Sección Especial: Óptica No Lineal / Special Section: Non-linear Optics
Spatial structures in nonlinear optical cavities at IFISC
Estructuras espaciales en cavidades ópticas no lineales en el IFISC
Pere Colet
(*), Damià Gomila, Adrian Jacobo, Roberta Zambrini
Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
(*) pere@ifisc.uib-csic.es
Recibido / Received: 30/10/2010. Aceptado / Accepted: 15/12/2010 ABSTRACT:
We present here a brief overview of the main topics studied in the Institute for Cross Disciplinary Physics and Complex Systems (IFISC) within the context of formation of spatial structures in nonlinear optical cavities, including both extended patterns as well as localized structures (dissipative solitons). We also study the effect of gradients and non local coupling arising either from feedback or from the presence of intracavity metamaterials as well as possible applications in optical processing of images and information.
Keywords: Non Linear Optics, Pattern Formation, Localized Structures, Dissipative Solitons, Kerr Media, Optical Parametric Oscillators, Second Harmonic Generation, Metamaterials, Non-Local Coupling, Drifting Structures, Optical Image Processing.
RESUMEN:
Aquí presentamos un breve resumen de los temas estudiados en el Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC) en el contexto de formación de estructuras en cavidades ópticas no lineales, incluyendo tanto patrones extendidos como estructuras localizadas (solitones disipativos). Se estudia también el efecto de gradientes y de acoplamientos no locales originados por realimentación o por la inclusión de metamateriales en la cavidad así como posibles aplicaciones en el procesado óptico de imágenes e información.
Palabras clave: Óptica No Lineal, Formación de Patrones, Estructuras Localizadas, Solitones Disipativos, Medios Kerr, Osciladores Ópticos Paramétricos, Generación de Segundo Harmónico, Metamateriales, Acoplamiento No Local, Efectos de Gradientes, Procesado Óptico de Imágenes.
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1. Introduction
Extended nonlinear photonic devices have the two main ingredients characteristic of complex systems: nonlinearity and spatial coupling. Nonlinearity is given by the response of the polarization of the medium to intense electric fields, which allows for the existence of multiple solutions. The spatial coupling is provided by the diffraction of light as it propagates, which spreads the information across the system. The combination of these two effects leads to the spontaneous formation of spatial structures (patterns) in systems driven out of equilibrium. This is the case of optical cavities filled with nonlinear (𝜒2) and (𝜒3) media and driven by an external field (Fig. 1). Since the seminal paper by Lugiato and Lefever in 1987 [1] optical patterns have been observed in many experimental setups with different nonlinear media.
Fig. 1:Sketch of a ring cavity with a nonlinear medium.
2. Pattern formation in nonlinear
optical cavities
At IFISC aspects of optical pattern formation have been studied from a theoretical point of view. In nonlinear optical cavities for low pump intensity values one usually has an homogeneous output. Increasing the pump strength this homogeneous solution becomes unstable and, typically, a spatial pattern appears. One peculiarity of the electric field is its vectorial nature, which plays an important role in
determining the existence of instabilities leading to pattern formation as well as in determining the patterns originated. We investigated this in cavities filled with second order nonlinear media (𝜒2) as well as with Kerr materials (𝜒3) [2,3].
Beyond this initial pattern forming instability, secondary instabilities can also take place if the pump strength is further increased. These secondary instabilities lead to more complex patterns. In some cases the new complex patterns are static, as for example in a sodium cell in a single-feedback mirror configuration [4]. In other cases, as for example in a self-focusing Kerr medium, the new patterns arising at secondary instabilities oscillate in time. Although in some parameter range the pattern can oscillate rigidly, one finds that typically this is not the case and the pattern breaks into sub-patterns, which oscillate in an alternate way [5,6]. Further increase of the pump strength leads to spatio-temporal chaotic regimes, whose dynamics is reminiscent of that encountered in Langmuir turbulence in plasmas [5,6].
In recent years, the continuous advances in the miniaturization and integration of optical components with electronics have driven the research in nonlinear optics to explore new materials and strategies to control light beyond what is possible with conventional optics. Thus photonic crystals and metamaterials have been incorporated as a new way to control the transverse properties of light in extended systems. In particular we proposed that undesired spatial instabilities of broad area devices could be controlled by including intracavity photonic crystals [7].
3. Dissipative solitons
systems. Roughly speaking, these solutions appear as a result of the balance between the spatial coupling (diffraction) and the nonlinearity, and the dissipation and the driving. These structures, also known as cavity solitons have to be distinguished from conservative solitons found, for example, in propagation in fibers, for which there is a continuous family of solutions depending on their energy. Instead, DS are unique once the parameters of the system have been fixed. DS have attracted a lot of attention in the nonlinear optics community, due to their potential as bits for all-optical memories and information processing applications. At IFISC we investigate the basic mechanisms leading to the formation of DS and their dynamics.
Very generally DS can be classified in two large groups: the ones appearing in regimes where a homogeneous solution coexists with a pattern (such as the one shown in Fig. 2), and those associated with the coexistence of two homogeneous solutions. In the first case we have studied the stability properties of DS and the different dynamical regimes they display [8].
Depending on parameter values, DS develop different kinds of instabilities. Some instabilities, such as azimuthal instabilities, lead to the formation of an extended pattern. More interesting are the instabilities that, while preserving its localized character, induce the DS to start moving, breathing, or oscillating [8]. Remarkably enough, since DS can be considered as individual entities, these instabilities may lead to dynamical regimes that appear not to be present in the local dynamics of the extended system. In particular, we have shown the existence of a regime of excitability mediated by DS [9,10] in optical cavities filled with a nonlinear Kerr medium (Fig. 3). Excitability is a concept arising originally from biology (e.g., neuroscience) found in a variety of systems. Typically a system is said to be excitable if, while it sits at a stable steady state, perturbations beyond a certain threshold induce a large response before coming back to the rest state, while smaller perturbations decay immediately. We have also shown that coherence resonance, a typical behavior encountered in excitable systems when noise is
Fig. 2: Field intensity profile of a dissipative soliton in a Kerr cavity. Units are arbitrary.
applied, is present under appropriate conditions [11]. We are currently investigating the possibility opened by the existence of this excitable regime of using the DS as all-optical reconfigurable logical gates.
While many fundamental properties of DS have been studied in prototypical models of passive nonlinear optical cavities, more and more attention is devoted recently to DS in semiconductor lasers, specially vertical cavity surface-emitting lasers (VCSELs), as these are very compact and available devices, integrable with semiconductor technology. At IFISC we also study the formation and dynamical properties of DS in VCSEL’s with frequency selective feedback [12,13].
The second class of DSs are those associated to the existence of two equivalent homogeneous solutions (usually called phases). In optics, these solutions may differ by a π phase shift or by their polarization state. In extended systems, regions occupied by different phases are called domains, which are delimited by fronts or interfaces, also called domain walls. The formation and stability properties of this kind of DS are intimately related to the general problem of domain growth and domain wall motion.
We developed a general theory [14] applicable to wide variety of systems, which explains the transition from a coarsening regime characterized by a 𝑡1/2 growth law to one of labyrinthine pattern formation. Such transition has been observed experimentally in reaction-diffusion and optical systems. We have applied our theory to models of optical parametric oscillators and vectorial Kerr cavities [15]. Furthermore the theory predicts a novel kind of localized structures, the stable droplets, large circular domain of one phase embedded in the other. Some of our results have also been observed experimentally in a sodium cell with a single feedback-mirror [16].
4. Drifting structures: gradients and
nonlocal coupling
Drift observed in some pattern forming systems is generally modeled by a gradient term 𝛼⃗·∇��⃗ with 𝛼⃗ velocity, which breaks the reflection
symmetry (𝑟⃗ → −𝑟⃗) and dramatically affects pattern selection and instabilities in spatially extended systems. In particular, drift is at the origin of convective instabilities, in which a localized perturbation grows while driven away, so that in the reference frame of the lab the perturbation decays. An interesting consequence is that in the convectively unstable regime a continuous source of perturbations, such as microscopic (for instance quantum) noise, may lead to the formation of macroscopic patterns downstream. The existence of convective instabilities and noise-sustained patterns in presence of transverse gradient terms has been recognized in several systems, from plasma to fluids or traffic, and our group has largely contributed to this subject and first showed convective instabilities in optics [17]. In nonlinear optical devices drift terms arise, for instance, in presence of tilt or misalignment or for birefringent crystals, due to walk-off between ordinary and extraordinary waves.
Gradient terms, not only lead to drifting patterns and convective instability; we have shown that, in wave mixing processes, a nonlinear-symmetry-breaking in the generated traveling waves can also arise [18]. Therefore, the "twin" beams that interfere forming the stripe pattern in an optical parametric oscillator are actually different when walk-off is taken into account, being their intensities imbalanced [19]. The classical and quantum nature of the spatial correlations in presence of drift have been considered in Refs. [19,20].
instability are found. The potential of the new system considered is that, just by varying the phase of the feedback field, features such as the transverse phase and group velocities as well as the amplification strength of local perturbations can be tuned. Indeed, light signals can be amplified while the background radiation in other regions of the system remains very low and can move across the cavity, being chirped and steered either towards or against the offset direction. A curious situation is found where a local signal can even be split into two counter-propagating components, one in the direction of the nonlocal coupling and the other in the opposite one, leading to a nonlinear signal splitter, shown in Fig. 4. As a matter of fact, two-point nonlocality is interesting not only because it introduces unexpected scenarios in the spatiotemporal dynamics of nonlinear optical devices but also because it opens new possibilities for light control and can underpin applications in optical communications, imaging, and micro-manipulation.
Besides the two point nonlocality discussed previously, nonlocal couplings represented by integral terms in partial differential equations may appear due to a number of different reasons. At IFISC we study from a general point of view the effects of such non-local couplings in the formation and dynamics of spatial structures in nonlinear optics. In particular, we have focused on the non-local effects induced by the finite size of the structural cells of metamaterials. These structured materials can
Fig. 4: Temporal evolution of an initial local perturbation of the non-lasing state, in the convectively unstable regime. For this choice of the phase of the feedback, the initial perturbation splits into two counter-propagating signals. Both the intensity and the real part (green stripe) of the field are shown. More details in Refs. [23,24].
be used to engineer diffraction, allowing for the reduction of the diffraction limit. Metamaterials at optical wavelengths would allow, for instance, reducing the size of the spatial features of optical patterns and DS. This would be very useful to achieve more compact all-optical devices. At IFISC we have studied the ultimate limit in the size of DS using metamaterials to eliminate diffraction. The nonlocal couplings introduced by the presence of metamaterials set a new limit in the size of DS [26]. This nonlocal coupling introduces also important changes in the stability and dynamics of DS [27], as well as in the interaction of fronts between different phases [28].
5. Image processing with nonlinear
optical cavities
Image processing is any form of information processing for which the input is an image, such as photographs or frames of video, and the output is usually the same image with some of its properties altered, or some of its features enhanced.
In all-optical image processing the image is represented in the transverse plane of a light beam and processed by means of the interaction of light with different elements. All-optical processing has some advantages over computer based image processing because it avoids pixelation and it is intrinsically parallel. We have studied a scheme for all optical image processing using Type II Second Harmonic Generation (SHG) in an optical cavity, first in a plane mirror configuration [29] and later in the case of spherical mirrors [30]. The proposed processing scheme is sketched in Fig. 5 the image is inserted in an optical cavity filled with a nonlinear medium. Inside the cavity, the image processing occurs because of the interaction of the light with the nonlinear medium. Finally the processed image is obtained at the output of the cavity. The use of a Type II SHG cavity allows for the transfer of the image from the fundamental to the second harmonic, contrast enhancement, contour recognition and image inversion.
In contrast to traditional all-optical processing techniques that consist of light propagating trough some medium or device, our proposal takes advantage of the properties of the
nonlinear optical cavity which introduces a nonlinear treatment of the image along with the possibility to tune the processing effects by means of the control of the thresholds introduced by the cavity.
The main feature of the system that allows for image processing is the bistability of the equations that describe it and the interplay between diffraction and nonlinearity. We have used the same basic ideas to process time series since a time series can be considered as a 1 dimensional image. We have shown that it is possible to use this techniques detect sudden jumps in time series of ecological data and we have implemented it in a software package [31]. Finally similar techniques can be used to decode messages encrypted in a chaotic carrier when the encoding method does not preserve the mean power [32].
Acknowledgements
