Observation of Geometric Parametric Instability Induced by the Periodic Spatial Self-Imaging of Multimode Waves

Cascaded geometric parametric process in a tapered air-silica graded-like multimode microstructure fiber

We experimentally study the spatial beam profile and the spectral broadening at the output of a multimode air-silica microstructure fiber taper, used along the direction of an increasing fiber diameter. By using a laser pump at 1064 nm emitting 60 ps Gaussian beam pulses, we observed a competition between Raman beam cleanup and Kerr beam self-cleaning: the multimode frequency conversion process permits to generate spectral sidebands with frequency detuning from the pump that are difficult to obtain in standard graded-index multimode fibers. The generated supercontinuum spans from 500 nm up to 2.5 µm.

High-gain far-detuned nonlinear frequency conversion in a few-mode graded-index optical fiber

We present theoretical and experimental evidence of high-gain far-detuned nonlinear frequency conversion, extending towards both the visible and the mid-infrared, in a few-mode graded-index silica fiber pumped at 1.064  μ m , and more specifically achieving gains of hundreds of dB per meter below 0.65  μ m and beyond 3.5  μ m . Interestingly, our findings highlight the potential of graded-index fibers for enabling high-gain wavelength conversion into the strong-loss spectral region of silica. Such advancements require an accurate interpretation of intramodal and intermodal four-wave mixing processes.

Peregrine solitons and resonant radiation in cubic and quadratic media

We present the fascinating phenomena of resonant radiation emitted by transient rogue waves in cubic and quadratic nonlinear media, particularly those shed from Peregrine solitons, one of the main wavepackets used today to model real-world rogue waves. In cubic media, it turns out that the emission of radiation from a Peregrine soliton can be attributed to the presence of higher-order dispersion, but is affected by the intrinsic local longitudinal variation of the soliton wavenumber. In quadratic media, we reveal that a two-color Peregrine rogue wave can resonantly radiate dispersive waves even in the absence of higher-order dispersion, subjected to a phase-matching mechanism that involves the second-harmonic wave, and to a concomitant difference-frequency generation process. In both cubic and quadratic media, we provide simple analytic criteria for calculating the radiated frequencies in terms of material parameters, showing excellent agreement with numerical simulations.

On the maximization of entropy in the process of thermalization of highly multimode nonlinear beams

We present a direct experimental confirmation of the maximization of entropy which accompanies the thermalization of a highly multimode light beam, upon its nonlinear propagation in standard graded-index (GRIN) optical fibers.

Supercontinuum generation in a graded-index multimode tellurite fiber

We report the generation of a broadband supercontinuum (SC) from 790 to 2900 nm in a tellurite graded-index (GRIN) multimode fiber with a nanostructured core. We study the SC dynamics in different dispersion regimes and observe near-single-mode spatial intensity distribution at high input energy values. Numerical simulations of the (3 + 1)D generalized nonlinear Schrödinger equation are in good agreement with our experiments. Our results open a new avenue for the generation of high-power mid-infrared SC sources in soft-glass fibers.

Spatial-spectral complexity in Kerr beam self-cleaning

We report on a comprehensive experimental investigation into the spatial-spectral complexity of the laser beam during Kerr-induced beam self-cleaning in graded-index multimode fibers. We demonstrate the self-cleaning of beams using both transform-limited and chirped femtosecond pulses. By utilizing the spectrally resolved imaging technique, we examine variations in beam homogeneity during the beam cleanup process and reveal correlations observed among spatial beam profiles at different wavelengths for the various cleaned pulses. Our results significantly advance our understanding of Kerr-induced self-cleaning with chirped ultrafast pulses and offer new possibilities for diverse applications.

Spatiotemporal dual-periodic soliton pulsation in a multimode fiber laser

Spatiotemporal mode-locked (STML) fiber lasers have become a new platform for investigating nonlinear phenomena. In this work, spatiotemporal dual-periodic soliton pulsation (SDSP) is firstly observed in an STML fiber laser. It is found that in the SDSP, the long-period pulsations (LPPs) of different transverse modes are synchronous, while the short-period pulsations (SPPs) exhibit asynchronous modulations. The numerical simulation confirms the experimental results and further reveals that the proportion of transverse mode components can manipulate the periods of the LPP and SPP but does not affect the synchronous and asynchronous pulsations of different transverse modes. The obtained results bring the study of spatiotemporal dissipative soliton pulsation into the multi-period modulation stage, which helps to understand the complex spatiotemporal dynamics in STML fiber lasers and discover new dynamics in high-dimensional nonlinear systems.

Spectral-temporal-spatial customization via modulating multimodal nonlinear pulse propagation

Multimode fibers (MMFs) are gaining renewed interest for nonlinear effects due to their high-dimensional spatiotemporal nonlinear dynamics and scalability for high power. High-brightness MMF sources with effective control of the nonlinear processes would offer possibilities in many areas from high-power fiber lasers, to bioimaging and chemical sensing, and to intriguing physics phenomena. Here we present a simple yet effective way of controlling nonlinear effects at high peak power levels. This is achieved by leveraging not only the spatial but also the temporal degrees of freedom during multimodal nonlinear pulse propagation in step-index MMFs, using a programmable fiber shaper that introduces time-dependent disorders. We achieve high tunability in MMF output fields, resulting in a broadband high-peak-power source. Its potential as a nonlinear imaging source is further demonstrated through widely tunable two-photon and three-photon microscopy. These demonstrations provide possibilities for technology advances in nonlinear optics, bioimaging, spectroscopy, optical computing, and material processing.

Periodically tunable multimode soliton pulsation in a spatiotemporal mode-locked fiber laser

Multimode fiber lasers have become a new platform for investigating nonlinear phenomena since the report on spatiotemporal mode-locking. In this work, the multimode soliton pulsation with a tunable period is achieved in a spatiotemporal mode-locked fiber laser. It demonstrates that the pulsation period drops while increasing the pump power. Moreover, it is found that different transverse modes have the same pulsation period, asynchronous pulsation evolution and different dynamical characteristics through the spatial sampling technique and the dispersive Fourier transform technique. To further verify the experimental results, we numerically investigate the influences of the gain and the loss on the pulsation properties. It is found that within a certain parameter range, the pulsation period drops and rises linearly with the increase of the gain and the loss, respectively. The obtained results contribute to understanding the formation and regulating of soliton pulsations in fiber lasers.

Nature of Optical Thermodynamic Pressure Exerted in Highly Multimoded Nonlinear Systems

The theory of optical thermodynamics provides a comprehensive framework that enables a self-consistent description of the intricate dynamics of nonlinear multimoded photonic systems. This theory, among others, predicts a pressurelike intensive quantity (p[over ^]) that is conjugate to the system's total number of modes (M)-its corresponding extensive variable. Yet at this point, the nature of this intensive quantity is still nebulous. In this Letter, we elucidate the physical origin of the optical thermodynamic pressure and demonstrate its dual essence. In this context, we rigorously derive an expression that splits p[over ^] into two distinct components, a term that is explicitly tied to the electrodynamic radiation pressure and a second entropic part that is responsible for the entropy change. We utilize this result to establish a formalism that simplifies the quantification of radiation pressure under nonlinear equilibrium conditions, thus eliminating the need for a tedious evaluation of the Maxwell stress tensor. Our theoretical analysis is corroborated by numerical simulations carried out in highly multimoded nonlinear optical structures. These results may provide a novel way in predicting and controlling radiation pressure processes in a variety of nonlinear electromagnetic settings.

Spatiotemporal mode-locking and dissipative solitons in multimode fiber lasers

Multimode fiber (MMF) lasers are emerging as a remarkable testbed to study nonlinear spatiotemporal physics with potential applications spanning from high energy pulse generation, precision measurement to nonlinear microscopy. The underlying mechanism for the generation of ultrashort pulses, which can be understood as a spatiotempoal dissipative soliton (STDS), in the nonlinear multimode resonators is the spatiotemporal mode-locking (STML) with simultaneous synchronization of temporal and spatial modes. In this review, we first introduce the general principles of STML, with an emphasize on the STML dynamics with large intermode dispersion. Then, we present the recent progress of STML, including measurement techniques for STML, exotic nonlinear dynamics of STDS, and mode field engineering in MMF lasers. We conclude by outlining some perspectives that may advance STML in the near future.

High-temperature wave thermalization spoils beam self-cleaning in nonlinear multimode GRIN fibers

In our experiments, we reveal a so-far unnoticed power limitation of beam self-cleaning in graded-index nonlinear multimode optical fibers. As the optical pulse power is progressively increased, we observed that the initial Kerr-induced improvement of the spatial beam quality is eventually lost. Based on a holographic mode decomposition of the output field, we show that beam spoiling is associated with high-temperature wave thermalization, which depletes the fundamental mode in favor of a highly multimode power distribution.

Three octave visible to mid-infrared supercontinuum generation seeded by multimode silica fiber pumped at 1064 nm

Hyperspectral spectroscopy requires light sources with wide spectral ranges from the visible to the mid-infrared. Here, we demonstrate the first fiber-based mid-infrared supercontinuum covering three octaves of frequency by leveraging 1-µm laser technology. The process consists in spectral broadening of a 1064-nm pump toward 0.48-2.5 µm in a graded-index multimode fiber, followed by a fluoro-indate fiber used to reach deeper into the near infrared (4.3 µm). Finally, an arsenic selenide chalcogenide fiber allows us to reach the 6-µm wavelength region, providing a 0.75-6-µm supercontinuum. We illustrate the potential of this light source by recording mid-infrared absorption spectra of organic compounds.

Modal phase-locking in multimode nonlinear optical fibers

Spatial beam self-cleaning, a manifestation of the Kerr effect in graded-index multimode fibers, involves a nonlinear transfer of power among modes, which leads to robust bell-shaped output beams. The resulting mode power distribution can be described by statistical mechanics arguments. Although the spatial coherence of the output beam was experimentally demonstrated, there is no direct study of modal phase evolutions. Based on a holographic mode decomposition method, we reveal that nonlinear spatial phase-locking occurs between the fundamental and its neighboring low-order modes, in agreement with theoretical predictions. As such, our results dispel the current belief that the spatial beam self-cleaning effect is the mere result of a wave thermalization process.

Quasi-periodic spectro-temporal pulse breathing in a femtosecond-pumped tellurite graded-index multimode fiber

We report on the multidimensional characterization of femtosecond pulse nonlinear dynamics in a tellurite glass graded-index multimode fiber. We observed novel multimode dynamics of a quasi-periodic pulse breathing which manifests as a recurrent spectral and temporal compression and elongation enabled by an input power change. This effect can be assigned to the power dependent modification of the distribution of excited modes, which in turn modifies the efficiency of involved nonlinear effects. Our results provide indirect evidence of periodic nonlinear mode coupling occurring in graded-index multimode fibers thanks to the modal four-wave-mixing phase-matched via Kerr-induced dynamic index grating.

Coherence properties of light in highly multimoded nonlinear parabolic fibers under optical equilibrium conditions

We study the coherence characteristics of light propagating in nonlinear graded-index (GRIN) multimode fibers after attaining optical thermal equilibrium conditions. The role of optical temperature on the spatial mutual coherence function and the associated correlation area is systematically investigated. In this respect, we show that the coherence properties of the field at the output of a multimode nonlinear fiber can be controlled through its optical thermodynamic properties.

Interplay of Thermalization and Strong Disorder: Wave Turbulence Theory, Numerical Simulations, and Experiments in Multimode Optical Fibers

We address the problem of thermalization in the presence of a time-dependent disorder in the framework of the nonlinear Schrödinger (or Gross-Pitaevskii) equation with a random potential. The thermalization to the Rayleigh-Jeans distribution is driven by the nonlinearity. On the other hand, the structural disorder is responsible for a relaxation toward the homogeneous equilibrium distribution (particle equipartition), which thus inhibits thermalization (energy equipartition). On the basis of the wave turbulence theory, we derive a kinetic equation that accounts for the presence of strong disorder. The theory unveils the interplay of disorder and nonlinearity. It unexpectedly reveals that a nonequilibrium process of condensation and thermalization can take place in the regime where disorder effects dominate over nonlinear effects. We validate the theory by numerical simulations of the nonlinear Schrödinger equation and the derived kinetic equation, which are found in quantitative agreement without using any adjustable parameter. Experiments realized in multimode optical fibers with an applied external stress evidence the process of thermalization in the presence of strong disorder.

Thermalization of Orbital Angular Momentum Beams in Multimode Optical Fibers

We report on the thermalization of light carrying orbital angular momentum in multimode optical fibers, induced by nonlinear intermodal interactions. A generalized Rayleigh-Jeans distribution of asymptotic mode composition is obtained, based on the conservation of the angular momentum. We confirm our predictions by numerical simulations and experiments based on holographic mode decomposition of multimode beams. Our work establishes new constraints for the achievement of spatial beam self-cleaning, giving previously unforeseen insights into the underlying physical mechanisms.

Multimode soliton collisions in graded-index optical fibers

In this work, we unveil the unique complex dynamics of multimode soliton interactions in graded-index optical fibers through simulations and experiments. By generating two multimode solitons from the fission of an input femtosecond pulse, we examine the evolution of their Raman-induced red-shift when the input pulse energy grows larger. Remarkably, we find that the output red-shift of the trailing multimode soliton may be reduced, so that it accelerates until it collides with the leading multimode soliton. As a result of the inelastic collision, a significant energy transfer occurs between the two multimode solitons: the trailing soliton captures energy from the leading soliton, which ultimately enhances its red-shift, thus increasing temporal separation between the two multimode solitons.

Nonlinear rotation of spin-orbit coupled states in hollow ring-core fibers

We experimentally demonstrate that when two spin-orbit coupled orbital angular momentum (OAM) modes of opposite topological charge co-propagate in the Kerr nonlinear regime in a hollow ring-core optical fiber, the vectorial mode superposition exhibits a unique power-dependent rotation effect. This effect is analogous to nonlinear polarization rotation in single-mode fibers, however, the added spatial dimension produces a visually observable rotation of the spatial pattern emerging from the fiber when imaged through a linear polarizer. A dielectric metasurface q-plate was designed and fabricated to excite the desired mode combination in a hollow ring-core fiber that supports stable propagation of OAM modes. The observed spatial patterns show strong agreement with numerical simulations of the vector coupled nonlinear Schrödinger equations. These results constitute the first measurements of what can be described as the spin-orbit coupled generalization of the nonlinear polarization rotation effect.

Quadratic Peregrine solitons resonantly radiating without higher-order dispersion

We show that two-color Peregrine solitary waves in quadratic nonlinear media can resonantly radiate dispersive waves even in the absence of higher-order dispersion, owing to a phase-matching mechanism that involves the weaker second-harmonic component. We give very simple criteria for calculating the radiated frequencies in terms of material parameters, finding excellent agreement with numerical simulations.

Multiple intermodal-vectorial four-wave mixing bands generated by selective excitation of orthogonally polarized LP01 and LP11 modes in a birefringent fiber

This study investigates the nonlinear frequency conversions between the six polarization modes of a two-mode birefringent fiber. The aim is to demonstrate that the selective excitation of different combinations of linearly polarized spatial modes at the pump wavelength initiates distinct intermodal-vectorial four-wave mixing processes. In particular, this study shows that exciting two orthogonally polarized LP₀₁ and LP₁₁ modes can lead to the simultaneous generation of up to three pairs of different spatial modes of orthogonal polarizations at different wavelengths. The role of the phase birefringence of the spatial modes in the phase matching of such a four-wave mixing process is explained. Moreover, the theoretical predictions are verified through numerical simulations based on coupled nonlinear Schrödinger equations, and are also confirmed experimentally in a commercially available birefringent fiber.

Two octave supercontinuum generation in a non-silica graded-index multimode fiber

The generation of a two-octave supercontinuum from the visible to mid-infrared (700-2800 nm) in a non-silica graded-index multimode fiber is reported. The fiber design is based on a nanostructured core comprised of two types of drawn lead-bismuth-gallate glass rods with different refractive indices. This yields an effective parabolic index profile and ten times increased nonlinearity when compared to silica fibers. Using femtosecond pulse pumping at wavelengths in both normal and anomalous dispersion regimes, a detailed study is carried out into the supercontinuum generating mechanisms and instabilities seeded by periodic self-imaging. Significantly, suitable injection conditions in the high power regime are found to result in the output beam profile showing clear signatures of beam self-cleaning from nonlinear mode mixing. Experimental observations are interpreted using spatio-temporal 3+1D numerical simulations of the generalized nonlinear Schrödinger equation, and simulated spectra are in excellent agreement with experiment over the full two-octave spectral bandwidth. Experimental comparison with the generation of supercontinuum in a silica graded-index multimode fiber shows that the enhanced nonlinear refractive index of the lead-bismuth-gallate fiber yields a spectrum with a significantly larger bandwidth. These results demonstrate a new pathway towards the generation of bright, ultrabroadband light sources in the mid-infrared.

Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers

Since its first demonstration in graded-index multimode fibers, spatial beam self-cleaning has attracted a growing research interest. It allows for the propagation of beams with a bell-shaped spatial profile, thus enabling the use of multimode fibers for several applications, from biomedical imaging to high-power beam delivery. So far, beam self-cleaning has been experimentally studied under several different experimental conditions. Whereas it has been theoretically described as the irreversible energy transfer from high-order modes towards the fundamental mode, in analogy with a beam condensation mechanism. Here, we provide a comprehensive theoretical description of beam self-cleaning, by means of a semi-classical statistical mechanics model of wave thermalization. This approach is confirmed by an extensive experimental characterization, based on a holographic mode decomposition technique, employing laser pulses with temporal durations ranging from femtoseconds up to nanoseconds. An excellent agreement between theory and experiments is found, which demonstrates that beam self-cleaning can be fully described in terms of the basic conservation laws of statistical mechanics.

Generation of Subpicosecond Pulse Trains in Fiber Cascades Comprising a Cylindrical Waveguide with Propagating Refractive Index Wave

A cylindrical waveguide structure with the running refractive index wave has been recently demonstrated as a means for the generation of high-repetition-rate pulse trains. The operation mechanism involves a proper combination of the frequency modulation and modulation instability simultaneously experienced by the input continuous wave (CW) signal as it propagates through the cylinder waveguide. Here, we explore the same idea but employ the cylindrical waveguide only as a part of the cascaded optical fiber configuration now comprising both passive and active optical fiber segments. The new system design enables the improved control of the pulse train formation process in the cascaded system elements, relaxes strong requirements for the CW signal power, and provides an additional optical gain for the advanced pulse peak power scaling. In particular, using a low-amplitude, weakly modulated, continuous wave as an input signal we explore and optimize the nonlinear mechanisms underlying its cascaded transformation into the train of kilowatt peak power picosecond pulses.

Spatial Beam Self-Cleaning in Bi-Tapered Multimode Fibers

We report the spatial beam self-cleaning in bi-tapered conventional multimode fibers (MMFs) with different tapered lengths. Through the introduction of the bi-tapered structure in MMFs, the input beam with poor beam quality from a high-power fiber laser can be converted to a centered, bell-shaped beam in a short length, due to the strengthened nonlinear modes coupling. It is found that the bi-tapered MMF with longer tapered length at the same waist diameter shows better beam self-cleaning effect and larger spectral broadening. The obtained results offer a new method to improve the beam quality of high-power laser at low cost. Furthermore, it may be interesting for manufacturing bi-tapered MMF-based devices to obtain the quasi-fundamental mode beam in spatiotemporal mode-locked fiber lasers.

High-Peak Power Frequency Modulation Pulse Generation in Cascaded Fiber Configurations with Inscribed Fiber Bragg Grating Arrays

We explored the dynamics of frequency-modulated (FM) pulses in a cascaded fiber configuration comprising one active and one passive optical fiber with multiple fiber Bragg gratings (FBGs) of different periods inscribed over the fiber configuration length. We present a theoretical formalism to describe the mechanisms of the FM pulse amplification and pulse compression in such fiber cascades resulting in peak powers up to ~0.7 MW. In combination with the decreasing dispersion fibers, the considered cascade configuration enables pico- and sub-picosecond pulse trains with a sub-terahertz repetition rate and sub-kW peak power generated directly from the continuous optical signal.

Modal dynamics in multimode optical fibers: an attractor of high-order modes

Multimode fibers (MMFs) support abundant spatial modes and involve rich spatiotemporal dynamics, yielding many promising applications. Here, we investigate the influences of the number and initial energy of high-order modes (HOMs) on the energy flow from the intermediate modes (IMs) to the fundamental mode (FM) and HOMs. It is quite surprising that random distribution of high-order modes evolves to a stationary one, indicating the asymptotic behavior of orbits in the same attraction domain. By employing the Lyapunov exponent, we prove that the threshold of the HOMs-attractor is consistent with the transition point of the energy flow which indicates the HOMs-attracotr acts as a "valve" in the modal energy flow. Our results provide a new perspective to explore the nonlinear phenomena in MMFs, such as Kerr self-cleaning, and may pave the way to some potential applications, such as secure communications in MMFs.

Spatiotemporal beam self-cleaning for high-resolution nonlinear fluorescence imaging with multimode fiber

Beam self-cleaning (BSC) in graded-index (GRIN) multimode fibers (MMFs) has been recently reported by different research groups. Driven by the interplay between Kerr effect and beam self-imaging, BSC counteracts random mode coupling, and forces laser beams to recover a quasi-single mode profile at the output of GRIN fibers. Here we show that the associated self-induced spatiotemporal reshaping allows for improving the performances of nonlinear fluorescence (NF) microscopy and endoscopy using multimode optical fibers. We experimentally demonstrate that the beam brightness increase, induced by self-cleaning, enables two and three-photon imaging of biological samples with high spatial resolution. Temporal pulse shortening accompanying spatial beam clean-up enhances the output peak power, hence the efficiency of nonlinear imaging. We also show that spatiotemporal supercontinuum (SC) generation is well-suited for large-band NF imaging in visible and infrared domains. We substantiated our findings by multiphoton fluorescence imaging in both microscopy and endoscopy configurations.

Origin of spontaneous wave mixing processes in multimode GRIN fibers

We show that geometric parametric instability (GPI) in graded-index multimode fibers is strongly influenced by higher-order dispersion. By measuring the output spectrum for different core radii, we distinguish peaks generated by GPI from other coexisting parametric processes using phase-matching arguments and numerical simulations. We highlight for the first time a non-degenerate GPI process involving two pumps at different wavelengths.

Scalable optical learning operator

Today's heavy machine learning tasks are fueled by large datasets. Computing is performed with power-hungry processors whose performance is ultimately limited by the data transfer to and from memory. Optics is a powerful means of communicating and processing information, and there is currently intense interest in optical information processing for realizing high-speed computations. Here we present and experimentally demonstrate an optical computing framework called scalable optical learning operator, which is based on spatiotemporal effects in multimode fibers for a range of learning tasks including classifying COVID-19 X-ray lung images, speech recognition and predicting age from images of faces. The presented framework addresses the energy scaling problem of existing systems without compromising speed. We leverage simultaneous, linear and nonlinear interaction of spatial modes as a computation engine. We numerically and experimentally show the ability of the method to execute several different tasks with accuracy comparable with a digital implementation.

Frequency-resolved spatial beam mapping in multimode fibers: application to mid-infrared supercontinuum generation

We present a new, to the best of our knowledge, spatial-spectral mapping technique permitting measurement of the beam intensity at the output of a graded-index multimode fiber (GIMF) with sub-nanometric spectral resolution. We apply this method to visualize the fine structure of the beam shape of a sideband generated at 1870 nm by geometric parametric instability (GPI) in a GIMF. After spatial-spectral characterization, we amplify the GPI sideband with a thulium-doped fiber amplifier to obtain a microjoule-scale picosecond pump whose spectrum is finally broadened in a segment of InF₃ optical fiber to achieve a supercontinuum ranging from 1.7 up to 3.4 µm.

Nonlinear spatial reshaping of pulsed beam in a step-index few-mode optical fiber

In this paper we demonstrate the spatial and spectral dynamics of pulse propagation in a step-index few-mode optical fiber, through an experimental and numerical analysis. The Kerr induced spatial self-cleaning is demonstrated by coupling a sub-nanosecond pulsed laser at 532nm into the fiber supporting above 10 modes. A bell-shaped and approximately single mode beam can be obtained for peak powers above 6kW and it remained relatively unchanged up to 25kW. But at significantly higher input peak powers, the spatial contents of spectral sidebands change dramatically, because of intermodal four wave mixing effect.

Modal perspective on geometric parametric instability sidebands in graded-index multimode fibers

In this paper, we investigated the geometric parametric instability (GPI) in graded-index multimode fibers through the multimode generalized nonlinear Schrödinger equation. Our results clearly and intuitively indicate that the generations of GPI sidebands are nearly synchronous in the spectrums of all modes, and the shapes of these spectrums are nearly the same. The numerical results show that the energies of the GPI sidebands come from the pump sideband, and these sidebands are carried by similar spatial beam profiles due to the similar modal components. We also found that the large modal dispersion has an influence for the symmetry of these GPI sidebands.

Spatiotemporal Mode-Locking in Lasers with Large Modal Dispersion

Dissipative nonlinear wave dynamics have been investigated extensively in mode-locked lasers with single transverse mode, whereas there are few studies related to three-dimensional nonlinear dynamics within lasers. Recently, spatiotemporal mode locking (STML) was proposed in lasers with small modal (i.e., transverse-mode) dispersion, which has been considered to be critical for achieving STML in those cavities because the small dispersion can be easily balanced. Here, we demonstrate that STML can also be achieved in multimode lasers with much larger modal dispersion, where we find that the intracavity saturable absorber plays an important role for counteracting the large modal dispersion. Furthermore, we observe a new STML phenomenon of passive nonlinear autoselection of single-mode mode locking, resulting from the interaction between spatiotemporal saturable absorption and spatial gain competition. Our work significantly broadens the design possibilities for useful STML lasers thus making them much more accessible for applications, and extends the explorable parameter space of the novel dissipative spatiotemporal nonlinear dynamics that can be achieved in these lasers.

Switchable and spacing tunable dual-wavelength spatiotemporal mode-locked fiber laser

We report a switchable and spacing tunable dual-wavelength spatiotemporal mode-locked (STML) laser based on the multimode interference filtering effect in an all-fiber linear cavity. The dual-wavelength STML operations combined with different pulse patterns are achieved. By adjusting the polarization controllers, the dual-wavelength STML pulses can be switched to single wavelength operation, which is tunable up to 35 nm under certain pump powers. Moreover, the dual-wavelength spacing can also be tuned from 8 nm to 22 nm. The obtained results contribute to understanding and exploring the spatiotemporal characteristics operating in the multi-wavelength regime of STML fiber lasers. All-fiber STML lasers with lasing wavelength tunability and flexibility may have applications in the fields of optical communications and optical measurements.

Discretized Conical Waves in Multimode Optical Fibers

Multimode optical fibers are essential in bridging the gap between nonlinear optics in bulk media and single-mode fibers. The understanding of the transition between the two fields remains complex due to intermodal nonlinear processes and spatiotemporal couplings, e.g., some striking phenomena observed in bulk media with ultrashort pulses have not yet been unveiled in such waveguides. Here we generalize the concept of conical waves described in bulk media towards structured media, such as multimode optical fibers, in which only a discrete and finite number of modes can propagate. Such propagation-invariant optical wave packets can be linearly generated, in the limit of superposed monochromatic fields, by shaping their spatiotemporal spectrum, whatever the dispersion regime and waveguide geometry. Moreover, they can also spontaneously emerge when a rather intense short pulse propagates nonlinearly in a multimode waveguide, their finite energy is also associated with temporal dispersion. The modal distribution of optical fibers then provides a discretization of conical emission (e.g., discretized X waves). Future experiments in multimode fibers could reveal different forms of dispersion-engineered conical emission and supercontinuum light bullets.

3D time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers

Characterization of the complex spatiotemporal dynamics of optical beam propagation in nonlinear multimode fibers requires the development of advanced measurement methods, capable of capturing the real-time evolution of beam images. We present a new space-time mapping technique, permitting the direct detection, with picosecond temporal resolution, of the intensity from repetitive laser pulses over a grid of spatial samples from a magnified image of the output beam. By using this time-resolved mapping, we provide, to the best of our knowledge, the first unambiguous experimental observation of instantaneous intrapulse nonlinear coupling processes among the modes of a graded index fiber.

Recent progress in all-fiber ultrafast high-order mode lasers

Ultrafast high-order mode (HOM) lasers are a relatively new class of ultrafast optics. They play a significant role in the fieldsof scientific research and industrial applications due to the high peak power and unique properties of spatial intensity and polarization distribution. Generation of ultrafast HOM beams in all-fiber systems has become an important research direction. In this paper, all-fiber mode conversion techniques, pulsed HOM laser strategies, and few-mode/multi-mode fiber (FMF/MMF) lasers are reviewed. The main motivation of this review is to highlight recent advances in the field of all-fiber ultrafast HOM lasers, for example, generating different HOM pulses based on fiber mode converters and mode-locking in the FMF/MMF lasers. These results suggest that mode selective coupler can be used as a broad bandwidth mode converter with fast response and HOM can be directly oscillated in the FMF/MMF laser cavity with high stability. In addition, spatiotemporal mode-locking in the FMF/MMF is also involved. It is believed that the development of all-fiber ultrafast HOM lasers will continue to deepen, thus laying a good foundation for future applications.

Correlation between geometric parametric instability sidebands in graded-index multimode fibers

The spectral analysis of the light propagating in normally dispersive graded-index multimode fibers is performed under initial noisy conditions. Based on the obtained spectra with multiple simulations in the presence of noise, we investigate the correlation in energy between the well-separated spectral sidebands through both the scattergrams and the frequency-dependent energy correlation map and find that conjugate couples are highly correlated while cross-combinations exhibit a very poor degree of correlation. These results reveal that the geometric parametric instability processes associated with each sideband pair occur independently from each other, which can provide significant insights into the fundamental dynamical effect of the geometric parametric instability and facilitate the future implementation of high-efficiency photon pair sources with reduced Raman decorrelations.

Classical Rayleigh-Jeans Condensation of Light Waves: Observation and Thermodynamic Characterization

Theoretical studies on wave turbulence predict that a purely classical system of random waves can exhibit a process of condensation, which originates in the singularity of the Rayleigh-Jeans equilibrium distribution. We report the experimental observation of the transition to condensation of classical optical waves propagating in a multimode fiber, i.e., in a conservative Hamiltonian system without thermal heat bath. In contrast to conventional self-organization processes featured by the nonequilibrium formation of nonlinear coherent structures (solitons, vortices,…), here the self-organization originates in the equilibrium Rayleigh-Jeans statistics of classical waves. The experimental results show that the chemical potential reaches the lowest energy level at the transition to condensation, which leads to the macroscopic population of the fundamental mode of the optical fiber. The near-field and far-field measurements of the condensate fraction across the transition to condensation are in quantitative agreement with the Rayleigh-Jeans theory. The thermodynamics of classical wave condensation reveals that the heat capacity takes a constant value in the condensed state and tends to vanish above the transition in the normal state. Our experiments provide the first demonstration of a coherent phenomenon of self-organization that is exclusively driven by optical thermalization toward the Rayleigh-Jeans equilibrium.

Coherent combining of self-cleaned multimode beams

A low intensity light beam emerges from a graded-index, highly multimode optical fibre with a speckled shape, while at higher intensity the Kerr nonlinearity may induce a spontaneous spatial self-cleaning of the beam. Here, we reveal that we can generate two self-cleaned beams with a mutual coherence large enough to produce a clear stable fringe pattern at the output of a nonlinear interferometer. The two beams are pumped by the same input laser, yet are self-cleaned into independent multimode fibres. We thus prove that the self-cleaning mechanism preserves the beams’ mutual coherence via a noise-free parametric process. While directly related to the initial pump coherence, the emergence of nonlinear spatial coherence is achieved without additional noise, even for self-cleaning obtained on different modes, and in spite of the fibre structural disorder originating from intrinsic imperfections or external perturbations. Our discovery may impact theoretical approaches on wave condensation, and open new opportunities for coherent beam combining.

Fragility of a soliton's shot-to-shot coherence

We show that a soliton in a high-order spatial mode of a multi-mode fiber can completely lose its shot-to-shot coherence due to a noise seed with energy orders of magnitude below that of the soliton. The total degradation of shot-to-shot coherence is caused by a very strong recently demonstrated intermodal nonlinear effect, soliton self-mode conversion. The results indicate that the robustness of solitons against perturbations is not entirely applicable in the presence of intermodal nonlinearities, and, more generally, that certain single-mode results cannot be trivially extrapolated to multi-mode fibers.

Nonlinear beam self-imaging and self-focusing dynamics in a GRIN multimode optical fiber: theory and experiments

Beam self-imaging in nonlinear graded-index multimode optical fibers is of interest for many applications, such as implementing a fast saturable absorber mechanism in fiber lasers via multimode interference. We obtain a new exact solution for the nonlinear evolution of first and second order moments of a laser beam of arbitrary transverse shape carried by a graded-index multimode fiber. We have experimentally directly visualized the longitudinal evolution of beam self-imaging by means of femtosecond laser pulse propagation in both the anomalous and the normal dispersion regime of a standard telecom graded-index multimode optical fiber. Light scattering out of the fiber core via visible photo-luminescence emission permits us to directly measure the self-imaging period and the beam dynamics. Spatial shift and splitting of the self-imaging process under the action of self-focusing are also revealed.

Multimode nonlinear simulation technique having near-linear scaling with mode number in circular symmetric waveguides

An efficient scheme for performing coupled-mode simulations of nonlinear propagation in multimoded waveguides having circular symmetry is presented. In contrast to currently established modal-expansion methods the scheme displays a nearly linear scaling of numerical complexity with mode number and may enable simulations with hundreds of guided modes.

Highly efficient few-mode spatial beam self-cleaning at 1.5µm

We experimentally demonstrate that spatial beam self-cleaning can be highly efficient when obtained with a few-mode excitation in graded-index multimode optical fibers. By using 160 ps long, highly chirped (6 nm bandwidth at -3dB) optical pulses at 1562 nm, we demonstrate a one-decade reduction of the power threshold for spatial beam self-cleaning, with respect to previous experiments using pulses with laser wavelengths at 1030-1064 nm. Self-cleaned beams remain spatio-temporally stable for more than a decade of their peak power variation. The impact of input pulse temporal duration is also studied.

Spatial Beam Self-Cleaning in Second-Harmonic Generation

We experimentally demonstrate the spatial self-cleaning of a highly multimode optical beam, in the process of second-harmonic generation in a quadratic nonlinear potassium titanyl phosphate crystal. As the beam energy grows larger, the output beam from the crystal evolves from a highly speckled intensity pattern into a single, bell-shaped spot, sitting on a low energy background. We demonstrate that quadratic beam cleanup is accompanied by significant self-focusing of the fundamental beam, for both positive and negative signs of the linear phase mismatch close to the phase-matching condition.

Statistical mechanics of weakly nonlinear optical multimode gases

By utilizing notions from statistical mechanics, we develop a general and self-consistent theoretical framework capable of describing any weakly nonlinear optical multimode system involving conserved quantities. We derive the fundamental relations that govern the grand canonical ensemble through maximization of the Gibbs entropy at equilibrium. In this classical picture of statistical photo-mechanics, we obtain analytical expressions for the probability distribution, the grand partition function, and the relevant thermodynamic potentials. Our results universally apply to any other weakly nonlinear multimode bosonic system.