Nanoscale orbital angular momentum beam instabilities in engineered nonlinear colloidal media

Jingbo Sun; Salih Z. Silahli; Wiktor Walasik; Qi Li; Eric Johnson; Natalia M. Litchinitser

doi:10.1364/OE.26.005118

1. Introduction

Almost a century ago, the work of Einstein [1] and Perrin [2] laid the foundations of modern physics of colloids—liquids containing structures on the scale of roughly 10 nm to 1 μm that are stable against sedimentation. Since then, colloids with well-defined particle size, shape and interaction lengths have been widely used as model systems in fundamental studies of statistical physics phenomena [3], phase transitions and optical trapping [4], as well as assembly techniques of nanostructures [5–7]. Propagation of light beams through some common colloidal media such as fog, clouds, smoke, paints, and milk finds increasingly important applications in science and technology, ranging from optical bar-coding for applications in genomics, proteomics and drug discovery [8], free-space communication technologies [9] and weather control [10] to security and defense [11].

Recent progress in the development of artificial materials, or metamaterials, with fundamentally new physical properties opens new opportunities for tailoring the properties of colloids. Metamaterials are built of resonant elements with dimensions much smaller than the wavelength of light, sometimes referred to as meta-atoms, enabling light-matter interactions that are difficult or impossible to realize using naturally available materials. A majority of photonic metamaterials that have been demonstrated to date were solid-state materials. However, the concept of meta-atoms can be extended further to realize artificial media with novel electromagnetic properties in liquid [12] or gaseous [13] phases at frequencies ranging from microwave to visible. In particular, at optical frequencies, engineered colloidal suspensions offer a promising platform for engineering polarizabilities and realization of large and tunable nonlinearities. Recent studies have shown that the nonlinearity of colloidal suspensions has exponential character and can be either supercritical, in case of particles with positive polarizability, or saturable, for negative polarizability particles [14–18].

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 [19] or imaging applications [20]. 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 [4,21–23].

In this work, 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. We decided to work with the negative polarizability suspension because solitons and necklace beams generated there are more stable and do not collapse as compared to the case of positive polarizability [16, 17]. 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 [17], the perturbation 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 [24–26] in naturally existing colloids such as fog and clouds. Consequently, our results can enable potential applications such as remote sensing techniques, light filament based long distance probing, and imaging/spectroscopic applications based on the light propagation in highly scattering biological and chemical media.

2. Materials and methods

2.1. Analytical model and numerical simulations

Let us consider an optical vortex beam propagating along the z-direction in a nano-colloidal 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, and 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 parameters of the vortex beam. This process is described by the nonlinear Schrödinger equation (NLSE) which governs the evolution of the slowly varying electric field envelope E and can be written as [14,17]:

i \frac{\partial E}{\partial z} + \frac{1}{2 k_{0} n_{b}} \nabla_{⊥}^{2} E + k_{0} (n_{b} - n_{p}) V_{p} ρ_{0} e^{\frac{α}{4 k_{B} T} {| E |}^{2}} E + \frac{i}{2} σ ρ_{0} e^{\frac{α}{4 k_{B} T} {| E |}^{2}} E = 0

where

\nabla_{⊥}^{2} = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}}

is the transverse Laplacian. The particle polarizability is denoted by α, and k_BT is the thermal energy, with the Boltzmann constant k_B and at temperature T, V_p is the volume of a particle, ρ₀ is the unperturbed particle concentration, σ is the scattering cross-section,

k_{0} = \frac{2 π}{λ_{0}}

is the wave number, and λ₀ is the free-space wavelength. Numerical results presented here were obtained by solving Eq. (1) using the split-step Fourier algorithm.

Fig. 1 Propagation of a charge one optical vortex beam in a colloidal solution with negative polarizability. In free space, the helical wave front (right) and the doughnut intensity profile of the beam (left) are schematically shown. Inside the colloidal medium, the input vortex beam transforms into a rotating necklace beam. The repulsion of the particles in the path of the high intensity beam leads to a local nonlinear index change.

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Following the standard linear stability analysis [17, 25], we assume that the high-intensity optical beam with a topological vortex charge m ∈ ℤ is accompanied by an azimuthal perturbation:

E (θ, z) = [| E_{0} | + a_{1} e^{- i (M θ + μ z)} + a_{2}^{*} e^{i (M θ + μ^{*} z)}] e^{i (m θ + λ z)}

where E₀ = E(r = r_m, z = 0) is the electric field amplitude of the rotationally invariant steady state solution of Eq. (1) with charge m, taken at the average radius r_m; a₁, a₂ are the amplitudes of the small perturbations, and M ∈ ℤ is the deviation from m of the perturbation charge. The topological charge of the perturbation is given by m ± M, λ is the propagation constant of the steady state solution, and μ is the propagation constant correction for the perturbation. The linear stability analysis allows us to calculate the MI gain for the perturbations with the charge m ± M imposed atop the main beam with the charge m and the corresponding averaged radius r_m. The gain is given by [17]:

Im (μ) = g_{m} (M) = \frac{| M |}{2 k_{0} n_{b} r_{m}} \times Im \sqrt{\frac{M^{2}}{r_{m}^{2}} - \frac{| α |}{2 k_{B} T L^{2}} {| E_{0} |}^{2} exp (\frac{α}{2 k_{B} T} {| E_{0} |}^{2})}

where

L^{2} = {(2 k_{0}^{2} n_{b} | n_{p} - n_{b} | V_{p} ρ_{0})}^{- 1}

.

Equation (3) 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 λ₀ = 532 nm, in the negative-polarizability suspension made of low refractive index polytetrafluoroethylene (PTFE) particles (n_p = 1.35) dispersed in glycerin water (n_b = 1.44) with the volume filling fraction of f₀ = ρ₀V_p = 0.7%. The radius of the particles is assumed to be 120 nm and the experiments were performed at room temperature. For this set of the parameters, Fig. 2(a) shows the gain curves g_m(M) as a function of the perturbation azimuthal index deviation M for different values of the vortex charge m. The analytical predictions are only valid when the perturbation intensity is significantly lower than that of the main beam. Above this limit the dynamics of MI has to be studied using numerical simulations of a three-dimensional NLSE given by Eq. (1).

Fig. 2 Azimuthal modulation instability gain. (a) Analytically computed instability gain g_m(M) as a function of the perturbation azimuthal index deviation M, for negative polarizability particle-based systems for different topological charges m of the initial steady-state vortex solution. (b)–(c) Inverse of the beam breakup distance recorded in numerical simulations of seeded MI. Dashed lines show analytical curves with rescaled magnitude that help guide the eye.

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In order to confirm the analytical prediction for the number of maxima and the shape of the gain curves, we have numerically solved the NLSE in the absence of scattering losses (σ = 0) using the split-step Fourier method [27,28]. First, based on the theoretical predictions, we have found the parameters of the stationary vortex solitons with charges m = 2 and m = 4. We have numerically confirmed that for a given parameters of the beam (power and average radius r_m) the vortex propagates in a stable manner, provided that the medium is lossless. Addition of the random noise on top of the stable solution resulted in the MI induced beam breakup into a necklace beam with the number of maxima predicted by the analytical results (N = 4 for the main vortex charge m = 2, and N = 7 for the main vortex charge m = 4). Simulations of the MI allow us to determine the rate at which the pattern with N maxima grows. In order to seed the growth of a pattern with N maxima, we add only the perturbation of charges m ± M = N. The distance z₀ at which the pattern with N maxima emerges is inversely proportional to the MI gain g_m(M). The distance z₀ is read from the light intensity maps I(r, θ, z), and its choice is somewhat arbitrary. We have chosen z₀ to be the distance at which the contrast between the N maxima and the minima in between them is the highest. The results of 1/z₀ for main vortex charges m = 2 and m = 4 are shown in Fig. 2(b), 2(c). We can see a great agreement with the rescaled analytical curves showing the MI gain 1/z₀.

2.2. Experimental setup

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 [17], the particles have negative polarizability. The beam profiles and the interference patterns shown in Fig. 4(a)–(f) were taken using the setup shown in Fig. 3(a). The necklace beam generated during the propagation through the nonlinear colloidal suspension were allowed to diffract in free space and projected onto a screen placed behind the cuvette. The images of the screen shown in Fig. 4(g)–(i) were taken with a camera oriented quasi-perpendicular to the screen, in order to minimize the image distortion.

Fig. 3 Experimental setup. (a) Interference setup used to test the vortex beams with different charges generated by the corresponding spiral phase plate. (b) Setup used to study (seeded) modulation instability of vortex beams in colloidal media. Collimated beam from Verdi V6 laser (λ₀ = 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 (f = 10 cm). 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.

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Fig. 4 Experimental results showing the formation of the necklace beam from an initial vortex beam propagating in a nonlinear colloidal suspension with negative polarizability particles. (a)–(c) Intensity profiles of the incident vortex beams of charges 1, 2, and 4. (d)–(f) Interference patterns corresponding to vortex beams with topological charges in (a)–(c), respectively. (g)–(i) Intensity distributions of the resulting necklace beams after the propagation in the colloidal medium corresponding to the incident beams (a)–(c), respectively. Necklace beam generated by: a single vortex beam with charge 1 (g); vortex beam with charge 2 with perturbation of charges 4 and 8 with power levels corresponding to 3% of the main beam (h); vortex beam with charge 4 with perturbation of charge 8 with power corresponding to 3% of the main beam (i).

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3. Results and discussion

First, we observed MI growing from noise, i.e. without a well-defined perturbation. Figure 4(a)–4(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)–4(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 the seed, as shown in Fig. 3. For the incident beam with m = 1 which was directly focused into the cuvette without adding any induced perturbation, the beam after passing through the colloidal solution splits into 2 bright spots, as shown in Fig. 4(g). In the case of seeded (or induced) MI, we investigated the propagation of the vortex beams of m = 2 and m = 4. In the experiment, firstly, we explored the case of focused beam with m = 2 in the presence of weak seeded perturbations. Perturbations of charge 4 and charge 8 were added separately and then together to the main beam. The intensity ratio between the perturbations of either charge 4 or charge 8 and the main beam was adjusted from 0 to 3%. Then, we added both charge 4 and charge 8 together to the main beam with total perturbations’ ratio ranging from 0 to 3%. By carefully testing all these cases with different perturbation charges and intensities, we find that in such a competition between the perturbations originating from the noise and those seeded by the low intensity beam, the final beam pattern on the screen always shows 3 maxima. This result is qualitatively consistent with our analytical predictions, revealing the fact that only the perturbation with the charge close to the maximum of the gain curve is amplified, as it grows faster than other perturbations, even if they are seeded. Secondly, a similar test was performed for the vortex with charge 4 in the presence of the perturbation of charge 8 with power ratio ranging from 0 to 3%. The final pattern observed on the screen shows a necklace beam including 7 maxima, which corresponds to the maximum of the gain curve shown in Fig. 2(c). Based on these results, we conclude that the final pattern is intrinsically determined by the charge of the incident beam itself.

4. Summary

In summary, we have experimentally and numerically studied both seeded and unseeded modulation instability in colloidal suspensions of negative polarizability nano-particles. The experimental results are in good agreement with the numerical predictions. In particular, in the case of seeded modulation instability, the observed necklace beam patterns were identical with the patterns obtained without the seed. This shows that the perturbation with the largest growth rate predicted in the analytical and numerical calculations prevails over all the other perturbations introduced to the beam either through the noise or as a seeded perturbation. These results are likely to enable a new platform for fundamental studies of nonlinear optical phenomena in engineered media as well as for imaging and light manipulation in scattering media, such as biological and chemical systems.

Funding

Army Research Office (ARO) (W911NF-11-1-0297, W911NF-15-1-0146).

Acknowledgments

The authors thank professor D. Christodoulides from University of Central Florida fruitful discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Nanoscale orbital angular momentum beam instabilities in engineered nonlinear colloidal media

Abstract

1. Introduction

2. Materials and methods

2.1. Analytical model and numerical simulations

2.2. Experimental setup

3. Results and discussion

4. Summary

Funding

Acknowledgments

Disclosures

References and links

Cited By

Figures (4)

Equations (3)

Optics Express