Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy

Speeding up liquid crystal SLMs using overdrive with phase change reduction

Nematic liquid crystal spatial light modulators (SLMs) with fast switching times and high diffraction efficiency are important to various applications ranging from optical beam steering and adaptive optics to optical tweezers. Here we demonstrate the great benefits that can be derived in terms of speed enhancement without loss of diffraction efficiency from two mutually compatible approaches. The first technique involves the idea of overdrive, that is the calculation of intermediate patterns to speed up the transition to the target phase pattern. The second concerns optimization of the target pattern to reduce the required phase change applied to each pixel, which in addition leads to a substantial reduction of variations in the intensity of the diffracted light during the transition. When these methods are applied together, we observe transition times for the diffracted light fields of about 1 ms, which represents up to a tenfold improvement over current approaches. We experimentally demonstrate the improvements of the approach for applications such as holographic image projection, beam steering and switching, and real-time control loops.

Three-dimensional subpixel estimation in holographic position measurement of an optically trapped nanoparticle

We propose three-dimensional (3D) subpixel estimation in the position measurement of a nanoparticle held in optical tweezers in water by using an in-line, low-coherence digital holographic microscope. The 3D subpixel estimation was performed with the addition of axial subpixel estimation to the lateral subpixel estimation introduced in our previous work [Appl. Opt.50, H183 (2011)]. The axial subpixel estimation allowed the step length in the diffraction calculation of a hologram to be increased to ~20 nm while keeping the axial resolution of ~3 nm. This drastically decreased the computation time of the diffraction calculation to less than 10% of the two-dimensional subpixel estimation.

Independent and simultaneous three-dimensional optical trapping and imaging

Combining imaging and control of multiple micron-scaled objects in three dimensions opens up new experimental possibilities such as the fabrication of colloidal-based photonic devices, as well as high-throughput studies of single cell dynamics. Here we utilize the dual-objectives approach to combine 3D holographic optical tweezers with a spinning-disk confocal microscope. Our setup is capable of trapping multiple different objects in three dimensions with lateral and axial accuracy of 8 nm and 20 nm, and precision of 20 nm and 200 nm respectively, while imaging them in four different fluorescence channels. We demonstrate fabrication of ordered two-component and three dimensional colloidal arrays, as well as trapping of yeast cell arrays. We study the kinetics of the division of yeast cells within optical traps, and find that the timescale for division is not affected by trapping.

An optically actuated surface scanning probe

We demonstrate the use of an extended, optically trapped probe that is capable of imaging surface topography with nanometre precision, whilst applying ultra-low, femto-Newton sized forces. This degree of precision and sensitivity is acquired through three distinct strategies. First, the probe itself is shaped in such a way as to soften the trap along the sensing axis and stiffen it in transverse directions. Next, these characteristics are enhanced by selectively position clamping independent motions of the probe. Finally, force clamping is used to refine the surface contact response. Detailed analyses are presented for each of these mechanisms. To test our sensor, we scan it laterally over a calibration sample consisting of a series of graduated steps, and demonstrate a height resolution of ∼ 11 nm. Using equipartition theory, we estimate that an average force of only ∼ 140 fN is exerted on the sample during the scan, making this technique ideal for the investigation of delicate biological samples.

Video-based and interference-free axial force detection and analysis for optical tweezers

For measuring the minute forces exerted on single molecules during controlled translocation through nanopores with sub-piconewton precision, we have developed a video-based axial force detection and analysis system for optical tweezers. Since our detection system is equipped with a standard and versatile CCD video camera with a limited bandwidth offering operation at moderate light illumination with minimal sample heating, we integrated Allan variance analysis for trap stiffness calibration. Upon manipulating a microbead in the vicinity of a weakly reflecting surface with simultaneous axial force detection, interference effects have to be considered and minimized. We measured and analyzed the backscattering light properties of polystyrene and silica microbeads with different diameters and propose distinct and optimized experimental configurations (microbead material and diameter) for minimal light backscattering and virtually interference-free microbead position detection. As a proof of principle, we investigated the nanopore threading forces of a single dsDNA strand attached to a microbead with an overall force resolution of ±0.5 pN at a sample rate of 123 Hz.

Partial synchronization of stochastic oscillators through hydrodynamic coupling

Holographic optical tweezers are used to construct a static bistable optical potential energy landscape where a brownian particle experiences restoring forces from two nearby optical traps and undergoes thermally activated transitions between the two energy minima. Hydrodynamic coupling between two such systems results in their partial synchronization. This is interpreted as an emergence of higher mobility pathways, along which it is easier to overcome barriers to structural rearrangement.

Optical shield: measuring viscosity of turbid fluids using optical tweezers

The viscosity of a fluid can be measured by tracking the motion of a suspended micron-sized particle trapped by optical tweezers. However, when the particle density is high, additional particles entering the trap compromise the tracking procedure and degrade the accuracy of the measurement. In this work we introduce an additional Laguerre-Gaussian, i.e. annular, beam surrounding the trap, acting as an optical shield to exclude contaminating particles.

Power spectrum and Allan variance methods for calibrating single-molecule video-tracking instruments

Single-molecule manipulation instruments, such as optical traps and magnetic tweezers, frequently use video tracking to measure the position of a force-generating probe. The instruments are calibrated by comparing the measured probe motion to a model of Brownian motion in a harmonic potential well; the results of calibration are estimates of the probe drag, α, and spring constant, κ. Here, we present both time- and frequency-domain methods to accurately and precisely extract α and κ from the probe trajectory. In the frequency domain, we discuss methods to estimate the power spectral density (PSD) from data (including windowing and blocking), and we derive an analytical formula for the PSD which accounts both for aliasing and the filtering intrinsic to video tracking. In the time domain, we focus on the Allan variance (AV): we present a theoretical equation for the AV relevant to typical single-molecule setups and discuss the optimal manner for computing the AV from experimental data using octave-sampled overlapping bins. We show that, when using maximum-likelihood methods to fit to the data, both the PSD and AV approaches can extract α and κ in an unbiased and low-error manner, though the AV approach is simpler and more robust.

Phase-transition-like properties of double-beam optical tweezers

We report on double-beam optical tweezers that undergo previously unknown phase-transition-like behavior resulting in the formation of more optical traps than the number of beams used to create them. We classify the optical force fields which produce multiple traps for a double-beam system including the critical behavior. This effect is demonstrated experimentally in orthogonally polarized (noninterfering) dual-beam optical tweezers for a silica particle of 2.32  μm diameter. Phase transitions of multiple beam trapping systems have implications for hopping rates between traps and detection of forces between biomolecules using dual-beam optical tweezers. It is an example of a novel dynamic system with multiple states where force fields undergo a series of sign inversions as a function of parameters such as size and beam separation.

Three-dimensional positioning of optically trapped nanoparticles

We firstly demonstrate the three-dimensional (3D) measurement of a nanometer-sized sphere held in optical tweezers in water using an in-line digital holographic microscope with a green light emitting diode. Suppressing the movement with optical tweezers enabled us to detect the three-dimensional position of a polystyrene sphere with a diameter of 200 nm. The positioning resolutions of the microscope were 3.2 nm in the transverse direction and 3.4 nm in the axial direction, from the standard deviation of measurements of the 200 nm sphere fixed on glass. Changes in the Brownian motion in response to a change in the trapping laser power were measured. We also demonstrated that this holographic measurement is an effective method for determining the threshold power of the optical trapping.

Holographic aberration correction: optimising the stiffness of an optical trap deep in the sample

We investigate the effects of 1(st) order spherical aberration and defocus upon the stiffness of an optical trap tens of μm into the sample. We control both these aberrations with a spatial light modulator. The key to maintain optimum trap stiffness over a range of depths is a specific non-trivial combination of defocus and axial objective position. This optimisation increases the trap stiffness by up to a factor of 3 and allows trapping of 1 μm polystyrene beads up to 50 μm deep in the sample.

Surface imaging using holographic optical tweezers

We present an imaging technique using an optically trapped cigar-shaped probe controlled using holographic optical tweezers. The probe is raster scanned over a surface, allowing an image to be taken in a manner analogous to scanning probe microscopy (SPM), with automatic closed loop feedback control provided by analysis of the probe position recorded using a high speed CMOS camera. The probe is held using two optical traps centred at least 10 µm from the ends, minimizing laser illumination of the tip, so reducing the chance of optical damage to delicate samples. The technique imparts less force on samples than contact SPM techniques, and allows highly curved and strongly scattering samples to be imaged, which present difficulties for imaging using photonic force microscopy. To calibrate our technique, we first image a known sample--the interface between two 8 µm polystyrene beads. We then demonstrate the advantages of this technique by imaging the surface of the soft alga Pseudopediastrum. The scattering force of our laser applied directly onto this sample is enough to remove it from the surface, but we can use our technique to image the algal surface with minimal disruption while it is alive, not adhered and in physiological conditions. The resolution is currently equivalent to confocal microscopy, but as our technique is not diffraction limited, there is scope for significant improvement by reducing the tip diameter and limiting the thermal motion of the probe.

Holographic optical tweezers and their relevance to lab on chip devices

During the last decade, optical tweezers have been transformed by the combined availability of spatial light modulators and the speed of low-cost computing to drive them. Holographic optical tweezers can trap and move many objects simultaneously and their compatibility with other optical techniques, particularly microscopy, means that they are highly appropriate to lab-on-chip systems to enable optical manipulation, actuation and sensing.

Measuring the complete force field of an optical trap

The use of optical traps to measure or apply forces on the molecular level requires a precise knowledge of the trapping force field. Close to the trap center, this field is typically approximated as linear in the displacement of the trapped microsphere. However, applications demanding high forces at low laser intensities can probe the light-microsphere interaction beyond the linear regime. Here, we measured the full nonlinear force and displacement response of an optical trap in two dimensions using a dual-beam optical trap setup with back-focal-plane photodetection. We observed a substantial stiffening of the trap beyond the linear regime that depends on microsphere size, in agreement with Mie theory calculations. Surprisingly, we found that the linear detection range for forces exceeds the one for displacement by far. Our approach allows for a complete calibration of an optical trap.

Multidepth, multiparticle tracking for active microrheology using a smart camera

The quantitative measurement of particle motion in optical tweezers is an important tool in the study of microrheology and can be used in a variety of scientific and industrial applications. Active microheology, in which the response of optically trapped particles to external driving forces is measured, is particularly useful in probing nonlinear viscoelastic behavior in complex fluids. Currently such experiments typically require independent measurements of the driving force and the trapped particle's response to be carefully synchronized, and therefore the experiments normally require analog equipment. In this paper we describe both a specialized camera and an imaging technique which make high-speed video microscopy a suitable tool for performing such measurements, without the need for separate measurement systems and synchronization. The use of a high-speed tracking camera based on a field programmable gate array to simultaneously track multiple particles is reported. By using this camera to simultaneously track one microsphere fixed to the wall of a driven sample chamber and another held in an optical trap, we demonstrate simultaneous optical measurement of the driving motion and the trapped probe particle response using a single instrument. Our technique is verified experimentally by active viscosity measurements on water-ethylene glycol mixtures using a phase-shift technique.

Defining the limits of single-molecule FRET resolution in TIRF microscopy

Single-molecule FRET (smFRET) has long been used as a molecular ruler for the study of biology on the nanoscale (∼2-10 nm); smFRET in total-internal reflection fluorescence (TIRF) Förster resonance energy transfer (TIRF-FRET) microscopy allows multiple biomolecules to be simultaneously studied with high temporal and spatial resolution. To operate at the limits of resolution of the technique, it is essential to investigate and rigorously quantify the major sources of noise and error; we used theoretical predictions, simulations, advanced image analysis, and detailed characterization of DNA standards to quantify the limits of TIRF-FRET resolution. We present a theoretical description of the major sources of noise, which was in excellent agreement with results for short-timescale smFRET measurements (<200 ms) on individual molecules (as opposed to measurements on an ensemble of single molecules). For longer timescales (>200 ms) on individual molecules, and for FRET distributions obtained from an ensemble of single molecules, we observed significant broadening beyond theoretical predictions; we investigated the causes of this broadening. For measurements on individual molecules, analysis of the experimental noise allows us to predict a maximum resolution of a FRET change of 0.08 with 20-ms temporal resolution, sufficient to directly resolve distance differences equivalent to one DNA basepair separation (0.34 nm). For measurements on ensembles of single molecules, we demonstrate resolution of distance differences of one basepair with 1000-ms temporal resolution, and differences of two basepairs with 80-ms temporal resolution. Our work paves the way for ultra-high-resolution TIRF-FRET studies on many biomolecules, including DNA processing machinery (DNA and RNA polymerases, helicases, etc.), the mechanisms of which are often characterized by distance changes on the scale of one DNA basepair.

Real-time particle tracking at 10,000 fps using optical fiber illumination

We introduce optical fiber illumination for real-time tracking of optically trapped micrometer-sized particles with microsecond time resolution. Our light source is a high-radiance mercury arc lamp and a 600 μm optical fiber for short-distance illumination of the sample cell. Particle tracking is carried out with a software implemented cross-correlation algorithm following image acquisition from a CMOS camera. Our image data reveals that fiber illumination results in a signal-to-noise ratio usually one order of magnitude higher compared to standard Köhler illumination. We demonstrate position determination of a single optically trapped colloid with up to 10,000 frames per second over hours. We calibrate our optical tweezers and compare the results with quadrant photo diode measurements. Finally, we determine the positional accuracy of our setup to 2 nm by calculating the Allan variance. Our results show that neither illumination nor software algorithms limit the speed of real-time particle tracking with CMOS technology.

Particle tracking stereomicroscopy in optical tweezers: control of trap shape

We present an optical system capable of generating stereoscopic images to track trapped particles in three dimensions. Two-dimensional particle tracking on each image yields three dimensional position information. Our approach allows the use of a high numerical aperture (NA=1.3) objective and large separation angle, such that particles can be tracked axially with resolution of 3 nm at 340 Hz. Spatial Light Modulators (SLMs), the diffractive elements used to steer and split laser beams in Holographic Optical Tweezers, are also capable of more general operations. We use one here to vary the ratio of lateral to axial trap stiffness by changing the shape of the beam at the back aperture of the microscope objective. Beams which concentrate their optical power at the extremes of the back aperture give rise to much more efficient axial trapping. The flexibility of using an SLM allows us to create multiple traps with different shapes.

Calibration of optically trapped nanotools

Holographically trapped nanotools can be used in a novel form of force microscopy. By measuring the displacement of the tool in the optical traps, the contact force experienced by the probe can be inferred. In the following paper we experimentally demonstrate the calibration of such a device and show that its behaviour is independent of small changes in the relative position of the optical traps. Furthermore, we explore more general aspects of the thermal motion of the tool.

Real time characterization of hydrodynamics in optically trapped networks of micro-particles

The hydrodynamic interactions of micro-silica spheres trapped in a variety of networks using holographic optical tweezers are measured and characterized in terms of their predicted eigenmodes. The characteristic eigenmodes of the networks are distinguishable within 20-40 seconds of acquisition time. Three different multi-particle networks are considered; an eight-particle linear chain, a nine-particle square grid and, finally, an eight-particle ring. The eigenmodes and their decay rates are shown to behave as predicted by the Oseen tensor and the Langevin equation, respectively. Finally, we demonstrate the potential of using our micro-ring as a non-invasive sensor to the local environmental viscosity, by showing the distortion of the eigenmode spectrum due to the proximity of a planar boundary.

Superresolution imaging in optical tweezers using high-speed cameras

High-speed cameras are reliable alternatives for the direct characterization of optical trap force and particle motion in optical tweezers setups, replacing indirect motion measurements often performed by quadrant detectors. In the present approach, subpixel motion data of the trapped particle is retrieved from a high-speed low-resolution video sequence. Due to the richness structure of motion diversity of microscopic trapped particles, which are subjected to a Brownian motion, we propose to also use the obtained motion information for tackling the inherent lack of resolution by applying superresolution algorithms on the low-resolution image sequence. The obtained results both for trapping calibration beads and for living bacteria show that the proposed approach allows the proper characterization of the optical tweezers by obtaining the real particle motion directly from the image domain, while still providing high resolution imaging.

Measuring storage and loss moduli using optical tweezers: broadband microrheology

We present an experimental procedure to perform broadband microrheological measurements with optical tweezers. A generalized Langevin equation is adopted to relate the time-dependent trajectory of a particle in an imposed flow to the frequency-dependent moduli of the complex fluid. This procedure allows us to measure the material linear viscoelastic properties across the widest frequency range achievable with optical tweezers.

Dynamic position and force measurement for multiple optically trapped particles using a high-speed active pixel sensor

A high frame rate active pixel sensor designed to track the position of up to six optically trapped objects simultaneously within the field of view of a microscope is described. The sensor comprises 520 x 520 pixels from which a flexible arrangement of six independent regions of interest is accessed at a rate of up to 20 kHz, providing the capability to measure motion in multiple micron scale objects to nanometer accuracy. The combined control of both the sensor and optical traps is performed using unique, dedicated electronics (a field programmable gate array). The ability of the sensor to measure the dynamic position and the forces between six optically trapped spheres, down to femtonewton level, is demonstrated paving the way for application in the physical and life sciences.

Thermal motion of a holographically trapped SPM-like probe

By holding a complex object in multiple optical traps, it may be harmonically bound with respect to both its position and its orientation. In this way a small probe, or nanotool, can be manipulated in three dimensions and used to measure and apply directed forces, in the manner of a scanning probe microscope. In this paper we evaluate the thermal motion of such a probe held in holographic optical tweezers, by solving the Langevin equation for the general case of a set of spherical vertices linked by cylindrical rods. The concept of a corner frequency, familiar from the case of an optically trapped sphere, is appropriately extended to represent a set of characteristic frequencies given by the eigenvalues of the product of the stiffness matrix and the inverse hydrodynamic resistance matrix of the tool. These eigenvalues may alternatively be interpreted as inverses of a set of characteristic relaxation times for the system. The approach is illustrated by reference to a hypothetical tool consisting of a triangular arrangement of spheres with a lateral probe. The characteristic frequencies and theoretical resolution of the device are derived; variations of these quantities with tool size and orientation and with the optical power distribution, are also considered.

Optical tweezers with millikelvin precision of temperature-controlled objectives and base-pair resolution

In optical tweezers, thermal drift is detrimental for high-resolution measurements. In particular, absorption of the trapping laser light by the microscope objective that focuses the beam leads to heating of the objective and subsequent drift. This entails long equilibration times which may limit sensitive biophysical assays. Here, we introduce an objective temperature feedback system for minimizing thermal drift. We measured that the infrared laser heated the objective by 0.7 K per watt of laser power and that the laser focus moved relative to the sample by approximately 1 nm/mK due to thermal expansion of the objective. The feedback stabilized the temperature of the trapping objective with millikelvin precision. This enhanced the long-term temperature stability and significantly reduced the settling time of the instrument to about 100 s after a temperature disturbance while preserving single DNA base-pair resolution of surface-coupled assays. Minimizing systematic temperature changes of the objective and concurrent drift is of interest for other high-resolution microscopy techniques. Furthermore, temperature control is often a desirable parameter in biophysical experiments.

Microrheology with optical tweezers

Microrheology is the study of the flow of materials over small scales. It is of particular interest to those involved with investigations of fluid properties within Lab-on-a-Chip structures or within other micron-scale environments. The article briefly reviews existing active and passive methods used in the study of fluids. It then explores in greater detail the use of optical tweezers as an emerging method to investigate rheological phenomena, including, for example, viscosity and viscoelasticity, as well as the related topic of flow. The article also describes, briefly, potential future applications of this topic, in the fields of biological measurement, in general, and Lab-on-a-Chip, in particular.

Multipoint viscosity measurements in microfluidic channels using optical tweezers

We demonstrate the technique of multipoint viscosity measurements incorporating the accurate calibration of micron sized particles. We describe the use of a high-speed camera to measure the residual motion of particles trapped in holographic optical tweezers, enabling us to calculate the fluid viscosity at multiple points across the field-of-view of the microscope within a microfluidic system.

Quantifying noise in optical tweezers by allan variance

Much effort is put into minimizing noise in optical tweezers experiments because noise and drift can mask fundamental behaviours of, e.g., single molecule assays. Various initiatives have been taken to reduce or eliminate noise but it has been difficult to quantify their effect. We propose to use Allan variance as a simple and efficient method to quantify noise in optical tweezers setups.We apply the method to determine the optimal measurement time, frequency, and detection scheme, and quantify the effect of acoustic noise in the lab. The method can also be used on-the-fly for determining optimal parameters of running experiments.

Touching the microworld with force-feedback optical tweezers

Optical tweezers are a powerful tool for micromanipulation and measurement of picoNewton sized forces. However, conventional interfaces present difficulties as the user cannot feel the forces involved. We present an interface to optical tweezers, based around a low-cost commercial force feedback device. The different dynamics of the micro-world make intuitive force feedback a challenge. We propose a coupling method using an existing optical tweezers system and discuss stability and transparency. Our system allows the user to perceive real Brownian motion and viscosity, as well as forces exerted during manipulation of objects by a trapped bead.

Interferometric 3D tracking of several particles in a scanning laser focus

High-Speed tracking of several particles allows measuring dynamic long-range interactions relevant to biotechnology and colloidal physics. In this paper we extend the successful technique of 3D back-focal plane interferometry to oscillating laser beams and show that two or more particles can be trapped and tracked with a precision of a few nanometers in all three dimensions. The tracking rate of several kHz is only limited by the scan speed of the beam steering device. Several tests proof the linearity and orthogonality of our detection scheme, which is of interest to optical tweezing applications and various metrologies. As an example we show the position cross-correlations of three diffusing particles in a scanning line optical trap.