Intangible pointlike tracers for liquid-crystal-based microsensors

We propose an optical detection technique for liquid-crystal-based sensors that is based on polarization-resolved tracking of optical singularities and does not rely on standard observation of light-intensity changes caused by modifications of the liquid crystal orientational ordering. It uses a natural two-dimensional network of polarization singularities embedded in the transverse cross section of a probe beam that passes through a liquid crystal sample, in our case, a nematic droplet held in laser tweezers. The identification and spatial evolution of such a topological fingerprint is retrieved from subwavelength polarization-resolved imaging, and the mechanical constraint exerted on the molecular ordering by the trapping beam itself is chosen as the control parameter. By restricting our analysis to one type of point singularity, $C$ points, which correspond to location in space where the polarization azimuth is undefined, we show that polarization singularities appear as intangible pointlike tracers for liquid-crystal-based three-dimensional microsensors. The method has a superresolution potential and can be used to visualize changes at the nanoscale.

Figure 1

Optical trapping setup at 1064 nm with polarization-resolved imaging at 532 nm. P, polarizer; DBS, dichroic beam splitter; IF, interferential filter at 532 nm; QWP, quarter-wave plate; CCD, charge-coupled device imaging device. The local polarization ellipse $\mathcal{E}$ in the observation plane is illustrated on the left part of the figure, where $\psi$ refers to its azimuth angle. Insets: optical axis distribution of an undistorted radial nematic liquid crystal droplet (panel 1) and its image under crossed polarizers (panel 2).

Figure 2

Left part: polarization-resolved images of radial nematic droplets as a function of the trapping power $P_0$, where $s_1$, $s_2$, and $s_3$ are the reduced Stokes parameters. The droplet diameter is $8.5\hspace{0.5em}\mu$m and the upper, middle, and bottom sets of data correspond to $P_0=35$, 270, and 540 mW, respectively. Right part: typical director distribution in the equatorial plane of the droplet. At low power the radial alignment is almost unperturbed, whereas it is all the more distorted when the power is increased. The circle marker indicates the location of the hedgehog defect for the director and the dashed circle refers to the region where the director tends to be aligned along the trapping beam polarization plane as a result of laser-induced liquid crystal reorientation.

Figure 3

Definition of the effective inhomogeneous birefringent parallel slab that aims at describing the optical behavior of a possibly distorted radial droplet. (a) Off-axis radial distribution of the optical axis of the slab in the case of a distorted droplet. (b) Definition of the reduced polar coordinates $(\rho ,\phi )$ and $({\rho }^{\prime },{\phi }^{\prime })$ associated with the location $M$ in the coordinate systems $(\mathit{xOy})$ and $({\mathit{xO}}^{\prime }y)$, respectively. The location $B$ refers to the border viewed from the angle ${\phi }^{\prime }$ in $({\mathit{xO}}^{\prime }y)$ and $C$ corresponds to $B$$({\phi }^{\prime }=\pi )$. (c) ${\Phi }_0$ is a function of $\rho$ in the case of an unperturbed radial droplet for the parameters that correspond to Fig. 4 (see text for values).

Figure 4

Left part: polarization-resolved images obtained from the inhomogeneous birefringent slab model for the droplet that corresponds to Fig. 2. Right part: spatial distribution of the optical axis of the effective slab is shown on the right part of the figure. (a) Unperturbed radial droplet. (b) Distorted droplet where the shaded area on the right panel refers to the local light-induced birefringence changes $\delta \Phi$ (see text for details).

Figure 5

Map of the azimuth angle $\psi$ for an ideal (a) and real (b) radial nematic droplet with $7\hspace{0.5em}\mu$m diameter. In the ideal case (a), a nongeneric $C$ point with an index $m=1$ is located at the origin (square symbol), which is surrounded by a degenerate ring $C$ line (white circle). Instead, in the experiment (b), we observe a set of generic $C$ points (circle symbols): a pair of $C$ points with an index $m=1/2$ near the origin (labeled as $\{a,b\}$) and two pairs of $C$ points with opposite indices $m=\pm 1/2$ (labeled as $\{1,2,3,4\}$) that qualitatively lie along the ghost of the phase dislocation that would have appeared in the ideal case (dashed circle).

Figure 6

Mapping of the orientation of the polarization ellipse $\mathcal{E}$ in the neighborhood of each of the $C$ points shown in Fig. 5 that are labeled {$a$,$b$,1,2,3,4}, respectively. The orientation of the major axis of $\mathcal{E}$ is evidenced by a unit vector and the insets refer to the local mapping for the $\psi$ value where the color map is the same as in Fig. 5. Diamonds, stars, and triangles refer to $C$ points of the “lemon” ($m=1/2$), “star” ($m=-1/2$), and “monstar” ($m=1/2$) type, respectively. Each box corresponds to a zoom on a $900\hspace{0.5em}\mathrm{nm}\times 900\hspace{0.5em}\mathrm{nm}$ area.

Figure 7

(a) Trajectories of the pair of lemon $C$ points as a function of the linearly polarized trapping beam power. Error bars correspond to $\pm 1$ pixel precision. The set of data corresponds to $P_0=35$, 90, 150, 210, 270, 330, 390, and 540 mW, respectively. Inset: image of the $8.5\hspace{0.5em}\mu$m diameter droplet. (b) Azimuth angle mapping of the trapped droplet at $P_0=35\hspace{0.5em}$mW. The two diamond symbols refer to the pair of lemons and the color map is the same as in Fig. 5. (c) Local mapping of the major axis of the polarization ellipse on a $3\lambda \times 3\lambda$ area, where $\lambda =532\hspace{0.5em}$nm is the imaging wavelength. The diamond symbols correspond to those shown in (a).