Modulational Instability of Optical Vortices in Engineered Saturable Media

1. Introduction

Since the discovery of well-defined orbital angular momentum (OAM) [1], beams carrying OAM proved to be an important tool in the ascension of photonics. Characterized by the presence of an azimuthal phase eimθ, where the index m is known as topological charge and θ is the azimuthal angle [2], OAM beams are poised to enable a number of applications, including information transfer [3], optical tweezers [4], [5], quantum teleportation [6], and computation [7]. Light possessing an OAM, also known as an optical vortex, usually possesses a ring-like intensity distribution, whereas its topological charge can be measured by a number of optical techniques, such as interferometry [8], diffraction through a slit [9], [10], [11], and the tilted-lens method [12], to cite a few.

One of the most common ways of generating optical vortices in a laboratory would be using a spatial light modulator (SLM) [13], [14]. This liquid-crystal-based device allows one to use such a computer-generated phase-mask as a hologram, which makes it quite simple to generate any sort of optical beam. Other options include so-called spiral phase plates (SPPs) [15], [16] and q-plates—where the latter is another type of liquid crystal device with inhomogeneous patterned distribution of the local optical axis in the transverse plane [17], [18]. However, in the domain of integrated optics, compact devices that can readily be integrated on a chip are needed in order to create optical beams carrying OAM. To address this need, the most recently proposed approach to generate optical vortices relies on metasurfaces, including both dielectric [19] and plasmonic [20], [21] structures.

Although most commonly used optical vortices are known to be ring-shaped, there are other groups of beams with different shapes that also carry an OAM. Bessel beams (BBs), which are characterized by an intensity distribution consisting of an infinite multi-ring pattern, is one of these groups [22], [23]. BBs are particularly interesting because of their self-healing property [24], [25], [26]. Moreover, higher-order BBs find applications for particle trapping [27], [28] and imaging systems [29], [30]. Considering other symmetries, elliptical vortices (EVs) are another group of beams possessing OAM. First studied by Bandres and Gutiérrez-Vega [31], [32] and Schwarz et al. [33], Ince–Gaussian (IG) beams were found by solving the free-space paraxial wave equation in elliptical coordinates. IG beams can be related to either Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) beams by changing their eccentricity parameter to 0 and ∞, respectively. On the other hand, EVs can be found by performing a change of variables in the LG mode [34]. Introducing the elliptical parameter η, which assumes values between 0 and 1, simplifies analytical studies of EVs. The elliptical analogue for BBs is the Mathieu beams (MBs), whose intensity profiles consist of an infinite number of concentric ellipses [35]. It is important to mention that exact BBs and MBs, strictly speaking, cannot be realized in the laboratory experiment due to their infinite transverse profile implying an infinite energy requirement. Nevertheless, truncated BBs and MBs can be generated and studied experimentally [36], [37].

A new group of asymmetrical beams was recently developed using an acousto-optical deflector (AOD) together with log-polar optics to generate higher-order Bessel beams integrated in time (HOBBIT) [38], [39]. This method enables the generation of rapidly tunable OAM beams, which reach switching speeds of up to tens of megahertz with a sub-microsecond response time and high-power laser systems, due to the AOD’s very high damage threshold. Such properties make HOBBITs suitable to probe turbulence by rapidly scanning OAM states along an optical path [40] and may be helpful in communication protocols that demand fast-switching OAM modes and high power levels.

One of the most fascinating research directions in the field of optical vortices is the study of such structured light–matter interactions in various linear and nonlinear media. In particular, nonlinear processes such as second-harmonic generation [41], optical Kerr effect [42], [43], self-focusing [44], [45], and optical-parametric oscillations [46] have been reconsidered in the presence of OAM beams. Moreover, rapid progress in nanophotonics opened up new ways of “engineering” the nonlinear medium itself in order to tailor many of these nonlinear light–matter interactions, including self-focusing, modulation instability (MI), and spatial soliton formation [47]. A stability analysis of these solitons can be realized by using a variational approach together with a perturbation method [48]. In particular, carefully engineered nanocolloidal suspensions facilitate new ways of tailoring linear and nonlinear propagation. Indeed, it was shown that liquid suspensions of spherical dielectric nanoparticles can exhibit very large optical nonlinearities [47]. The nonlinearity of nanoparticle suspensions originates from the fact that in the presence of a continuous wave, optical field dielectric nanoparticles experience an optical dipole force proportional to the particle polarizability in the liquid. In the case of particles of higher refractive index np than the surrounding liquid nb, the polarizability is positive and the particles experience an electrostrictive volume force that attracts them into the spatial regions of high intensity thus increasing the local density and the local refractive-index. If optical vortices are considered, it has been predicted and experimentally demonstrated that an azimuthal MI may lead to different regimes of nonlinear beam shaping depending on the properties of the medium and the initial parameters of the OAM beam. In particular, a so-called necklace beam (NB) [49], [50], [51]formation has been demonstrated.

A majority of previous studies on nonlinear light–matter interactions in colloidal suspensions focused on symmetric OAM beams, including the formation of NBs originating from symmetric optical vortices [49], [51]. The dynamics of those NBs were also studied, focusing on a deeper discussion about stability, trajectories, and the formation of new optical beam structures [50]. Here, we report the behavior of several families of complex-shaped OAM beams in negative-polarized nanocolloidal suspensions.

Our paper is structured as follows. In Section 2, we review various types of OAM beams, including LG beams, EV beams, and HOBBIT. In Section 3, we describe all-dielectric as well as plasmonic-particles-based engineered colloidal media with saturable nonlinearity. In Section 4, we perform linear stability analysisfollowed by full numerical simulations for each family of beams. Finally, in Section 5, we summarize the results of our studies of nonlinear OAM beam propagation in saturable nonlinear nanocolloidal media.

2. Optical beams carrying OAM

2.1. Laguerre–Gaussian modes

Let us consider the orthogonal set of OAM-carrying beams defined by LG modes, which are characterized by the indices p and m, which respectively refer to the radial order and topological charge. For the case where p = 0, these optical modes can be written as [1], [2](1) ${LG}_{m} (r, θ, z) = \frac{1}{\sqrt{π |m|!}} \frac{1}{w (z)} {[\frac{r \sqrt{2}}{w (z)}]}^{|m|} e^{- \frac{r^{2}}{w^{2} (z)}} e^{i [m θ + \frac{k r^{2}}{2 R (z)} - Ψ_{m} (z)]}$ where $w (z) = w_{0} \sqrt{1 + {(z / z_{R})}^{2}}$ is the beam width, $R = z [1 + {(z_{R} / z)}^{2}]$ is the wavefront radius of curvature, $Ψ_{m} (z) = (|m| + 1) \arctan (z / z_{R})$ is the Gouy phase, and zR is the Rayleigh range. This set of optical modes is a family of solutions to the Helmholtz equation under the paraxial approximation in cylindrical coordinates and Fig. 1 shows the intensity distribution for some values of m. If one is specifically interested in studying the behavior, interaction, or characterization of the OAM, it is common to consider z = 0 and work with the mth order optical vortex [49]:(2) ${LG}_{m} (r, θ, z = 0) = A_{m} {(\frac{r}{w_{0}})}^{|m|} e^{- \frac{r^{2}}{w_{0}^{2}}} e^{i m θ}$

Other beams that can also carry an OAM include higher-order Bessel [52] and circular Airy beams [53], which constitute the solutions of the Helmholtz equation in cylindrical coordinates and the paraxial wave equation, respectively.

2.2. Elliptical vortices

From an experimental standpoint, it has been shown that elliptic beams can be generated using an oblique incidence of an axially symmetric beam onto an optical element such as a conical axicon or a binary diffractive axicon [54], [55]. From a theoretical viewpoint, exact solutions of the Helmholtz equation can be obtained for the MBs [35], which possess self-healing properties similar to that of the BBs. On the other hand, under the paraxial approximation, the IG modes arise as a family of solutions [31]. Their eccentricity parameter ε adjusts the ellipticity of the transverse structure of the beam, in which the transition to an LG (HG) mode occurs when ε tends to zero (infinite). Even though these are well-behaved solutions, they are not easily handled analytically. For this reason, alternative methods of generating EVs were developed. It was demonstrated that, by just adding an ellipticity parameter η (0 ≤ η ≤ 1), we have the following expression for an mth order elliptical optical vortex [34]:(3) $U (r, θ; η) = A_{m, η} {(\frac{η r}{\sqrt{\cos^{2} θ + η^{2} \sin^{2} θ}})}^{|m|} e^{- η^{2} r^{2} / [2 w_{m, λ}^{2} (\cos^{2} θ + η^{2} \sin^{2} θ)]} e^{i m θ} e^{i λ z}$ where Am,η denotes the amplitude, wm,λ is the beam width, and λ is the propagation constant. By making use of the elliptical coordinates in Eq. (3), where $\tilde{r} = \frac{r}{\sqrt{\cos^{2} θ + η^{2} \sin^{2} θ}}$ and $θ = \arctan [y / (η x)]$ , the angular dependence of the radial variable is explicitly displayed. Fig. 2 shows the intensity distribution of EVs for various ellipticity parameters η. Due to the similarity to regular cylindrical optical vortices, one can make use of this approach to analytically study optical effects as the cylindrical symmetry is broken into an elliptical symmetry.

2.3. Higher-order Bessel beams integrated in time

The HOBBIT system consists of a series of optical devices designed to prepare the input beam impinging on a pair of log-polar optical elements. It converts each Gaussian beam in an array to an asymmetric higher-order Bessel–Gaussian beam after multiple transformations. This results in a superposition of co-propagating higher-order Bessel–Gaussian beams possessing OAM. This technique may be useful to multiple applications, including quantum communication protocols, beam shaping, filamentation, and sensing methods for atmospheric turbulence and underwater systems.

The near-field output of a HOBBIT system with topological charge m at z = 0 can be expressed as [38], [39], [40](4) $U (r, θ) = A_{m, λ} e^{- {(r - r_{0})}^{2} / (2 w_{m, λ}^{2})} e^{- θ^{2} / (β^{2} π^{2})} e^{- i m θ}$ where r0 controls the ring radius, β is the asymmetry parameter, and Am,λ is the amplitude. This leads to an asymmetric ring-shaped beam, where the asymmetry is controlled by β. Fig. 3 [38] shows the amplitude distribution as well as the phase pattern for HOBBITs with the topological charges m = ±3, ±1.2, and 0. As reported in Ref. [38], the efficiency of the AOD coupled with the log-polar system is up to 60%. This means that, for an input power of 30 W, the output would be approximately 18 W.

3. Saturable nonlinear media

3.1. Self-focusing saturable nonlinearity

Initially, saturable nonlinearities were introduced as a correction to the cubic Schrödinger equation (CSE), which is a generic equation describing slow-varying envelopes in conservative, dispersive systems [56]. Optical fieldsinteracting with such systems are governed by the normalized equation [57], [58]:(5) $i \partial_{z} φ + \frac{1}{2} \nabla_{⊥}^{2} φ + f ({|φ|}^{2}) φ = 0$ where $\nabla_{⊥}^{2} = \partial_{x} + \partial_{y}$ is the Laplacian and $f ({|φ|}^{2})$ is the function related to the nonlinear response of the system. For instance, self-focusing saturable nonlinearity can be modeled by(6) $f ({|φ|}^{2}) = \frac{{|φ|}^{2}}{1 + α_{s} {|φ|}^{2}}$ where αs is the saturation parameter. Notice that, for αs = 0, the Kerr limit is achieved. Through this approach, it was demonstrated the existence of soliton solutions carrying OAM [57], the self-trapping effect [58], and NBs [59], among other findings. In Fig. 4, the authors demonstrate the azimuthal instability development in solitons carrying OAM propagating in a saturable self-focusing medium [57].

3.2. Engineered colloidal suspensions

The nonlinear response of nanoparticle suspensions was first studied by El-Ganainy et al. [47] in 2007. Starting from the particle current continuity equation $\partial_{t} ρ + \nabla \cdot J = 0$ , one can make use of the Nernst–Planck equation and obtain the expression for the particle current density [47], [60]:(7) $J = ρ v - D \nabla ρ$ where the diffusion coefficient is denoted by D, the particle convective velocityby v, and the particle concentration by ρ. Here, v is related to the optical force Facting on the nanoparticles as v = μF, where μ is the particle’s mobility. In this model, particle–particle interactions are neglected and highly diluted mixtures are assumed. After combining these expressions, we get the Smoluchowski equation [47], [60]:(8) $\partial_{t} ρ + \nabla \cdot (ρ v - D \nabla ρ) = 0$

Now, some assumptions are needed in order to solve this equation. First, let us consider steady-state solutions (∂t = 0). Second, with the system under equilibrium, diffusion (J = 0) is the responsible to balance the particle’s movement. Finally, if we consider the Rayleigh regime (wavelength is large compared to the particle aspect ratio), the external optical force can be obtained within the dipole approximation and is expressed as F = α∇I/4, where I = |φ|2 is the light intensity and α is the particle polarizability [47]. Still under the dipole approximation, one can express the polarizability of a spherical particle with refractive index np as [47], [61](9) $α = 3 V_{p} ε_{0} n_{b}^{2} (\frac{m_{r}^{2} - 1}{m_{r}^{2} + 2})$ where Vp is the volume of the particle, ε0 is the permittivity in free-space, and mr = np/nb is the ratio of the particle’s refractive index np to the refractive index of the background nb. Notice that, if mr > 1 (mr < 1), the polarizability is positive (negative). After solving Eq. (8) and making use of the Maxwell–Garnett formula, one can find the local index change [47], [61], [62]. In a relatively small index contrast regime (|mr < 1|), we find the optical nonlinearity of the nanosuspensions to be [47](10) $Δ n_{NL} = (n_{p} - n_{b}) V_{p} ρ_{0} (e^{\frac{α}{4 k_{B} T} I} - 1)$

The scattering losses can also be included into the system. Under the Rayleigh regime, the scattering cross section can be written as(11) $σ = \frac{128 π^{5} a^{2} n_{b}^{4}}{3} {(\frac{a^{4}}{λ})}^{4} {(\frac{m_{r}^{2} - 1}{m_{r}^{2} + 2})}^{2}$ with a representing the particle radius. Now, we can obtain the components to write the beam evolution equation through a nanosuspension system. After modifying the Helmholtz equation, $\nabla^{2} φ + k_{0}^{2} n_{eff}^{2} φ = 0$ , where k0 = 2π/λ, the nonlinear Schrödinger equation (NLSE) for this system reads [47](12) $i \partial_{z} φ + \frac{1}{2 k_{0} n_{b}} \nabla_{⊥}^{2} φ + k_{0} (n_{p} - n_{b}) V_{p} ρ_{0} e^{\frac{α}{4 k_{B} T} {|φ|}^{2}} φ + \frac{i}{2} σ ρ_{0} e^{\frac{α}{4 k_{B} T} {|φ|}^{2}} φ = 0$

Here, for either positive or negative polarizabilities, the nonlinear response is self-focusing. For the positive case, the refractive index of the particle is higher than the background (np > nb), leading the particles to move toward the light and increasing the scattering losses. On the other hand, negative polarizability regimes (np < nb) make the particles to move outward from the beam, reducing the scattering losses and leading to a more stable propagation through the system. Fig. 5 shows a sketch of the particle dynamics for a nanosuspension system with both positive and negative polarizability [47].

3.3. Plasmonic suspensions

In the case of the all-dielectric-based saturable nonlinear media described above, high power illumination was usually required to ignite the nonlinear response. However, replacing the dielectric nanoparticles with metallic ones allows relaxing this requirement for the input powers of continuous-wave (CW) lasers [63]. Using various metallic structures, including gold nanorods, silica–gold nanoshells, and gold and silver spheres, the authors of Ref. [63]demonstrate that the nonlinear dynamics are governed by thermal responses, scattering, and optical forces acting on the particles. For this system, the NLSE can be extended by including the thermal effects and, after some algebra, it can be written as [63](13) $i \partial_{z} φ + \frac{1}{2 k_{0} n_{b}} \nabla_{⊥}^{2} φ + k_{0} (n_{p} - n_{b}) V_{p} ρ φ - k_{0} |Δ n_{T}| φ + \frac{i}{2} σ ρ φ = 0$ where ρ is the particle concentration and ΔnT is the refractive index change mediated by the thermal effects. Here, the interplay between thermal effects and nonlinear colloidal responses leads to a nonlinearity compared to cubic–quintic saturable nonlinear media.

In Ref. [63], the authors experimentally demonstrate a self-trapping behavior in Gaussian beams propagating in both positive and negative metallic nanosuspensions. Fig. 6 presents the beam self-trapping in a negative polarizability nanosuspension system composed of gold nanorods, while Fig. 7displays the interactions within positive polarizability media filled with gold (Figs. 7(a)–(d)) and silver (Figs. 7(e)–(h)) particles [63].