Three-dimensional force calibration of optical tweezers

Abstract

We demonstrate a method for 3-dimensional force calibration of optical tweezers by recording the trapping dynamics of polystyrene beads. This is realized by time-resolved detection of the horizontal and vertical position of a bead which is drawn to the focus of a laser beam. The method provides real time characterization of the force profile of an optical trap in all directions.     

Optical trap detector for calibration of optical fiber powermeters: coupling efficiency

The optical trap detector is based on two, 1 cm x 1 cm silicon photodiodes and a spherical mirror contained in a package that is highly efficient for measuring light diverging from the end of an optical fiber. The mathematical derivation of the coupling efficiency relies on the integral directional response weighted by the angular intensity distribution of an idealized parabolic optical beam. Results of directional-uniformity measurements, acquired with the aid of a six-axis industrial robotic arm, indicate that the trap has a collection efficiency greater than 99.9% for a fiber numerical aperture of 0.24. Spatial uniformity measurements indicate that the variation of detector response as a function of position is less than 0.1%. The detector's absolute responsivity at 672.3, 851.7, and 986.1 nm is also documented by comparison with other optical detectors and various input conditions and indicates that the design is well suited for laser and optical fiber power measurements.     

Probing fluid flow using the force measurement capability of optical trapping

Interest in microfluidics is rapidly expanding and the use of microchips as miniature chemical reactors is increasingly common. Microfluidic channels are now complex and combine several functions on a single chip. Fluid flow details are important but relatively few experimental methods are available to probe the flow in confined geometry. We use optical trapping of a small dielectric particle to probe the fluid flow. A highly focused laser beam attracts particles suspended in a liquid to its focal point. A particle can be trapped and then repositioned. From the displacement of the trapped particle away from its equilibrium position one estimates the external force acting on the particle. The stiffness (spring constant) of the optical trap is low thus making it a sensitive force measuring device. Rather than using the optical trap to position and release a particle for independent velocimetry measurement, we map the fluid flow by measuring the hydrodynamic force acting on a trapped particle. The flow rate of a dilute aqueous electrolyte flowing through a plastic microchannel (W × H × L = 5 mm × 0.4 mm × 50 mm) was mapped using a small silica particle (1 μm diameter). The fluid velocity profile obtained experimentally is in very good agreement with the theoretical prediction. Our flow mapping approach is time efficient, reliable and can be used in low-opacity suspensions flowing in microchannels of various geometries.     

Detector effects in photothermal deflection experiments

We report on the theoretical analysis of a detector type influence on the normal deflection signal in photothermal experiments. Two cases are examined. In the first, the quadrant photodiode was considered as the detector; in the second the signal from the position detector, which measures the central moment displacement of the probe beam, was analyzed. Both analyses were carried out within the framework of the complex ray theory. The normal photodeflection signal was found to depend on the type of detector used in the photothermal deflection experiments for some parameters of its setup.     

Measurement of localized heating in the focus of an optical trap

Localized heating in the focus of an optical trap operating in water can result in a temperature rise of several kelvins. We present spatially resolved measurements of the refractive-index distribution induced by the localized heating produced in an optical trap and infer the temperature distribution. We have determined a peak temperature rise in water of 4 K in the focus of a 985-nm-wavelength 55-mW laser beam. The localized heating is directly proportional to power and the absorption coefficient. The temperature distribution is in excellent agreement with a model based on the heat equation.     

Dose Imaging Detectors for Radiotherapy Based on Gas Electron Multipliers

New techniques in charged particle therapy and widespread use of modern dynamic beam delivery systems demand new beam monitoring devices as well as accurate 2D dosimetry systems to verify the delivered dose distribution. We are developing dose imaging detectors based on gas electron multipliers (GEM) with the goal of improving dose measurement linearity, position and timing resolution, and to ultimately allow pre-treatment verification of dose distributions and dose delivery monitoring employing scanning beam technology. A prototype 10×10 cm(2) double-GEM detector has been tested in the 205 MeV proton beam using electronic and optical readout modes. Preliminary results with electronic cross-strip readout demonstrate fast response and single-pixel (4 mm) position resolution. In optical readout mode, the line spread function of the detector was found to have σ=0.7 mm. In both readout modes, the detector response was linear up to dose rates of 50 Gy/min, with adequate representation of the Bragg peak in depth-dose profile measurements.     

Iterative full-bandwidth wavefront reconstruction from a set of low-tilt Fizeau interferograms for high-numerical-aperture surface characterization

Full-bandwidth phase-shifting methods as well as band-limited fringe carrier techniques are both problematic when testing high-NA spherical surfaces in Fizeau interferometers. Phase stepping is usually performed by moving a sample and reference sphere relative to each other along the optical axis. At a high NA the method suffers from phase-shift inhomogeneity across the sample surface. Fringe carrier techniques rely on a minimum fringe frequency and call for an off-axis position of the sample, which in turn introduces condenser aberrations. Distortion of the imaging optics generates further apparent aberrations. We propose to combine both principles. The phase shifts are replaced by a set of very low tilts such that the sample is virtually on axis. Initial wavefront estimates are generated by a fringe carrier method. An adaptive Misell-type algorithm combines the interferometric data and iteratively improves the reconstructed wavefront until full spatial bandwidth is achieved.     

Optical absorption measurements with parametric down-converted photons

It has been known for some time that correlated detection of pairs of photons generated by parametric down-conversion can eliminate several sources of error that occur in single-beam measurements. In the correlated photon measurements, the down-converted photons are separated into two beams with one photon of a pair in each beam. The absolute detection efficiency of a detector in one beam can be determined from the count rate of a detector in the other beam and the coincidence rate for the two detectors. These ideas can be used to measure the optical absorbance of a sample placed in front of one of the detectors. Errors due to stray light and dark counts are substantially reduced and fluctuations in pump intensity largely eliminated.     

Trapping forces in a multiple-beam fiber-optic trap

Transverse and axial trapping forces are calculated in the ray optics regime for a multiple-beam fiber-optic light-force trap for dielectric microspheres located both on and off axis relative to the beam axis. Trap efficiencies are evaluated as functions of the effective index of refraction of the microspheres, normalized sphere radius, and normalized beam waist separation distance. Effects of the linear polarization of the electric field and of beam focusing through microlenses are considered. In the case of a counterpropagating two-beam fiber-optic trap, using microlenses at the distal ends of the fiber to focus the beams may somewhat increase the trapping volume and the axial stability if the fiber spacing is sufficiently large but will greatly reduce the stiffness of the transverse force. Trapping forces produced in a counterpropagating two-beam fiber-optic trap are compared with those generated in multiple-beam fiber-optic gradient-force traps. Multiple-beam fiber-optic traps use strong gradient forces to trap a particle; therefore they stabilize the particles much more firmly than do counterpropagating two-beam traps.     

Measuring microscopic viscosity with optical tweezers as a confocal probe

We demonstrate, what is to the best our knowledge, a new method for studying the motion of a particle trapped by optical tweezers; in this method the trapping beam itself is used as a confocal probe. By studying the response of the particle to periodic motion of the tweezers, we obtain information about the medium viscosity, particle properties, and trap stiffness. We develop the mathematical model, demonstrate experimentally its validity for our system, and discuss advantages of using this method as a new form of scanning photonic force microscopy for applications in which a high spatial and temporal resolution of the medium viscosity is desired.     

高精度抛物面镜光轴偏角分析仪

针对大数值孔径旋转抛物面反射镜对倾斜入射光线较大的散焦和离焦问题,本文设计一种用于大数值孔径旋转抛物面反射镜光轴偏角测量的系统,用于对大数值孔径旋转抛物面反射镜光轴与结构轴的校准.该方法根据旋转抛物面反射镜成像原理,采用CCD 作为成像器件,分析轴偏所带来的光斑偏移量.实验结果表明,系统达到偏角角度分辨率0.06 ′,最大可测角度3.65 ′,该参数可以满足抛物面反射镜用作接收光学系统的应用.     

Development of a diffraction-type optical triangulation sensor

We propose a diffraction-type optical triangulation sensor based on the diffraction theorem and the laser triangulation method. The advantage of the proposed sensor is that it obtains not only the linear displacement of a moving object but also its three angular motion errors. The developed sensor is composed mainly of a laser source, two quadrant detectors, and a reflective diffraction grating. The reflective diffraction grating can reflect the incident laser beam into several diffractive rays, and two quadrant detectors were set up for detecting the position of 0- and + 1-order diffraction rays. According to the optical triangulation relationship between the spatial incident angles of a laser beam and the output coordinates of two quadrant detectors, the displacement and the three angular motion errors of a moving object can be obtained simultaneously.     

Optimization of probe-laser focal offsets for single-particle tracking

In optical tweezers applications, tracking a trapped particle is essential for force measurement. One of the most popular techniques for single-particle tracking is achieved by analyzing the forward and backward light pattern, scattered by the target particle trapped by a trap laser beam, of an additional probe-laser beam with different wavelength whose focus is slightly apart from the trapping center. However, the optimized focal offset has never been discussed. In this paper, we investigate the tracking range and sensitivity as a function of the focal offset between the trapping and the probe-laser beams. As a result, the optimized focal offsets are a 3.3-fold radius ahead and a 2.0-fold radius behind the trapping laser focus in the forward tracking and the backward tracking, respectively. The experimental result agrees well with a theoretical prediction using the Mie scattering theory.     

Calibration of sound forces in acoustic traps

A two-dimensional or transverse acoustic trapping and its capability to noninvasively manipulate micrometersized particles with focused sound beams were experimentally demonstrated in our previous work. To apply this technique, as in optical tweezers, for studying mechanical properties of and interactions among biological particles such as cells, the trapping forces must be calibrated against known forces, i.e., viscous drag forces exerted by fluid flows. The trapping forces and the trap stiffness were measured under various conditions and the results were reported in this paper. In the current experimental arrangement, because the trapped particles were positioned against an acoustically transparent mylar membrane, the ultrasound beam intensity distribution near the membrane must be carefully considered. The total intensity field (the sum of incident and scattering intensity fields) around the droplet was thus computed by finite element analysis (FEA) with the membrane included, and it was then used in the ray acoustics model to calculate the trapping forces. The membrane effect on trapping forces was discussed by comparing effective beam widths with and without the membrane. The FEA results found that the broader beam width, caused by the scattered beams from the neighboring membrane and the droplet, resulted in the lower intensity, or smaller force, on the droplet. The experimental results showed that the measured forces were as high as 64 nN. The trap stiffness, approximated as a linear spring, was estimated by linear regressions and found to be 1.3 to 4.4 nN/μm, which was on a larger scale than that of optical trapping estimated for red blood cells, a few tenths of piconewtons/nanometer. The experimental and theoretical results were in good agreement.     

An overview and a contribution to the optical measurement of lineardisplacement

The present work is a contribution to the field of linear displacement measurements by optical means. For that purpose, a brief overview of some existing solutions is presented and two systems for axial linear displacement measurement based on light intensity detection are introduced. The systems have redundancy and were designed with the purpose of achieving identification and automatic correction of errors arising from inadvertent angular variations between the sensor and the light beam positions     

Improved axial position detection in optical tweezers measurements

We investigate the axial position detection of a trapped microsphere in an optical trap by using a quadrant photodiode. By replacing the photodiode with a CCD camera, we obtain detailed information on the light scattered by the microsphere. The correlation of the interference pattern with the axial position displays complex behavior with regions of positive and negative interference. By analyzing the scattered light intensity as a function of the axial position of the trapped sphere, we propose a simple method to increase the sensitivity and control the linear range of axial position detection.     

Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave

We study the axial force acting on dielectric spherical particles smaller than the trapping wavelength that are placed in the Gaussian standing wave. We derive analytical formulas for immersed particles with relative refractive indices close to unity and compare them with the numerical results obtained by generalized Lorenz-Mie theory (GLMT). We show that the axial optical force depends periodically on the particle size and that the equilibrium position of the particle alternates between the standing-wave antinodes and nodes. For certain particle sizes, gradient forces from the neighboring antinodes cancel each other and disable particle confinement. Using the GLMT we compare maximum axial trapping forces provided by the Gaussian standing-wave trap (SWT) and single-beam trap (SBT) as a function of particle size, refractive index, and beam waist size. We show that the SWT produces axial forces at least ten times stronger and permits particle confinement in a wider range of refractive indices and beam waists compared with those of the SBT.     

一种高灵敏度传感器：—四棱镜光点检测器

本文介绍一种用于激光准直高精度直线度测量的高灵敏度光电信号检测器,该检测器根据激光束腰处聚焦光斑点小的特点,采用一种四棱镜光点会聚式检测器来接收激光束信息,它的检测灵敏度比传统的四棱镜式四象限位敏传感器高出一个数量级以上;该检测器对激光束平行漂移与角漂移表现出不同的检出特征,因而尤适合于做为激光束漂移补偿检测和直线度误差检测;该检测器还具有结构简单、使用方便等优点。     

Semiconfocal optical disk readout with one linear-spread beam

A novel, to our knowledge, optics pickup for semiconfocal optical disk is proposed. Five tracks are imaged simultaneously by use of one linear-shaped beam that extends over the tracks in the radial direction with the focusing and the tracking position controlled. We also show how the cross talk between tracks is eliminated automatically. The approximately diffraction-limited width of the beam and the small-signal detector elements make semiconfocal detection possible in the tangential direction.     

Detection of nanoparticles using optical gradient forces

Abstract

We present a detection scheme for nanoscale particles based on the gradient force and torque near a tightly focused laser beam. The focus affects the path of nanoparticles passing by and a quadrant detector records the particle trajectory. A feedback system continuously adjusts the laser power and thereby prevents the particles from being trapped. Particle size and shape can be assessed by evaluating the time-trace of the quadrant detector signal.