Spatial structures in nonlinear optical cavities at IFISC

Transcription

1 ÓPTICA PURA Y APLICADA. 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 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. REFERENCES AND LINKS [1]. L. A. Lugiato, R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, (1987). [2]. M. Hoyuelos, D.Walgraef, P. Colet, M. San Miguel, Patterns arising from the interaction between scalar and vectorial instabilities in two-photon resonant Kerr cavities, Phys. Rev. E 65, (2002). [3]. M. Santagiustina, E. Hernández-García M. San Miguel, A. J. Scroggie, G.-L. Oppo, Polarization patterns and vectorial defects in type II optical parametric oscillators, Phys. Rev. E 65, (2002). [4]. D. Gomila, T. Ackemann, E. Grosee Westhoff, P. Colet, W. Lange, Secondary bifurcations of hexagonal patterns in a nonlinear optical system: alkali metal vapor in a single-mirror arrangement, Phys. Rev. E 69, (2004). [5]. D. Gomila, P. Colet, Transition from hexagons to optical turbulence, Phys. Rev. A 68, (R) (2003). Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

2 [6]. D. Gomila, P. Colet, Dynamics of hexagonal patterns in a self-focusing Kerr cavity, Phys. Rev. E 69, (2004). [7]. D. Gomila, R. Zambrini, G.-L. Oppo, Photonic band-gap inhibition of modulational instabilities, Phys. Rev. Lett. 92, (2004). [8]. W. J. Firth, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, Dynamical properties of two dimensional Kerr cavity solitons, J. Opt. Soc. Am. B 19, (2002). [9]. D. Gomila, M. A. Matías, P. Colet, Excitability mediated by localized structures in a dissipative nonlinear optical cavity, Phys. Rev. Lett. 94, (2005). [10]. A. Jacobo, D. Gomila, M.A. Matías, P. Colet, Effects of a localized beam on the dynamics of excitable cavity solitons, Phys. Rev. A 78, (2008). [11]. A. Jacobo, D. Gomila, M.A. Matías, P. Colet, Effects of noise on excitable dissipative solitons, Eur. Phys. J. D 59, (2010). [12]. P.V. Paulau, D. Gomila, P. Colet, M. Matías, N. Loiko, W. Firth, Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback, Phys. Rev. A 80, (2009). [13]. P. V. Paulau, D. Gomila, P. Colet, N. Loiko, N. N. Rosanov, T. Ackemann, W. Firth, Vortex solitons in lasers with feedback, Opt. Express 18, (2010). [14]. D. Gomila, P. Colet, G.-L. Oppo, M. San Miguel, Growth laws and stable droplets close to the modulational instability of a domain wall, Phys. Rev. Lett. 87, (2001). [15]. D. Gomila, P. Colet, M. San Miguel, A. J. Scroggie, G.-L. Oppo, Stable droplets and dark ring cavity solitons in nonlinear optical devices, IEEE J. Quantum Elect. 39, (2003). [16]. M. Pesch, W. Lange, D. Gomila, T. Ackemann, W. J. Firth, G.-L. Oppo, Two-dimensional front dynamics and spatial solitons in a nonlinear optical system, Phys. Rev. Lett. 99, (2007). [17]. M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, Noise-sustained convective structures in nonlinear optics, Phys. Rev. Lett. 79, (1997). [18]. R. Zambrini, M. San Miguel, C. Durniak, M. Taki, Convection-induced nonlinear symmetry breaking inwave Mixing, Phys. Rev. E 72, (R) (2005). [19]. R. Zambrini, M. San Miguel, Twin beams, non linearity and walk-off in optical parametric oscillators, Phys. Rev. A 66, (2002). [20]. R. Zambrini, S. M. Barnett, P. Colet, M. San Miguel, Macroscopic quantum fluctuations in noisesustained optical patterns, Phys. Rev. A 65, (2002). [21]. F. Papoff, R. Zambrini, Convective instability induced by nonlocality in nonlinear diffusive systems, Phys. Rev. Lett. 94, (2005). [22]. R. Zambrini, F. Papoff, Convective instability induced by two-points nonlocality, Phys. Rev. E 73, (2006). [23]. R. Zambrini, F. Papoff, Signal amplification and control in optical cavities with off-axis feedback, Phys. Rev. Lett. 99, (2007). [24]. F. Papoff, R. Zambrini, Lasers with nonlocal feedback, diffraction, and diffusion, Phys. Rev. A 79, (2009). [25]. R. Zambrini, F. Papoff, Nonlocal feedback in nonlinear systems, Eur. Phys. J. D 58, (2010). [26]. L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, J. Danckaert, Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures, Phys. Rev. A 75, (2007). [27]. L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, M.A. Matías, Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials, Phys. Rev. A 77, (2008). [28]. L. Gelens, D. Gomila, G. Van der Sande, M.A. Matías, P. Colet, Nonlocality-induced front interaction enhancement, Phys. Rev. Lett. 104, (2010). [29]. P. Scotto, P. Colet, M. San Miguel, All-optical image processing with cavity type-ii second harmonic generation, Opt. Lett. 28, (2010). Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

3 [30]. A. Jacobo, P. Scotto, P. Colet, M. San Miguel, Use of nonlinear properties of intracavity type II second harmonic generation for image processing, Appl. Phys. B 81, (2005). [31]. A. Jacobo, P. Colet, E. Hernandez-García, ThEnhancer: A computer program to detect jumps in ecological time-series, [32]. A. Jacobo, M. C. Soriano, C. R. Mirasso, P. Colet, Chaos-based optical communications: Encryption vs. nonlinear filtering, IEEE J. Quantum Elect. 46, 499 (2010). 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 (χ 2 ) and (χ 3 ) 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 (χ 2 ) as well as with Kerr materials (χ 3 ) [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 A related, yet different topic is the formation of dissipative solitons (DS), small-localized structures arising in some nonlinear dissipative Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

4 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. Fig. 2: Field intensity profile of a dissipative soliton in a Kerr cavity. Units are arbitrary. 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. 3: Field intensity profile at different times for an excitable excursion of a dissipative soliton in a Kerr cavity. The solid line shows a transverse cut of the structure. Units are arbitrary. Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

5 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 t 1/2 growth law to one of labyrinthine pattern formation. Such transition has been observed experimentally in reactiondiffusion 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 α with α velocity, which breaks the reflection symmetry (r r ) 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]. In presence of off-axis external feedback loops, a drift term is not a good approximation of the system. Shifted feedback needs to be modeled by a two-point nonlocality, meaning that each point in the transverse plane (x) of the light beam is coupled with a shifted one (x + x). We have considered the effect of this nonlocality in a broad class of systems, including diffusive field (for instance in liquid crystal light valves) [21,22] and diffractive fields (nonlinear optical cavities, including semiconductor lasers) [23,24]. The analysis has been recently extended to two transverse dimensions [25]. A rather important effect is that new instability regimes arise due to non-locality, and comparing with the case of drift, huge regions of convective Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

6 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 counterpropagating 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, twopoint 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 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. 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]. Fig. 5: Image processing by intracavity second harmonic generation. The image is inserted in the y-polarized field along with an homogeneous x-polarized field. The processed images are obtained in the fundamental fields and in the second harmonic field. Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica

7 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 We acknowledge financial support from MICINN (Spain) and FEDER (EU) through Grant FIS (FISICOS). Opt. Pura Apl. 44 (3) (2011) Sociedad Española de Óptica