Tailoring bistability in optical tweezers with vortex beams and spherical aberration (2024)

Authorization Required

×

×

• 

  Figure 1

  Illustration of the experimental setup. The laser beam goes through a polarizing beam splitter (PBS) and is directed to the spatial light modulator (SLM). The first order of diffraction propagates through a quarter-wave plate (QWP) and, for values of $\ell$ up to 5, through a beam expander (lenses L1 and L2) toward the microscope (dotted frame). Inside the microscope, the beam is reflected by a dichroic mirror (DM) and then focused by an oil-immersion objective. The inset shows a magnified view of the sample region with the glass slide lying at its bottom. The refractive index mismatch between the glass slide (index $n_i$) and the aqueous medium filling the sample (index $n_1$) leads to the introduction of a spherical aberration phase upon refraction at the interface between the slide and the sample. Such a phase is proportional to the height $L$ of the objective focal plane with respect to the slide, whose position is controlled by a piezoelectric nanopositioning stage (not shown). The resulting nonparaxial focused beam is indicated by the density plot of the electric energy density for $L=10\mu \mathrm{m}$ (red). The inset also depicts a trapped microsphere of refractive index $n_2$.

  Reuse & Permissions

• 

  Figure 2

  Transverse trap stiffness per unit power ${\kappa }_{\rho }/P$ as a function of the sphere radius for topological charges $\ell =1$ (blue), 5 (red), and 10 (green). We consider an aberration-free trapping beam. As the particle size increases, ${\kappa }_{\rho }$ changes sign from negative to positive at the critical radius $R_{\mathrm{on}}$.

  Reuse & Permissions

• 

  Figure 3

  For an aberration-free trapping beam, the parameter space spanned by the microsphere radius and the topological charge $\ell$ is divided into three different regions. For radii smaller than the critical radius $R_{\mathrm{on}}\left(\ell \right)$ (green), on-axis trapping is excluded, and the particle rotates around the optical axis along a circular trajectory. For radii larger than the critical radius $R_{\mathrm{off}}\left(\ell \right)$ (blue), orbital trapping is excluded, and the particle is trapped at a position along the optical axis. On-axis and orbital states coexist in the (colored) bistable stripe bounded by the plots of $R_{\mathrm{on}}\left(\ell \right)$ and $R_{\mathrm{off}}\left(\ell \right).$ Such a region corresponds to radii close to the maximum of electric energy density ${\tilde{r}}_{\ell }$ (circle). When an oil-immersion objective is employed, the spherical aberration introduced by refraction at the glass slide opens the way to switch between trapping regimes by changing the height $L$ of the objective focal plane. However, experimental bistability is possible only for radii and topological charges (red stars) close to or within the orange colored stripe.

  Reuse & Permissions

• 

  Figure 4

  Radial optical force component $Q_{\rho }$ as a function of radial position $\rho$ for $\ell =11$ and radii $a=1.5\mu \mathrm{m}$ (blue), $a=2.25\mu \mathrm{m}$ (red), and $a=3.5\mu \mathrm{m}$ (green). The fill plot indicates the electric energy density variation with distance to the optical axis. Its maximum is located at ${\tilde{r}}_{\ell }=2.1\mu \mathrm{m}$

  Reuse & Permissions

• 

  Figure 5

  Experimental realization of bistability with a polystyrene microsphere of radius $a=1.5\mu \mathrm{m}$ and a vortex beam with $\ell =8$. (a)–(d)show frames of the trapped microsphere at times $t_{\left(\mathrm{a}\right)}=85\mathrm{s},t_{\left(\mathrm{b}\right)}=90\mathrm{s},t_{\left(\mathrm{c}\right)}=95\mathrm{s}$, and $t_{\left(\mathrm{d}\right)}=100\mathrm{s}$. The outline (dash) of the trapped microsphere of diameter $a=3\mu \mathrm{m}$ is superimposed on each image to represent the scale. When the microsphere hops to an off-axis location (b)and (c), it circulates around the axis. The black arrows indicate the direction of motion in (b)and (c). The difference between the image patterns in (a) and (d) and in (c) and (d) indicates that the trapping height changes when hoping from axial equilibrium to orbital motion. (e) Time series for the radial coordinate. The vertical green lines indicate the times corresponding to the frames shown in (a)–(d). The horizontal dashed line represents the mean radial coordinate ${\rho }_{\exp }=(1.04\pm 0.05)\mu \mathrm{m}$ for the orbital state. (f) Color map of the energy distribution $U(x,y)/\left(k_{\mathrm{B}}T\right)$ across the $xy$ plane as derived from the position distribution density $p(x,y)$. (g) Power spectrum density (PSD) of the $x$ coordinate showing a peak at $f=1.82\mathrm{Hz}$, which corresponds to an orbital period $T_{\exp }=(0.55\pm 0.03)\mathrm{s}.$

  Reuse & Permissions

• 

  Figure 6

  Theoretical results for the variation of the optical force field with spherical aberration. The distance $L$ between the objective focal plane (for paraxial rays) and the glass slide is increased from left to right, thus enhancing the spherical aberration introduced by refraction at the interface between the slide and the sample. We consider a vortex beam with $\ell =8$ and left-handed circular polarization ($\sigma =+1$) at the objective entrance port and a polystyrene microsphere with radius $a=1.5\mu \mathrm{m}$. (a)From top to bottom, density plots representing the electric energy density $E^2$ and the axial and radial force components $Q_z$ and $Q_{\rho },$ respectively, on a meridional plane. The force components are normalized by Eq.(6) and their values are indicated by the color bar. The roots of $Q_z=0$ (blue line) and $Q_{\rho }=0$ (green line) as well as the positions of axial (brown dot) and orbital (magenta dot) stable equilibria are also indicated. Optical potential (in units of the thermal energy $k_{B}T$) vs radial distance $\rho$ for (b)$L=0\mu \mathrm{m}$, (c)$L=1.75\mu \mathrm{m}$, and (d)$L=3.5\mu \mathrm{m}$.

  Reuse & Permissions

• 

  Figure 7

  Radius ${\tilde{r}}_{\ell }$ of the focal annular spot (solid, left axis) and filling factor $A_{\ell }$ representing the fraction of total power transmitted to the sample chamber (dot, right axis) as functions of the radius $r_{\ell }$ of the paraxial vortex beam (in units of the objective entrance radius $R_p=2.8\mathrm{mm}$). The topological charges are $\ell =1,5,10,\text{and}15$. We consider an oil-immersion objective lens with numerical aperture $\mathrm{NA}=1.4$.

  Reuse & Permissions

×