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Publications of the Optical Pattern Formation Group
Review articles about patterns in sodium with single
mirror feedback
- Appl. Phys. B 72, 21 (2001)
Optical pattern formation in alkali metal vapors: Mechanisms, phenomena and use
T. Ackemann, W. Lange
The phenomena in optical pattern formation are reviewed using a single mirror feedback experiment as an example. The nonlinear,medium is sodium metal vapor. We discuss the mechanism responsible for the spontaneous formation of spatial structures and the selection between simple periodic patterns as hexagons and squares. Introducing a polarization changing element in the feedback loop quasiperiodic patterns with a twelve-fold rotational symmetry are obtained and their properties are studied by methods of filtering in Fourier space. Finally, we discuss the formation of localized states as spatial solitons due to a self-induced lensing effect. Their interaction is studied and the formation of clusters provides a link between localized states and extended patterns. Genuine optical features as the polarization degrees of freedom, the experimental accessibility of the Fourier space, and the possibility to control the optical susceptibility of atomic vapors by means of an external magnetic field provide very powerful tools to investigate the principles of pattern formation.
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Chaos, Solitons & Fractals 10, 617 (1999)
Atomic vapors - a versatile tool in studies
of optical pattern formation.
W. Lange, T. Ackemann, A. Aumann, E. Büthe, Yu. A. Logvin
Pattern formation in a single-mirror system containing an atomic vapor
as the nonlinear medium is discussed. It is shown in the case of sodium
atoms that the properties of the medium can drastically be modified by
the use of an oblique magnetic field which induces a light-shift-induced
levelcrossing. The phenomena observed in the system include a transition
from negative to positive hexagons and a secondary bifurcation from hexagons
to ultrahexagons. When linearly or elliptically polarized light is used
the pattern formation involves a decomposition of the light field into
its circularly polarized components. A transition from positive to negative
hexagons occurs in one polarization component, when the ellipticity of
the polarization passes zero; the other component displays the opposite
behavior. The transition is mediated by squares. The hexagons obtained
for weakly elliptical polarization at threshold undergo a secondary bifurcation
to squares if the power of the input beam is increased.
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Asian J. Phys. 7, 439 (1998)
Alkaline vapors with single-mirror feedback – a model
system for pattern formation
W. Lange, and T. Ackemann
We review our experiments involving a sodium-filled cell and a single
mirror providing feedback for a laser beam and give an interpretation of
the results. The experimental observations include transitions between
negative and positive hexagons, a secondary bifurcation in which hexagons
are unstable against their own spatial harmonics and another secondary
bifurcation involving a transition from hexagons to squares. In some of
the experiments the polarizational degrees of freedom are of crucial importance
for the pattern selection process. Further we report on a drift-instability
which displays a locking-phenomenon caused by a global inhomogeneity. Possible
modifications of the experimental scheme including the introduction of
polarization changing elements and applications of Fourier filtering are
shortly discussed.
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Quantum Semiclass. Opt. 10, R23 (1998)
Polarization patterns in alkaline vapours
W. Lange, A. Aumann, T. Ackemann, and E. Büthe
The occurrence of polarization instabilities in alkaline vapor and
their coupling with spatial degrees of freedom is considered. We discuss
experiments in which a laser beam is transmitted through a vapor cell without
external feedback or with feedback by a single mirror. In both cases a
lateral separation of the light field into domains of orthogonally circular
polarization is found. Without feedback, relatively simple structures like
rings or two spots are obtained. With feedback, more complicated polarization
patterns result from spontaneous symmetry breaking. We emphasize the crucial
role of the polarization ellipticity of the input beam. Changing this control
parameter we observe a transition from positive to negative hexagons in
one of the circular polarization components, while simultaneously the opposite
transition occurs in the other one. The transition is mediated by square
patterns occurring for linear input polarization. This metamorphosis is
governed by symmetry principles and therefore provides an example for the
ease of influencing the symmetry properties of systems consisting of multilevel
atoms interacting with a vector light field.
Articles about Nonlinear
Beam Propagation
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Opt. Photonics News 14, no 12, 41(2003)
Nonlinear effects and patterns in a cold atomic cloud
G. Labeyrie, T. Ackemann, B. Klappauf, M. Pesch, G.L. Lippi, and R. Kaiser
Our experiment demonstrates, for the first time to our knowledge, the appearance of transverse optical structures
in the field distribution of a high intensity laser beam transmitted through a cold atomic cloud. The measurements
are taken in the far field of the transmitted probe for varying probe power.
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Eur. Phys. J. D22, 473483 (2003)
Nonlinear beam shaping by a cloud of cold Rb atoms
G. Labeyrie, T. Ackemann, B. Klappauf, M. Pesch, G.L. Lippi, and R. Kaiser
First experimental investigations are reported on nonlinear beam shaping due to the interaction
between an intense laser beam and a cloud of laser cooled rubidium atoms. Resonant excitation of the F = 3 . F = 4 hyperfine transition is considered. The single-pass interaction through the cold vapor causes an increase in the laser beam intensity in the forward direction (zero transverse wavevector component) when observed in Fourier space, for sufficiently high values of saturation. A qualitative explanation of the observations based on a two-level model for a resonantly excited transition proves acceptable. The observations are compatible with an interpretation based on nonlinear index-induced focusing of an incident beam with curved wavefront, as is used in z-scan measurements. Simple physical considerations allow us to deduce the conditions for the observability of optical patterns in the beam transmitted by a cold atomic cloud.
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Chaos, Solitons & Fractals 10, 675 (1999)
Observing pattern dynamics in nonlinear optical systems
using the video-sampling method
M. Möller, J.P. Seipenbusch, T. Ackemann, B. Schäpers, A.
Aumann, A. Gahl, H.-J. Bruns, and W. Lange
We present a method of synchronizing a shutter for a video camera that
allows an instantaneous slow-motion visualization of fast repetitive processes
with a time scaling factor of up to 1000000 or more. The method uses standard,
unmodified CCIR or EIA video equipment, including video cassette recorders
for low-cost recording. A number of examples showing spatio-temporal dynamics
in nonlinear optical systems is shown, including magneto-optical beam-splitting,
drifting hexagons and phase synchronization of coupled solid-state lasers.
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J. Opt. B: Quantum Semiclass. Opt. 1, 121 (1999)
Lateral beam splitting in sodium vapour: magneto-optical
origin and polarization competition effects
A. Gahl, A. Aumann, M. Möller, and W. Lange
An almost linearly polarized laser beam focused into a sodium vapor
cell may split into spatially separated components of opposite circular
polarization. The symmetry of the emerging structures is controlled by
static magnetic fields of the order of the earth's field. The mechanism
of the decomposition of the light field into its two spin components is
clarified by numerical simulations revealing the evolution of the spatial
distribution of the light field during the propagation through the sodium
vapor. In this context, the experimentally observed occurrence of a negative
differential power transmission can be explained.
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J. Opt. B: Quantum Semiclass. Opt. 1,
90 (1999)
Modulational instability and beam splitting in the
nonlinear light propagation in sodium vapour
T. Ackemann, A. Aumann, and Yu. A. Logvin
A linearly polarized laser beam propagating through sodium vapor is
known to break-up in its circular polarization components ("beam-splitting").
We clarify the underlying mechanism by showing that spatially periodic
perturbations of the polarization state of a plane wave will be amplified
exponentially during propagation due to a modulational instability. For
a Gaussian beam in one transverse spatial dimension we find even as well
as odd active eigenmodes for polarization perturbations. Their two-dimensional
generalizations elucidate the reason for the radial and lateral splitting
that is observed in a number of experiments. In the limiting case of a
cubic nonlinearity the splitting can also be expected from a variational
approach.
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Opt. Comm. 147, 411 (1998)
Self-lensing in sodium vapor: influence of saturation,
atomic diffusion and radiation trapping
T. Ackemann, T. Scholz, Ch. Vorgerd, J. Nalik, L.M. Hoffer, G.L. Lippi
Beam shaping processes, such as self-lensing, strongly influence many
experiments in nonlinear optics. Concentrating on the self-defocusing
case, we present a detailed experimental and theoretical investigation
of the lens-like behavior of an atomic vapor optically pumped by a Gaussian
laser beam and discuss the validity of a description of the vapor as a
thin, "ideal" lens. Through comparison between experimental results and
theoretical predictions, we show how the simultaneous presence of "local"
effects (saturation) and "nonlocal" effects (atomic and radiation diffusion)
contribute to the radial distribution of the index of refraction in the
vapor. The separate contributions of atomic and radiation diffusion are
experimentally identified by measurements with different buffer gases.
A good qualitative agreement between the experimental results and
the theoretical predictions is obtained, although hyperfine and propagation
effects are neglected.
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Opt. Commun. 115, 339 (1995)
Phase singularities via nonlinear beam propagation
in sodium vapor
T. Ackemann, E. Kriege and W. Lange
The creation of phase singularities in a nonlinear optical system without
feedback is observed experimentally after a single-pass of a focused laser
beam through a high-density sodium vapor. Their appearance is reproduced
in a numerical model which allows for a slight astigmatism of the incoming
beam and can be qualitatively understood by the astigmatic deformation
of the phase fronts, to first order by means of self-induced astigmatic
lensing. Alternatively, the phase singularities can be interpreted to result
from the interference of symmetry broken modes whose mutual phases are
determined by the astigmatic lens.
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Phys.Rev. A 50, R917 (1994)
Self-induced planar and cylindrical splitting of a
laser beam in sodium vapor
A. Gahl, J.P. Seipenbusch, A. Aumann, M. Möller and W. Lange
An almost linearly polarized laser beam focused into a sodium vapor
cell may split into different patterns ranging from two spots to a spot
and a ring and a ring of opposite circular polarization. The patterns are
controlled by static magnetic fields of the order of the earth's field.
The behavior is explained, allowing for a longitudinal component of the
dieletric polarization in the paraxial wave equation. Numerical results
reveal the delicate balance between light shift, influence of the magnetic
field, and the creation of ground-state orientation by the competing polarization
components.
Articles about Driven Nonlinear Cavities
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J. Opt. B: Quantum Semiclass. Opt. 1, 161
(1999)
Bistability and transients in CO2 laser
patterns
G.L. Lippi, H. Grassi, T. Ackemann, A. Aumann, B. Schäpers, J.P.
Seipenbusch, J.R. Tredicce
We present experimental results on the selection of transverse patterns
and their dynamics in a CO2 laser with an intracavity grating.
Depending on the parameter values, we observe different single-frequency
patterns with possible bistability between them. Turning on the laser within
the bistable region yields a switch-on to a metastable pattern, which eventually
decays into the other, stable, pattern. In the bistable region, noise
appears to play an important role in the pattern dynamics over long time
scales.
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J. Mod. Opt. 45, 1913 (1998)
Fabry-Pérot and ring cavity configurations
and transverse optical patterns.
M. Möller, L.M. Hoffer, G.L. Lippi, T. Ackemann, A. Gahl, and
W. Lange
The properties of travelling- and standing-wave resonators and their
influence on the modal composition and propagation of transverse patterns
are analysed and a prescription for constructing a travelling-wave resonator
'equivalent' to a standing-wave resonator is given. It is shown that the
presence of an intracavity nonlinear medium invalidates the equivalence
between the two kinds of cavity, even when the interference between the
counterpropagating waves can be neglected. Therefore great care has to
be taken when comparing experimental and theoretical results obtained in
different types of cavity, since in general their properties differ. Some
implications concerning the appearance of phase singularities in the two
kinds of resonator are discussed.
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Phys. Rev. A 57, 4026 (1998)
Different Selection rules for transverse-mode excitation
in nonlinear ring and Fabry-Pérot resonators
A. Gahl, T. Ackemann, W. Große-Nobis, G.L. Lippi, L.M. Hoffer,
M. Möller, and W. Lange
We establish that the energy transfer between transverse modes in nonlinear
Fabry-Pérot resonators with curved mirrors, having degenerate or
nearly degenerate modes, is governed by selection rules. Although these
rules are derived in a perturbative limit, assuming a thin medium centered
in the resonator, we have found numerically and experimentally that they
still account for the otherwise unexpected enhancement or suppression of
certain modes in more realistic cases. The existence of selection rules
constitutes a fundamental difference between standing wave and traveling
wave cavities and therefore prevents in principle the possibility of describing
experiments on transverse effects performed in Fabry-Pérot cavities
with (simpler) ring resonator models.
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Chaos, Solitons & Fractals 4, 1409 (1994)
Transverse structures in a sodium-filled Fabry-Pérot
resonator. I. Experimental results: symmetries and the role of the incoupling
conditions.
G.L. Lippi, T. Ackemann, L.M. Hoffer, and W. Lange
Transverse patterns occurring in a sodium-filled Fabry-Pérot
resonator are studied in a carefully controlled experiment. The transitions
among them, observed as a function of cavity tuning, can be continuous
or discontinuous (Transverse Optical Bistability), depending on the experimental
parameters. For a variety of suitably chosen input conditions, in which
only a few excited modes participate in the interaction, the classification
of the patterns according to their topological characteristics reveals
underlying symmetries in the sign of the nonlinearity, the plane of observation,
and, most importantly, the mode-matching conditions. We show that the crucial
parameters controlling the structures are the phase coefficients of the
resonator modes excited by the input beam. The properties of the nonlinear
cavity spectrum are also used to characterize further the observations.
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Chaos, Solitons & Fractals 4, 1433 (1994)
Transverse structures in a sodium-filled Fabry-Pérot
resonator. II. Interpretation of experimental results.
G.L. Lippi, T. Ackemann, L.M. Hoffer, and W. Lange
We provide a heuristic physical interpretation of experimental results
given in preceding paper in this issue, on pattern formation in a nonlinear
confocal Fabry-Pérot resonator. The basic physical principles governing
a resonator with a self-induced, intracavity lens, well applicable to a
nonlinear medium dominated by diffusion, are shown to be sufficient to
explain the salient features of the structures that can be formed in the
nonlinear system for weak nonlinearities, and give useful guidance in the
case of stronger nonlinearities as well. This interpretation highlights
the basics of the interplay between the nonlinear medium and the optical
cavity, thereby elucidating the mechanisms that control pattern formation
in a passive nonlinear resonator. The analysis of the experimentally observed
symmetries, as a function of the beam incoupling and of the other experimental
conditions, shows that the crucial parameters that control the topological
properties of patterns are the modal phases, especially the excitation
phases.
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Phys. Rev. A 48, R4043 (1993)
Interplay of linear and nonlinear effects in the formation
of optical vortices in an nonlinear resonator.
G.L. Lippi, T. Ackemann, L.M. Hoffer, A. Gahl, and W. Lange
We study the role played by nonlinearities in the formation of optical
vortices when the boundary conditions determine the pattern formation in
a Fabry-Pérot resonator. Particular care is taken to avoid trivial
vortices coming from particular Gaussian modes or from the cavity alignment.
We show that, while the existence of optical vortices is a generic linear
property of the superpositions of Gaussian modes, their actual appearance
in the transmitted patterns is controlled by the nonlinearity itself.
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Phys. Rev. A 48, R4237 (1992)
Transverse optical bistability and formation of transverse
structures in a sodium-filled Fabry-Pérot resonator.
J. Nalik, L.M. Hoffer, G.L. Lippi, Ch. Vorgerd, and W. Lange
The authors report the observation of transverse optical bistability
(TOB) in a sodium-filled Fabry-Pérot resonator, pumped by circularly
polarized light, nearly resonant with the D2 line, in the presence
of a buffer gas, for low vapor density. A sequence of circularly symmetric
patterns, ascribable to superpositions of resonator modes, is obtained.
The appearance of TOB can be explained in terms of a self-induced intensity-dependent
lens formed in the medium by the interaction with the laser light. A simple
model that considers the transverse diffusion of the atoms inside the cell
provides predictions in good qualitative agreement with the observations.
Articles about hexagons, flowers and dynamical phenomena
in a quasi-scalar situation
- Ukr. J. Phys. 49, 358 (2004)
Self-organized intensity spirals and target
patterns in a single-mirror feedback system
F. Huneus, T. Ackemann, and W. Lange
We report on experimental and theoretical investigations on
spatio-temporal structures in a single-mirror feedback system
using sodium vapor as the nonlinear medium. Target patterns
and spirals with different numbers of arms are observed. A single
spiral arm has an archimedian shape. If the feedback mirror is
tilted the patterns transform to drifting segments of circles. It
is shown that the appearance of these patterns depends on selfinduced
and externally applied spatial phase variations. For high
order spirals the oscillation amplitude vanishes in beam center.
Simulations indicate that the oscillating perturbations include a
phase singularity in beam center.
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Appl. Phys. B 76, 191 (2003)
Optical target and spiral patterns in a single-mirror feedback scheme
F. Huneus, B. Schäpers, T. Ackemann, and W. Lange
Spatio-temporal structures with a regular time-dependence are
observed in a nonlinear optical system consisting of a sodium
vapour cell and a single feedback mirror. Target and spiral
patterns appear spontaneously in the intensity of the transmitted
light field as a result of a self-organization process. In
contrast to the results obtained in most other pattern forming
systems, the radial motion of the waves is directed towards the
center. The emergence of oscillatory structures is traced back to
a Hopf-bifurcation at a finite wave number. The preference for a
radial drift motion is due to a nonlinear guiding effect which
results from phase gradients created by the inhomogeneous pump
profile. The direction of the drift can be reversed by externally
applied phase gradients. The system is a striking example for a
case in which the radial variation of the pump profile has a
decisive influence on pattern formation.
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J. Opt. B: Quantum Semiclass. Opt. 1, 70 (1999)
Investigations of pattern forming mechanism by Fourier
filtering: Properties of hexagons and the transition to stripes in an anisotropic
system
T. Ackemann, B. Giese, B. Schäpers, and W. Lange
The properties of patterns in an anisotropic system are studied experimentally
and numerically in an optical pattern forming system with single-mirror
feedback. A controlled anisotropy is introduced by a slit filter in Fourier
space. If the slit width is reduced, a transition between hexagons and
stripes via nonequilateral hexagons and rhomboids is observed. The scenario
is reproduced in numerical simulations. The bifurcation to stripes is found
to be supercritical whereas the one to hexagons is subcritical in accordance
with expectation. The results illustrate the potential of Fourier filtering
as a tool for investigating mechanisms of pattern formation.
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Phys. Rev. E 58, 1654 (1998)
Interaction between Hopf and static instabilities
in a pattern-forming optical system
Yu. A. Logvin, T. Ackemann
Optical pattern formation in a spin-1/2 atomic system
is theoretically studied in the situation of interaction between the Hopf
and static instabilities. Variation of the parameters of the system
permits to change drastically the ratio between the spatial wavenumbers
of the Hopf and the static modes. If the mode coupling satisfies spatiotemporal
phase matching conditions, regular patterns in the form of triadic Hopf-static
patterns or oscillating patterns with hexagonal symmetry ("winking hexagons")
are formed. Otherwise, the simultaneous excitement of the Hopf and static
modes leads to the development of spatiotemporal turbulence that is in
agreement with results obtained in other systems.
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Europhys. Lett. 38, 583 (1997)
Winking hexagons
Yu. A. Logvin, T. Ackemann and W. Lange
A new kind of nonstationary hexagonal dissipative structure is found
in simulations of an optical pattern forming system. Since the brightness
of the individual constituents creating the hexagonal lattice oscillates
with time, these structures may be named winking hexagons. They are formed
by the resonant interaction of two hexagonal triads of spatial modes. One
of the triads is excited due to a Hopf bifurcation, the other one due to
a static instability.
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Erratum to Logvin et al., Europhys. Lett. 38, 583 (1997) (download).
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Phys. Rev. A 55, 4538 (1997)
Subhexagons and ultrahexagons as a result of a secondary
instability
Yu. A. Logvin, T. Ackemann and W. Lange
Simulating 2D optical pattern formation in sodium vapor we show that
hexagonal patterns can be unstable against their own spatial harmonics
or subharmonics. The instability results in ultrahexagons or subhexagons,
respectively. The formation of ultrahexagons is due to interaction with
a second solution branch with a wavenumber nearly resonant to the harmonics
of the primary hexagon. The mechanism of the instabilities is discussed
on the basis of order parameter equations obtained from symmetry considerations.
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Phys. Rev. A 56, 2321 (1997)
Light-shift induced level crossing and resonatorless
optical bistability in sodium vapour
T. Ackemann, A. Heuer, Yu. A. Logvin, and W. Lange
It is demonstrated experimentally and theoretically that a light-shift
induced level crossing in an oblique magnetic field results in a nonmonotonous
dependency of the transmission of a sodium vapour cell on the input power.
If feedback by a single plane mirror is added, bistability of the transmission
and of the beam profile (resonatorless transverse optical bistability)
is observed. The origin of the bistable behaviour and the parameter dependencies
are explained by the properties of the homogeneous solution. Numerical
simulations with a Gaussian incident beam reproduce well the switching
behaviour of the beam profile found in the experiment.
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Physica D 96 (1996) 230-241
Spontaneous optical patterns in an atomic vapor: observation
and simulation
W. Lange, Yu. A. Logvin, T. Ackemann
Optical pattern formation in an experiment with single mirror feedback
is described. The nonlinear medium is sodium vapor in a buffer gas atmosphere.
A microscopic model is given and a stability analysis and numerical simulations
are performed. Good agreement between the results of the experiment and
the simulation is obtained. By numerical treatment of the model for the
case of a plane incident wave (large aspect ratio), the results obtained
with a narrow Gaussian beam (small aspect ratio) are traced.
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Erratum to Lange et al., Physica D 96 (1996) 230 (download).
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Phys. Rev. Lett. 75, 3450 (1995)
Transition between positive and negative hexagons
in optical pattern formation
T. Ackemann, Yu. A. Logvin, A. Heuer and W. Lange
We report the observation of the transition between positive and negative
hexagons by using optical pattern formation in a sodium vapor cell with
a feedback mirror. We show that because of a secondary instability of developed
hexagon against its spatial ultra-harmonics, a new pattern (ultra-hexagon)
appears. Pattern formation for an incident Gaussian light beam takes place
in combination with switching effects that result in flower-like structures.
The data of analytical investigations and numerical simulations based on
a microscopic theory are in good accordance with experiment.
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Physical Review A, 50, R4468 (1994)
Non- and nearly-hexagonal patterns in sodium vapor
generated by single-mirror feedback
T. Ackemann, and W. Lange
In the transmitted beam of a sodium cell with a single feedback mirror
highly modulated patterns appear. The patterns possess the dihedral symmetries
D6, D2 and D3. More complex patterns consisting
of more than one elementary cell show a local hexagonal arrangement. The
topological characteristics are very similar to the one of boundary affected
patterns predicted in a thin Kerr slice. The instability is due to the
competition of diffraction, diffusion and optical pumping which is influenced
by a weak magnetic field. For higher cell-to-mirror distances the observation
of an additional length scale indicates a competition between the feedback
and a further instability mechanism, probably the counterpropagating beam
instability.
Articles about Polarization Patterns
- Ann. Phys. 13, No. 7-8, 379-390 (2004)
Selection between hexagonal, square and stripe patterns
in a polarization instability: an experimental investigation
A. Aumann, T. Ackemann, and W. Lange
The paper reports on experimental results on pattern selection in a nonlinear optical system based on a
single-mirror feedback-scheme. Zeeman pumping in sodium vapor is utilized as optical nonlinearity. Above
a certain power threshold the unstructured state with defined polarization becomes simultaneously unstable
against a pattern forming and a polarization instability. In the resulting patterns the right- and left-hand
circular polarization components of the light field tend to separate in space. The pattern selection depends
crucially on the polarization ellipticity of the input beam. Transitions between positive and negative hexagons
via stripes or squares are observed. They are determined by the symmetry of the interaction between the spin
of the light field and the atomic spin and are considered as experimental demonstrations of general principles
of pattern formation.
- J. Opt. B: Quantum Semiclass. Opt. 3, S124 (2001)
Polarization degrees of freedom in optical pattern forming systems: alkali metal vapor in a single-mirror arrangement
T. Ackemann, A. Aumann, E. Große Westhoff, Yu. A. Logvin, W. Lange
The use of polarization degrees in optimizing the conditions for optical pattern formation is investigated by discussing the particular example of single-mirror feedback systems. It is demonstrated that in a quasi-scalar saturable medium pattern formation may be completely suppressed by the interplay of saturation and diffusion. This effect takes place for the optical pumping nonlinearity between Zeeman ground states in alkaline vapor cells. It is shown that these limitations can be circumvented by introducing additional polarization components either in the input beam or in the feedback beam. The latter is achieved by introducing polarization changing elements like a quarter-wave plate in the feedback loop. In the optimized scheme pattern formation is obtained with rather low thresholds. Thus investigations with a rather high aspect ratio and far beyond threshold become feasible.
download erratum
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Int. J. Bif. Chaos 11, 2789 (2001)
Transition to spatio-temporally irregular states in a single-mirror feedback system
A. Aumann, T. Ackemann, E. Große Westhoff, W. Lange
In an optical pattern forming system (sodium vapor with feedback from a single plane mirror) different scenarios for the destabilization of stationary patterns in favor of states that are irregular in space and in time are studied experimentally and numerically. Secondary bifurcations from hexagons to squares occur via a time dependent state whose origin is traced back to noise-driven hexagon-square competition in a region of bistability between the two states. Squares can display a transition to another state with irregular - probably intrinsic - time dependence in which a whole band of wave vectors is excited. In real space pentagons are a prominent feature.
download erratum
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J. Opt. B: Quantum Semiclass. Opt. 2, 426 (2000)
Magnetic Field Control over Microscopic Symmetry Properties of an Optical Pattern Forming System: Theory
Yu. A. Logvin, A. Aumann, M. Tegeler, T. Ackemann, W. Lange
Recent experimental work demonstrates a rich transition scenario between patterns of different symmetry (hexagons and squares) close and beyond threshold in a sodium vapour cell with optical feedback in dependence of the magnitude of an external magnetic field. We reproduce the observations in numerical simulations and explain them by calculating the quadratic coupling coefficient between inhomogeneous modes which determines the pattern selection close to threshold. The observed scenario is in accordance with expectations based on general symmetry principles and should occur in many systems, but apparently only in optical systems it is possible to tune the parameters continuously over a sufficiently large area to observe the complete bifurcation scenario.
Link
to the Journal's web site at IOP
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J. Opt. B: Quantum Semiclass. Opt. 2, 421 (2000)
Magnetic Field Control over Microscopic Symmetry Properties of an Optical Pattern Forming System: Experiment
A. Aumann, E. Große Westhoff, T. Ackemann, W. Lange
We study the selection between hexagonal and square polarisation patterns in a sodium vapour cell with optical feedback provided by a single mirror. Scanning an external magnetic field, a transition sequence from squares via negative hexagons, squares, and positive hexagons again to squares is observed close to the instability onset. Well above threshold the hexagons give way to squares in a secondary bifurcation. These observations demonstrate how the macroscopic symmetries of optical patterns can be controlled by manipulating the microscopic symmetries of the light matter interaction. The key to this control are the vectorial degrees of freedom of the optical pattern forming system.
Link
to the Journal's web site at IOP
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J. Opt. B: Quantum Semiclass. Opt. 2, 386 (2000)
Pattern formation in the presence of an intrinsic
polarization instability
E. Große Westhoff, V. Kneisel, Yu.A. Logvin, T. Ackemann, W.
Lange
The interaction of pattern formation with an intrinsic polarization
instability is studied in a modified single feedback mirror experiment.
We report on hexagonal patterns in an inversion symmetric system which
occur due to a symmetry-breaking polarization instability and on domain
patterns which are stabilized by the modulational instability. For parameter
values for which no symmetry breaking bifurcation occurs, we observe rhombic
and triangular patterns.
Link
to the Journal's web site at IOP
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Quantum Semiclass. Opt. 10, R23 (1998)
Polarization patterns in alkaline vapours
W. Lange, A. Aumann, T. Ackemann, and E. Büthe
The occurrence of polarization instabilities in alkaline vapor and
their coupling with spatial degrees of freedom is considered. We discuss
experiments in which a laser beam is transmitted through a vapor cell without
external feedback or with feedback by a single mirror. In both cases a
lateral separation of the light field into domains of orthogonally circular
polarization is found. Without feedback, relatively simple structures like
rings or two spots are obtained. With feedback, more complicated polarization
patterns result from spontaneous symmetry breaking. We emphasize the crucial
role of the polarization ellipticity of the input beam. Changing this control
parameter we observe a transition from positive to negative hexagons in
one of the circular polarization components, while simultaneously the opposite
transition occurs in the other one. The transition is mediated by square
patterns occurring for linear input polarization. This metamorphosis is
governed by symmetry principles and therefore provides an example for the
ease of influencing the symmetry properties of systems consisting of multilevel
atoms interacting with a vector light field.
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Phys. Rev. A 56, R1709 (1997)
Polarized patterns in sodium vapor with single mirror
feedback
A. Aumann, E. Büthe, Yu. A. Logvin, T. Ackemann and W. Lange
In a simple nonlinear optical system consisting of a sodium vapor cell
and a single feedback mirror, the spatial and polarization degrees of freedom
are simultaneously unstable if the incident light is nearly linearly polarized.
The instability results in the formation of spatially separated domains
of opposite elliptical polarization. For initially linearly polarized light
we find square patterns which give way to positive or negative hexagons
if the polarization is elliptical. This confirms a recent theoretical prediction
based on a purely dispersive model (Scroggie and Firth, Phys. Rev. A 53,
2752(1996)). However, absorption has to be accounted for to reproduce the
experimentally observed threshold behavior.
Articles about drift instability
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Chaos, Solitons & Fractals 10, 665 (1999)
Drifting and locking behaviour of optical patterns
– an investigation using Fourier filtering.
T. Ackemann, B. Schäpers, J.P. Seipenbusch, Yu.A. Logvin, W. Lange
Stable hexagonal patterns have been observed by many groups in a nonlinear
optical system with single-mirror feedback. In contrast, a linear stability
analysis predicts drifting patterns if there is an arbitrarily small
misalignment of the feedback mirror. Studying the situation of small tilt
angles we observe that the patterns remain stationary up to a critical
value alpha_c of the angle (locking) and start to drift abruptly
as a whole if alpha_c is exceeded. Above this threshold the drift velocity
depends linearly on the tilt angle. The locking is reproduced by
numerical simulations with a Gaussian input beam and traced back to the
boundary conditions imposed by the spatially limited input beam. Filtering
of the feedback beam in Fourier space is used to generate stripe patterns
which are normally unstable in the system. The investigation of the stripe
patterns allows the verification of a dispersion relation predicted before
and the confirmation that the locking behavior is not restricted to hexagons.
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J. Opt. B: Quantum Semiclass. Opt. 1, 58 (1999)
Nonequilateral drifting hexagons in a strongly misaligned
single-mirror system
B. Schäpers, T. Ackemann, J.P. Seipenbusch, and W. Lange
We report on the appearance of a new type of patterns in an optical
pattern forming system consisting of sodium vapor as a nonlinear medium
and a single feedback mirror. These patterns are drifting and stationary
nonequilateral hexagons which appear if the feedback mirror is strongly
tilted. They are due to the resonant coupling of static and oscillatory
wavevectors lying in different instability balloons and thus constitute
a special example for the recently predicted Hopf-static structures.
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Phys. Rev. A 56, R4401 (1997)
Drift instability and locking behavior of optical
patterns
J.P. Seipenbusch, T. Ackemann, B. Schäpers, B. Berge, W. Lange
In a nonlinear optical system with single-mirror feedback, several
experiments displayed stationary hexagonal patterns. A linear stability
analysis, however, predicts drifting patterns even for an arbitrarily small
misalignment of the mirror. Studying the situation of small tilt angles
we observe that the patterns remain stationary up to a critical value alpha
[sub c] of the angle (locking). Above this threshold they discontinuously
start to drift with a velocity that depends linearly on the tilt angle.
Both features are reproduced by numerical simulations with a Gaussian input
beam. The existence of the locking region is traced back to the boundary
conditions imposed by the spatially limited input beam.
Articles about formation of superlattices and quasiperiodic
patterns
- Phys. Rev. E 69, 036205 (2004)
Secondary bifurcations of hexagonal patterns in a nonlinear optical system:
Alkali metal vapor in a single-mirror arrangement
D. Gomila, T. Ackemann, E. Große Westhoff, P. Colet, and W. Lange
Secondary bifurcations of hexagonal patterns are analyzed in a model of a single-mirror arrangement with an
alkali metal vapor as the nonlinear medium. A stability analysis of the hexagonal structures is performed
numerically. Different instabilities are predicted in dependency on the wave number of the hexagons. Some of
the instabilities take place at a finite wave number and result in the formation of structures with 12 spatial
modes. These structures are compared with those observed experimentally.
download pdf-file
- Phys. Rev. E 68, 016209 (2003)
Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system
M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange
We determine the limits of stability of the homogeneous state of a pattern forming optical system in
dependency on the wave number by experimental means. The measurement becomes feasible by adopting a
scheme based on a Fourier filtering technique. The system under study is a single-mirror feedback arrangement
using sodium vapor as the nonlinear medium. The experiment confirms the existence of multiple instability
regions of the homogeneous state expected by theory. The measurements do not agree quantitatively with the
marginal stability curve determined by a linear stability analysis of an infinitely extended homogeneous
system. We study the system numerically and demonstrate that the results of the simulations for the case of a
Gaussian beam can be reproduced by a simple modification of the linear stability analysis which accounts for
the finite diameter of the input beam. This explains the wave number dependent systematic deviations between
the experiment and the linear stability analysis of the infinitely extended system.
download pdf-file
- Phys. Rev. E 67, 025203 (2003)
Self-organized superlattice patterns with two slightly differing wave numbers
E. Große Westhoff, R. Herrero, T. Ackemann, and W. Lange
We report on the observation of superlattices that occur spontaneously in a nonlinear optical system with
O(2) symmetry. A secondary bifurcation from hexagons yields patterns formed by twelve wave vectors.
Besides irregular patterns these may either be quasiperiodic patterns or superlattices built from two classes of
wave vectors differing slightly in their length. Both classes of wave vectors stem from only one patternforming
instability. The wave vectors fit on a hexagonal or a square grid. In the former case the set of wave
vectors can be decomposed into two hexagonal triads, whereas in the case of the square grid squeezed triads
occur
(download pdf-file)
-
Proc. SPIE. 4751, 370 (2002)
Optical pattern formation far beyond threshold
T. Ackemann, E. Große Westhoff, M. Pesch, D. Rudolph, and W. Lange
Optical pattern formation is studied far beyond threshold in a single-mirror feedback scheme using sodium vapors as the nonlinear medium. Patterns with twelve fundamental wave vectors arise from hexagons in a secondary bifurcation. Besides irregular patterns, quasipatterns and superlattices are obtained. Even after a tertiary bifurcation the patterns remain stationary. Fourier filtering experiments show that the harmonics of the fundamental wave vectors are essential for the stability of the secondary and tertiary patterns. A novel Fourier filtering technique is used for a measurement of the neutral stability curve and proves experimentally the existence of multiple instability regions existing due to the periodicity of the Talbot effect.
(download pdf-file)
-
Phys. Rev. Lett. 82, 4627 (1999)
Twelvefold quasiperiodic patterns in a nonlinear optical
system with continuous rotational symmetry
R. Herrero, E. Große Westhoff, A. Aumann, T. Ackemann, Yu. A.
Logvin, and W. Lange
Twelvefold quasiperiodic structures are observed in an autonomous optical
pattern forming system with continuous rotational symmetry. These quasipatterns
arise from a primary hexagonal structure. In dependence on the experimental
parameters the bifurcation can be sub- or supercritical. In the supercritical
case, the transition from hexagons to quasipatterns is mediated by a new
kind of patterns with different amplitudes in its fundamental modes. It
is proven by an optical Fourier filtering technique that the observed quasipatterns
can only exist in the presence of harmonics of the fundamental unstable
modes. This finding supports the general theoretical result that quasipatterns
can be stabilized via resonant quadratic interaction of harmonics. The
experimental results agree fairly well with numerical solutions of a
microscopic model.
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J. Opt. B: Quantum Semiclass. Opt. 1, 166
(1999)
Interplay of dispersion and absorption in a new optical
pattern-forming system
A. Aumann, E. Große Westhoff, R. Herrero, T. Ackemann, and W.
Lange
We present a new experimental optical pattern forming system that allows
a continuous transition from the purely dispersive to the purely absorptive
limiting case. This system is based on a single feedback mirror arrangement
with an alkaline vapor and a polarization changing element inside the feedback
loop. We present a microscopic model for this system that accounts for
both absorption and dispersion and successfully describes the observed
phenomena by means of a linear stability analysis.
download erratum
Localized Structures
- Phys. Rev. Lett. 95, 143906 (2005)
Observation of a Discrete Family of Dissipative Solitons in a Nonlinear Optical System
M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange
We report on the observation of a discrete family of spatial dissipative solitons in a simple optical
pattern forming system, which is based on a modified single-mirror feedback arrangement. After a
pitchfork bifurcation the system possesses two (nearly) equivalent coexisting states of different polarizations.
The spatial solitons correspond to excursions from one of the two states serving as a background
state towards the other one. The members of the soliton family differ in the number of high-amplitude
radial oscillations. The observations are in good agreement with numerical simulations and general
expectations for dissipative solitons.
download pdf-file
- IEEE Journal of Quant. Electronics 39, 227 (2003)
Properties of feedback solitons in a single-mirror experiment
B. Schäpers, T. Ackemann, W. Lange
We report on the observation of localized states (LSs)
in a single-mirror feedback system using sodium vapor as the nonlinear
medium. By combining experimental results with simulations
based on a microscopic model and with results of a linear stability
analysis of the homogeneous state, we discuss the properties
of the LSs, the mechanism of their formation, their domain of existence
in parameter space, their interaction with the amplitude and
phase gradient in the Gaussian beam of the driving laser, and the
formation of “molecules” and “clusters” of LSs. We elucidate the
dominant role of diffraction in the interaction of the LSs and discuss
relations between the clusters of LSs and extended patterns.
- J. Opt. Soc. Am. B 19, 707 (2002)
Robust control of switching of localized structures and ist dynamics in a single mirror-feedback scheme
B. Schäpers, T. Ackemann, W. Lange
Experimental results on the switching dynamics of bistable localized states in a single mirror-feedback system with sodium vapor as the nonlinear medium are presented. The type of switching ('on' or 'off') is determined by the polarization state of the addressing beam which gives a robust - since phase-insensitive - way of control. Critical and non-critical slowing down are demonstrated.
- Proc. SPIE. 4271, 130 (2001)
Characteristics and possible applications of localized structures in an optical pattern-forming system
B. Schäpers, T. Ackemann, W. Lange
We report on the observation of dissipative localized structures in an optical pattern-forming system. After an experimental and theoretical analysis of the mechanism which stabilizes these structures we focus on the demonstration of possible applications of localized structures for information processing.
(download pdf-file)
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Phys. Rev E 61, 4622-4625 (2000)
Stationary and drifting localized structures near a multiple bifurcation point
Yu.A. Logvin, B. Schäpers, T. Ackemann
Localized states embedded in a patterned background
are found in numerical simulations of spontaneous pattern
formation in a spin-1/2 atomic system with optical feedback.
In the vicinity of a parameter region with bistability between two
homogeneous states large amplitude peaks as well as dark holes exist
as stable localized states on a hexagonal background. Moreover,
resonant interaction between oscillatory and stationary
inhomogeneous modes produces a nonstationary background which may force
the localized states to drift.
(download pdf-file)
-
Phys. Rev. Lett. 85, 748-751 (2000)
Interaction of localized structures in an optical
pattern-forming system
B.Schäpers, M.Feldmann, T.Ackemann and W.Lange
We report on the observation and interaction of dissipative localized structures in an
optical pattern forming system. Single localized structures are found to have oszillatory
decaying tails originating from diffraction. We observe bound states of two or more
constituents. These clusters contain serveral preferred mutual distances. Numerical
simulations show that the corresponding interactions are mediated by the oscillatory tails.
(download pdf-file)
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