Long-Range One-Dimensional Longitudinal Optical Binding

We create extended longitudinally optically bound chains of microparticles with the use of counterpropagating “nondiffracting” light fields, the so-called Bessel beams. The beam homogeneity and extended propagation range allow the creation of $200\mu \mathrm{m}$ long chains of organized microparticles. We observe short-range multistability within a single chain and long-range multistability between several distinct chains. Our observations are supported by theoretical results of the coupled dipole method.

Figure 1

CDM calculated optical force acting upon the right-hand side particle $R$ in a two-particle setup (upper scheme). Left plot shows overall shape of the force revealing waves related to the long-range multistability denoted by vertical dashed lines separated by about ${\lambda }_{\mathrm{W}}$. Wavelets are apparent in the detailed view in the right plot and cause short-range multistability marked by vertical dotted lines separated by about ${\lambda }_{\mathrm{w}}$. Polystyrene spheres of diameter 802 nm are placed in two counterpropagating incoherent identical Bessel beams with core radii ${\rho }_0$ (calculated for two values of ${\rho }_0$: $1.787\mu \mathrm{m}$—green thick, $2.380\mu \mathrm{m}$—blue thin). The following parameters were used for the CDM calculation: refractive index of particles $n_p=1.5755+0.0006i$ and surrounding medium ${\mathrm{D}}_2\mathrm{O}$ $n_m=1.320$; laser wavelength in vacuum 1070 nm. The on-axis optical intensity of one beam $3.86\mathrm{mW}/\mu {\mathrm{m}}^2$ corresponded well to the experimental value [8].

Figure 2

Two principal mechanisms responsible for short- and long-range multistability. The force influencing particle $R$ due to the light scattering from particle $L$ depends on the propagation direction of the incoming Bessel beam (BB). The dashed vertical lines show the wave fronts of the incoming Bessel beam with the axial component of the wave vector $k_z$ and the full curves denote parts of the scattered spherical wave fronts of wave vector $k$ on the axis. Left scheme. The BB comes from the left and propagates in the same direction as scattered field from particle $L$. The interference of copropagating fields causes force modulation driven by a long period ${\lambda }_W$ (so-called waves). Right scheme. The BB comes from the right and counterpropagates with respect to the field scattered from particle $L$. Therefore a short-period force modulation ${\lambda }_w$ exists (wavelets). If both particles are illuminated by two counterpropagating incoherent BBs, both mechanisms determine the total force acting on particle $R$ as well as on particle $L$.

Figure 3

Top: Theoretical dependencies of optical force acting on particle $R$ in a chain of two identical polystyrene spheres of diameters 802 nm (left column) and 1070 nm (right column). The equilibrium positions are localized at the separations with zero total force and negative slope (indicating a stable equilibria with the nearest separations between them close to ${\lambda }_{\mathrm{w}}$). Bottom: Plots of calculated probability densities of interparticle separation enable easier recognition of multistability events separated from each other by ${\lambda }_{\mathrm{w}}=409\mathrm{nm}$ and also comparison with experimental results (see the stairlike black curves). The same parameters as in Fig. 1 were used for calculations except the Bessel beam core radii ${\rho }_0=1.77\mu \mathrm{m}$ (blue dashed), ${\rho }_0=1.80\mu \mathrm{m}$ (green full) and ${\rho }_0=1.83\mu \mathrm{m}$ (red dash-dot).

Figure 4

Short-range self-arrangement of multiple identical polystyrene particles in a single chain. Left and right column corresponds to Bessel beam core radii ${\rho }_0=1.8\mu \mathrm{m}$ and ${\rho }_0=2.4\mu \mathrm{m}$, respectively. The upper (lower) row of pictures shows particles of 802 nm (1070 nm) in diameter. The black spots in the individual pictures represent the snapshots of a single particle (mean background intensity subtracted for each image). The horizontal black bar shows the standard deviation of particle positions. The $\mathrm{\text{red}}\times \mathrm{\text{denote}}$ the calculated position of the particle center (the same parameters used as in Fig. 3), $\mathrm{\text{clusters of}}\times \mathrm{\text{indicate}}$ the short-range multistability events for the cases we could theoretically model.

Figure 5

Top: Formation of chains with more particles of diameter 802 nm (left) or 1070 nm (right). Bottom: Self-arrangement of 1070 nm particles in wider BB cores of radii ${\rho }_0=2.4\mu \mathrm{m}$ and ${\rho }_0=3.7\mu \mathrm{m}$.