\mu$s. The situations considered in the previous subsection with
PS occurring for constant $\Delta$ can be experimentally accessed
by applying injection current pulses of duration smaller than the
thermal response time \cite{RegaladoAPL1997}.

In Fig. \ref{fig:tswich} we show the time evolution of the
intensities $I_x$ and $I_y$ of the linearly polarized field
components when the detuning changes according to eq.
(\ref{eq:vardet}). The parameters chosen are those of Fig.
\ref{fig:salvador}(b). The device is prepared with an initial
negative detuning $\Delta_0<\Delta_m$. A switching
($\hat{x}\rightarrow\hat{y}$) is seen to occur with a thermal
roll-over in the polarization components due to the change of the
threshold current with temperature.  The point of PS is the one
predicted by the linear analysis shown in Fig.
\ref{fig:salvador}(b).
%
\bfig{h}
\bcen \begin{tabular}{c}
\psfig{file=tswich.eps,width=8.5cm,clip=true}
\end{tabular}\ecen
\caption{Thermally induced switching. The detuning increases
according to eq. (\ref{eq:vardet}). Parameters of Fig.
\ref{fig:salvador}(b).}
\label{fig:tswich}
\efig

%------------------------------------------------------------------------------------
\subsection{Polarization resolved power spectrum}

Generally speaking the power spectrum
gives important information of the device operation
\cite{Ebeling}. More specifically, polarization resolved power
spectra yield information on polarization partition noise
which might cause a degradation of the signal-to-noise ratio. The
properties of these spectra and their change through a PS also
give interesting information on PS mechanisms. We present here
results for power spectra corresponding to the PS of Fig.
\ref{fig:scantcte}(a).

Let $I_{u}$, $I_{v}$ be the intensity of each orthogonal
polarization component ($u=+$ , $v=-$ in the circular basis or
$u=x$, $v=y$ in the linear basis). The power spectrum of each
component is defined by:
\begin{equation}
  S_{u,v}(\omega)=<|\delta\tilde{I}_{u,v}(\omega)|^{2}>,
\label{eq:spectuv}
\end{equation}
and the power spectrum for the total intensity is:
\begin{equation}
\label{eq:spectut}
  S_{I}(\omega)=<|\delta\tilde{I}_{u}(\omega)+\delta\tilde{I}_{v}(\omega)|^{2}>,
\end{equation}
where $\langle\cdot\rangle$ means the average over different
realizations of the noise.
%
\bfig{h}
\bcen \begin{tabular}{c}
\psfig{file=rines.eps,width=16.0cm,clip=true}
\end{tabular}\ecen
\caption{Power spectrum in the linear and circular basis
corresponding to the states $\alpha$, $\beta$, $\delta$ in
Fig. \ref{fig:scantcte}(a). The symbols $t,x,y,\pm$ refer to
$S_I(\omega),S_x(\omega),S_y(\omega)$, and $S_\pm(\omega)$
respectively.} \label{fig:rines} \efig


In Fig. \ref{fig:rines}, we show the power spectra corresponding
to the states labeled as ($\alpha$), ($\beta$), ($\delta$) in
\ref{fig:scantcte}(a). State $\alpha$ is a $\hat{y}$-polarized
state far below the polarization switching. From panels
\ref{fig:rines}(a) and \ref{fig:rines}(b) we observe that
different useful information can be extracted from the two
orthogonal polarization basis that we consider. The (+/-) basis is
the natural physical basis for the ideal case of an isotropic
cavity. In this basis a peak associated with the birefringence
appears in the power spectrum of each component, since
birefringence gives a linear coupling of the two circularly
polarized components. This peak is missing in the (x/y) basis.
Moreover, intensity fluctuations of the two polarization
components in the (+/-) basis are highly anticorrelated at low
frequencies, while this anticorrelation is less apparent in the
(x/y) basis. Anticorrelations disappear in both basis for frequencies
 close to the relaxation oscillation (RO) frequency where a positive
correlation between the two components is maximum.

Figs. \ref{fig:rines}(e) and \ref{fig:rines}(f) (state $\delta$ of
Fig. \ref{fig:scantcte}(a)) correspond to a  $\hat{x}$-polarized
state far above the polarization switching. The general features
are similar to those of the state $\alpha$ described above, except
that the anticorrelations observed at low frequency are now
strongly reduced. On the contrary the positive correlation at the
peak of the RO frequency is maintained. The RO peak occurs at
higher frequencies due to the larger injection current.

The $\beta$ state, Figs. \ref{fig:rines}(c) and
\ref{fig:rines}(d), displays more complex spectra since it is
associated with a state of complicated sustained polarization
dynamics. The intensity of both polarization components are
oscillating in time. This kind of dynamical states appear close to
the polarization switching. The corresponding power spectra
exhibit peaks at different frequencies associated with the
irregular time evolution. A very large anticorrelation appears at
low frequency due to polarization partition noise. The peak
at the RO frequency with a maximum positive correlation is
maintained.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3
\section{TRANSVERSE DYNAMICS IN GAIN-GUIDED VCSELs}

Our analysis so far implicitly uses as dynamical variables the
amplitudes of the different polarizations of a fundamental
transverse mode. For gain guided VCSELs the mode profile is not
precisely defined and spatial transverse effects might be
important. Inclusion of these effects is also necessary to study
the emergence of transverse modes of higher order. A
generalization of the SFM taking into account both a
frequency-dependent QW semiconductor complex susceptibility and
the spatial dependence on the region of the active plane is given
by
\begin{eqnarray}
\partial _{t}E_{\pm }(x,y;t) &=&-\kappa E_{\pm }-i\frac{a\Gamma }{2}P_{\pm}
-i\frac{c^{2}}{2n_e n_g \Omega}\bigtriangledown_{\bot }^{2}E_{\pm}
-(\gamma _{a}+i\gamma _{p})E_{\mp}+\sqrt{\beta D_{\pm }}\xi
_{\pm }(x,y;t) \label{eq:v2de} \\
%
\partial _{t}D_{\pm }(x,y;t) &=&{\mathcal J}(x,y)\mu (t)/2-AD_{\pm }-BD_{\pm
}^{2}-\gamma _{j}(D_{\pm }-D_{\mp })
+{\mathcal D}\bigtriangledown_{\bot }^{2}D_{\pm
}-\frac{a}{2}(P_{\pm }E_{\pm }^{*}-c.c.) \label{eq:v2dn}
\\
%
P_{\pm }(x,y;t) &\simeq&\varepsilon_0\chi(\Omega +i\frac{\partial _{t}E_{\pm }}{%
E_{\pm }},D_{\pm })E_{\pm }.  \label{eq:v2dp}
\end{eqnarray}

$E_{\pm}$, $D_{\pm}$, and $P_{\pm}$ have the same meaning as in
section \ref{sec:model1D}. Diffraction of the optical field is
modeled by the transverse laplacian term in eq. (\ref{eq:v2de}).
The laplacian term in eq. (\ref{eq:v2dn}) models the carrier
diffusion with constant ${\mathcal D}$. The current profile,
${\mathcal J}(x,y)$, together with the carrier diffusion
determines the shape of the carrier distribution below threshold.
A realistic shape for the current profile for relatively small
VCSELs is a supergaussian ${\mathcal J}(r)=\exp[-(2r/\phi)^6]$,
where $\phi$ the active region diameter. $\mu(t)$ allows to apply
current steps or ramps. The rest of parameters were previously
defined in Table \ref{tab:table1}. The numerical integration of
eqs. (\ref{eq:v2de})-(\ref{eq:v2dp}) is delicate and time
consuming. We use FFT routines to calculate the diffraction and
diffusion terms and an Euler method with noise to update the
dynamical variables.

In Fig. \ref{fig:compon2d}, we show an example of a numerical
integration of eqs. (\ref{eq:v2de})-(\ref{eq:v2dp}) for a laser of
diameter $\phi$=12.5 $\mu$m. The laser operates at the positive
detuning side ($\Delta>\Delta_m$) and the
$\hat y$-solution is selected at threshold. We first show how raising 
the current from below to
above threshold, at $t=0$, the laser switches-on in the $\hat y$-polarized
solution. At time $t=10$ ns the injection current is increased by
a step and the $\hat y$-polarized solution switches to the $\hat
x$-polarized solution. The near field images demonstrate that the
overall process takes place in the fundamental transverse mode.
Although spatial hole burning can modify the linear stability
analysis, the main qualitative features remain unchanged with
respect the results presented before when neglecting spatial
transverse effects. We are presently using the integration of eqs.
(\ref{eq:v2de})-(\ref{eq:v2dp}) to analyze the emergence of higher
order transverse modes and  the correlations between polarization
and transverse mode selection.
%
\bfig{h}
\bcen \begin{tabular}{c}
\psfig{file=2dcomp2.eps,width=14.5cm,clip=true}
\end{tabular}\ecen
\caption{A PS in the fundamental transverse mode. Near field
images show the transverse beam profile during the PS. The parameters
used are $\Delta=1$, $\gamma_a=0$, $\gamma_p/A=1$, $\gamma_j/A=150$ 
and $a/A=1.3\times 10^4$.}
\label{fig:compon2d} \efig


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{ACKNOWLEDGMENTS}
Work partially supported by the European Commission (TMR Project
FMRX-CT96-0066 and IHP-RTN1-1999-00279) and by DGICYT and CICYT
(Spain) Projects PB94-1167 and TIC99-0645.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
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\end{document}