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Nonlinear optics and quantum optics
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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.

  • 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.

  • 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.

  • 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

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.
  • 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.

  • 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.

  • 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.

  • 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.
  • Erratum to Logvin et al., Europhys. Lett. 38, 583 (1997) (download).


  • 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.

  • 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.

  • 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.
  • Erratum to Lange et al., Physica D 96 (1996) 230 (download).


  • 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.

  • 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

  • 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


  • 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

  • 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

  • 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

  • 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.

  • 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

  • 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.

  • 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.

  • 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.

  • 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)

  • 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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AG Lange & Ackemann
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