The paper presents a critical analysis of publications devoted to one of the methods for production of the so-called "slow light" (the light with an anomalously low group velocity) due to extremely high local steepness of the dispersion curve of the medium. The method in point employs for this purpose the effect of coherent population oscillations accompanied by burning of a narrow spectral hole in the homogeneously broadened absorption spectrum. Physical model of the effect proposed in the studies under consideration is based on analysis of response of a nonlinear medium to a low-frequency intensity modulation of the incident light beam. We show that all the observations described in these papers can be easily interpreted in the framework of the simplest model of saturable absorber and have nothing to do with the effects of hole burning and group velocity reduction.
A series of benchmark tests of slow light in saturable absorbers is proposed. Stage I concerns experimental tests of saturable absorption, which can mimic slow light in saturable media. Stage II outlines the more demanding requirements for practical observation of spectral hole-burning in the absorption line, which is responsible for the reduction in group velocity.
The influence of nonlinear properties of semiconductor saturable absorbers on ultrashort pulse generation was investigated. It was shown, that linewidth enhancement, quadratic and linear ac Stark effect contribute essentially to the mode locking in cw solid-state lasers, that can increase the pulse stability, decrease pulse duration and reduce the mode locking threshold
We introduce a spin-flip model for a broad-area Vertical-Cavity Surface-Emitting Laser (VCSEL) with a saturable absorber. We demonstrate simultaneous existence of orthogonally linearly polarized and elliptically polarized cavity solitons. We show that polarization degree of freedom leads to period doubling route to spatially localized chaos of the elliptically polarized cavity solitons.
After advent of graphene as a saturable absorber many experiments have been conducted to produce short pulse duration pulses. Here, we have measured the properties of flake-graphene saturable absorber mirrors of various flake sizes dependent on fabrication technique. These mirrors enabled us to obtain a large mode-locking bandwidth of 16nm in an erbium-doped fiber laser. Mirrors with large flake size and multi-layered thickness induce strong pulse shaping and reflect mode-locked train of pulses with very large bandwidths.
Based on self - consistent field theory we study a soliton generation in cw solid-state lasers with semiconductor saturable absorber. Various soliton destabilizations, i.e. the switch from femtosecond to picosecond generation (''picosecond collapse''), an automodulation regime, breakdown of soliton generation and hysteresis behavior, are predicted.
We demonstrate saturable absorber behavior of n-type semiconductors GaAs, GaP and Ge in THz frequency range at room temperature using nonlinear THz spectroscopy. The saturation mechanism is based on a decrease in electron conductivity of semiconductors at high electron momentum states, due to conduction band nonparabolicity and scattering into satellite valleys in strong THz fields. Saturable absorber parameters, such as linear and non-saturable transmission, and saturation fluence, are extracted by fits to a classic saturable absorber model. Further, we observe THz pulse shortening, and an in…
We experimentally demonstrate a passively Q-switched erbium-doped fiber laser (EDFL) operation by using a saturable absorber based on Fe3O4 nanoparticles (FONP) in magnetic fluid (MF). As a kind of transition metal oxide, the FONP has a large nonlinear optical response with a fast response time for saturable absorber. By depositing MF at the end of optical fiber ferrule, we fabricated a FONP-based saturable absorber, which enables a strong light-matter interaction owing to the confined transmitted optical field within the single mode fiber. Because of large third-order optical nonlinearities o…
Two-dimensional mono-elemental material is an excellent saturable absorber candidate with low saturation intensity, large modulation depth, high nonlinearities, and fast recovery time of excited carriers. Typically, these mono-elemental material with two-dimensional structure possesses tunable bandgap from metallic to semiconducting according to different number of layers. The successful application of these materials as the saturable absorber has exploited the development of mode-locked fiber lasers. Therefore, this review is intended to provide an up-to-date information to the development of…
Generally speaking, the self-sweeping effect relies on the dynamical grating formed in a gain fiber. Here, the normal self-sweeping was generated in a pump-free ytterbium-doped fiber which serves as a fiber saturable absorber and is introduced to the laser cavity by a circulator in this experiment. The sweeping rate and the sweeping range alter as usual, both of which can be controlled by the pump power. Further, a new self-pulse signal is observed and discussed in this work, which shows the difference of the self-sweeping effects between active fiber and fiber saturable absorber.
Saturable absorbers are a key component for mode-locking femtosecond lasers. Polymer films containing graphene flakes have recently been used in transmission as laser mode-lockers, but suffer from high nonsaturable loss, limiting their application in low-gain lasers. Here we present a saturable absorber mirror based on a film of pure graphene flakes. The device is used to mode lock an erbium-doped fiber laser, generating pulses with state-of-the-art, sub-200-fs duration. The laser characteristic indicate that the film exhibits low nonsaturable loss (13% per pass) and large absorption modulatio…
It attracts wide interest to seek universe saturable absorber covering wavelengths from near infrared to mid-infrared band. Multilayer black phosphorus, with variable direct bandgap (0.3-2 eV) depending on the layer number, becomes a good alternative as a universe saturable absorber for pulsed lasers. In this contribution, we first experimentally demonstrated broadband saturable absorption of multilayer black phosphorus from 1 μm to 2.7 μm wavelength. With the as-fabricated black phosphorus nanoflakes as saturable absorber, stable Q-switching operation of bulk lasers at 1.03 μm, 1.93 μm, 2.72 …
In connection with the experiments recently achieved on doped crystals, biological samples, doped optical fibers and semiconductor heterostructures, we revisit the theory of the propagation of a pulse-modulated light in a saturable absorber. Explicit analytical expressions of the transmitted pulse are obtained, enabling us to determine the parameters optimizing the time-delay of the transmitted pulse with respect to the incident pulse. We finally compare the maximum fractional delay or figure of merit so attainable to those which have been actually demonstrated in the experiments.
Black phosphorus, a newly emerged two-dimensional material, has attracted wide attention as novel photonic material. Here, multi-layer black phosphorus is successfully fabricated by liquid phase exfoliation method. By employing black phosphorus as saturable absorber, we demonstrate a passively Q-switched Er-doped ZBLAN fiber laser at the wavelength of 2.8 μm. The modulation depth and saturation fluence of the black phosphorus saturable absorber are measured to be 15% and 9 μJ/cm2, respectively. The Q-switched fiber laser delivers a maximum average power of 485 mW with corresponding pulse energ…
To date, MXene has been discovered for its viability as alternatives to conventional saturable absorber such as carbon nanotube and graphene. The characteristics of high nonlinear saturable absorption, astounding modulation depth, flexible bandgap tunability, and high electron density near Fermi level are the fundamentals of the MXene as an excellent saturable absorber candidate. In particular, the research effort contributed to MXene in nonlinear ul-trafast optics are extensively growing because MXene comprises one of the largest families in 2D nanomaterials that provides huge combination pos…
The self-starting dynamics of a model for passively mode-locked lasers with saturable absorber, in which the optical amplifier has a saturable nonlinearity, is examined. The basic assumption is that the laser will operate in the mode-locked regime when the continuous-wave regime becomes unstable. Within the framework of the modulational-instability analysis, a global map for the laser self-starting criteria is constructed. According to this map, the saturable nonlinearity enhances the input-field intensity required for laser self-starting. Analytical expression of the zero modulation-frequency…
We study extreme events occurring in the transverse $(x,y)$ section of the field emitted by a broad-area semiconductor laser with a saturable absorber. The spatio-temporal events on which we perform the statistical analysis are identified as maxima of the field intensity in the 3D space $(x,y,t)$. We identify regions in the parameter space where extreme events are more likely to occur and we study the connection of those extreme events with the cavity solitons that are known to exist in the same system, both stationary and self-pulsing.
We theoretically and experimentally demonstrate a novel mode-locked ytterbium-doped fiber laser with a saturable absorber based on nonlinear Kerr beam cleanup effect. The saturable absorber was formed by a 2-m graded-index multimode fiber and a single-mode fiber segment served as a diaphragm. With an all-normal-dispersion-fiber configuration, the laser generated dissipative soliton pulses with pulse duration of 26.38 ps and pulse energy more than 0.25 nJ; output pulses could be compressed externally to 615.7 fs. Moreover, the self-starting mode-locking operation of this laser exhibited a high …
We demonstrate femtosecond performance of an ultra-broadband high-index-contrast saturable Bragg reflector consisting of a silicon/silicon-dioxide/germanium structure that is fully compatible with CMOS processing. This device offers a reflectivity bandwidth of over 700 nm and sub-picosecond recovery time of the saturable loss. It is used to achieve mode-locking of an Er-Yb:glass laser centered at 1540 nm, generating 220 fs pulses, with the broadest output spectrum to date.
We numerically investigate the deterministic generation of a perfect soliton crystal (PSC) in an optical microresonator functionalized with a saturable absorber (SA). The SA allows the direct formation of a PSC from an initial, periodic Turing roll. It prevents passage through a chaotic state, which induces a stochastic nature as regards the number of generated dissipative Kerr solitons. We show that PSCs form deterministically, and the number is controlled by adjusting the input power and SA parameter. Our work provides a simple approach for obtaining a stable PSC that offers an ultra-high re…
In this work, we identify and characterize intra-pulse intensity noise shaping by saturable absorbers applied in mode-locked lasers and ultra-low noise nonlinear fiber amplifiers. Reshaped intra-pulse intensity noise distributions are shown to be inevitably interconnected with self-amplitude modulation, the fundamental physical mechanism for initiation and stabilization of ultra-short pulses in the steady-state of a mode-locked laser. A theoretical model is used to describe the ultrafast saturation dynamics by an intra-pulse noise transfer function for widely-applied slow and fast saturable ab…
In this letter, we demonstrate the first all-fiber all-normal dispersion ytterbium-doped oscillator with nonlinear multimodal interference based saturable absorber capable to generate ultrashort dissipative soliton pulses. Additional to functioning as a saturable absorber, the use of multimode fiber segments between single mode fibers also ensures the bandpass filtering via multimode interference reimaging necessary to obtain dissipative soliton mode-locking. The oscillator generates dissipative soliton pulses at 1030 nm with 5.8 mW average power, 5 ps duration and 44.25 MHz repetition rate. P…
We introduce a new type of synthetic saturable absorbers based on quantum inspired photonic arrays whose linear light transport characteristics can be derived via bosonic algebra. We demonstrate that the interplay between optical Kerr nonlinearity, interference effects and non-Hermiticity through radiation loss leads to a nonlinear optical filtering response with two distinct regimes of small and large optical transmissions. More interestingly, we show that the boundary between these two regimes can be very sharp. The threshold optical intensity that marks this abrupt "phase transition" and it…
We present a numerical study of the collective dynamics in a population of coupled excitable lasers with saturable absorber. At variance with previous studies where real-valued (lossy) coupling was considered, we focus here on the purely imaginary coupling (evanescent wave coupling). We show that evanescently coupled excitable lasers synchronize in a more efficient way compared to the lossy coupled ones. Furthermore we show that out-of-diagonal disorder-induced localization of excitability takes place for imaginary coupling too, but it can be frustrated by nonvanishing linewidth enhancement fa…
We demonstrate a ytterbium (Yb) and an erbium (Er)-doped fiber laser Q-switched by a solution processed, optically uniform, few-layer tungsten disulfide saturable absorber (WS2-SA). Nonlinear optical absorption of the WS2-SA in the sub-bandgap region, attributed to the edge-induced states, is characterized by 3.1% and 4.9% modulation depths with 1.38 and 3.83 MW/cm2 saturation intensities at 1030 and 1558 nm, respectively. By integrating the optically uniform WS2-SA in the Yb- and Er-doped laser cavities, we obtain self-starting Q-switched pulses with microsecond duration and kilohertz repetit…
In this work we demonstrate comprehensive studies on graphene oxide (GO) and reduced graphene oxide (rGO) based saturable absorbers (SA) for mode-locking of Er-doped fiber lasers. The paper describes the fabrication process of both saturable absorbers and detailed comparison of their parameters. Our results show, that there is no significant difference in the laser performance between the investigated SA. Both provided stable, mode-locked operation with sub-400 fs soliton pulses and more than 9 nm optical bandwidth at 1560 nm center wavelength. It has been shown that GO might be successfully u…
We demonstrate generation of linearly polarized pulses from a passively Q-switched Erbium-doped fiber laser. The cavity was designed using only polarization maintaining (PM) fibers and components, resulting in linearly polarized output beam with degree of polarization (DOP) at the level of 97.6%. Reduced graphene oxide (rGO) was used as a saturable absorber for Q-switched operation. The laser was capable of delivering 1.85us pulses with 125nJ pulse energy at 115kHz repetition rate.
Black phosphorus (BP) with its enticing electric and optical properties is intensely researched in the field of optoelectronics. In this paper, Q-switched pulses at 1550 nm and 2 um wavelengths are obtained by inserting bulk-structured BP based saturable absorber (SA) into an erbium-doped fiber laser (EDFL) and an thulium/holmium-doped fiber laser (THDFL), respectively. The BP-SA was prepared by depositing powered BP material on to the flat side of a side-polished single mode fiber. Q-switched 1550 nm pulses with width tuned from 9.35 to 31 us were obtained for the EDFL. For the THDFL, over 10…
The two most commonly used models for passively modelocked lasers with fast saturable absorbers are the Haus modelocking equation (HME) and the cubic-quintic modelocking equation (CQME). The HME corresponds to a special limit of the CQME in which only a cubic nonlinearity in the fast saturable absorber is kept in the model. Here, we use singular perturbation theory to demonstrate that the CQME has a stable high-energy solution for an arbitrarily small but non-zero quintic contribution to the fast saturable absorber. As a consequence, we find that the CQME predicts the existence of stable model…
We consider a wide-aperture surface-emitting laser with a saturable absorber section subjected to time-delayed feedback. We adopt the mean-field approach assuming a single longitudinal mode operation of the solitary VCSEL. We investigate cavity soliton dynamics under the effect of time- delayed feedback in a self-imaging configuration where diffraction in the external cavity is negligible. Using bifurcation analysis, direct numerical simulations and numerical path continuation methods, we identify the possible bifurcations and map them in a plane of feedback parameters. We show that for both t…
A novel mode-locked fiber laser based on offset splicing technology between two kinds of graded index multimode fibers is proposed and experimentally demonstrated. The offset splice spot structure acts as a saturable absorber via nonlinear multimodal interference mechanism. The absorption modulation depth of the device is 2.8%, with a saturation intensity of 2.5 MW/cm2. The fiber laser operates at a stable mode-locked state when the pump power reaches 114 mW. The laser oscillates at 1039 nm with a pulsewidth of 17.6 ps, a repetition rate of 32.619 MHz, and a high signal noise ratio of 65 dB. T…
A comparative analysis of two approaches to description of the light modulation pulse delay in a saturable absorber is presented. According to the simplest model, the delay of the optical pulse is a result of distortion of its shape due to absorption self-modulation in the nonlinear medium. The second model of the effect, proposed at the beginning of our century, connects the pulse delay with the so-called "slow light" resulting from the group velocity reduction under conditions of the coherent population oscillations. It is shown that all the known experimental data on the light pulse delay i…
The optical conductance of monolayer graphene is defined solely by the fine structure constant. The absorbance has been predicted to be independent of frequency. In principle, the interband optical absorption in zero-gap graphene could be saturated readily under strong excitation due to Pauli blocking. Here, we demonstrate the use of atomic layer graphene as saturable absorber in a mode-locked fiber laser for the generation of ultrashort soliton pulses (756 fs) at the telecommunication band. The modulation depth can be tuned in a wide range from 66.5% to 6.2% by varying the thickness of graphe…
Intersubband (ISB) transitions in semiconductor multi-quantum well (MQW) structures are promising candidates for the development of saturable absorbers at terahertz (THz) frequencies. Here, we exploit amplitude and phase-resolved two-dimensional (2D) THz spectroscopy on the sub-cycle time scale to observe directly the saturation dynamics and coherent control of ISB transitions in a metal-insulator MQW structure. Clear signatures of incoherent pump-probe and coherent four-wave mixing signals are recorded as a function of the peak electric field of the single-cycle THz pulses. All nonlinear sign…
We demonstrate that the intrinsic properties of monolayer graphene allow it to act as a more effective saturable absorber for mode-locking fiber lasers compared to multilayer graphene. The absorption of monolayer graphene can be saturated at lower excitation intensity compared to multilayer graphene, graphene with wrinkle-like defects, and functionalized graphene. Monolayer graphene has a remarkable large modulation depth of 95.3%, whereas the modulation depth of multilayer graphene is greatly reduced due to nonsaturable absorption and scattering loss. Picoseconds ultrafast laser pulse (1.23 p…
Passive Q-switching of an ytterbium-doped fiber (YDF) laser with few-layer topological insulator (TI) is, to the best of our knowledge, experimentally demonstrated for the first time. The few-layer TI: Bi2Se3 (2-4 layer thickness) is fabricated by the liquid-phase exfoliation method, and has a low saturable optical intensity of 53 MW/cm2 measured by the Z-scan technique. The optical deposition technique is used to induce the few-layer TI in the solution onto a fiber ferrule for successfully constructing the fiber-integrated TI-based saturable absorber (SA). By inserting this SA into the YDF la…
This technical note describes the physical model, numerical implementation, and validation of multilevel atomic media for lasers and saturable absorbers in Meep: a free/open-source finite-difference time-domain (FDTD) software package for electromagnetics simulation. Simulating multilevel media in the time domain involves coupling rate equations for the populations of electronic energy levels with Maxwell's equations via a generalization of the Maxwell--Bloch equations. We describe the underlying equations and their implementation using a second-order discretization scheme, and also demonstrat…
The enhanced nonlinear optical response of a one-dimensional (1D) photonic crystal (PC) made from polymers and graphene composites is observed. The graphene PC was fabricated by spin-coating. It shows obvious bandgaps at two wavelengths in transmittance. Femtosecond Z-scan measurement at 515 nm and 1030 nm reveals a distinct enhancement in the effective nonlinear absorption coefficient Beff (and imaginary part of third-order dielectric susceptibility ImX(3)) for graphene nanoflakes embedded in the PC, when compared with the bulk graphene-polymer dispersion. The effect is studied in a wide rang…
We present a superconducting metamaterial saturable absorber at terahertz frequencies. The absorber consists of an array of split ring resonators (SRRs) etched from a 100nm YBaCu3O7 (YBCO) film. A polyimide spacer layer and gold ground plane are deposited above the SRRs, creating a reflecting perfect absorber. Increasing either the temperature or incident electric field (E) decreases the superconducting condensate density and corresponding kinetic inductance of the SRRs. This alters the impedance matching in the metamaterial, reducing the peak absorption. At low electric fields, the absorption…
AgInSe2 nanorods (NRs) with diameter of 15 nm have been investigated for their nonlinear optical responses by using Z-scan and transient absorption techniques with femtosecond laser pulses of photon energy greater than the bandgap. At excitation irradiance of 20 GW/cm2, AgInSe2 NRs reveal saturation in the nonlinear absorption and optical Kerr nonlinearity with a recovery time determined to be a few ten picoseconds. Such large saturable absorption and Kerr nonlinearity exhibit a third-order susceptibility of 1.2 x 10-8 esu and a figure of merit of 0.04 esu.cm.s-1, making AgInSe2 NRs a promisin…
The newly-emergent two-dimensional topological insulators (TIs) have shown their unique electronic and optical properties, such as good thermal management, high nonlinear refraction index and ultrafast relaxation time. Their narrow energy band gaps predict their optical absorption ability further into the mid-infrared region and their possibility to be very broadband light modulators ranging from the visible to the mid-infrared region. In this paper, a mid-infrared mode-locked fluoride fiber laser with TI Bi2Te3 nano-sheets as the saturable absorber is presented. Continuous wave lasing, Q-swit…
Statistics and dynamics of 2D rogue waves in a broad-area semiconductor laser with an intracavity saturable absorber are numerically investigated under the effect of transverse carrier diffusion. We show that lateral diffusion of carriers alters the statistics of rogue waves by enhancing their formation in smaller ratios of carrier lifetimes in the active and passive materials while suppressing them when the ratio is larger. Temporal dynamics of the emitted rogue waves is also studied and shown that finite nonzero transverse carrier diffusion coefficient gives them a longer duration. To furthe…
Low-dimensional materials, due to their versatile properties are very interesting for numerous electronics and optoelectronics applications. Recently rediscovered black phosphorus, with a graphite-like structure can be exfoliated up to the single atomic layer. In contrary to graphene it possesses a direct band gap controllable by the number of stacked atomic layers. For those reasons, it is now intensively investigated. Here we demonstrate, that black phosphorus can serve as a broadband saturable absorber and can be used for ultrashort optical pulse generation. The mechanically exfoliated ~300…
We present an experimental and theoretical study of modal nonlinear dynamics in a specially designed dual-mode semiconductor Fabry-Perot laser with a saturable absorber. At zero bias applied to the absorber section, we have found that with increasing device current, single mode self-pulsations evolve into a complex dynamical state where the total intensity experiences regular bursts of pulsations on a constant background. Spectrally resolved measurements reveal that in this state the individual modes of the device can follow highly symmetric but oppositely directed spiralling orbits. Using a g…
Few-layer black phosphorus (BP), as the most alluring graphene analogue owing to its similar structure as graphene and thickness dependent direct band-gap, has now triggered a new wave of research on two-dimensional (2D) materials based photonics and optoelectronics. However, a major obstacle of practical applications for few-layer BPs comes from their instabilities of laser-induced optical damage. Herein, we demonstrate that, few-layer BPs, fabricated through the liquid exfoliation approach, can be developed as a new and practical saturable absorber (SA) by depositing few-layer BPs with micro…
We fabricate a free-standing molybdenum diselenide (MoSe2) saturable absorber by embedding liquid-phase exfoliated few-layer MoSe2 flakes into a polymer film. The MoSe2-polymer composite is used to Q-switch fiber lasers based on ytterbium (Yb), erbium (Er) and thulium (Tm) gain fiber, producing trains of microsecond-duration pulses with kilohertz repetition rates at 1060 nm, 1566 nm and 1924 nm, respectively. Such operating wavelengths correspond to sub-bandgap saturable absorption in MoSe2, which is explained in the context of edge-states, building upon studies of other semiconducting transit…
Interacting resonators can lead to strong non-linearities but the details can be complicated to predict. In this work, we study the non-linearities introduced by two nested microcavities that interact with nitrogen vacancy centers in a diamond waveguide. Each cavity has differently designed resonance; one in the green and one in the infrared. The magnetic-field dependence of the nitrogen vacancy center absorption rates on the green and the recently observed infrared transitions allows us to propose a scalable on-chip magnetometer that combines high magnetic-field sensitivity and micrometer spa…
Photonic neuromorphic computing has emerged as a promising avenue toward building a low-latency and energy-efficient non-von-Neuman computing system. Photonic spiking neural network (PSNN) exploits brain-like spatiotemporal processing to realize high-performance neuromorphic computing. However, the nonlinear computation of PSNN remains a significant challenging. Here, we proposed and fabricated a photonic spiking neuron chip based on an integrated Fabry-Pérot laser with a saturable absorber (FP-SA) for the first time. The nonlinear neuron-like dynamics including temporal integration, threshold…
There are two opposing models in the analysis of the slow transmission of light pulses through saturable absorbers. The canonical incoherent bleaching model simply explains the slow transmission by combined effects of saturation and of non-instantaneous response of the medium resulting in absorption of the front part of the incident pulse larger than that of its rear. The second model, referred to as the coherent-population-oscillations (CPO) model, considers light beams whose intensity is slightly pulse modulated and attributes the time delay of the transmitted pulse to a reduction of the gro…
The ultrashort pulse laser operation is crucially defined by engaging explicit modulators. Conventionally, they are laborious to incorporate into an all-fibre design or involve the design of sophisticated cavities. We exploit Thulium-doped fibre as the gain and simple, highly integrated saturable absorber with a modulation depth of 23% and a high thermal damage threshold. Hence, pulses as short as 350fs carrying 1nJ energy at a repetition rate of 45MHz have been demonstrated. The laser average output power showed long-term stability with ~1%. The utilised concept of all-fibre saturable absorbe…
We fabricate a saturable absorber mirror by coating a graphene film on an output coupler mirror. This is then used to obtain Q-switched mode-locking from a diode pumped linear cavity waveguide laser inscribed in Ytterbium-doped Bismuthate Glass, with high slope and optical conversion efficiencies. The laser produces mode-locked pulses at 1039nm, with 1.5GHz repetition rate at an average 202mW output power. This performance is due to the combination of the graphene saturable absorber with the high quality laser glass.
Extreme events (sometimes also called optical rogue waves), in the form of pulses of extraordinary intensity, are easily observed in its chaotic regime if the Fresnel number of the cavity is high. This result suggests that the nonlinear interaction among transverse modes is an essential ingredient in the formation of extreme events in this type of lasers, but there is no theoretical description of the phenomenon yet. We report here a set of experimental results on the regularities of these extreme events, to provide a basis for the development of such a description. Among these results, we poi…
