The discovery of the spontaneous mode-locking of lasers, i.e., the synchronous oscillation of electromagnetic modes in a cavity, has been a milestone of photonics allowing the realization of oscillators delivering ultra-short pulses. This process is so far known to occur only in standard ordered lasers with meter size length and only in the presence of a specific device (the saturable absorber). Here we demonstrate that mode-locking can spontaneously arise also in random lasers composed by micronsized laser resonances dwelling in intrinsically disordered, self-assembled clusters of nanometer-sized particles. Moreover by engineering a novel mode-selective pumping mechanism we show that it is possible to continuously drive the system from a configuration in which the various excited electromagnetic modes oscillate in the form of several, weakly interacting, resonances to a collective strongly interacting regime. By realizing the smallest mode-locking device ever fabricated, we open the way to novel generation of miniaturized and all-optically controlled light sources.Random lasers[1] (RLs) are made by disordered highly scattering materials able to amplify light when externally pumped. The simultaneous presence of structural disorder and nonlinearity makes these devices a fertile ground to connect photonics with advanced theoretical paradigms[2] like chaos [3], non Gaussian statistics[4], complexity [5] and also the physics of Bose Einstein condensation [6]. Historically there has been a bridge in the RL interpretation. In pioneering experiments a smooth, single-peaked emission was produced by pumping finely ground laser crystals [7], or titania particles dispersed in a dye-doped solution [8,9]. This phenomenon has been dubbed RL with incoherent feedback (IFRL) because it may be explained in the framework of the diffusion approximation [10] that neglects interference and treats light rays as the trajectories of random walking particles. However this theoretical framework does not explain another kind of RL exhibitting subnanometre sharp spectral peaks [11][12][13] associated with high-Q resonances[14-17] and labeled resonant feedback random laser(RFRL).Standard multimode lasers without disorder and characterized by equispaced resonances may be driven to a synchronous regime through the so called mode-locking transition [18,19], which so far has only been shown to occur spontaneously in the presence of a saturable absorber and allows to generate ultra-short light pulses [20,21]. We show that the same transition occurs in RLs and allows to lock modes of a RFRL casting its emission in the typical IFRL spectrum and demonstrating the inherently coherent nature of the random lasing phenomenon.The system we consider is an isolated micrometer sized cluster of titania nanoparticles immersed in a rhodamine dye solution (see supplementary information (SI)). In our novel setup we use the amplified spontaneous emission (ASE) from the surrounding dye to pump the cluster. The the ASE areas are defined by shaping the beam of an e...

The complex processes underlying the generation of a coherent-like emission from the multiplescattering of photons and wave-localization in the presence of structural disorder are still mostly un-explored. Here we show that a single nonlinear Schroedinger equation, playing the role of the Schawlow-Townes law for standard lasers, quantitatively reproduces experimental results and threedimensional time-domain parallel simulations of a colloidal laser system. PACS numbers:Random lasers (RL) are a rapidly growing field of research, with implications in soft-matter physics, light localization and photonic devices [1,2]. Since the pioneering investigations [3,4], different groups reported on experimental observations, from paint pigments to human tissue [5,6,7,8,9]. In all of these cases a coherent-like narrow spectral line emerges from the fluorescence as the pump energy is increased and, in some instances, several spectral peaks have been reported [9,10]. In standard single-mode lasers, without structural disorder, the emission linewidth is linked to the electromagnetic energy stored in the cavity by the so-called Schawlow-Townes (ST) law [11,12]. An equivalent law for RL is missing. Nevertheless various issues (like the statistical properties and the link with spin-glass theory [9,13,14,15,16,17]), were theoretically analysed, while the leading model (quantitatively compared with experiments) is that based on the light-diffusion approximation [18,19,20,21], which however overlooks the ondulatory character of the involved photons. Within a different perspective, RL is due to several localized electromagnetic (EM) states put into oscillations in a disordered environment (as, e.g., in [9, 17, 22, 23]). In this framework, it is expected that the number of involved modes increases with the pump energy and, correspondingly, the spectrum widens. However, exactly the opposite happens and this is also accompanied by the shortening of the emitted pulse [24,25,26]. In addition, the fact that strong (or Anderson) localization of light sustains the RL action is still debated. Ab-initio computational studies were limited to 1D and 2D geometries [27,28], not accounting for the critical character of three-dimensional (3D) localization [29]. Monte-Carlo simulations neglect interference effects [30,31,32]. Here we report on an original theoretical formulation; we quantitatively compare its predictions with experiments and with the first ever reported 3D+1 ab-initio MaxwellBloch simulations. We show that the RL linewidth is ruled by a nonlinear differential-equation, which is the equivalent of the ST-law, and is formally identical to the nonlinear Schroedinger, or Gross-Pitaevskii (GP), equation governing ultra-cold atoms [33]. There is hence a strict connection between photons in RL and ultra-cold bosons; the spectral narrowing observed in RL is thus ascribed to a condensation process [34] of the involved electromagnetic resonances. Simulations -We consider a vectorial formulation of the Maxwell-Bloch (MB) equations [35,36]. 21 nonl...

We demonstrate an experimental technique that allows to achieve a robust control on the emission spectrum of a micro random laser and to select individual modes with sub-nanometer resolution. The presented approach relies on an optimization protocol of the spatial profile of the pump beam. Here we demonstrate not only the possibility to increase the emission at a wavelength, but also that we can "isolate" an individual peak suppressing unwanted contributions form other modes.Standard laser sources, allow to generate light with a high degree of coherence, precise directionality, and gaussian beam profile. Thanks to the progress of nonlinear optics, modern devices allow also femtosecond pulse compression[1] and wavelength tunability. [2,3] These technological achievements always rely on a precise control of light paths in a macroscopic (usually of the order of the meter) cavity for which any imperfection (like material damage, optical impurity, mechanical instability or misalignment) result in a severe decrease in device efficiency: any form of disorder is detrimental. On the other hand it has been recently demonstrated that disorder may be compensated by various approaches involving adaptive optics: in practice, by introducing an appropriate phase delay to each point of a laser wavefront warped by disorder, it is possible to restore it to its pristine state. [4,5] Applications range from the possibility to focus through opaque media both in space and in time [6][7][8] to subwavelenght microscopy.[9] Merging nonlinear optics with adaptive techniques is much more difficult for the inherent instability of nonlinear processes. However one-dimensional numerical simulations[10] predict that such an approach is possible for Random Lasers (RLs).RLs [11] are coherent light sources in which stimulated emission is generated in a "disordered" cavity, with a certain degree of localization, [12][13][14] infiltrated with a gain material. Here disordered means that the cavity has not been previously designed but is randomly selected by the light diffusion process inside the random medium.[15] Therefore the position, direction and wavelength at which the lasing action occurs are unpredictable ex-ante. From the fundamental point of view RLs are paradigmatic systems that mix intriguing features like nonlinearity, disorder and complexity, but are becoming increasingly more attractive also from the applied point of view. [16,17] In trying to control the RL emission spectrum various approaches have been proposed to date such as the design of individual scattering elements [18] or the engineering of the absorption. [19] In this paper we demonstrate another strategy that allows an active, sub-nanometer precise control over the spectrum emitted by the RL. The procedure relies on the selective excitation of modes through pumping engineering. The hypothesis behind this work is that pumping configurations can be found that couple best to a (preselected) target mode so that its emission is enhanced whereas other modes' is suppressed. To demo...

A disordered structure embedding an active gain material and able to lase is called random laser (RL). The RL spectrum may appear either like a set of sharp resonances or like a smooth line superimposed to the fluorescence. A recent letter [1] accounts for this duality with the onset of a mode locked regime in which increasing the number of activated modes results in an increased inter mode correlation and a pulse shortening ascribed to a synchronization phenomenon. An extended discussion of our experimental approach together with an original study of the spatial properties of the RL is reported here.PACS numbers: 42.55. Zz, 42.60.Fc The possibility of obtaining lasing into a scattering medium has been predicted in the sixties by Letokov [2] and experimentally realized for bulk lasing material and scattering particles dispersed in liquid dye in the last decades [3][4][5]. These pioneering experiments on random lasers (RL) have been characterized by a line narrowed frequency spectrum (from tens to some nanometers) on the top of the active medium fluorescence spectrum. This phenomenon, denoted as Intensity Feedback Random Lasers (IFRL), may be explained in the framework of diffusion approximation [6] in which all the properties of the photons propagation are defined by the transport mean free path ℓ. This approach, which neglects the wavelike nature of light, is particulary useful in different scattering regimes where it may be used to predict particular properties of RL, like fluctuations [7], or the appearance of spikes in the spectrum at random positions [8]. On the other hand, the diffusion approximation does not allow to predict the existence of RL peaks that have been more recently measured in numerous experiments [9]. These peaks, which appear at a fixed wavelength and whose light is emitted from localized portions of the disordered structure, may be associated with the presence of Anderson localized states [10] or with highly scattering resonances dwelling in the disordered structures [11], that are commonly referred to as Resonant Feedback Random Lasers (RFRL). This kind of lasing emission has been observed in very strong scattering systems (Zinc Oxide or Gallium Phosphide) with very small spatial extension, very small pump spot, or also with stripe shaped spot [12]. In particular it has been shown [13] that by increasing the pump spot of a RFRL a smooth spectrum is recovered.Up to now different theoretical approaches have been suggested to account for the different aspects of RLs. The RL spectrum has been investigated through the strong nonlinear interaction of RL individual lossy modes [14], while an analogy with condensed matter physics a spinglass-like model [15] predicts the existence of various thermodynamic phases including a mode locked condition, in which lasing resonances are synchronized. A different approach valid in the condition in which the This review follows a recent letter [1] in which it has been demonstrated that the particular regime (so far called either resonant or diffusive) in...

We show that the degree of localization for the modes of a random laser (RL) is affected by the inter mode interaction that is controlled by shaping the spot of the pump laser. By experimentally investigating the spatial properties of the lasing emission we infer that strongly localized modes are activated in the low interacting regime while in the strongly interacting one extended modes are found lasing. Thus we demonstrate that the degree o localization may be finely tuned at the micrometer level.PACS numbers: 42.25.Dd 07.05.Fb Keywords: Light localization, Random lasing RL are among the most complex systems in photonics, encompassing structural disorder and nonlinearity, 1 and ranging from micron sized optical cavities 2 to kilometerlong fibers 3 . Attention on this systems has been constantly growing as the number of potential applications, ranging from object coding 4 to speckle-free illumination 5 .First-principle time domain simulations show that modes of a RL arise from electromagnetic states [6][7][8] , that may appear in localized or extended fashion. Several experiments confirmed this view 2,9,10 , and tried to address the connection between the structure 11 pumping condition, or gain and the degree of localization 12-14 . The The pulsed laser (532 nm, repetition frequency 10 Hz and fluence 0.1nJ/µm 2 ), whose spot is shaped by a SLM in amplitude configuration (by using the two crossed polarizers P1 and P2), pumps a single titanium dioxide cluster (diameter between 5 and 12 µm). The RL emission is collected by a microscope objective (OBJ) to be imaged by using a beamsplitter (BS) in two different image planes. In one of them lies a fiber controlled by translators with nanometric resolution that allows the measurement of the spatio-spectral map. In this way it is possible to scan a magnified (50×) image of the sample and measure the spectra emitted from a single point. The fibre core (50 µm in diameter) collects spectra originating in an area of 1µm of diameter of the sample. The other light path allows imaging the sample on a CCD. presence of many modes may give rise to unique phenomena: in a linear system extended necklace states spread over the sample via multiple (localized) resonances 15 while in a system with gain the inter-mode coupling affects the whole spectrum generating mode repulsion 16 directly connected to the nonlinear interaction 17 .On the other hand it has been recently demonstrated that the inter-mode coupling plays a critical role in the onset of two fundamentally different RL regimes, distinguished by the shape of the emission: a "resonant feedback random laser" (RFRL) 18 , which appears as a set of sharp peaks oscillating independently at fixed spectral positions, and the "intensity feedback random laser" (IFRL) 19 , characterized by a smooth single line narrowed spectrum. By using a tailored spatial shape of the pump area 20,21 , a switching between the two can be achieved. In fact a RFRL is observed when activating a set of weakly interacting resonances while IFRL is produced un...

Anderson localization of light is traditionally described in analogy to electrons in a random potential. Within this description the disorder strength -and hence the localization characteristicsdepends strongly on the wavelength of the incident light. In an alternative description in analogy to sound waves in a material with spatially fluctuating elastic moduli this is not the case. Here, we report on an experimentum crucis in order to investigate the validity of the two conflicting theories using transverse-localized optical devices. We do not find any dependence of the observed localization radii on the light wavelength. We conclude that the modulus-type description is the correct one and not the potential-type one. We corroborate this by showing that in the derivation of the traditional, potential-type theory a term in the wave equation has been tacititly neglected. In our new modulus-type theory the wave equation is exact. We check the consistency of the new theory with our data using a field-theoretical approach (nonlinear sigma model).

We study with numerical simulation the possible limit behaviors of synchronous discrete-time deterministic recurrent neural networks composed of N binary neurons as a function of a network's level of dilution and asymmetry. The network dilution measures the fraction of neuron couples that are connected, and the network asymmetry measures to what extent the underlying connectivity matrix is asymmetric. For each given neural network, we study the dynamical evolution of all the different initial conditions, thus characterizing the full dynamical landscape without imposing any learning rule. Because of the deterministic dynamics, each trajectory converges to an attractor, that can be either a fixed point or a limit cycle. These attractors form the set of all the possible limit behaviors of the neural network. For each network we then determine the convergence times, the limit cycles' length, the number of attractors, and the sizes of the attractors' basin. We show that there are two network structures that maximize the number of possible limit behaviors. The first optimal network structure is fully-connected and symmetric. On the contrary, the second optimal network structure is highly sparse and asymmetric. The latter optimal is similar to what observed in different biological neuronal circuits. These observations lead us to hypothesize that independently from any given learning model, an efficient and effective biologic network that stores a number of limit behaviors close to its maximum capacity tends to develop a connectivity structure similar to one of the optimal networks we found.

We report the experimental observation of the interaction and attraction of many localized modes in a two-dimensional system realized by a disordered optical fiber supporting transverse Anderson localization. We show that a nonlocal optically nonlinear response of thermal origin alters the localization length by an amount determined by the optical power and also induces an action at a distance between the localized modes and their spatial migration. Evidence of a collective and strongly interacting regime is given.