Modulation instability and pattern formation in spatially incoherent light beams

Fiber cross-sectional imaging by manually controlled low coherence light sources

We couple a variable-coherence light beam into a multimode optical fiber and observe the fiber cross-sectional images. The variation in the fiber imaging is explored as we change the degree of optical coherence of the incident light. Low coherence light is shown to be capable of improving the quality of the fiber images. Various mode patterns of a multimode optical fiber are also shown numerically and experimentally to elucidate the fiber coupling characteristics.

Optical spatial solitons and modulation instabilities in transparent entangled polymer solutions

Fully transparent nondilute polydiene solutions exhibit optical nonlinearities when irradiated by a low-power cw laser in the visible. The formation of optical spatial solitons is imaged through phase contrast microscopy. Both (2+1)D and (1+1)D modulational instabilities are evidenced as 2D and 1D arrays of linear filaments formed by beam defocusing or by using a cylindrical lens. The origin of the nonlinearity remains elusive.

Spatial frequency combs and supercontinuum generation in one-dimensional photonic lattices

We experimentally demonstrate the formation of spatial supercontinuum and of spatial frequency combs in nonlinear photonic lattices. This process results from multiple four-wave mixing initiated by launching two Floquet-Bloch modes into a one-dimensional lattice. The dynamics of the waves is sensitively dependent on the transverse momentum difference between the two initial modes: when this momentum difference is commensurable with the lattice momentum the waves evolve into a frequency comb, whereas when it is incommensurable the waves evolve into a supercontinuum of spatial frequencies.

Coupled modulational instability of copropagating spin waves in magnetic thin films

This letter reports the first results on the coupled modulational instability of copropagating spin waves in a magnetic film. Strong instability was observed for the two waves with either attractive or repulsive nonlinearity. If the two waves have attractive nonlinearity, the instability leads to the formation of bright solitons. If the two waves have repulsive nonlinearity, the process results in the formation of black solitons. The instability was also observed for the two waves in separated attractive-repulsive nonlinearity regimes.

Spontaneous pattern formation upon incoherent waves: from modulation-instability to steady-state

We study the long-range propagation of incoherent light following the modulation instability (MI) process in non-instantaneous nonlinear Kerr-type media. We find that the system eventually reaches a steady-state characterized by a lower degree of coherence than in the initial state, with small fluctuations around a pronounced mean value. We find that the average values of the spatial correlation distance at steady-state and the fluctuations around it, which are obtained either through ensemble averaging, or by spatial averaging, or via temporal averaging, are all identical. This feature may be viewed as indication of ergodic behavior, which occurs in the long-time evolution following incoherent MI. Finally, we find that the steady-state properties of the system depend on the initial coherence but not on the nonlinearity strength, although the system evolves faster to steady-state as the strength of the nonlinearity is increased.

Observation of all-optical bump-on-tail instability

We demonstrate an all-optical bump-on-tail instability by considering the nonlinear interaction of two partially coherent spatial beams. For weak wave coupling, we observe momentum transfer with no variation in intensity. For strong wave coupling, modulations appear in intensity and evidence appears for wave (Langmuir) collapse at large scales. Borrowing plasma language, these limits represent regimes of weak and strong spatial optical turbulence. In both limits, the internal spectral energy redistribution is observed by recording and reconstructing a hologram of the evolving dynamics. The results are universal and can appear in any wave-kinetic system with short-wave-long-wave coupling.

Optical beam instabilities in nonlinear nanosuspensions

We investigate the modulation instability of plane waves and the transverse instabilities of soliton stripe beams propagating in nonlinear nanosuspensions. We show that in these systems the process of modulational instability depends on the input beam conditions. On the other hand, the transverse instability of soliton stripes can exhibit new features as a result of 1D collapse caused by the exponential nonlinearity.

Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics

<p> <a href="http://oe.osa.org/virtual_issue.cfm?vid=36">Focus Serial: Frontiers of Nonlinear Optics</a> </p> This concise review is aimed at providing an introduction to the kinetic theory of partially coherent optical waves propagating in nonlinear media. The subject of incoherent nonlinear optics received a renewed interest since the first experimental demonstration of incoherent solitons in slowly responding photorefractive crystals. Several theories have been successfully developed to provide a detailed description of the novel dynamical features inherent to partially coherent nonlinear optical waves. However, such theories leave unanswered the following important question: Which is the long term (spatiotemporal) evolution of a partially incoherent optical field propagating in a nonlinear medium? In complete analogy with kinetic gas theory, one may expect that the incoherent field may evolve, owing to nonlinearity, towards a thermodynamic equilibrium state. Weak-turbulence theory is shown to describe the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Precisely, the theory describes an irreversible evolution of the spectrum of the field towards a thermodynamic equilibrium state. The irreversible behavior is expressed through the H-theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann's H-theorem of kinetic gas theory. It is shown that thermal wave relaxation to equilibrium may be characterized by the existence of a genuine condensation process, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. In spite of the formal reversibility of optical wave propagation, the condensation process occurs by means of an irreversible evolution of the field towards a homogeneous plane-wave (condensate) with small-scale fluctuations superimposed (uncondensed particles), which store the information necessary for the reversible propagation. As a remarkable result, an increase of entropy ("disorder") in the optical field requires the generation of a coherent structure (plane-wave). We show that, beyond the standard thermodynamic limit, wave condensation also occurs in two spatial dimensions. The numerical simulations are in quantitative agreement with the kinetic wave theory, without any adjustable parameter.

Laser beam filamentation in fractal aggregates

We investigate filamentation of a cw laser beam in soft matter such as colloidal suspensions and fractal gels. The process, driven by electrostriction, is strongly affected by material properties, which are taken into account via the static structure factor, and have impact on the statistics of the light filaments.

Incoherent white-light solitons in nonlinear periodic lattices

We predict the existence of lattice solitons made of incoherent white light: lattice solitons made of light originating from an ordinary incandescent light bulb. We find that the intensity structure and spatial power spectra associated with different temporal frequency constituents of incoherent white-light lattice solitons (IWLLSs) arrange themselves in a characteristic fashion, with the intensity structure more localized at higher frequencies, and the spatial power spectrum more localized at lower frequencies; the spatial correlation distance is larger at lower frequency constituents of IWLLSs. This characteristic shape of incoherent white-light lattice solitons reflects the fact that diffraction is stronger for lower temporal frequency constituents, while higher frequencies experience stronger effective nonlinearity and deeper lattice structure.

Observation of modulational instability in discrete media with self-defocusing nonlinearity

We report what we believe is the first observation of modulation instability in the anomalous-diffraction regions of a photonic lattice. The experiments were carried out in a 1D waveguide array fabricated in a lithium niobate crystal displaying the photovoltaic self-defocusing nonlinearity, and our results are confirmed numerically by simulating the nonlinear beam propagation.

Incoherent solitons in instantaneous nonlocal nonlinear media

We predict random-phase spatial solitons in instantaneous nonlocal nonlinear media. The key mechanism responsible for self-trapping of such incoherent wave packets is played by the nonlocal (rather than the traditional noninstantaneous) nature of the nonlinearity. This kind of incoherent soliton has profoundly different features than other incoherent solitons.

Incoherent light-induced self-organization of molecules

Although coherent light is usually required for the self-organization of regular spatial patterns from optical beams, we show that peculiar light-matter interaction can break this evidence. In the traditional method of recording laser-induced periodic surface structures, a light intensity distribution is produced at the surface of a polymer film by an interference between two coherent optical beams. We report on the self-organization followed by propagation of a surface relief pattern. It is induced in a polymer film by using a low-power and small-size coherent beam assisted by a high-power and large-size incoherent and unpolarized beam. We demonstrate that we can obtain large size and well-organized patterns starting from a dissipative interaction. Our experiments open new directions to improving optical processing systems.

Transmission of digital images consisting of white-light dark solitons

A distortion-free digital image is transmitted through parallel propagation of white-light photovoltaic dark solitons. The waveguide channels induced by white-light dark solitons can guide both the laser beam and the white-light beam well. We determine experimentally the critical separations of the dark solitons in the directions parallel and perpendicular to the crystalline c axis for the given crystal thickness.

Modulational instability and unstable patterns in the discrete complex cubic Ginzburg-Landau equation with first and second neighbor couplings

The generation of nonlinear modulated waves is investigated in the framework of hydrodynamics using a model of coupled oscillators. In this model, the separatrices between each pair of vortices may be viewed as individual oscillators and are described by a phenomenological one-dimensional discrete complex Ginzburg-Landau equation involving first- and second-nearest neighbor couplings. A theoretical approach based on the linear stability analysis predicts regions of modulational instability, governed by both the first and second-nearest neighbor couplings. From numerical investigations of different wave patterns that may be driven by the modulational instability, it appears that analytical predictions are correctly verified. For wave number in the unstable regions, an initial condition whose amplitude is slightly modulated breaks into a train of unstable patterns. This phenomenon agrees with the description of amplification of the spectral component of the perturbation and its harmonics, as well.

Observation of power-dependent walk-off via modulational instability in nematic liquid crystals

We investigate power-controlled angular steering of light filaments produced by transverse modulational instability in an anisotropic nematic liquid-crystal cell.

Incoherent modulation instability in instantaneous nonlinear Kerr media

We demonstrate theoretically and experimentally in an optical fiber system that partially temporally incoherent light exhibits modulational instability during its propagation in an instantaneous response nonlinear medium. We show that the modulation frequency and gain are substantially increased with respect to the corresponding values of coherent modulational instability.

Self-organization of spatial solitons

We present experimental results on the transverse modulation instability of an elliptical beam propagating in a bulk nonlinear Kerr medium, and the formation and self-organization of spatial solitons. We have observed the emergence of order, self organization and a transition to an unstable state. Order emerges through the formation of spatial solitons in a periodic array. If the initial period of the array is unstable the solitons will tend to self-organize into a larger (more stable) period. Finally the system transitions to a disordered state where most of the solitons disappear and the beam profile becomes unstable to small changes in the input energy.

Induced spatiotemporal modulation instability in a noninstantaneous self-defocusing medium

We demonstrate theoretically and experimentally that induced spatiotemporal modulation instability can exist in a self-defocusing medium if the nonlinearity is noninstantaneous. We predict the growth rate as a function of the spatial and temporal frequencies of the modulation and the response time of the nonlinearity and confirm it by our experiments.

Interplay between nonlocality and nonlinearity in nematic liquid crystals

Spatial nonlocality has a crucial role in the optical response of bulk liquid crystals. With specific reference to one-dimensional modulation instability, we demonstrate the tuning of transverse nonlocality in the nonlinear response of nematics.

Routing of anisotropic spatial solitons and modulational instability in liquid crystals

In certain materials, the spontaneous spreading of a laser beam (owing to diffraction) can be compensated for by the interplay of optical intensity and material nonlinearity. The resulting non-diffracting beams are called 'spatial solitons' (refs 1-3), and they have been observed in various bulk media. In nematic liquid crystals, solitons can be produced at milliwatt power levels and have been investigated for both practical applications and as a means of exploring fundamental aspects of light interactions with soft matter. Spatial solitons effectively operate as waveguides, and so can be considered as a means of channelling optical information along the self-sustaining filament. But actual steering of these solitons within the medium has proved more problematic, being limited to tilts of just a fraction of a degree. Here we report the results of an experimental and theoretical investigation of voltage-controlled 'walk-off' and steering of self-localized light in nematic liquid crystals. We find not only that the propagation direction of individual spatial solitons can be tuned by several degrees, but also that an array of direction-tunable solitons can be generated by modulation instability. Such control capabilities might find application in reconfigurable optical interconnects, optical tweezers and optical surgical techniques.

Spontaneous pattern formation with incoherent white light

We present the first experimental observation of modulation instability and spontaneous pattern formation with incoherent white light emitted from an incandescent light bulb. We show experimentally that modulation instability of white light propagating in a noninstantaneous self-focusing medium is a collective effect, where the entire temporal spectrum of the light beam becomes unstable at the same threshold value and collectively forms a pattern with a single periodicity. We experimentally demonstrate that the temporal spectrum of the evolving perturbation self-adjusts to match the collective pattern formation phenomenon.

Spontaneous pattern formation in a cavity with incoherent light

We present the first experimental observation of spontaneous pattern formation in a nonlinear optical cavity in which the circulating light is both spatially and temporally incoherent.

Random-phase solitons in nonlinear periodic lattices

We predict the existence of random phase solitons in nonlinear periodic lattices. These solitons exist when the nonlinear response time is much longer than the characteristic time of random phase fluctuations. The intensity profiles, power spectra, and statistical (coherence) properties of these stationary waves conform to the periodicity of the lattice. The general phenomenon of such solitons is analyzed in the context of nonlinear photonic lattices.

Experimental observation of discrete modulational instability

We report the first experimental observation of modulation instability in a discrete optical nonlinear array.

Optical multisoliton generation in nematic liquid crystals

The nonlocal nonlinearity stemming from molecular reorientation in nematic liquid crystals supports the formation of multiple solitary waves, following the onset of spatial modulational instability from both wide and focused input beams. We report experimental observations of such phenomena at power levels of hundreds of milliwatts in planar cells with a voltage bias.

Spatial versus temporal deterministic wave breakup of nonlinearly coupled light waves

We investigate experimentally the competition between spatial and temporal breakup due to modulational instability in chi((2)) nonlinear mixing. The modulation of the wave packets caused by the energy exchange between fundamental and second-harmonic components is found to be the prevailing trigger mechanism which, according to the relative weight of diffraction and dispersion, leads to the appearance of a multisoliton pattern in the low-dimensional spatial or temporal domain.

Optical modulational instability in a nonlocal medium

We report the first observation of modulational instability in nematic liquid crystals encompassing a nonlocal behavior. The experimental results are quantitatively compared with the theory revealing the fundamental features associated to a nonlocal nonlinearity.

Cavity pattern formation with incoherent light

We study the propagation dynamics of an incoherent light beam circulating in a passive cavity containing noninstantaneous nonlinear media. It is shown that patterns form in this cavity in spite of spatial incoherence of the light. We show that the pattern formation process is always associated with two consecutive thresholds. The first (instability) threshold is unaffected by the cavity boundary conditions, whereas the second threshold is induced by the feedback through the interplay of nonlinear gain and cavity loss.

Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity

The onset of instability of a multiple-scattering speckle pattern in a random medium with Kerr nonlinearity is significantly affected by the noninstantaneous character of the nonlinear medium's response. The fundamental time scale of the spontaneous speckle dynamics beyond the instability threshold is set by the larger of times TD and tau(NL), where TD is the time required for the multiply scattered waves to propagate through the random sample and tau(NL) is the relaxation time of the nonlinearity. The inertial nature of the nonlinearity should complicate the experimental observation of the instability phenomenon.

Induced modulation instability of partially spatially incoherent light with varying perturbation periods

We demonstrate experimentally that a periodic perturbation on a partially spatially incoherent optical beam induces modulation instability that depends strongly on the perturbation periods as well as on the strength of the nonlinearity and the degree of spatial coherence. At a fixed value of the nonlinearity and coherence, the incoherent modulation instability has a maximum growth at a preferred perturbation period (or spatial frequency), leading to the formation of ordered patterns. While the nonlinearity in our photorefractive system is inherently anisotropic, pattern control and pattern switching with anisotropic coherence is readily realized. Our experimental observations are in good agreement with theoretical predictions.

Spatial soliton pixels from partially incoherent light

We report what is to our knowledge the first observation of pixellike spatial solitons from partially spatially incoherent light. We created an array of as many as 32x32 soliton pixels by launching a spatially modulated incoherent light beam into a noninstantaneous self-focusing photorefraction medium. These solitons were stable and robust, forming a steady-state two-dimensional waveguide array in which optical coupling and control of local waveguide channels could be realized.

Pattern formation in a cavity longer than the coherence length of the light in it

We study, theoretically and experimentally, the evolution of patterns in a passive nonlinear cavity that is longer than the coherence length of the light circulating in it. The patterns exhibit spatial line narrowing as the feedback is increased, resembling the line narrowing in lasers.

Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media

We develop a theory describing propagation of spatially and temporally incoherent light in noninstantaneous nonlinear media, and predict the existence of modulation instability of "white" light. We find that the modulation instability of white light is fundamentally a collective effect, where all the temporal frequencies participate in the formation of a pattern, and self-adjust their respective contributions.

Clustering of solitons in weakly correlated wavefronts

We demonstrate theoretically and experimentally the spontaneous clustering of solitons in partially coherent wavefronts during the final stages of pattern formation initiated by modulation instability and noise.

Spatiotemporal optical modulation instability of coherent light in noninstantaneous nonlinear media

We report on the theoretical derivation and experimental observation of spatiotemporal modulation instability (MI) of a coherent light beam in noninstantaneous nonlinear media. We obtain analytically the MI growth rate as a function of the spatial and temporal frequencies of the perturbation and the material response time. In the experiment, we observe that the varying speed of the MI patterns increases with the decreased material response time. We also observe that increasing the material response time can arrest the MI, agreeing with our theoretical derivation.

Pattern formation via symmetry breaking in nonlinear weakly correlated systems

We study pattern formation initiated by modulation instability in nonlinear partially coherent wave fronts and show that anisotropic noise and/or anisotropic correlation statistics can lead to ordered patterns.

Mathematical frontiers in optical solitons

Solitons are localized concentrations of field energy, resulting from a balance of dispersive and nonlinear effects. They are ubiquitous in the natural sciences. In recent years optical solitons have arisen in new and exciting contexts that differ in many ways from the original context of coherent propagation in a uniform medium. We review recent developments in incoherent spatial solitons and in gap solitons in periodic structures.

Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system

We have investigated the amplification of a spatially periodic perturbation applied to a wide fundamental beam launched near phase matching for second-harmonic generation in a lithium niobate film waveguide. We measured the gain coefficient for the modulational instability of quadratic eigenmodes as a function of periodicity, intensity, and wave-vector mismatch. Excellent agreement with theory was obtained.

Transmission of images through highly nonlinear media by gradient-index lenses formed by incoherent solitons

We experimentally demonstrate image transmission through a noninstantaneous self-focusing medium. A partially spatially incoherent soliton is used to form a multimode waveguide in a photorefractive crystal, and the modes of that waveguide are used to transmit an incoherent image through this nonlinear medium.

Dynamic soliton-like modes

Incoherent optical spatial solitons require noninstantaneous nonlinearity, i.e., the local intensity fluctuation of the solitons must be faster than the medium can respond. Observing partially incoherent bicomponent solitons, we find that there exists a threshold speed. When the fluctuation of the soliton intensity, resulting from the time-varying interference of its constituent modes, is below the threshold, the soliton beam and its induced waveguide oscillate violently. Just above the threshold, the soliton-induced waveguide is observed to be dragged by the soliton beam.

Experiments on induced modulational instability of an incoherent optical beam

We report the observation of modulational instability (MI) of a partially spatially incoherent beam induced by seeding noise through cross-phase modulation. We show experimentally that a threshold exists for such induced incoherent MI to occur that depends on the degree of spatial coherence as well as on the strength of the nonlinearity. Above threshold, the induced MI leads to the formation of ordered and disordered patterns of incoherent light.

Eliminating the transverse instabilities of kerr solitons

We show analytically, numerically, and experimentally that a transversely stable one-dimensional [(1+1)D] bright Kerr soliton can exist in a 3D bulk medium. The transverse instability of the soliton is completely eliminated if it is made sufficiently incoherent along the transverse dimension. We derive a criterion for the threshold of transverse instability that links the nonlinearity to the largest transverse correlation distance for which the 1D soliton is stable.