\documentstyle[opex,color]{article}

\begin{document}

\title{Growth dynamics of noise-sustained structures in 
nonlinear optical resonators}

\author{Marco Santagiustina, Pere Colet, Maxi San Miguel, Daniel Walgraef}
\address{Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB), \\
E-07071 Palma de Mallorca, Spain}
\email{marco@hp1.uib.es \\
{\color{blue} http://www.imedea.uib.es/PhysDept}
}

\begin{abstract}%
The existence of macroscopic noise-sustained structures in nonlinear optics
is theoretically predicted and numerically observed, in the regime
of convective instability.
The advection-like term, necessary to turn the instability to convective for the
parameter region where advection overwhelms the growth, can stem from pump beam
tilting or birefringence induced walk-off.
The growth dynamics of both noise-sustained and deterministic patterns is
exemplified by means of movies. This allows to observe the process of formation
of these structures and to confirm the analytical predictions. 
The amplification of quantum noise by several orders of magnitude is predicted.
The qualitative analysis of the near- and far-field is given. It
suffices to distinguish noise-sustained from deterministic structures;
quantitative informations can be obtained in terms of the statistical
properties of the spectra.
\end{abstract}
\ocis{(190.4420) Nonlinear optics, transverse effects in; (270.2500)
Fluctuations, relaxations, and noise}


\begin{OERefLinks}

\item\label{lugi96} L.A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G.L. 
Oppo, H. Wiedemann, ``Quantum aspects of nonlinear optical patterns,''
Coherence and Quantum Optics VII (Plenum Press), 5 (1996).

\item\label{lugi97} A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G-
L. Oppo, S.M. Barnett, ``A Langevin approach to quantum fluctuations and optical
patterns formation,'' \pra {\bf 56}, 877 (1997); A. Gatti, 
L.A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, A. Berzanskis, ``From quantum
to classical images,'' \opex {\bf 1}, 21 (1997).

\item\label{lugiPRL97} I. Marzoli, A. Gatti,  L.A. Lugiato, ``Spatial quantum
signatures in parametric downconversion,'' \prl {\bf 78}, 2092-2095 (1997).

\item\label{pra} M. Hoyuelos, P. Colet, M. San Miguel, ``Fluctuations and
correlations in polarization patterns of a Kerr medium,'' submitted (1998).

\item\label{deis85} R.J. Deissler,``Noise-sustained structure, intermittency, and the
Ginzburg-Landau equation,'' J. Stat. Phys. {\bf 40} 376 (1985).  

\item\label{prl} M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, ``Convective
noise-sustained structures in nonlinear optics,'' \prl {\bf 79}, 3363 (1997).

\item\label{lugi87} L.A. Lugiato, R. Lefever, ``Spatial dissipative structures in
passive optical systems,'' \pra {\bf 58}, 2209- (1987).

\item\label{firt92} W.J. Firth, A.G. Scroggie, G.S. McDonald, L.A. 
Lugiato, ``Hexagonal patterns in optical bistability,'' \pra {\bf 46}, R3609 (1992).

\item\label{lugi92} L.A. Lugiato, F. Castelli, ``Quantum noise reduction in a spatial
dissipative structure,'' \prl {\bf 68}, 3284 (1992).

\item\label{oppo94} K. Staliunas, ``Optical vortices during three-wave nonlinear
coupling,'' \oc {\bf 91}, 82 (1992); G-L. Oppo, M. Brambilla, L.A. Lugiato, ``Formation and evolution
of roll patterns in optical parametric oscillators,'' \pra {\bf 49},
2028 (1994).

\item\label{kimb87} L.A. Wu, H.J. Kimble, J. Hall, H. Wu, ``Generation of
squeezed states by parametric down conversion,'' \prl {\bf 57}, 2520 (1986).

\item\label{pre} M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef,
``Two-dimensional noise-sustained structures in optical parametric oscillators,''
submitted (1998).

\item\label{halt92} M. Haelterman, G. Vitrant, ``Drift instability and spatiotemporal
dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,''
\josab {\bf 9}, 1563- (1992).

\item\label{gryn94} G. Grynberg, ``Drift instability and light-induced spin waves
in an alkali vapor with feedback mirror,'' \oc {\bf 109}, 483- (1994);
A. Petrossian, L. Dambly, G. Grynberg, ``Drift instability for a laser beam transmitted
through a rubidium cell with feedback mirror,'' Europhys. Lett. {\bf 29}, 209-
(1995).

\item\label{bloe65} N. Bloembergen, ``Nonlinear Optics,'' Benjamin Inc. Publ. (1965);
Y.R. Shen, ``The principles of nonlinear optics,'' Wiley (1984). 

\item\label{drum80} P.D. Drummond, K.J. Mc Neil, D.F. Walls, ``Non-equilibrium
transitions in sub/second harmonic generation: semiclassical theory,''
Opt. Acta {\bf 27}, 321 (1980).


\end{OERefLinks}

\section{Introduction}
Optical patterns offer the very attractive possibility of studying the 
interface between classical and quantum patterns. Macroscopic and 
spatially structured manifestations of quantum correlations are 
foreseen to occur in these patterns [\ref{lugi96}]. 
Such correlations are expected since, at a microscopic level, the 
physical mechanism behind the pattern formation process is often 
the simultaneous emission of twin photons, four wave mixing 
processes or other processes involving highly correlated photons. 
Correlations are easily observed in the far field and should encode 
specific features of quantum statistics. 
\newline
In the search for the manifestations of quantum noise in optical 
patterns it is natural to look for situations in which noise is 
enhanced or amplified. Critical fluctuations close to an instability 
point are one of these situations recently considered [\ref{lugi97}, 
\ref{lugiPRL97},\ref{pra}].
The noisy precursor, observed just below threshold anticipates
the pattern to be formed beyond the instability. This precursor occurs 
because fluctuations with the wavenumber to be selected above 
threshold become weakly damped as the threshold is approached.
\newline
A more dramatic manifestation of noise occurs above a convective 
instability threshold [\ref{deis85}]. Here fluctuations are amplified
(instead of being weakly damped) while being advected away from the system.
This gives rise to macroscopic structures that are
continuously regenerated by noise and hence the name of noise 
sustained patterns. This phenomenon acts as a microscope, with 
amplification factors of several orders of magnitude, to observe noise
and its spatially 
dependent manifestations [\ref{prl}]. In this paper we give two 
examples of noise sustained optical patterns. Our emphasis here is 
in showing how these patterns grow dynamically from noise, 
invading part of the system and being there maintained by noise.
\newline
The two examples to be considered are paradigmatic in the field of 
optical pattern formation and quantum noise properties. The first 
is a cavity filled by a Kerr type nonlinear medium and pumped 
by an external laser beam [\ref{prl}]. This system was a prototype 
model for pattern formation in optics [\ref{lugi87},\ref{firt92}] and it has 
also been where the question of quantum fluctuations in patterns 
was first addressed [\ref{lugi92}]. The second
example is an optical parametric oscillator (OPO), also a paradigm for studies of 
pattern formation [\ref{oppo94}] and generation of squeezed and 
nonclassical light [\ref{kimb87}]. A necessary condition for 
the existence of a convective instability is the presence of an advection-like
term in the governing equations; this term can have different 
origins. In our first example this originates in any pump 
misalignement; we will study this example in a simple transverse 
one-dimensional geometry to clarify the main concepts. For the OPO 
we consider a type-I phase matching in a uniaxial crystal. Here, the 
advection term originates in the walk-off between the ordinary and 
extraordinary rays, due to birefringence.
\newline
Thus, the outline of the paper is as follows. In Section 2 we briefly recall
the definition of the convectively unstable regime and the linear stability
analysis which allows to determine the different regimes.
In Section 3,
we describe the convective instabilities and noise sustained structures 
in Kerr nonlinear resonators with one transverse dimension. We show in two 
movies the growth dynamics of the pattern in the convectively and absolutely 
unstable regimes. The role of the noise in sustaining the structure in 
the convective regime is clearly exemplified. 
The distinction between these two regimes manifests qualitatively
in the time evolution of the far field of the pattern.
Section 4 is devoted to noise-sustained structures in OPO with two 
transverse dimensions. We show the diagram in parameter space where the 
zero solution becomes unstable either convectively or absolutely. Two movies,
displaying the growth dynamics of the noise-sustained and the deterministic
patterns, are also presented, for the near- and the far-field. Finally we
compare a snapshot of the pattern formed in each of the two cases with the
noisy precursor below threshold. 
Conclusions are presented in section V.

\section{Definitions and linear stability analysis}
We start by briefly recalling the notion of convective and absolute instabilities;
readers may refer to [\ref{deis85},\ref{prl},\ref{pre}] for more details.
The steady-state of a generic system is defined to be absolutely stable
(unstable) when a perturbation decays (grows) with time. However, a third
possibility is that the perturbation grows (i.e. is unstable) but at the
same time is advected so quickly that, at a fixed position, it actually decays.
In this case the state is called convectively unstable. Note that the definition
is unambiguous only if a fixed frame of reference is defined; in the cases
we consider here the fixed frame corresponds to the pump beam.
\newline
The usual linear stability analysis of the steady-state can be re-formulated in
order to take into account the above distinction of convectively and absolutely
unstable regimes.
In general, the calculation of the pump amplitude thresholds of the instability
for the systems we are considering entails the evaluation of the 
linearized asymptotic behaviour of a generic perturbation of the steady-state
[\ref{deis85},\ref{pre}]. The convective threshold 
turns out to correspond to the instability threshold which can be calculated as if the
advection was not present, i.e. $\Re[\lambda({\vec q})]>0$ where
$\lambda({\vec q})$ is the linearized eigenvalue of largest real part and
${\vec q}$ the wave-vector of the perturbation.
This means that at threshold all unstable modes are convectively unstable.
As regards the threshold of the absolutely unstable regime, its determination
reduces to the calculation of the pump amplitude which satisfies the following
conditions:
\begin{eqnarray}
\label{marginal}
\Re[\lambda({\vec k}_s)]>0 \nonumber \\
\Re[\nabla_{\vec k}^2 \lambda({\vec k}={\vec k}_s)] \geq 0
\end{eqnarray}
where the complex vector ${\vec k}_s$ defined by
\begin{equation}
\label{saddle}
\nabla_{\vec k} \lambda({\vec k}={\vec k}_s)=0 
\end{equation}
is a saddle point for $\Re[\lambda({\vec k}_s)]$ in the complex vector space.
\newline
A detailed mathematical explication of the procedure to get the above conditions
can be found in [\ref{deis85},\ref{prl},\ref{pre}]. Here, for the sake of simplicity we just mention
that wave-vectors ${\vec q}$ are extended to the complex space ${\vec k}$ in order
to evaluate the integral, in the wave-vector space, which determines the asymptotic
linearized evolution of a perturbation.

\section{Kerr resonators}
For a nonlinear resonator containing a Kerr medium
[\ref{lugi87}] an advection-like term can stem from the input pump
beam tilting [\ref{halt92}]. A one-dimensional (1D), transversal
model is used in order to simplify the analysis and to clarify the main concepts.
The 1D assumption can be also justified from an experimental viewpoint
[\ref{gryn94}].
\newline
The equation governing the electric field $A(x,t)$ is
[\ref{prl},\ref{lugi87},\ref{halt92}]:
\begin{equation}
\label{master0}
\partial_t A - 2\alpha_0\partial_x A = 
i \partial^2_x A - [1 + i\eta(\Delta - \vert A\vert ^2)]A + E_0 +
\sqrt{\epsilon} \xi(x,t) \, ,
\end{equation}
where: $\alpha_0$ represents the tilt angle, $\eta$ the sign of the
nonlinearity, $\Delta$ the cavity detuning and $E_0$ the pump.
Diffraction is represented by the first term on the r.h.s. and mirror losses by
the first in the squared brackets (exact definitions can be found in
[\ref{prl}]).
We have introduce a complex additive
noise $\xi(x,t)$, Gaussian, with zero-mean and correlation
$\left< \xi(x,t), \xi^*(x',t') \right> =2 \delta(x-x') \delta(t-t')$,
which is a standard semiclassical model of noise. For the linearized version
of the Langevin equations of the optical parametric oscillator a similar
term describes quantum noise in the Wigner representation, as considered in
[\ref{lugi97}]. In our case it can also account for thermal and input field
fluctuations.
\newline
Through conditions (\ref{marginal},\ref{saddle}) applied to the linearized
eigenvalue of eq. (\ref{master0}) we have estimated the threshold of the
absolute instability for a fixed set of parameters for the steady-state $A_0$,
solution of $A_0[1 + i\eta(\Delta - \vert A_0\vert ^2)]=E_0$.
For the same parameters we have integrated the equation for a pump
amplitude slightly above this threshold and slightly below, in the regime of
convective instability.
\newline
The time evolution of the near-field ($A$) and far-field (the space Fourier transform
of $A$) of eq. (\ref{master0}) confirm that different, unstable regimes actually
exist.
In movie 1 the near field intensity time evolution (left side) can be observed, for
the pump amplitude above the threshold of absolute instability.
\vskip 0.05in
%\hskip 0.625in \epsfxsize=3in \epsfbox{m1.ps}
\begin{center}
{\footnotesize Movie 1. Near field (left) and far field (right) growth dynamics
in the absolutely unstable regime.}
\end{center}
\vskip 0.125in
The initial condition is the steady-state plus a weak perturbation: noise is
not applied ($\epsilon=0$) because it is not necessary to generate the pattern.
After a certain transitory a drifting structure is generated. In spite of
the drift, the pattern tends to invade all the flat top region where the pump
is above the threshold. The evolution of the
far-field (right) shows that after the transitory, well defined harmonics are
generated (due to the multiple wave mixing).
Their linewidth is scarcely influenced by the presence of noise as demonstrated 
in [\ref{prl}] in the equivalent time analysis.
\vskip 0.05in
%\hskip 0.625in \epsfxsize=3in \epsfbox{m2.ps}
\begin{center}
{\footnotesize Movie 2. Near field (left) and far field (right) growth dynamics
in the convectively unstable regime.}
\end{center}
\vskip 0.125in
In the convective regime (movie 2) we applied noise ($\epsilon \simeq 10^{-5}$) and
a pattern forms again. However, note that the structure, even for long times
does not invade all the system but rather gathers its saturated value
at a random spatial position. This is due to the fact that noise needs to
drift for enough time in order to be amplified.
When the noise source is turned off the pattern (after the delay due to the
drifting) eventually disappears.
In the far-field, the noticable broadening of the spectral lines with respect to the
previous case confirms the different, noisy, nature of the pattern observed.

\section{Parametric oscillators}
In the optical parametric oscillator, i.e. when the
nonlinear medium inside the resonator has a quadratic response,
the advection-like term stems naturally from the birefringence
of the nonlinear crystal, which is exploited to phase-match the nonlinear
interaction. In fact, in a 
birefringent medium the ordinary and extraordinary polarizations can be subject
to a transversal walk-off [\ref{bloe65}]. In particular,
we consider, a degenerate, type I OPO (scattered photons are thus frequency and
polarization degenerate). The pump ($A_0(x,y,t)$ at frequency $2 \omega_0$) and the
signal ($A_1(x,y,t)$ at frequency $\omega_0$) evolution is described by the following
set of coupled equations [\ref{oppo94},\ref{drum80}]:
\begin{eqnarray}
\label{master}
\partial_t A_0 &=& \gamma_0 [ -(1+i \Delta_0) A_0 + E_0 + i a_0 \nabla^2 A_0
+ 2 i K_0 A_1^2] + \sqrt{\epsilon_0} \xi_0(x,y,t) \nonumber \\
\partial_t A_1 &=& \gamma_1 [ -(1+i \Delta_1) A_1 + \rho_1 \partial_y A_1 + 
i a_1 \nabla^2 A_1 + i K_0 A_1^* A_0] + \sqrt{\epsilon_1} \xi_1(x,y,t)
\end{eqnarray}
where: $\gamma_{0,1}$ represent the losses, $\Delta_{0,1}$ the detunings,
$a_{0,1}$ the diffraction, $K_0$ the nonlinearity, $\rho_1$ the walk-off,
$E_0$ the pump (see [\ref{oppo94},\ref{pre}] for details). Noise terms
$\xi_{0,1}$ have the same characteristics of the Kerr case and are uncorrelated.
\newline
The uniform steady-state, whose stability we are interested in, is:
$A_0=E_0/(1+i\Delta_0) \; , \; A_1=0$. It turns out that it can become
unstable along the signal component of the eigenvector, $A_1$, and thus it
is necessary to consider only one linearized equation. 
We can calculate the predicted absolute instability
thresholds through conditions (\ref{marginal},\ref{saddle}) with $\lambda$
determined from eqs. (\ref{master}). In summary, the stability diagram for the
OPO is presented in figure 1 as a function of the signal detuning $\Delta_1$.
\vskip 0.05in
\hskip0.625in \epsfxsize=3in \epsfbox{threshcol.ps}
\begin{center} 
{\footnotesize Fig. 1. Stability diagram for the OPO. Shadowed regions are:
stable (green), convectively unstable (red). The white region indicates the
absolute instability. Absolute threshold shifts upwards by increasing
$\alpha_0$ (red dashed curves).}
\end{center}
\vskip0.125in
The linear analysis also reveals that the
first mode to become unstable satisfies $q_x=0$, i.e. is parallel to the x
axis. This stems from the breaking of the rotational system due to
the walk-off. The walk-off does not affect the growth rate
but rather the spreading velocity of the perturbation. The first mode to
become absolutely unstable is that which balances the advection with spreading
[\ref{pre}].
\newline
The growth dynamics for $A_1$ is shown in movie 3, for the absolutely unstable, and in
movie 4 for the convectively unstable regime. The pump $A_0$ was a supergaussian
beam and we show in the movie the central region of space where the pump was
flat.
\vskip 0.05in
%\hskip 0.625in \epsfxsize=3in \epsfbox{m3.ps}
\begin{center}
{\footnotesize Movie 3. Near field (left) and far field (right) growth dynamics
in the absolutely unstable regime.}
\end{center}
In the initial stage, noise generates a randomly oriented pattern in both
cases; later the two evolutions start to differ. In the former case the stripes
generated are parallel to the x-axis, as predicted. The deterministic pattern
invades the whole region of pumping, stripes are well defined and no defects of
the horizontal orientation can be observed waiting a long enough time
(see also figure 2, top, which corresponds to the snapshot of the final
time of the evolution).
\newpage
\vspace*{0.5in}
\centerline{\epsfxsize=5in \epsfbox{maxi2.ps}}
\vskip 3.5in
%2in
\begin{center}
{\footnotesize Fig. 2. Snapshots of the near(far)-field at time $t=2000$
on the left (right) hand side. Parameteres of the top, middle and bottom images
correspond respectively to (*, +, X) of Fig. 1.}
\end{center}
\vskip 0.125in
In the convective regime (movie 4) the pattern is continuously
generated by noise with random orientation at the bottom of the window and
amplified while drifting.
During this process stripes get parallel to the x-axis.
Note that the dynamical orientation is less marked than in the previous
case and defects can be still seen. The location where the
pattern gathers its saturated value is random as in the Kerr case (see figure
2, middle). The average space delay and its variance depends on the noise
intensity.
\vskip 0.05in
%\hskip 0.625in \epsfxsize=3in \epsfbox{m4.ps}
\begin{center}
{\footnotesize Movie 4. Near field (left) and far field (right) growth dynamics
in the convectively unstable regime.}
\end{center}
\vskip 0.125in
The far field observation suffices to distinguish the two different
regimes. At the first stage all modes on a ring of radius
$q_c=\sqrt{-\Delta_1/a_1}$
are excited; later, in the absolute regime two narrow spots form, in
correspondance with the first mode that become absolutely unstable ($q_x=0$),
in the convective regime two broadenend arcs of the ring remain visible even
for very large times (see figure 2b).
\newline
Quantitative results which help to sharply distinguish the two regimes can
be obtained by means of a time spectral analysis [\ref{pre}].
To summarize we present three situations in figure 2, i.e. from the top to the
bottom: absolutely unstable, convectively unstable and absolutely stable (close to
threshold). The first is a deterministic pattern, the second a noise-sustained
one and the last is a noisy-precursor we have referred to in the introduction.
Signatures of a deterministic pattern are: the high intensity, the pattern
orientation (if 2D), orthogonal to the drift direction due to the symmetry
breaking, the fact that it invades all the system, narrow spatial dispersion in 
the far field.
Noise-sustained structures show: high intensity due to large noise magnification
factors, preferential selection of the stripe orientation (in 2D), although
defects are clearly observable, only partial and random occupancy of the system,
broadened far fields.
Noisy precursors below the instability threshold are characterized by: low
intensities and random orientation (in 2D), because noise is only selectively
enhanced by the filtering effect of the nonlinearity, and very broaden far
field.

\section{Conclusions}
We have theoretically predicted the existence of macroscopic, 
noise-sustained transversal structures in nonlinear optical
resonators. Numerical solutions confirm the qualitative and quantitative predictions. Noise-sustained structures can be found in the regime of convective instability which
can be induced either by a tilt in the input pump beam or by the walk-off due
to birefringence.
The growth dynamics of noise-sustained as well as deterministic patterns is
presented and helps to distinguish the nature of the structures.
\vskip 0.05in
\noindent
This work is supported by QSTRUCT (Project ERB FMRX-CT96-0077). Financial
support from DGICYT (Spain) Project PB94-1167 is also acknowledged.
\end{document}