Orbital angular momentum beam instabilities in engineered nonlinear colloidal media

In this letter, we experimentally demonstrate the evolution of the optical vortex beams of different topological charges propagating in engineered nano-colloidal suspension of negative polarizability with saturable nonlinearities. Due to the high power of the incident beam, the modulation instability leads to an exponential growth of weak perturbations and thus splits the original vortex beam into a necklace beam consisting of several bright spots. The number of observed bright spots is intrinsically determined by the topological charge of the incident beam and agrees well with the predictions of our linear stability analysis and numerical simulations. Besides contributing to the fundamental science of light-matter interactions in engineered soft-matter media, this work opens new opportunities for dynamic optical manipulation and transmission of light through scattering media as well as formation of complex optical patterns and light filamentation in naturally existing colloids such as fog and clouds.

has exponential character and can be either supercritical, in case of particles with positive polarizability, or saturable, for negative polarizability particles. [11][12][13][14] To date, such engineered colloidal systems have been studied using simple Gaussian beams. However, recent progress in structuring amplitude and phase properties of optical beams opens new remarkable opportunities for manipulating and controlling light-matter interactions in such engineered media. Compared to the conventionally used Gaussian beams, optical vortices that, are characterized by the doughnut-shaped intensity profile and a helical phase front, offer even more degrees of freedom for optical trapping 15 or imaging applications. 16 Optical vortices can be used to trap and circulate colloidal particles, constituting a model test-bed for studying many-body hydrodynamic coupling and instabilities in mesoscopic, many-particle systems with potential applications in lab-on-a-chip systems. 4,[17][18][19] In this letter, we experimentally investigate the evolution of the optical vortex beams of different topological charges in engineered nano-colloidal suspensions with saturable nonlinearities, in which the particles with negative polarizability are repelled away from the high-intensity region. As the high-intensity vortex beam propagates in such a medium, the modulation instability (MI) phenomenon leads to an exponential growth of weak perturbations. As we predicted in our linear stability analysis and numerical simulations, 20 the perturbations with an orbital angular momentum (OAM) of a particular charge is amplified leading to the formation of a necklace beam with a well-defined number of peaks. The experimental results are in excellent agreement with the analytical and numerical predictions.
Besides contributing to the fundamental science of light-matter interactions in engineered soft-matter media, our work might bring about new possibilities for dynamic optical manipulation and transmission of light through scattering media as well as formation of complex optical patterns and light filamentation [21][22][23] in naturally existing colloids such as fog and clouds.
Let us consider an optical vortex beam propagating along the z-direction in a nanocolloidal system consisting of dielectric particles with refractive index n p lower than the refractive index of the background medium n b . If n p < n b , the colloidal suspension has a negative polarizability, as schematically illustrated in Fig. 1, the nano-particles are driven away from the high intensity region of the beam, resulting in a change of the local refractive index in the suspension, which exhibits a focusing nonlinearity. For large input intensities, the beam becomes unstable due to the well-known phenomenon of MI. This effect reveals itself as the exponential growth of weak perturbations or noise in the presence of an intense pump beam propagating in a nonlinear medium. As a result of the MI, the original vortex beam of a doughnut shape may split into a necklace-like beam with several bright spots, whose number is intrinsically determined by the topological charge of the vortex beam. This process is described by the nonlinear Schrödinger equation (NLSE) (see Methods). Following the standard linear stability analysis 20,22 we assume that the high-intensity optical beam with a topological vortex charge m ∈ Z is accompanied by an azimuthal perturbation: where Here, the particle polarizability is denoted by α, and k B T is the thermal energy, with the Boltzmann constant k B and at temperature T . V p is the volume of a particle, ρ 0 is the unperturbed particle concentration, k 0 = 2π λ 0 is the wave number, and λ 0 is the free-space wavelength.
Equation (2) is used in the following to predict the MI gain for vortices propagating in the nano-colloidal media. We study the propagation of light with the free-space wavelength   are shown in Fig. 2(b), (c). We can see a great agreement with the rescaled analytical curves showing the MI gain 1 z 0 . Figure 3: Experimental setup used to study (seeded) modulation instability of vortex beams in colloidal media. Collimated beam from Verdi V6 laser (λ 0 = 532 nm) is initially split into two beams using beam splitters with reflectivity varying in the range from 0.6% to 8% of the total power. The high intensity beam is transmitted through a spiral phase plate (SPP) to generate the main vortex beam with lower charge. In the seeded configuration, the low intensity beam is transmitted through a SPP with a higher charge to generate the perturbation beam. The beams are then recombined at the second beam splitter and focused onto the cuvette by a lens. The longitudinal beam profile inside the cuvette and the transverse beam profile behind the cuvette are recorded by a camera and shown in the insets.
In our experiments, the beam from a 532 nm, 6 W, continuous wave Coherent Verdi 6 laser was first converted into an optical vortex beam using a spiral phase plate and then focused inside a 10-mm-long cuvette filled with the colloidal suspension consisting of PTFE particles [Laurel, Ultraflon AD-10] dispersed in glycerin/water solution (3:1, v/v), as shown in Fig. 3.
The filling ratio of the PTFE is 0.7%. Since the refractive index of the PTFE particles is lower than that of glycerin water, 20 the particles have negative polarizability. First, we observed MI growing from noise, i.e. without a well-defined perturbation. Figure 4(a)-(c) shows different optical vortices of charge 1, 2, and 4, generated using the spiral phase plates.
Interference experiments were performed to confirm the topological charges of the generated vortex beams, as shown in Fig. 4(d)-(f). Due to the MI, the original doughnut-shaped beam after passing through the colloidal suspension splits into several bright spots, depending on its initial charge. Here, we performed two series of experiments with and without

Methods
The nonlinear Schrödinger equation governing the evolution of the slowly varying electric field envelope E can be written as: 11,20 i ∂E ∂z + where ∇ 2 ⊥ = 1 r ∂ ∂r r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 is the transverse Laplacian. The particle polarizability is denoted by α, and k B T is the thermal energy, with the Boltzmann constant k B and at temperature T , cross-section, k 0 = 2π λ 0 is the wave number, and λ 0 is the free-space wavelength. This equation was analyzed in detail using the linear stability analysis 20 and solved numerically using the split-step Fourier algorithm.