Tutorials - Optical Tweezers

Back focal plane interferometry

Multiple detection techniques have been developed that allow calibration and quantification. In general, one can distinguish between imaging and interferometric detection. In the first method, the trapped particle is directly imaged onto a detector, monitoring the particle's movement. For simple imaging, a camera mounted onto a port of the microscope is used. The lateral position resolution can be as high as 5 nm, owing to centroid-finding algorithms, but due to the video acquisition frame rates, the temporal resolution is rather limited, seldom exceeding 100 Hz. This limitation can be overcome by use of quadrant photodiodes (QPD), increasing both the bandwidth and resolution.

Back focal plane interferometry is the fastest and most sensitive technique for measuring the exact position of a trapped particle with respect to the trap center. Consequently, it has become the standard in the field of force-sensing optical tweezers. The laser light scattered from the trapped particle can be used to yield information on the accurate position of the particle inside the trap, as well as the external force exerted on it. By imaging the far-field interference of the illuminating laser light with the scattered light from the trapped particle, shifts of the object from the centre of the trap can be monitored. For the interferometric detection, either the light from the trapping laser itself or from a second laser can be used. The latter approach yields a somewhat bigger linear range of detection owing to the use of a less collimated beam, but is more complex and less practicable to maintain.

Calibration and detector response

There are several techniques to calibrate the detector response from volts into displacement and/or force. When recording of data is only possible at low bandwidth - as with video detection - the displacement of a micrometer-sized bead from the trap center by mechanical manipulation is the method of choice. This can either be performed by moving a particle attached to the surface relative to the trap, or by imposing a viscous drag onto a trapped particle. The first method is only capable of calibrating the displacement, while the second also includes force. When more bandwidth for data recording is available by using photodiodes, the method of choice for an accurate calibration is to make use of the thermal motion of a particle in a trap.
The diffusive thermal motion of a particle will be distorted by the influence of the external restoring optical force, confining its motion to within the laser focus. The power spectral density (PSD) of the displacement fluctuations of a trapped particle has Lorentzian shape:

Here, ƒ_c ≡ κ/2πγ is the characteristic corner frequency of the trap, at which the power spectrum amplitude changes from a constant value to falling as 1/ƒ². With the knowledge of the viscosity γ of the medium, the stiffness of the trap κ can be derived from the corner frequency, yielding the information for the conversion from displacement to force.

There is a finite range in which the calibration of the detector response is linear. With a maximally focused beam, this range spans about ±150 nm in the lateral and ±250 nm in the axial directions. Beyond this range, the force associated with the displacement slowly converges towards its maximal value, the escape force of the particle.

Bead size, stiffness and escape force

Both the stiffness of the trap and the escape force (maximal force applicable to the trapped bead) depend on several parameters, including the particle size and the laser power. This makes the particle size one of the most important parameters to choose at the beginning of an experiment. If a particularly high stiffness and escape force are required, then the particle should be near the optimal size for the wavelength of the trapping laser.

The optimal bead diameter for a high trap stiffness for 1064 nm infrared light is 800 nm, i.e. the wavelength of this laser in water.
The absolute strength of the trap depends on the refractive index mismatch between the bead material and the surrounding liquid. Accordingly, polystyrene particles (n = 1.59) experience higher trapping forces than silica particles (n = 1.43-1.46). For pulling experiments with a maximal escape force, the particle is to be chosen as big as possible, being in the micrometer range (2-5 µm). It is also possible to trap particles much smaller than the optimal size (below 50 nm), which is particularly interesting for tracking experiments where high forces are not required.

A tunable laser power offers the possibility to vary the stiffness of the trap and the escape force of the bead during the experiment. This renders experiments flexible as regarding the force range exerted onto a particle. It can be easily switched between two experiment modes: a manipulating trap with a high stiffness and a "soft" trap exerting little influence on the trapped particle, allowing tracking of unhindered motion.