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Optical Tweezers Optical Tweezers Experiment OT - sjh,rd University of Florida | Department of Physics PHY4803L | Advanced Physics Laboratory Objective K. Berg-Sorensen and H.Flyvbjerg, Power spectrum analysis for optical tweezers, An optical tweezers apparatus uses a tightly Rev. Sci. Instr. 75 594-612 (2004). focused laser to generate a trapping force that can capture and move small particles under S. Chattopadhyay, R. Moldovan, C. Yeung, a microscope. Because it can precisely and X.L. Wu, Swimming efficiency of bac- non-destructively manipulate objects such as terium Escherichiacoli, Proc. Nat. Acad. individual cells and their internal components, Sci. USA 103 13712-13717 (2006). the optical tweezers is extremely useful in bi- Daniel T. Gillespie The mathematics of ological physics research. In this experiment Brownian motion and Johnson noise, the student will use optical tweezers to trap Amer. J. of Phys. 64 225 (1996). small silica spheres immersed in water. By analyzing the dynamics of particles held in Daniel T. Gillespie Fluctuation and dissipa- the trap, including the frequency spectrum of tion in Brownian motion, Amer. J. of their Brownian motion and their response to Phys. 61 1077 (1993). hydrodynamic drag, the student will charac- terize the physical parameters of the optical S.F. Tolic-Norrelykke et al., Calibration of trap with high precision. The apparatus can optical tweezers with positional detection then be used to measure a microscopic bio- in the back focal plane, Rev. Sci. Instr. logical force, such as the force that propels a 77 103101 (2006). swimming bacterium or the force generated by a transport motor operating inside a plant cell. Introduction The key idea of optical trapping is that a laser References beam brought to a sharp focus generates a restoring force that can pull particles into that D.C. Appleyard et al., Optical trapping for focus. Arthur Ashkin demonstrated the prin- undergraduates, Am. J. Phys. 75 5-14 ciple in 1970 and reported on a working ap- (2007). paratus in 1986. The term optical trapping often refers to laser-based methods for hold- A. Ashkin, Acceleration and Trapping of Par- ing neutral atoms in high vacuum, while the ticles by Radiation Pressure, Phys. Rev. term optical tweezers (or laser tweezers) typ- Lett. 24 156-159 (1970). ically refers to the application studied in this OT - sjh,rd 1 OT - sjh,rd 2 Advanced Physics Laboratory experiment: A microscope is used to bring a low absorption rate and because the particle laser beam to a sharp focus inside an aqueous has good thermal conductance with the sur- sample, such that microscopic, non-absorbing rounding water, effects of heating should be particles such as small beads or individual cells modest. can become trapped at the beam focus. Op- The laser force arises almost entirely from tical tweezers have had a dramatic impact on the elastic scattering of laser photons by the the field of biological physics, as they allow particle whereby the particle alters the direc- experimenters to measure non-destructively tion of the photon momentum without absorb- and with high precision the tiny forces gen- ing any of its energy. The laser force is typ- erated by individual cells and biomolecules. ically decomposed into two components: (1) This includes propulsive forces generated by a scattering force proportional to the laser in- swimming bacteria, elastic forces generated by tensity and (2) a gradient force proportional deformation of biomolecules, and the forces to the gradient of laser intensity. The scatter- generated by processive enzyme motors op- ing force is in the direction of the laser beam, erating within a cell. Experimenting with while the gradient force depends on the spatial an apparatus capable of capturing, transport- dependence of the laser intensity and near the ing, and manipulating individual cells and or- focus provides the Hooke's law force respon- ganelles provides an intriguing introduction to sible for trapping the particle. The origin of the world of biological physics. both forces is similar: the particle elastically A photon of wavelength λ and frequency scatters a photon and alters its momentum. f = c/λ carries an energy E = hf and a Momentum conservation implies that the scat- momentum of magnitude p = h/λ in the di- tered photon imparts an equal and opposite rection of propagation.1 Note that our laser momentum change to the particle. The net power|around 100 mW|focused down to force on the particle is a vector equal and op- a few square microns, imply laser intensi- posite the net rate of change of momentum of ties around 107 W/cm2 at the beam focus. the laser photons. Particles that absorb more than a tiny frac- The theory and practice of laser tweezers are tion of the incident beam will absorb a large highly developed and numerous excellent re- amount of energy relative to their volume views, tutorials, examples, and other resources rather quickly. In fact, light-absorbing par- on the subject are available2. For particles ticles can be quite rapidly vaporized (opti- with diameters d large compared to λ, a ray cuted) by the trapping laser. (Incidentally, optics model gives a good description of the your retina contains many such particles - see physics that generates the force due to the Laser Safety below). While the scatterer and laser. Reflection and refraction of the rays at surrounding fluid always absorbs some energy, the surface of the sphere alter the direction of our infrared laser wavelength (λ = 975 nm) is the rays and thus the momentum they carry. specifically chosen because it is where absorp- The laser force is the cumulative effect from all tion in water and most biological samples is the scattered photons. For smaller particles of lowest. The absorption rate is also near a min- diameter d λ, Rayleigh scattering describes imum for the silica spheres you will study. You the interaction: The particle acts as a point should keep an eye out for evidence of heating dipole, scattering the incident beam in a spa- in your samples, but because of the relatively tially dependent fashion that depends on the 1h = Planck's constant, c = speed of light. 2Google: optical tweezers May 28, 2013 Optical Tweezers OT - sjh,rd 3 location of the particle, leading to a trapping tion. Because the laser beam and focusing op- force on the particle. tics are cylindrically symmetric around the z- The center of the trap will be taken as axis, the trap has the same properties in the (r = 0). For any small displacement (any di- x-direction as in the y-direction. We need only rection) away from the trap center the parti- consider the equations for the x motion of the cle is subject to a Hooke's law restoring force, particle, and a similar set of equations will de- i.e., proportional to and opposite the displace- scribe the motion in the y-direction. However ment. Detailed calculations show that the the trapping force that acts along the z direc- force constants are sensitive to the shape and tion is different than for x and y, as the laser intensity of the laser field, the size and shape intensity in the focal region is clearly not a of the trapped particle, and the optical prop- spherically symmetric pattern. The width of erties of the particle and surrounding fluid. the beam focus in its radial (xy) dimension Consequently the Hooke's law force is diffi- is very narrow. It is limited by wave diffrac- cult to predict. Furthermore, our apparatus tion to roughly one wavelength (λ ∼ 1µm), operates in an intermediate regime of particle whereas this is not the case in z. Hence the sizes where neither the ray optics nor Rayleigh restoring force in z is not necessarily as strong models are truly appropriate. The diameter d as in xy. If the beam is not very tightly fo- of the silica spheres (SiO2) range in size from cused along the z-direction, the trapping force 0.5-5 µm. Thus with the laser wavelength of in the z-direction may be too weak. If the λ = 975 nm, we have d ∼ λ. Fortunately we focal \cone" has too shallow an angle (tech- do not need to calculate or predict the Hooke's nically, a large f-number or small numerical law force constants. We only need to under- aperture), particles may be trapped in the xy stand their physical origin. In this lab you direction but not trapped along z: the laser will learn how to calibrate the trap by measur- beam will tend to pull small particles in to- ing the force constants indirectly on a trapped ward the central optical axis, propel them to- particle in a fluid. With an accurate force cali- ward the focus, and keep pushing them out bration method at hand, one can measure and the other side of the trap. By employing a explore biomolecular forces quantitatively. large numerical aperture, our apparatus pro- vides excellent trapping in all three directions xyz. Physics of the trapped particle We will investigate the motions of the par- The design, operation, and calibration of our ticle in the xy directions only. Consequently, laser tweezers draws on principles of optics, in the discussion that follows, when forces, im- mechanics and statistical physics. We begin pulses, velocities or other vector quantities are with an overview of the physics that is relevant written without vector notation (F instead of to generating and calibrating the trap. We F), they represent the x-component of the cor- then describe the design of the apparatus and responding vector quantity. the procedure for calibration. The ray optics in Figure1 illustrate how Consider the motion and forces in terms of laser beam refraction generates a trapping their components.
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