Laser trapping of low refractive index colloids in a nematic liquid crystal

We describe and analyze laser trapping of small colloidal particles in a nematic liquid crystal, where the index of refraction of colloids is smaller compared to the indices of the liquid crystal. Two mechanisms are identified that are responsible for this anomalous trapping: (i) below the optical Fréedericksz transition, the trapping is due to the anisotropic dielectric interaction of the polarized light with the inhomogeneous director field around the colloid, (ii) above the optical Fréedericksz transition, the optical trapping is accompanied by the elasticity-mediated interaction between the optically distorted region of a liquid crystal and the colloid. In the majority of the experiments, the trapping above the Fréedericksz transition is highly anisotropic. Qualitative agreement is found with a numerical analysis, considering the nematic director elastic distortion, dielectric director-light field coupling and optical repulsion due to low refraction index colloid in high index surroundings.

Figure 1

(a) Polarization micrograph of a $4\mu \mathrm{m}$ diameter silica colloid treated with DMOAP in a homeotropic cell of 5CB. The dipolar field distribution around the colloid cannot be resolved, because the dipolar axis is along the director, i.e., perpendicular to the plane of the paper. The hedgehog is, therefore, either positioned above or below the colloid. The scale corresponds to $5\mu \mathrm{m}$. (b) The same type of colloid in a planar 5CB cell clearly shows the dipolar director distribution. The same type of ordering is observed for $0.97\mu \mathrm{m}$ colloids.

Figure 2

Sequences of micrograph images under crossed polarizers, showing two different trapping events in 5CB, in a $50\mu \mathrm{m}$ thick cell. The scale bar is $10\mu \mathrm{m}$ and is equal for both images. The time steps are equal for both images. Note the apparent size of a $0.97\mu \mathrm{m}$ diameter colloid under crossed polarizers. (a) Trapping at low power of $25\mathrm{mW}$, which is just below the power that is necessary to induce the local optical Fréedericksz transition (in this case, the threshold power is $30\mathrm{mW}$). The cross indicates the position of the optical trap, which is not visible. (b) Trapping at $35\mathrm{mW}$, which is above the optical Fréedericksz transition. The bright spot is due to a local birefringent area, created by the laser-light-induced distortion of the director field.

Figure 3

(a) Recorded trajectory during the Brownian motion of a silanated $0.97\mu \mathrm{m}$ silica colloid in a homeotropic cell filled with 5CB. (b) and (c) represent histograms of particle’s displacements between two consecutive positions, separated by a time interval of $61\mathrm{ms}$, along the two perpendicular directions. The solid lines are Gaussian fits with $2\sigma =34(1\pm 0.01)\mathrm{nm}$. There is no difference between the $x$ and $y$ directions.

Figure 4

(a) Spiderlike trajectories during the optical trapping of a $0.97\mu \mathrm{m}$ silica colloid in a $15\mu \mathrm{m}$ homeotropic 5CB cell. The laser power is $35\mathrm{mW}$ (i.e., above the optical Fréedericksz transition) and the starting positions are labeled 1–8. (b) The set of spiderlike trajectories rotates as a whole, as the polarization of the laser light is rotated by 90°. (c) Trapping of the same type of colloid at the same laser power, in a thicker $50\mu \mathrm{m}$ cell, that is observed in approximately 20% of the experiments. The spiderlike trajectories correspond to dipole director field around the colloid; whereas, the starlike trajectories shown in (c) indicate quadrupole symmetry of the director field around the colloid.

Figure 5

(a) Optical micrograph of a pair of $4\mu \mathrm{m}$ silanated colloids in a planar cell of $7\mu \mathrm{m}$ thickness. The colloid on the left side is of dipolar type, whereas the colloid on the right side has a nearly fully open defect, which is close to the saturn ring configuration. (b) The same pair of colloids as seen under crossed polarizers.

Figure 6

Schematic of the interaction of a Gaussian light beam with a colloid in a homeotropic nematic cell. The beam induces an optical Fréedericksz transition, which creates a region of a distorted director field inside the cell. This can be considered as a ghost colloid that interacts with a real colloid. For convenience, the position of the colloid is set in the center of the cell; whereas, in the experiment, they can be found everywhere in the cell except close to the surface. Panel (a) shows the dipolar symmetry of the colloid and (b) shows the quadrupolar one.

Figure 7

(a) Recorded time dependence of the separation between a silica particle and the laser trap during the trapping event in a $15\mu \mathrm{m}$ homeotropic 5CB cell at a laser power of $50\mathrm{mW}$, starting from the “easy position” labeled 1 in Fig. 4a. The colloid is trapped at a time labeled $t_{\max }$. The inset shows the log-log plot of the same data showing clear power-law dependence with an exponent $\alpha =0.33(1\pm 0.02)$. (b) The laser trapping potential derived from the data in (a). The solid line is the best fit to $W\left(r\right)=W_0∕r$ with $W_0=5980(1\pm 0.002)k_{B}T\mu \mathrm{m}$.

Figure 8

The strength of the trapping potential along the easy direction as a function of the optical power.

Figure 9

The effective pair potential for the colloid in the vicinity of the laser trap at large laser power, i.e., above the local Fréedericksz transition. The trap is centered at $X=Y=0$, the light polarization is along the $x$ direction. The cell thickness is $20\mu \mathrm{m}$, $K=6\times {10}^{-12}\mathrm{N}$, ${ϵ}_{\Vert }=2.9$, ${ϵ}_{\perp }=2.3$, ${\mathbf{E}}_0=(8\times {10}^6\mathrm{V}∕\mathrm{m}){\mathbf{e}}_{\mathbf{x}}$, $w_0=0.5\mu \mathrm{m}$, $\lambda =1064\mathrm{nm}$, $a=0.5\mu \mathrm{m}$, and ${ϵ}_C=1.7$.

Figure 10

The effective potential for the colloid approaching the laser trap below the local Fréedericksz transition. The trap is centered at $X=Y=0$, the light polarization is along the $x$ direction. The following parameters were used in numerical calculation: $K=6\times {10}^{-12}\mathrm{N}$, ${ϵ}_{\Vert }=2.9$, ${ϵ}_{\perp }=2.3$, ${\mathbf{E}}_0=(3\times {10}^6\mathrm{V}∕\mathrm{m}){\mathbf{e}}_{\mathbf{x}}$, $w_0=0.5\mu \mathrm{m}$, $\lambda =1064\mathrm{nm}$, $a=0.5\mu \mathrm{m}$, and ${ϵ}_C=1.7$

Figure 11

Contributions of different interactions [see Eq. (1)] to the total effective potential that acts on a colloidal particle within the electric field of the optical trap for laser power. Results are presented in the easy direction for laser powers: (a) above the optical Fréedericksz transition (OFT) value, (b) below the OFT value. Note the dominating contribution of the dielectric interaction at larger separations.