Stiffness of Optical Traps: Quantitative Agreement between Experiment and Electromagnetic Theory

The first quantitative agreement between measured and calculated stiffnesses of optically trapped particles in the subwavelength regime is presented. It is shown for all three dimensions that the measured extent of harmonic optical trapping potentials for dielectric spheres comes very close to the theoretically predicted extent, provided all known instrumental parameters are considered. The recently predicted strong asymmetry of the trapping potential due to the electric field’s linear polarization has been verified in all three directions. This effect vanishes for spheres with diameters $d\approx \lambda$, which exhibit the strongest trap stiffnesses.

Figure 1

Asymmetric intensity distribution $I(x,y,z=0)$ in the focal plane for linear polarized light and $\mathrm{NA}=1.2$. The sum of the intensities of $|E_x{|}^2$ (left) and $|E_z{|}^2$ (center) is $I=I_x+I_z$ (right). The negligibly small component $I_y$ is not shown. The extents of the plots are $2\lambda$ by $2\lambda$.

Figure 2

Trap stiffnesses along $x$, $y$, and $z$ as a function of laser power in an $x$-polarized beam. The symbols correspond to experimental data, whereas the lines describe the theoretical force constants.

Figure 3

Change of the force constant per unit laser power for various polystyrene spheres ($\lambda =800\mathrm{nm}$ in water). White markers indicate theoretical values, black markers experimentally obtained stiffnesses, experimental error bars are all below 1%. The lines are drawn to guide the eye. Right: the trapping volumes of a $0.22\mu \mathrm{m}$ and $1.03\mu \mathrm{m}$ bead indicate the different ratios of axial to lateral stiffness.