Power spectrum analysis for optical tweezers

Abstract

The force exerted by an optical trap on a dielectric bead in a fluid is often found by fitting a Lorentzian to the power spectrum of Brownian motion of the bead in the trap. We present explicit functions of the experimental power spectrum that give the values of the parameters fitted, including error bars and correlations, for the best such 2 fit in a given frequency range. We use these functions to determine the information content of various parts of the power spectrum, and find, at odds with lore, much information at relatively high frequencies. Applying the method to real data, we obtain perfect fits and calibrate tweezers with less than 1% error when the trapping force is not too strong. Relatively strong traps have power spectra that cannot be fitted properly with any Lorentzian, we find. This underscores the need for better understanding of the power spectrum than the Lorentzian provides. This is achieved using old and new theory for Brownian motion in an incompressible fluid, and new results for a popular photodetection system. The trap and photodetection system are then calibrated simultaneously in a manner that makes optical tweezers a tool of precision for force spectroscopy, local viscometry, and probably other applications.

Design strategies for optimizing holographic optical tweezers set-ups

“…New applications are being proposed in many fields ranging from microfluidics [10,11] to nanotechnology [12] and biophysics [13]. However, contrary to singlebeam technology, which has been thoroughly documented in its many facets [14][15][16][17][18], holographic optical tweezers systems remain comparatively poorly described. The inclusion of the spatial light modulator in the optical setup has important design implications that are specific to this technology.…”

Brownian Motion after Einstein: Some New Applications and New Experiments

“…It differs from Einstein's theory in a manner that matters in practise with the precision that optical tweezers have achieved recently [21,22]; see Fig. 1.…”

“…1. The power spectrum of Brownian motion in an optical trap according to Einstein's theory, PLorentz, divided by the hydrodynamically correct power spectrum for the same motion, P Hydro ; see [21] for explicit expressions for the two spectra. Fully drawn line: Trap with Hooke's constant 3.8·10 −2 pN/nm for a micro-sphere with diameter 1 µm.…”

“…Instead of measuring a photon correlation function for laser light scattered off a suspension of micro-spheres, developments in instrumentation [22] and data analysis [21] for optical tweezers have made it possible now to measure directly, with accuracy and precision, on a single micro-sphere [24]. Thus it just might be possible to observe directly the "color" of the thermal noise, the frequency dependence of the non-white power spectrum, in a very challenging single-particle experiment with optical tweezers [30].…”

“…This method resembles an old method of calibration that moves the bead back and forth periodically with constant speed, but harmonic motion has a number of technical advantages. One is that the precision of power spectral analysis demonstrated in [21] can be maintained, while adding the advantage of not having to know the bead's radius, nor its distance to a nearby surface, nor the fluid's viscosity and temperature. On the contrary, the combination of these parameters that occurs in the expression (3) for the bead's diffusion coefficient, is determined experimentally from its Brownian motion, so, e.g., the bead's radius is measured to the extent the other parameter values are known.…”