1. Introduction

A detailed understanding of light-matter interactions is crucial for many areas of physics, chemistry and biology. One such area was pioneered by Arthur Ashkin and co-workers who dealt with the exchange of momentum of light with micro and nanoobjects since the 70s of the last century [1] and their effort led e.g. to laser trapping and cooling of atoms [2] or optical tweezers [3]. Among many other results, they also reported that initially a homogeneous suspension of dielectric nanoparticles forms a colloidal structure (CS) under the influence of laser illumination which acts as a non-linear Kerr medium and self-focuses a laser beam passing through [4–7]. Qualitatively, when a laser beam propagates through a colloidal suspension, particles optically denser than the ambient medium tend to flow to the region with the highest optical intensity of the beam under the influence of the so-called optical gradient force. The growth of concentration of optically denser particles in such a high-intensity region increases the effective refractive index therein. Similarly, as in the case of non-linear optical Kerr effect, self-focusing of the beam occurs and potentially an optical spatial soliton can be formed which guides the laser beam over long distances without noticeable laser beam diffraction [8–11].

Several theoretical studies [12–17] describe this phenomenon considering a CS as a continuous medium where the effective refractive index depends on the optical intensity and a gradient of refractive index is formed within the CS as a results of the balance between the particles in-flow $\textbf{J}_{\textrm {drift}}$, due to optical force, and out-flow $\textbf{J}_{\textrm {diff}}$, due to diffusion: $\textbf{J}_{\textrm {drift}}+\textbf{J}_{\textrm {diff}} = \rho \textbf{v} - D\nabla \rho =0$ where the drift velocity $\textbf{v} = \mu \textbf{F}_{\textrm {opt}}$, with $\mu$ denoting the particle mobility, $D$ the diffusion constant, and $\rho$ the density of particles. Since the particles are considered to be much smaller compared to the laser beam wavelength, the optical force acting upon a particle $\textbf{F}_{\textrm {opt}}$ can be expressed within the framework of Rayleigh approximation [18–21] as the sum of the gradient force $\textbf{F}_{\textrm {grad}}$ and scattering force $\textbf{F}_{\textrm {scat}}$: $\textbf{F}_{\textrm {opt}}=\textbf{F}_{\textrm {grad}}+\textbf{F}_{\textrm {scat}}$. Considering nanoparticles, the gradient force is proportional to $a^3$, points in the direction of optical intensity gradient (for a particle optically denser than the ambient medium) and, in focused laser beams, it is typically stronger than the scattering force, which is proportional to $a^6$ and points along the Poynting vector (i.e. typically along the beam propagation). Generally, the influence of the scattering force on the formation of CS has been ignored, however, recent observations [11,22,23] explained an essential role of the scattering force on the beam propagation through a biological suspension where it propels larger particles axially [11,22,23]. Moreover, illumination of many objects leads to multiple scattering of light by them and results in forces acting between the objects – referred to as the optical binding forces [24]. These forces are responsible for self-organization of objects into static [25] or dynamic [26] optically bound microstructures.

2. Results and discussion

In majority of previously reported experiments a single and relatively wide Gaussian beam (beam waist $w_0 > 10\,\mu$m) was focused into a colloidal medium to observe formation of a CS, laser beam guiding, and self-focusing [8–11,22,23]. In contrast, we used an optical trap formed from two counter-propagating laser beams with smaller beam waists $w_0 < 4\,\mu$m [27] that were linearly polarized in orthogonal directions, see Fig. 1(a). Here, the scattering forces coming from both beams are compensated and the gradient force drags nanoparticles towards the beams focal regions where the effective refractive index of the CS increases. The spatial distribution of particles in the structure is dictated only by gradient force and by the diffusion and thus it can be easily externally controlled by modification of the laser beam waist or by the laser power [29,30].

 

Fig. 1. a) A dual-beam optical trap [27,28] formed by two orthogonally linearly polarized counter-propagating laser beams (wavelength 1064 nm, beam waist $w_0 = 1.6\,\mu$m). Increased concentration of polystyrene nanoparticles of radius $a \simeq 50$ nm in high-intensity region of the beams increases locally an effective refractive index of the medium and consequently creates a gradient refractive index medium. b) Propagation of a weak probe beam (543 nm) through the colloidal structure which was simulated by the multiple Mie scattering theory and the effective refractive medium. c) and d) Intensity patterns of the probe beam scattered into $xz$ and $xy$ planes for the laser power varied between 20 and 70 mW giving different widths $\sigma _x$ of the formed colloidal structures. e) Comparison of transmission coefficients $T$ obtained from the experiment and the theoretical models described in the text.

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To investigate the optical properties of the CS we added a weak probe beam (vacuum wavelength $\lambda _0=543\,$nm, $w_0\approx 40\,\mu$m, power 5 mW) that illuminated the CS coaxially along the counter-propagating trapping beams, see Fig. 1(a), and we recorded its intensity pattern scattered from the CS into two perpendicular planes $xz$ and $xy$, see Figs. 1(b). Figures 1(c) and (d) reveal that the CS serves for the probe beam as a gradient index lens (GRIN) which is externally tunable by the magnitude of the laser power of the counter-propagating trapping beams [30,31].

To qualitatively characterize the transmission properties of the CS, we introduce the transmission coefficient $T=\langle I(x,y) \rangle /I_0$, defined as the mean value of the intensity pattern $I(x,y)$ scattered by the CS to the central region of the $xy$ intensity profile (denoted by a solid black circle in Fig. 1(d) and divided by the intensity of the probe beam $I_0$ without particles taken from the same region of the camera pattern. Thus $T>1$ indicates the probe beam tends to be guided by the CS and has an intensity maximum on the beam axis, while $T<1$ shows the beam is deflected off the beam axis [30].

We investigate the optical properties of the CS also using two theoretical models. The first model simulated the full three-dimensional scattering of the probe beam by the CS using multiple Mie scattering theory [32], employing the CELES [33] and MATLAB/CUDA MEX implementation of the multi-sphere T-matrix method. Since we use laser beams with Gaussian lateral intensity profiles, we assume that the position distribution of nanoparticles in the CS follows Gaussian distributions in each axis $f_{\textrm {G}}(x,y,z) = f_{\textrm {norm}} \exp [-x^2/(2\sigma _x^2) -y^2/(2\sigma _y^2) -z^2/(2\sigma _z^2)]$, where $\sigma _i$ denotes the width of the distribution along $i$-th axis and $f_{\textrm {norm}}$ is a normalization constant. The corresponding values were obtained from the fit to the experimental $xz$ intensity profiles by 2D Gaussian function obtaining $\sigma _x = \sigma _y \in (0.4, 2.2)\,\mu$m and $\sigma _z = 7\,\mu$m.

The second, more simple and commonly used theoretical model considers the CS as an effective refractive medium [34] forming a bi-directional gradient index (GRIN) lens [30]. GRIN lens calculations were performed using commercial Finite Element Method software Comsol Multiphysics. Light propagation was modeled by the beam envelopes method that solves the Helmholtz equation using the assumption of a slowly spatially varying amplitude (envelope) of the propagating light. The local value of the effective refractive index was calculated from the simplified Maxwell-Garnett formula [34] $n_{\textrm {eff}} = (1-f) n_b + f n_p,$ where $n_b = 1.33$ and $n_p= 1.59$ is the refractive index of water and polystyrene nanoparticles, respectively, and $f = N V_p f_{\textrm {G}}$ denotes the volume filling factor with number of particles $N$ in the structure and volume $V_p = 4/3\pi a^3$ of a particle with radius $a = 99/2$ nm. The Gaussian beam that would propagate through the undisturbed medium was incident upon the front face of the computational domain. Figures 1(e) compares the transmission coefficients $T$ obtained from these two theoretical models with the experimental data for several widths $\sigma _x$. The best agreement with the experiment is observed for 6400 particles forming the CS within the multiple Mie scattering model. However, both theoretical models predict a minimum value of the transmission coefficient near the experimentally observed value ($\sigma _x = 0.45\,\mu$m) and follow the growing trend of the experimental results. However, for larger value $\sigma _x>1.3\,\mu$m both theoretical models tend to decrease the $T$ values with increasing $\sigma _x$ while the experimental value keeps constant $T\simeq 1.2$. This discrepancy can be caused by non-symmetry of the CS lateral spatial profile caused by a slow flow of our colloidal sample which became significant for the lower value of the laser power.

Figures 2(a)–2(c) illustrate the modification of the intensity pattern of the probe beam scattered into $xz$ plane as the number of nanoparticles in the CS grows. The intensity profiles along $z$ axis do not follow any more a simple Gaussian distribution, instead the intensity maximum shifts from the rear to the front part of the structure and gradually forms two intensity maxima. Similar behavior is observed in a bi-directional gradient-index medium if its gradient of refractive index is increased [30,31,35]. The formation of a series of intensity maxima and minima on the optical axis typically occurs also in multimode graded-index fibers [36]. Higher-order modes and lower-order modes propagate with different propagation constants, interfere, and create on-axial intensity maxima and minima (see Fig. 2(c)). Figure 2(d) shows axial intensities of the probe beam (543 nm) calculated using the theoretical model based on the Mie scattering theory revealing qualitatively similar results. Here, we assumed the CS formed from various number of nanoparticles (from 10 000 to 25 000) with particle spatial distribution given by 3D Gaussian distribution with widths $\sigma _x = \sigma _y = 400$ nm and $\sigma _z = 20\,\mu$m. The same behavior one can assume for other wavelengths including the trapping one which has important implications for the CS stability. Figure 2(e) shows a slight shift of the maximum of the axial intensity of the left-hand trapping beam (1064 nm, $w_0 = 1.6\,\mu$m) for the same CS spatial configurations.

 

Fig. 2. Evolution of the scattered intensity pattern of the probe beam as the number $N$ of nanoparticles in the colloidal structure grows. The axial maximum of the scattered intensity moves from the right part of the structure a) to the front one b), and gradually forms two successive maxima c). d) and e) (from numerical simulation) show axial intensities of the probe beam (543 nm, $w_0 \approx 40\,\mu$m) and one of the trapping beam (1064 nm, $w_0 = 1.6\,\mu$m) propagating in the colloidal structure composed from various number of particles, respectively. The dashed black curves indicate the axial intensity profiles of the original probe and trapping beams.

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Figure 3 analyzes the axial intensity of the trapping beam coming from the left and optical force acting on 50 000 nanoparticles spatially distributed with the same Gaussian distribution as in Fig. 2. The initial Gaussian profile of the beam ($w_0 = 0.91\,\mu$m) (see Fig. 3(a)) is modified by the formed CS into a beam intensity profile shown in Fig. 3(b). Comparing to Fig. 2(e), Figs. 3(c) considers 2-5 $\times$ more nanoparticles in the CS and thus the modification of the trapping beam is stronger. Since the particle spatial distribution is influenced by the gradient force, the particles are attracted to the shifted intensity maximum in front of the original trapping beam focus ($z = 0\,\mu$m), see Figs. 3(c) and 3(d), which leads to axial redistribution of the effective refractive index of the CS. Moreover, the trapping beam is also modified laterally and a new high-intensity ring is formed off the beam axis, as the comparison of intensity and force profiles in Figs. 3(b), 3(e), and 3(f) reveals.

 

Fig. 3. Calculated modification of the intensity and force profile of a single trapping beam coming from left after passing the colloidal structure. a) Log-scaled spatial intensity profile of the incident Gaussian beam (beam waist $w_0=910$ nm, vacuum wavelength 1064 nm). b) Modified intensity profile due to the propagation through the colloidal structure placed symmetrically around the axis origin with 3D Gaussian distribution widths $\sigma _x = \sigma _y = 400$ nm and $\sigma _z = 20\,\mu$m). c) Intensity profile along optical axis $z$ modified by the CS (blue) and incident beam intensity profile (black). d) Optical force acting on the nanoparticles along optical axis. Nanoparticles are pushed to the equilibrium position overlapping with the intensity maximum and denoted by the orange circle. e) Lateral intensity profile modified by the CS (blue) and incident intensity profile (black). f) Corresponding lateral force profile with new off-axial trapping regions and original central one marked by orange circles.

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Such a high-intensity ring attracts nanoparticles and contributes to the off-axial formation of the CSs, as a series of camera images illustrates in Fig. 4. At the begin, one CS is formed along the beam axis and its length is about $80\times$ longer then the Gaussian beam Rayleigh length $z_{\textrm {R}}=\pi w_0^2/\lambda$, where $\lambda$ denotes the beam wavelength in the medium without particles. Note, that the relative length of this CS is comparable to relative lengths observed in previously reported experiments with approximately $10\times$ larger beam waist [8–11]. As the number of particles in the CS grows, the off-axial intensity maxima are formed and the CS grows in off-axial directions. This behavior resembles a multi-directional formation of photo-polymerized waveguides using higher laser power [37] or planar colloidal assembly out of the beam focus [38], where the underlying three-dimensional geometry and physics are similar. However, it differs from observations of Reece et al. [39] where parallel CSs were formed in a plane parallel with the surface illuminated by a relatively wide evanescent field.

 

Fig. 4. Bright-field images of colloidal structure formed from polystyrene nanoparticles (radius $\simeq 50$ nm) in off-axial direction in two counter-propagating trapping beams (beam waist of each $w_0 = 0.91\,\mu$m, $P = 2$ W) in the course of 80 s.

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3. Conclusion

In summary, we employ a counter-propagating optical trap to study the formation and optical properties of colloidal structures (CS) composed from $\sim$ 10 000 polystyrene nanoparticles dispersed in water. Using the weak probe beam propagating through the CS and corresponding theoretical models (multiple Mie scattering, effective refractive index medium) we illustrate that the beam transforms similarly to continuous bi-directional GRIN lenses. We demonstrate external tunability of the optical properties of this GRIN lens by the power of the incident trapping beams. Using the theoretical model based on multiple Mie scattering we link together more complex axial and lateral redistribution of more than 10 000s of nanoparticles in the CS, observed experimentally, and a strong modification of the axial and lateral spatial intensity profiles of the trapping beams.