\documentstyle[pra,aps,epsf,twocolumn]{revtex}
%\input epsf.sty

\begin{document}

\title{Dynamical Polarization States above the instability threshold
for
circularly polarized laser emission}
\author{C.A. Schrama$^{\dag *}$, M.A. van Eijkelenborg$^{\dag}$, J.P.
Woerdman$^{\dag}$,
and M. San Miguel$^+$}
\address{$^\dagger$Huygens Laboratory, Leiden University, P.O. Box
9504,
2300 RA Leiden, The Netherlands\\
$^+$Departament de Fisica, Universitat
de les Illes Baleares, and\\
Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB)\\
E-07071 Palma de Mallorca, Spain\\
{\rm\small(\today)}}
\thanks{Present Address: Netherlands Measurements Institute, P.O. Box
654, 2600 AR Delft, The Netherlands}
\address{
\onecolumn
\begin{abstract}
{\em
We show that in a laser with a transversally isotropic resonator
operating
on a $J\!\!=\!\!1\!\!\rightarrow\!\!J\!\!=\!\!0$ transition the
instability of Circularly
Polarized states leads to Linearly Polarized states with a rotating
direction of polarization.
This transition occurs through intermediate states which are
Elliptically
Polarized with rotating azimuth. For the range of pump values for which
the intermediate Rotating Elliptically Polarized states occur, the
total
output intensity is constant, and all the energy is pumped into the
coherence between the $m_{\mathrm z} = 1 $ and $m_{\mathrm z} = -1$
state.
\newline
\hspace*{\fill}
\newline
Pacs number: 42.55.f
\hspace*{\fill}}
\end{abstract}
\twocolumn}
\maketitle
If the pump rate $\sigma$ of a laser is increased the system eventually
crosses a threshold above which it is unstable~\cite{grasyuk,haken}.
Above this threshold the output intensity is oscillating. In a laser
where
the
polarization state of the field is not fixed by cavity anisotropies,
this
boundary can be
considerably lower, and not only influences the output intensity but
also
the state of polarization.
Beyond the second threshold a richness of time dependent polarization
states can be found~\cite{puccioni,abraham1,matlin,abraham2,redondo}.
In this paper we report on the states occurring beyond this second
instability
for a laser which for low pump rates emits Circularly Polarized light.
The instability leads to states with time dependent polarization
properties in which there is simultaneous emission of the two
circularly
polarized components with equal intensities but with opposite
shifts of the optical frequency.
The salient feature is that these states are reached, as pumping is
increased, through intermediate states for which the total
output intensity $I$ is independent of pumping.
In these intermediate states the two circularly polarized components
of the laser field have different intensities and different
frequencies.

Already in the early days of laser physics the polarization state of
laser
radiation
in an isotropic cavity
was understood in terms of the angular
momentum values of the laser levels~\cite{vanHaeringen}.
The theory used to describe the
lasers was based on third-order Lamb theory. Within this framework most
laser transitions could be well described, except for the
$J\!=\!1\!\leftrightarrow\!J\!=\!0$
and the $\!J\!=\!1\!\leftrightarrow \!J\!=\!1\!$ transitions. In the
experiments these transitions displayed a certain polarization
preference, whereas
theory predicted that in a transversally isotropic resonator they
should behave neutrally stable; every state of polarization
should be possible. Examples of gas lasers operating on these
transitions are the
HeNe 1.523 $\mu$m line~\cite{Tomlinson}
and the HeXe 2.652 $\mu$m line~\cite{Culshaw}.
The state selection of the
$J\!=\!1\!\leftrightarrow\!J\!=\!0$ transition close to the laser
threshold was explained~\cite{abraham1,vanHaeringen,Lenstra}
in terms of the ratio $\gamma_{\mathrm c}/\gamma_{\mathrm J}$, where
$\gamma_{\mathrm
c}$ is the decay rate of the coherence $C$ between $|J\!=\!1,m_{\mathrm
z}\!=\!1>$
and $|J\!=\!1,m_{\mathrm z}\!=\!-1>$ level, and
$\gamma_{\mathrm J}$ is the decay rate of the population difference
between these two levels.
The parameters $\gamma_{\mathrm J}$ and $\gamma_{\mathrm c}$ are more
commonly referred to as the relaxation rate of, respectively, the
orientation and the alignment of the upper state~\cite{Wang}.
If
$\gamma_{\mathrm c}/\gamma_{\mathrm J}>1$ the laser field is linearly
polarized, if
$\gamma_{\mathrm c}/\gamma_{\mathrm J}<1$ the field is circularly
polarized, and if
$\gamma_{\mathrm c}/\gamma_{\mathrm J}\!=\!1$ the field can show any
polarization
state (neutrally stable). We will focus on the situation
$\gamma_{\mathrm c}/\gamma_{\mathrm J}<1$, i.e. circular polarization
near
threshold, since this is the situation that occurs in
practice~\cite{Wang};
it is found in all reported experiments on a
$J\!=\!1\!\leftrightarrow\!J\!=\!0$ laser transition, including
the above mentioned HeNe and HeXe laser
lines~\cite{Tomlinson,Culshaw}.
Recently the $J\!=\!1\!\leftrightarrow\!J\!=\!0$ laser transition has
gained renewed interest in the context of dynamical instabilities
because
of the new phenomena
associated with the vectorial properties of the laser field
\cite{puccioni,abraham1,matlin,abraham2,redondo,vilaseca}.
The description of the second laser threshold and associated dynamical
phenomena
requires a theory that goes beyond the
third order Lamb theory, taking into account the coupled dynamical
equations for the vector electric field and the population levels and
coherences of
the atomic levels involved in the laser transition. The general problem
of
polarization
state selection has been revisited in this framework and the threshold
for
the
instability of Linearly (LP) and Circularly Polarized (CP) states have
been
obtained~\cite{abraham1}.
The situation in which there is preference for LP emission has been
analyzed in
detail in~\cite{matlin} including the effect of cavity detuning and
anisotropies. A numerical
analysis exploring a variety of time dependent states beyond the second
laser threshold
has been given in~\cite{abraham2}.

The basis of the discussion in this paper
is given by the set of equations which
describe the $J\!=\!1\!\rightarrow\!J\!=\!0$ laser transition as
derived
in~\cite{abraham1}, with the resonator tuned to atomic linecentre
and the assumption of a polarization-isotropic resonator. The situation
including resonator
anisotropies is treated numerically at the end of this paper.
For the $J\!=\!1$ state there are three decay
rates for the three tensor orders of the density matrix of this state.
For the $J\!=\!0$ state there is only the scalar population which
implies
only one decay rate. To simplify the discussion it is assumed that the
decay
rate of the total population of the $J\!=\!1$ state and the $J\!=\!0$
state is
equal, and that there is no spontaneous decay between these two
states.
Further, the population of the $|J\!=\!1,m_{\mathrm z}=\!0>$ state,
which is
mixed into
the dynamics through the collisions, is adiabatically eliminated.
The equations of
motion in rescaled variables~\cite{puccioni} for a cavity resonantly
tuned
to the atomic
frequency are given by
\begin{eqnarray}
\frac{{\mathrm d} E_{\mathrm R}}{ {\mathrm d} t} & = &
-\kappa E_{\mathrm R} + \kappa P_{\mathrm R}, \nonumber \\
\frac{{\mathrm d} E_{\mathrm L}}{ {\mathrm d} t} & = &
-\kappa E_{\mathrm L} + \kappa P_{\mathrm L}, \nonumber \\
\frac{{\mathrm d} P_{\mathrm R}}{ {\mathrm d} t} & = &
-\gamma_\perp P_{\mathrm R} + \gamma_\perp E_{\mathrm
R}\frac{D_++D_-}{2} +
\gamma_\perp E_{\mathrm L} C, \nonumber \\
\frac{{\mathrm d} P_{\mathrm L}}{ {\mathrm d} t} & = &
-\gamma_\perp P_{\mathrm L} + \gamma_\perp E_{\mathrm L}
\frac{D_+-D_-}{2} +
\gamma_\perp E_{\mathrm R} C^*, \nonumber \\
\frac{{\mathrm d} C}{ {\mathrm d} t} & = &
-\gamma_{\mathrm c} C - \frac{\gamma_\|}{4}
\left(E_{\mathrm L}^*P_{\mathrm R}+E_{\mathrm R}P_{\mathrm L}^*\right),
\nonumber \\
\frac{{\mathrm d} D_+}{ {\mathrm d} t} & = & -\gamma_\| (D_+ - 2\sigma)
-\frac{3\gamma_\|}{2} {\mathrm Re}\left(E_{\mathrm R}^*P_{\mathrm
R}+E_{\mathrm
L}^*P_{\mathrm L}\right),
\nonumber \\
\label{eq:start}
\frac{{\mathrm d} D_-}{ {\mathrm d} t} & = & -\gamma_{\mathrm J} D_-
-\frac{\gamma_\|}{2}{\mathrm Re}\left(E_{\mathrm R}^*P_{\mathrm
R}-E_{\mathrm
L}^*P_{\mathrm L}\right),
\end{eqnarray}
where $E_{\mathrm R}$ and $E_{\mathrm L}$ are the right and left
circularly polarized
complex field amplitudes,
$P_{\mathrm R}$ and $P_{\mathrm L}$ are the amplitudes of
dipole moments on the circularly polarized transitions, $C$ is the
coherence between the $m_{\mathrm z}=1$ and $m_{\mathrm z}=-1$ state,
and
$(D_+\pm D_-)/2$ is the
inversion on the right/left circularly polarized transition; $\kappa$
is
the cavity decay rate~\cite{remark1}, $\gamma_\perp$ is the decay rate
of the
atomic dipole moment and  $\gamma_\|$ is the population decay rate.

Above the laser threshold $\sigma=1$ these equations have steady state
solutions as discussed
above~\cite{abraham1}. There is a CP solution in which either
$I_{\mathrm R}\!=\!|E_{\mathrm R}|^2\!=\!0$ or
$I_{\mathrm L}\!=\!|E_{\mathrm L}|^2\!=\!0$ and a LP solution with
$I_{\mathrm R}\!=\!I_{\mathrm L}$. In both solutions
the field amplitudes are resonantly tuned to the atomic frequency. For
CP
solutions
there is bistability between right and left polarization, while for the
LP
solution the polarization angle is undetermined. A linear stability
analysis of these
solutions indicates that, close to threshold, CP is stable for
$\gamma_{\mathrm c}/\gamma_{\mathrm J}\!<\!1$, while LP is stable for
$\gamma_{\mathrm c}/\gamma_{\mathrm J}\!>\!1$.
The case $\gamma_{\mathrm c}/\gamma_{\mathrm J}\!=\!1$ is `neutral' and
any Elliptically
Polarized (EP) state in which
two polarization components coexist is possible:
The ellipticity defined as
$\tan\chi = (|E_{\mathrm R}|\!-\!|E_{\mathrm L}|)/(|E_{\mathrm
R}|+|E_{\mathrm L}|)$ takes on
arbitrary values and the azimuth giving the orientation
of the dominant linearly polarized component is undetermined.
The linear stability analysis also identifies the second laser
threshold
as the critical value of the pump
rate above which these particular solutions are unstable. A Hopf
bifurcation occurs at the critical pump
rate given by~\cite{abraham1}
\begin{equation}
\label{eq:crit}
\sigma_{1} = 1 + \frac{
n_{\mathrm gr}\gamma_<(\kappa+\gamma_\bot+\gamma_<)(3 \gamma_> +
\gamma_\|)}
{\gamma_\|\left[2\kappa(\kappa+\gamma_\bot)+\gamma_>(\kappa-\gamma_< -
\gamma_\bot)\right]},
\end{equation}
where $\gamma_<$ is the smaller of $\gamma_{\mathrm c}$ and
$\gamma_{\mathrm J}$, and
$\gamma_>$ is the larger of $\gamma_{\mathrm c}$ and $\gamma_{\mathrm
J}$.
The group index
$n_{\mathrm gr}$ is given by $1+\kappa/\gamma_\bot$.
The total intensity $I=I_{\mathrm R}+I_{\mathrm L}$ at this
critical value can be found from
\begin{equation}
I = 4 \gamma_> (\sigma -1)/(3\gamma_>+\gamma_\|).
\label{eq:inten1}
\end{equation}

Above $\sigma_{1}$ this symmetric description ceases to hold. For
$\gamma_{\mathrm c}/\gamma_{\mathrm J}\!>\!1$ the total intensity and
the
ellipticity
oscillate
anharmonically~\cite{abraham1,matlin}. In this paper we focus on the
case
$\gamma_{\mathrm c}/\gamma_{\mathrm J}<1$ for which a rather different
behaviour with time
independent total
intensity is found. Guided by numerical phenomenology~\cite{abraham2}
we search for steady state solutions of Eq.~(\ref{eq:start})
which involve two frequencies in the form
\begin{eqnarray}
E_{R,L} & = & \epsilon_{R,L} e^{ \mp i \Omega_{R,L} t}, \nonumber \\
P_{R,L} & = & p_{R,L} e^{ \mp i \Omega_{R,L} t}, \nonumber \\
C & = & c e^{ - i (\Omega_{\mathrm R} + \Omega_{\mathrm L}) t}.
\end{eqnarray}
One then finds
\begin{mathletters}
\label{eq:real}
\begin{eqnarray}
\label{eq:reala}
& & \left(\sigma -1 + \frac{\Omega_{\mathrm R}^2}{\gamma_\bot\kappa}
-\frac{3}{4}(I_{\mathrm R}+I_{\mathrm L}) -
\frac{\gamma_\|}{4\gamma_{\mathrm
J}}(I_{\mathrm R}-I_{\mathrm L}) \right.
\nonumber\\
& & \left. \hspace*{1.5cm}
-\frac{\gamma_\|}{4 \kappa}\,\frac{2\kappa\gamma_{\mathrm c} +
(\Omega_{\mathrm R}+\Omega_{\mathrm L})^2}
{\gamma_{\mathrm c}^2 + (\Omega_{\mathrm R}+\Omega_{\mathrm L})^2}
I_{\mathrm L}
\right) I_{\mathrm R} = 0, \\
\label{eq:realb}
& & \left(\sigma -1 + \frac{\Omega_{\mathrm L}^2}{\gamma_\bot\kappa}
-\frac{3}{4}(I_{\mathrm R}+I_{\mathrm L}) +
\frac{\gamma_\|}{4\gamma_{\mathrm J}}(I_{\mathrm R}-I_{\mathrm L})
\right.
\nonumber\\
& & \left. \hspace*{1.5cm}
-\frac{\gamma_\|}{4 \kappa}\,\frac{2\kappa\gamma_{\mathrm c} +
(\Omega_{\mathrm R}+\Omega_{\mathrm L})^2}
{\gamma_{\mathrm c}^2 + (\Omega_{\mathrm R}+\Omega_{\mathrm L})^2}
I_{\mathrm R}
\right) I_{\mathrm L} = 0,
\end{eqnarray}
\end{mathletters}
and
\begin{mathletters}
\label{eq:im}
\begin{eqnarray}
\label{eq:ima}
& & \left(\Omega_{\mathrm R} n_{\mathrm gr} -
\frac{\gamma_\|}{4} \frac{(2\kappa - \gamma_{\mathrm
c})(\Omega_{\mathrm
R} +
\Omega_{\mathrm L})}
{\gamma_{\mathrm c}^2 + (\Omega_{\mathrm R} + \Omega_{\mathrm L})^2}
I_{\mathrm L}
\right) I_{\mathrm R} = 0, \\
\label{eq:imb}
& & \left(\Omega_{\mathrm L} n_{\mathrm gr}
- \frac{\gamma_\|}{4} \frac{(2\kappa - \gamma_{\mathrm
c})(\Omega_{\mathrm
R} +
\Omega_{\mathrm L})}
{\gamma_{\mathrm c}^2 + (\Omega_{\mathrm R} + \Omega_{\mathrm L})^2}
I_{\mathrm R}
\right) I_{\mathrm L} = 0.
\end{eqnarray}
\end{mathletters}
Eqs.~(\ref{eq:im}) admit several solutions which obviously include the
ones
known to be stable for $\sigma<\sigma_1$. It follows from
~(\ref{eq:im})
that if $I_{\mathrm R}\!=\!0$ then $\Omega_{\mathrm L}\!=\!0$, and if
$I_{\mathrm L}\!=\!0$ then
$\Omega_{\mathrm R}\!=\!0$. These two cases correspond to the left and
right CP solutions
described above. If both $I_{\mathrm R}\!\neq\!0$ and $I_{\mathrm
L}\!\neq\!0$ then either
$\Omega_{\mathrm R}\!=\!\Omega_{\mathrm L}\!=\!0$, or $\Omega_{\mathrm
R}\!\neq\!0$
and $\Omega_{\mathrm L}\!\neq\!0$. The
first case corresponds to the, above described, LP solutions.
The last
case corresponds to the new solutions we are looking for. We will
distinguish two cases: $I_{\mathrm R}\!=\!I_{\mathrm L}$ with
$\Omega_{\mathrm R}\!=\!\Omega_{\mathrm L}$ (Rotating LP, RLP), and
$I_{\mathrm R}\!\neq\!I_{\mathrm L}$, (Rotating EP, REP).

Before discussing these two types of solutions we first give a relation
valid in
both cases.
By adding Eqs.~(\ref{eq:im}) one
finds
\begin{equation}
\label{eq:rot-I}
(\Omega_{\mathrm R}+\Omega_{\mathrm L})^2 =
\frac{\gamma_\|(2\kappa-\gamma_{\mathrm
c})}
{4 n_{\mathrm gr} }(I_{\mathrm R}+I_{\mathrm L}) - \gamma_{\mathrm
c}^2.
\end{equation}
This equation reveals that these solutions can only
exist if $2\kappa>\gamma_{\mathrm c}$. This condition
can be called the polarization bad cavity condition~\cite{Charles}.

{\sl Rotating Linear Polarization, RLP}:
$I_{\mathrm R}\!=\!I_{\mathrm L}\!=\!I/2$,
$\Omega_{\mathrm R}\!=\!\Omega_{\mathrm L}\!=\!\Omega $. In these
solutions the two circularly polarized components
of the field coexist with equal strength, which corresponds to a
linearly
polarized
field whose angle of polarization (azimuth) is given by the phase
difference of the two complex amplitudes.
Here the phase difference is given by $ 2 \Omega t$,
which implies a rotation of the polarization direction with
frequency $2\Omega$. From Eqs.~(\ref{eq:real}) and~(\ref{eq:im})
one immediately obtains that
\begin{equation}
\label{eq:lin-I}
I = 4 n_{\mathrm gr}
\left\{
\frac{4 \kappa \gamma_\bot(\sigma -1) -\gamma_{\mathrm
c}(\gamma_{\mathrm
c} +2 \kappa
+2
\gamma_\bot)}{12 \kappa(\kappa+\gamma_\bot) + \gamma_\|(\gamma_{\mathrm
c}
+ 2\gamma_\bot)}
\right\}.
\end{equation}
The frequency $\Omega$ is easily obtained replacing (\ref{eq:lin-I}) in
(\ref{eq:rot-I}).
The requirements that $I\!>\!0$ and $\Omega^2\!>\!0$ identify the range
of
pump rates for
which these
solutions exist. The condition giving a higher pump rate is $\Omega^2
>0$,
implying
that RLP solutions only exist for $\sigma>\sigma_0$, where
\begin{equation}
\sigma_0= 1 + \frac{\gamma_{\mathrm c} (3 \gamma_{\mathrm c}
+\gamma_\|)
n_{\mathrm gr}}
{\gamma_\| (2\kappa - \gamma_{\mathrm c})}.
\end{equation}
In practical cases $\sigma_0\!<\!\sigma_1$~\cite{remark2} so that there
is
a range of pump rates in which the CP states are stable and for which
RLP
states already exist.
However, as we see next, the RLP states are stable only for
$\sigma>\sigma_2$ and the system chooses not to switch
from CP to RLP with a discontinuity
in the value of $I_{\mathrm R}$ and $I_{\mathrm L}$, but rather the
transition from CP to RLP appears mediated with continuity through REP
states.

{\sl Rotating Elliptical Polarization, REP:}
$I_{\mathrm R}\!\neq\!I_{\mathrm L}$ and
$\Omega_{\mathrm R}\!\neq\!\Omega_{\mathrm L}$. The two circularly
polarized components
coexist with unequal but time
independent strength. This gives rise to an EP state with time
independent
ellipticity. However, the azimuth of the ellipse is given by the phase
difference of the
two complex amplitudes and therefore it rotates with frequency
$\Omega_{\mathrm
R}+\Omega_{\mathrm L}$.
The total intensity and rotating frequency is obtained from
Eqs.~(\ref{eq:real}) and~(\ref{eq:im}).
It follows from the difference of Eq.~(\ref{eq:reala})
and
Eq.~(\ref{eq:realb}), while using Eqs.~(\ref{eq:im}) and
(\ref{eq:rot-I}), that
\begin{equation}
\label{eq:I-el}
I=I_{\mathrm R}+I_{\mathrm L} = \frac{4 \gamma_{\mathrm
c}\gamma_{\mathrm
J}(\kappa+\gamma_\bot+\gamma_{\mathrm c})n_{\mathrm gr}}
{\gamma_\|\left[2\kappa(\kappa+\gamma_\bot)+\gamma_{\mathrm
J}(\kappa-\gamma_{\mathrm
c} - \gamma_\bot)\right]}.
\end{equation}
Substituting Eq.~(\ref{eq:I-el}) into Eq.~(\ref{eq:rot-I}) yields
\begin{equation}
\label{eq:rot-el}
(\Omega_{\mathrm R} + \Omega_{\mathrm L})^2 =
\frac{2\kappa\gamma_{\mathrm c} (\gamma_{\mathrm J} - \gamma_{\mathrm
c})(\kappa +
\gamma_\bot)}
     {2\kappa (\kappa +\gamma_\bot) + \gamma_{\mathrm J}(\kappa
     -\gamma_{\mathrm c}-\gamma_\bot)}.
\end{equation}
 Note that $I$ coincides with the intensity at the
instability threshold $\sigma_1$ for the CP solutions and that it is
independent
of pump rate. Note also that
$(\Omega_{\mathrm R}+\Omega_{\mathrm L})^2$ coincides with the
frequency
of the Hopf
bifurcation at
this threshold~\cite{abraham1}  and it is also independent of pump
rate.
The existence of these solutions require
$(\Omega_{\mathrm R} + \Omega_{\mathrm L})^2 \!>\!0 $ which is
fulfilled
for $\gamma_{\mathrm J}>\gamma_{\mathrm c}$ and $I>0$, but it also
requires
$I_{\mathrm R}>0$ and $I_{\mathrm L}\!>\!0$.
For the latter conditions we see that from the sum of
Eq.~(\ref{eq:reala})
and Eq.~(\ref{eq:realb}) one finds
\begin{eqnarray}
\label{eq:cos}
\cos^2 (2\chi) & = & (\sigma -1)  \frac{2 \left[2 \kappa
(\kappa+\gamma_\bot)  + \gamma_{\mathrm J}(\kappa-\gamma_{\mathrm c}
-\gamma_\bot)\right]}
{\gamma_{\mathrm c}(\gamma_{\mathrm J}-\gamma_{\mathrm c})n_{\mathrm
gr} }
\nonumber
\\
& & - \frac{2(\gamma_\|+3\gamma_{\mathrm J})(\gamma_\bot +
\gamma_{\mathrm
c}
+\kappa)}
{\gamma_\|(\gamma_{\mathrm J}-\gamma_{\mathrm c})}.
\end{eqnarray}
Solving $\cos^2(2\chi)\!=\!0$ and $\cos^2(2\chi)\!=\!1$ with
$\gamma_{\mathrm J}>\gamma_{\mathrm c}$ gives the
range of pump rates of existence of REP solutions.
The first condition reproduces the instability threshold
Eq.~(\ref{eq:crit}), while the second identifies another threshold
value
\begin{equation}
\label{eq:crit2}
\sigma_{2} = \sigma_{1} +
\frac{\gamma_{\mathrm c} (\gamma_{\mathrm J} - \gamma_{\mathrm
c})n_{\mathrm gr}}
     {2\left[
     2\kappa (\kappa +\gamma_\bot) + \gamma_{\mathrm J}(\kappa
     -\gamma_{\mathrm c}-\gamma_\bot)
     \right]}.
\end{equation}
At $\sigma=\sigma_1$ the CP solution coincides with a
REP solution and in turn, this coincides with the RLP solution at
$\sigma=\sigma_2$.
The individual frequencies $\Omega_{\mathrm R}$ and $\Omega_{\mathrm
L}$
for these
solutions can be found from
Eqs.~(\ref{eq:im}) while using the identity $I_{\mathrm R} = I (1
+\sin(2\chi))/2$.

\begin{figure}
\centerline{\epsfxsize=75mm\epsffile{fig1.eps}}
\caption{\label{fg:trans} A,B: total intensity versus $\sigma$. The
inset B is an enlargement of the transition region. The inset C shows
$\cos^2(2\chi)$ versus $\sigma$. These plots have been made
with the parameters $\kappa=\gamma_\|=\gamma_{\mathrm
J}=\gamma_\perp=1$
and $\gamma_{\mathrm c}=0.5$~[17].}
\end{figure}

The behaviour of the laser in the transition from CP to RLP
states is shown in Fig.~\ref{fg:trans}, where the total intensity as
function of $\sigma$ has been plotted.
For $\sigma < \sigma_1,\; \sigma_1 < \sigma < \sigma_2$ and
$\sigma > \sigma_2$ the intensity $I$ is calculated from
Eq.~(\ref{eq:inten1}), Eq.~(\ref{eq:I-el}) and Eq.~(\ref{eq:lin-I})
respectively. A plateau in $I$ appears between
the vertical lines which correspond to $\sigma_1$ and
$\sigma_2$. In the insets the transition region has been enlarged.
In B, again, $I$ as function of $\sigma$ has been plotted.
In C $\cos^2(2\chi)$ versus $\sigma$ is drawn as calculated
for $\sigma_1 < \sigma < \sigma_2$ from Eq.~(\ref{eq:cos}).
The latter inset shows the change in ellipticity
during this transition, going from a circle to a line as degenerate
cases
of the ellipse.
In all three plots the circles are results of
numerical integrations of Eqs.~(\ref{eq:start}).
Note that ${\mathrm d}I/{\mathrm d}\sigma$ is different before, during,
and
after the transition. In the CP state ( stable for $\sigma<\sigma_1$)
the
total intensity, which equals the intensity of the nonvanishing
component
of the field, grows linearly with $\sigma$ and with unit slope (for the
parameters noted in the caption of Fig.~\ref{fg:trans}).
In the RLP solution (stable for $\sigma>\sigma_2$), the intensity of
both components also grows linearly with a slope smaller than $1$, but
the
total intensity has a slope larger than $1$.
This solution has an unstable branch with the same slope for
$\sigma<\sigma_2$ down to $\sigma=\sigma_0$ where
the solution disappears.

During the transition region ($\sigma_1\!<\!\sigma\!<\!\sigma_2$) the
total intensity does not depend on $\sigma$ (see Eq.~(\ref{eq:I-el})).
Therefore during the transition from CP to RLP states the total
intensity has zero slope,
while the intensities of the two components giving rise to
these REP states grow and decrease with $\sqrt{\sigma}$ to become equal
at
$\sigma\!=\!\sigma_2$.
The energy pumped into the laser for $\sigma_1\!<\!\sigma\!<\!\sigma_2$
does not result in higher output power but it goes into the atomic
coherences.
\begin{figure}
\centerline{\epsfxsize=75mm\epsffile{fig2.eps}}
\caption{\label{fg:gain} Mode frequencies with respect to the gain
profile.  A:
$\sigma\leq\sigma_1$;  B: $\sigma_1<\sigma<\sigma_2$;
C: $\sigma\geq\sigma_3$. }
\end{figure}

The emergence of laser states with time dependent polarization
properties,
while keeping a total intensity constant, is possible because of the
existence of only two frequencies in the dynamical solution. In general
two frequencies give rise to polarization selfpulsations as it becomes
evident if the RLP or REP solutions are described in terms of the
linear
components $E_{\mathrm x}$ and $E_{\mathrm y}$ of the field
($E_\pm= (E_{\mathrm x} \pm i E_{\mathrm y})/\sqrt{2}$).
The intensities of the $x$- and $y$-components become
periodic in time. In our problem the circular components are the
natural basis in which the emergence of two frequencies can be easily
understood as visualized in Fig.~\ref{fg:gain} where the mode
frequencies
with respect to the gain profile are shown.
For $\sigma\leq\sigma_{1}$ only one mode is lasing.  For
$\sigma_1<\sigma<\sigma_2$ both circular modes are
lasing . They are however not equally detuned with respect to
gain maximum. This causes the elliptical polarization.  For
$\sigma>\sigma_2$ the modes are equally detuned, hence the field is
RLP.

The polarization time dependent states which we have found beyond the
instability
of CP states admit an interesting representation on a Poincar\'{e}
sphere (Fig.~\ref{fg:sphere}) which is parametrized by the two relevant
polarization angles,
ellipticity and azimuth.
The two poles of the sphere describe the two CP states. A point in
the equator is a linearly polarized state with direction of
polarization
given by
the azimuth on the equator. An arbitrary point in the sphere
corresponds
to an
elliptically polarized state. The point on the pole becomes unstable
for
$\sigma>\sigma_1$
and for $\sigma>\sigma_2$ ends on a point that circles the sphere along
the equator
(RLP). The intermediate states in this transition are states in
which the representative point rotates along a parallel circle with a
frequency $\Omega_{\mathrm
R} +
\Omega_{\mathrm L}$ (REP). The rotating frequency at $\sigma=\sigma_1$
is the frequency of the Hopf bifurcation in which the point in the pole
becomes unstable.

In summary, we have analyzed a laser operating on a
$J\!=\!1\!\rightarrow\!J\!=\!0$ transition under the conditions of
polarization bad cavity ($2\kappa>\gamma_{\mathrm c}$) and preference
to
start
lasing with circularly polarized states ($\gamma_{\mathrm
J}>\gamma_{\mathrm c}$).
In these circumstances there is a Hopf bifurcation of the CP states.
Beyond this bifurcation at $\sigma=\sigma_1$ there appear states
with time dependent polarization properties characterized by two
frequencies and constant total intensity. The instability
leads from CP states to LP states with a rotating direction of
polarization.
This transition occurs continuously through intermediate states which
are
EP with rotating azimuth.
For the range of pump values $\sigma_1\!<\!\sigma\!<\!\sigma_2$
for which the intermediate
REP states occur, the total output intensity is
independent of pump rate. The frequency of rotation of the ellipse is
also
constant and it is given by the Hopf frequency at the instability
point.


\begin{figure}
\centerline{\epsfxsize=75mm\epsffile{fig3.ps}}
\caption{\label{fg:sphere} Poincar\'{e} sphere representation of
polarization states.
A) Pole: Circularly  Polarized (CP),
B) Parallel Circle: Rotating Elliptically Polarized (REP),
C) Equator:Rotating Linearly Polarized (RLP).}
\end{figure}

Experimental work on the $J\!=\!1\!\rightarrow\!J\!=\!0$ transition is
of
great importance since this transition is the only
transition for which the rigorous laser models can be handled.
In our theoretical treatment we have assumed an isotropic resonator.
However, practical laser resonators always contain some residual
anisotropies.
In principle, these can be cancelled by adding intentional intracavity
anisotropies of opposite sign. However, since such cancellation is
never
perfect one might still worry whether the presence of anisotropies
would
spoil the theoretical results reported in this paper.
We have therefore investigated the effects of linear dichroism $\alpha$
and linear birefringence $\beta$ ($\alpha, \beta \ll \kappa$) by
numerical
integration of Eqs.~(\ref{eq:start}) in the same manner as was done
in ref.~\cite{matlin}. This gives the following results.
Just above threshold  ($\sigma \stackrel{\textstyle
\mbox{\raisebox{-4pt}{$>$}}}{\mbox{\raisebox{-3pt}{$\sim$}}} 1$)
a dichroism $\alpha$ induces linearly polarized steady state solutions,
whereas a birefringence $\beta$ induces oscillations in the ellipticity
$\chi$. Further above threshold (but still below $\sigma_1$) the
influence
of the medium on the polarization becomes stronger, and the
steady states become identical to the states found without anisotropies
(CP).
Above the threshold $\sigma_1$ both $\alpha$ and $\beta$ induce
small oscillations in both $\chi$ and $I$. The time-averaged behavior
is
still given by the above polarization isotropic situation
(Fig.~\ref{fg:trans}). The modulation depth of the oscillations in
$\chi$ and $I$ is
roughly of the order of $|\alpha + i \beta|$ (which experimentally can
be
brought to below $10^{-4}$~\cite{eijkel}). Thus, although residual
anisotropies
induce small oscillations around the steady state for values of $\sigma
>
\sigma_1$, the oscillations remain very small and the time averaged
behavior of the laser is unaffected. Therefore our theoretical results
may
be expected to be valid in practical cases.

Additional motivation for the work on the non-linear dynamics of gas
lasers is recent work on VCSELs (Vertical Cavity Surface Emitting
Lasers)~\cite{bveld,strain}.
In order to avoid condensed matter complexity, existing theory for
polarization behaviour of VCSELs is structured along the lines of gas
laser theory~\cite{sanmig}.
Questions in relation to such VCSEL theory and its improvement may be
elucidated by the gas-laser results.

This work is part of the research program of the ``Stichting voor
Fundamenteel Onderzoek der Materie (FOM)''. We acknowledge financial
support from the European Union under the ESPRIT project 20029 ACQUIRE
and
the TMR Network ERB4061PL951021.
We are grateful for helpful discussions of this problem with N.B.
Abraham.

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\end{document}