Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
Department of Mathematics, University of Arizona, Tucson, Arizona, 85721
I. Aranson , D. Hochheiser , J.V. Moloney
Pattern selection in wide aperture lasers is provided by active sources of traveling waves. Finite absorbing transverse boundaries select near-field lasing patterns from a one-parameter continuous family of allowed modes. The selected pattern on the boundary invades the interior consuming any spiral waves that are formed in this region. As a result, only simple stationary patterns are established in non-stiff lasers.
Large aspect ratio lasers offer an excellent paradigm for pattern formation in nonvariational systems. Variational pattern forming systems, such as the real Swift-Hohenberg equation, have an associated Liapunov functional and hence static patterns (passive defects, rolls and square patterns, grain boundaries, etc.) tend to be realized near threshold. Typically, an optimal (or critical) wavenumber is selected near threshold. Traveling wave near-field solutions are ubiquitous in large aperture two-level lasers and correspond to off-axis far-field lasing emission[1]. The relative ease with which the near- and far-field can be experimentally measured in optics offers a unique opportunity to monitor the time space behavior of the complex order parameter. Active point defects as spiral waves and target patterns [3, 2], linear defects as zipper states between oppositely traveling waves [4], alternating rolls, etc. all appear to be dynamic manifestations of the nonvariational diffraction term appearing in laser systems. Traveling waves are advected with nonzero group velocity and may carry information from the active sources emitting these waves [5]. Waves emitted by such sources may eventually invade the entire domain. This is in marked contrast to the variational real or complex Swift-Hohenberg equation where the group velocity is zero and information does not propagate in a homogeneous system. Boundary effects are therefore expected to be important in a whole class of externally pumped wide aperture lasers where absorbing boundaries act as natural sources of traveling waves which are then advected to the interior. Typically, a wavenumber selected by an active source does not coincide with the optimal one. A natural question to ask then is how pattern wavenumber selection induced by a boundary can compete with pattern selection arising from other mechanisms within the bulk.
The two-level laser can be described by the set of Maxwell-Bloch equations (MBE) , written here in the complex Lorenz notation
as in [6, 7].
The complex variables e,p are scaled envelopes of the electric and
polarization fields, n is the
scaled deviation of the population inversion from threshold. The parameters
are respectively the decay rates of the
electric field and of the population inversion, the detuning 
is the scaled difference between the atomic line and the cavity frequency,
and r characterizes the pump amplitude. For a detailed description
of this scaling see [6, 7]
Near lasing threshold 
, one can systematically reduce the MBE
description to a set of two coupled equations for the electric field and the
population inversion [7]
For the purpose of the present analysis it is convenient to introduce the following scaling of the variables
bringing equations (2) to the form
In the so-called non-stiff limit defined by the condition 
and close to threshold, one can drop 
(the population inversion
becomes a slaved variable). One then obtains a single complex Swift-Hohenberg
equation
Near threshold, equations (4,5) show explicitly that waves are
advected with the group velocity 
.
Therefore, we can expect that active sources will
have invasive (or aggressive) character, invading the entire domain
[8, 9].
Equations (4,5) possess a family of traveling wave solutions
In an infinitely extended system, these waves form a continuum in k-space. However boundary conditions typically select a unique wavenumber from the family. Other sources of traveling waves also select a unique wavenumber. If more than one wavenumber selection mechanism is present there is competition. In contrast, a sink (an object absorbing traveling waves) may exist for arbitrary wavenumber k [3, 5].
We know of two main types of sources: point sources, having the structure of a spiral wave, the target pattern is known to be unstable in a homogeneous system, and line sources. Among line sources one can distinguish between a one dimensional source and a domain wall separating traveling waves with different orientation (zipper state). Spiral waves are nucleated spontaneously in nonvariational systems from an initially disordered state. Line sources can be associated with boundary sustained effects. It is fairly unclear which kind of pattern will dominate the long time evolution. In this paper we discuss the importance of these sources in selecting patterns in large aspect ratio lasers.
The spiral wave solution of Eqs. (4,5) is given by
where 
are polar coordinates,
the functions 
have the following asymptotic
behavior 
and 
for 
. The asymptotic wavenumber k is uniquely selected by the parameters
and, 
and in general has to be obtained numerically.
For 
numerical solution of the spiral wavenumber selection problem
gives 
[4] .
The stability problem for the spiral wave consists of two parts: stability of
the continuous spectrum and stability of the discrete spectrum.
The stability problem for the continuous spectrum is not different from that
of the traveling waves because, asymptoticly, the radiation field of the
spiral corresponds to traveling waves with the selected wavenumber k. From
our knowledge of the stability of traveling waves [7, 10] the
stability criterion is that k should lay inside the stable Busse balloon and
(no zigzag instability).
The stability problem for the discrete spectrum is more complicated. For 
we can use the method developed in [2]. A general
statement is that spirals suffer a core instability resulting in the
spontaneous acceleration of the core. The domain of the core instability
is widespread in 
parameter space and, in the stiff
limit where the Busse balloon vanishes [7], the instability is strongly
enhanced.
For Eq. (4) possesses a type of Galilean invariance
resulting in the existence of a family of spirals moving with the velocity v
(see for details [2])
where 
.
For 
the symmetry is broken
due to diffusion terms 
and the
nonlocal relation between n and 
. As a result one has acceleration
.
Using 
we obtain a modified CSHE
In the reference frame moving with velocity v,
the last term in (9) becomes
and renormalizes the acceleration.
Following the lines of the analysis of [2], developed primarily for the
complex Ginzburg-Landau equation, we obtain the neutral curve
for the core instability depicted in the Fig. 1.
The stable region for the core instability, described above, lies inside the zig-zag instability regime [7] of the traveling waves. One can therefore conclude that spirals are of very limited relevance as a stationary source of traveling waves. Of course, a single spiral cannot vanish due to topological reasons, but the core instability makes it impossible to establish a radiation field and accelerated spirals vanish at the boundaries. Numerical simulations with periodic boundary conditions show that inside the core unstable region ``small islands'' exist where the core instability appears to saturate at some finite level. In this case the core of the spiral rotates around a center at some finite distance. This is reminiscent of stable meandering of spirals in reaction-diffusion systems. Approaching the neutral curve in Fig. 1 from below, the radius of the meander diverges.
Typically large aperture lasers have a kind of absorbing boundary
arising from fast decay of the pumping amplitude away from the beam axis. In
an ideal situation (very sharp decay) one can assume that 
at the
boundary. For large aspect ratio one can consider the boundary to be locally
flat, then the problem becomes quasi-one-dimensional.
Consider the selection of the wavenumber in the one dimensional situation.
We assume
at the edges. These zeros of the field act as sources of traveling waves.
For the solution for the half-plane can be written analytically:
where the wavenumber is 
and
.
This type of solution first obtained by A. Newell, was shown by
Nozaki and Bekki to belong to a family of moving hole solutions [11].
Typically, Nozaki-Bekki holes are unstable because the depression of the
amplitude vanishes in the dynamics. In our case it is stabilized by the
imposed boundary conditions.
For the wavenumber is renormalized at order .
Numerical simulations show a kind of "wavenumber pulling effect",
i.e. for 
the selected k grows and decreases for 
. In the limit of
the correction to the selected wavenumber is obtained analytically by
substituting an anzatz
into the Eq. (4). Here we assume 
,
and 
are free parameters.
At first order in we obtain
Let now consider the more realistic situation when the beam has a sharp but
finite decay length. In one dimension, for a finite interval with
at the edges, one has two sources
at the edges separated by a sink in the middle of the domain.
To study this in our computations we consider
Eq.(2) with pumping of the form
, where r is the pump
amplitude.
For 
the profile approaches the step-function:
r(x)=r for x>0 and 
for x<0.
This gives rise to the boundary conditions by taking 
and .
The finite pumping region provides us with a means of selecting a wavenumber
from within the continuum of traveling wave solutions on the infinite line.
The numerically computed dependence of the wavenumber on 
, and 
is shown in Fig. 2.
One sees that the stronger the decay of the beam, as a function of x, the larger
the selected wavenumber. Strong beam decay corresponds to a smaller width
( ) of the decay region, and larger decay (bigger ).
For and one has an asymptotic expression
for the selected wavenumber
, where C is some positive constant.
The curves shown in
Fig 2.
verify this 
dependence, and give
good agreement with the analytic expression for k Eq.(12).
In two dimensions, one
expects that the boundary will emit traveling waves with the predicted
wavenumber and that these waves would collide in the interior of the domain
forming a sink (shock). In general the shock can include a defect with
nonzero topological charge.
Although the shock includes the topological defect, the latter is not expected
to select a wavenumber because it plays a passive roll. An
interesting question in 2 dimensions is the coexistence of a spiral
(point) and a line (absorbing boundary) source. Each of these sources
have a wavenumber selection mechanism, but one expects the waves
emitted by the line source to invade the entire domain pushing the spiral
away [12].
From the analysis above one expects the spiral to select a smaller wavenumber
( 
) than the line source ( 
).
In order to test these expectations, we have carried out extensive
numerical simulations for a variety of ramped absorbing boundaries,
simultaneously on the complex Swift-Hohenberg (2) and the original
Maxwell Bloch equations. Figures 3 and 4 display the real part of for
a circular absorbing boundary, and a square absorbing boundary.
In Figure 3(a)
the final asymptotic state is a target pattern which was
seeded with an initial condition of a constant
plus small amplitude random noise.
Figure 3(b) shows a one-armed spiral
trapped
in the center. This one armed spiral arises from initial seeding using a
one-armed logarithmic
spiral with small amplitude random noise.
Figure 3(c) shows the initial
condition for a four-armed logarithmic spiral with same amplitude random
noise, and the final asymptotic state of this initial condition is shown in
Figure 3(d), consisting of a four-armed spiral trapped in the center of the
domain.
Figure 4(a) shows a transient state arising from random initial conditions,
note that the spiral in this figure is at the critical wavenumber which is
significantly different from the boundary induced wavenumber which is present in
the final asymptotic state shown in Figure 4(b).
Figure 4(c) and (d) are for a square ramped region.
Figure 4(c) shows a transient for this region. One can see the waves
emitted by the boundary but now in a square pattern. Traveling waves emanating
from the corners eventually dominate. The final asymptotic
state is shown in Figure 4(d). This corresponds to a zipper-like state where
the roll orientation is enforced by the boundary. Waves emanating from the
corners collide along a cross-like interior boundary which forms an extended
sink.
In all cases, the wavenumber of the final asymptotic state is
that selected by the boundary. Remarkably, simulations on the 2-level
Maxwell-Bloch and Complex Swift-Hohenberg equations are in excellent
agreement even when the system is pumped well beyond threshold. This confirms
the robustness of the Complex Swift-Hohenberg as a generalized rate equation
description of the 2-level Maxwell-Bloch equations.
In summary, we have shown that highly absorbing boundaries serve to trap otherwise unstable spiral cores and enforce a selected pattern wavenumber by invading the interior of the domain. Moreover we have shown from numerical simulations that the Complex Swift Hohenberg equations (5) and the original Maxwell-Bloch equations from which they are derived shows excellent agreement even well beyond lasing threshold. This confirms the analytic predictions made in reference [7]. Starting with an initial condition with a given winding number (i.e. a give number of defects inside the ramped region), one can see the development of spiral patterns in 2D. Our numerical simulations indicate that multiarmed (passive) spirals are stabilized by the absorbing boundary conditions.
This work was supported under contract AFOSR-F49620-94-1-0144DEF and
AFOSR-F49620-94-1-0463. D.H. and J.V.M. would like to thank NSF for
their support,
under grant NSF-INT-9404732.
I.A. acknowledges the hospitality of the Arizona Center for Mathematical
Sciences.
The work of I.A. was supported in part by Raschi foundation and
Israeli Science Foundation.
Figure 1: Neutral curve of the core instability for Eq. (5)
Figure 2: Selected wavenumber for Eq. (5)
corresponding to an inhomogeneous pump profile for different values of ,
for 
, 
and
.
Figure 3:Stationary patterns for a circular domain (a) target pattern arising
from a constant plus small amplitude white noise initial condition (b) spiral
pattern arising from a 1-armed logarithmic spiral plus same amplitude white
noise initial condition (c) 4-armed logarithmic spiral plus same amplitude
white noise initial condition (d) 4-armed spiral asymptotic state arising
from initial conditions in picture (c)
a=0.4267, b=0.8, 
, r=1.711, and 
.
The domain size is 125 
125
Figure 4: Stationary patterns for circular and square domains (a) transient
state arising from low amplitude white noise initial condition (b) final
asymptotic state arising from low amplitude white noise initial condition
(c) transient state arising from a constant plus low amplitude white noise
initial condition in a square domain (d) final asymptotic state arising
from a constant plus low amplitude white noise in a square domain.
