Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture.

Back-focal-plane interferometry is a method capable of determining the three-dimensional position of a particle with high precision ($<3 \mathrm{nm}$) at high sampling rates (1 MHz). We investigated theoretically the performance of such a system for dielectric spheres with diameters $D=0.53–3 \mathit{\mu }\mathrm{m}$ and for metallic spheres with $D\le 300 \mathrm{nm}$. Good sensitivity and linearity were achieved for a detection angular aperture $sin\left(\alpha \right)$ of no more than 0.5. A value of $sin\left(\alpha \right)>0.7$ should be used only for dielectric spheres with diameters approximately equal to the laser wavelength. Harmonic optical traps can be calibrated by measurement of the thermal motion of the sphere. We performed Brownian dynamics simulations and subsequent thermal noise analyses to prove that the wrong $sin\left(\alpha \right)$ incorrectly suggests an increased and nonharmonic axial trapping potential.